*Article* **Deformed Mathematical Objects Stemming from the** 𝒒**-Logarithm Function**

**Ernesto P. Borges 1,\* and Bruno G. da Costa <sup>2</sup>**


**Abstract:** Generalized numbers, arithmetic operators, and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of 𝑞logarithm/𝑞-exponential inverse functions. Some of the objects were previously described in the literature, while others are newly defined. Commutativity, associativity, and distributivity, and also a pair of linear/nonlinear derivatives, are observed within each class. Two entropic functionals emerge from the formalism, and one of them is the nonadditive Tsallis entropy.

**Keywords:** deformed numbers; deformed algebras; deformed calculus; nonadditive entropy

#### **1. Introduction**

Extensivity of an entropy is expressed as 𝑆 being proportional to the number 𝑁 of elements of the system. The hypervolume Ω of the phase space of a system composed by independent subsystems increases with the product of the hypervolumes 𝜇<sup>𝑖</sup> of the corresponding subspaces of its elements (𝜇<sup>𝑖</sup> > 1). For identical and independent subsystems, the phase space exponentially increases with the number of elements, Ω = 𝜇 𝑁 1 , and thus the Boltzmann entropy is proportional to 𝑁: 𝑆 = 𝑘 ln Ω = 𝑁 𝑘 ln 𝜇1, i.e., it is extensive. Correlations between subsystems make the hypervolume of the phase space smaller than that of the product of the hypervolumes of its subsystems, and particular kinds of strong correlations make the phase space asymptotically increase as a power law, at a much slower rate than the exponential law; in these cases the Boltzmann entropy is no longer extensive. For such special cases, —and there are plenty of observational, experimental, and numerical examples— the nonadditive entropy 𝑆<sup>𝑞</sup> [1] becomes proportional to 𝑁, recovering extensivity, which is a central property for connecting with thermodynamics (see details and further implications of extensivity in Ref. [2]). The mathematical property that plays this role is a generalized multiplication operator defined in Ref. [3]. The present paper identifies four classes of generalized algebras associated with the nonextensive formalism in a broader point of view. One of them contains the above-mentioned generalized multiplication. These developments hopefully help to understand the underlying mathematical structures that support the nonextensive statistical mechanics.

The Tsallis nonadditive entropy 𝑆<sup>𝑞</sup> has induced investigations on deformed mathematical structures aiming to represent relations of the nonextensive framework through expressions formally similar to the standard Boltzmann-Gibbs (BG) statistical mechanics. The definition of the generalized logarithm function (the 𝑞-logarithm) [4]

$$\ln\_q x \equiv \frac{x^{1-q} - 1}{1-q}, \quad (x > 0), \tag{1}$$

**Citation:** Borges, E.P.; da Costa, B.G. Deformed Mathematical Objects Stemming from the 𝑞-Logarithm Function. *Axioms* **2022**, *11*, 138. https://doi.org/10.3390/ axioms11030138

Academic Editor: Hans J. Haubold

Received: 26 January 2022 Accepted: 6 March 2022 Published: 16 March 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

allowed to rewrite 𝑆<sup>𝑞</sup> ≡ 𝑘 (𝑞 − 1) −1 1 − Í𝑊 𝑖 𝑝 𝑞 𝑖 (in its discrete version) as

$$\begin{aligned} S\_q &= -k \sum\_i^W p\_i^q \ln\_q p\_{i\nu} \\ &= -k \sum\_i^W p\_i \ln\_q (1/p\_i) \end{aligned} \tag{2}$$

(sum over 𝑊 microstates, each one labeled 𝑖, with their corresponding probabilities 𝑝<sup>𝑖</sup> , 𝑘 is a positive constant, 𝑞 ∈ R is the generalizing entropic index). Ordinary formalism is recovered as 𝑞 → 1 (ln<sup>1</sup> 𝑥 = ln 𝑥; 𝑆<sup>1</sup> = 𝑆BG = 𝑘 Í𝑊 𝑖 𝑝𝑖 ln 1/𝑝𝑖), equiprobability yields 𝑆<sup>𝑞</sup> [𝑝<sup>𝑖</sup> = 1/𝑊] = 𝑘 ln<sup>𝑞</sup> 𝑊. The 𝑞-logarithm presents the limiting cases

lim 𝑥→0 + ln<sup>𝑞</sup> 𝑥 = ( −1 1−𝑞 , 𝑞 < 1, −∞, 𝑞 ≥ 1, (3)

$$\lim\_{\chi \to \infty} \ln\_q x \quad = \quad \begin{cases} \infty, & q \le 1, \\ \frac{1}{q - 1}, & q > 1. \end{cases} \tag{4}$$

Its inverse, the 𝑞-exponential, is

$$\exp\_q(\mathbf{x}) = \begin{cases} \left[1 + (1 - q)\mathbf{x}\right]^{\frac{1}{1 - q}} \theta\left(\mathbf{x} + \frac{1}{1 - q}\right), & q < 1, \\\mathbf{e}^{\mathbf{x}}, & q = 1, \\\mathbf{e} = \frac{1}{\left[1 - (q - 1)\mathbf{x}\right]^{\frac{1}{q - 1}} \theta\left(\frac{1}{q - 1} - \mathbf{x}\right)}, & q > 1, \end{cases} \tag{5}$$

(𝜃(𝑥) is the Heaviside step function) that is more compactly written as exp<sup>𝑞</sup> (𝑥) = [1 + (1 − 𝑞)𝑥] 1/(1−𝑞) + , with the symbol [·]<sup>+</sup> ≡ max{0, ·}, — the subscript symbol <sup>+</sup> encompasses the Heaviside function. The Heaviside step function 𝜃(𝑥) defines the cutoff condition: the 𝑞-exponential is set to zero for 𝑞 < 1 and 𝑥 < −1/(1 − 𝑞), and diverges for 𝑞 > 1 and 𝑥 > 1/(𝑞 − 1). In the following we use either notations exp<sup>𝑞</sup> (𝑥) or e 𝑥 𝑞 , equivalently. Some properties of 𝑞-logarithm and 𝑞-exponential functions may be found in [2,5–7].

The 𝑞-logarithm of a product splits into a nonadditive form for 𝑞 ≠ 1:

$$
\ln\_q(xy) = \ln\_q x + \ln\_q y + (1-q)\ln\_q x \ln\_q y. \tag{6}
$$

This property triggered the definition of new generalized arithmetic operators: (*i*) what if the right hand side (r.h.s.) of this expression is viewed as the definition of a generalized addition of 𝑞-logarithms? Answer: Equation (4) of [3], Equation (7) of [8], Equation (25) of the present work. (*ii*) What should be the argument of the 𝑞-logarithm of the left hand side (l.h.s.) of (6) if its r.h.s. were an ordinary addition, instead of the generalized addition just defined? Answer: Equation (7) of [3], Equation (8) of [8], Equation (48) of the present work. Since then, these operators have usually been referred to as 𝑞-addition and 𝑞-multiplication, or, more colloquially, 𝑞-sum and 𝑞-product. This 𝑞-multiplication is the one that makes 𝑆<sup>𝑞</sup> extensive, as mentioned previously, and it is not distributive with respect to the 𝑞addition, and Nivanen et al. [8] identified additional deformed operators, recovering the distributivity [their Equations (24)–(28)]. In an extension of that work by the same authors with collaborators [9], the 𝑞-multiplication and the 𝑞-addition were identified as belonging to two different classes, and further operators were defined.

Examples of mathematical developments along these lines include: spiral generalizations of the trigonometric and hyperbolic functions through the extension of Euler's formula to the complex domain [10], generalization of derivative operators [3,11], generalizations of Fourier transforms, representations of the Dirac delta function [12,13], two parameter extensions for the logarithm and exponential and their related algebras [14,15] etc. Other deformed mathematical structures were introduced, particularly the Kaniadakis formalism [16–19], from which some of the developments within the nonextensive context

have been inspired. Generalization of algebras has been recently proposed [20], conforming to the group entropy theory [21].

Examples of physical systems described by nonextensive statistical mechanics include: anomalous diffusion of cold atoms in dissipative optical lattices [22], anomalous diffusion in granular matter [23], experimental high energy physics [24], and observational high energy physics in cosmic rays [25]. An up-to-date bibliography may be found at the site [26].

The present paper revisits generalized algebras and calculus motivated by the nonextensive formalism in a broader point of view. It identifies the basic arithmetic operators for four complementary classes, and defines a pair of linear/nonlinear derivative for each one. A connection with entropic functionals is established. The starting point is the definition of the generalized numbers.

The paper is organized as follows. Section 2 introduces four deformed numbers, by combining the pair of the inverse logarithm/exponential functions and their generalized forms. Section 3 explores each class of deformed arithmetics, derived from the generalized numbers. Section 4 is dedicated to the deformed calculus emerged from the infinitesimal deformed differences. Two possibilities are focused: a linear and a nonlinear deformed derivative. A connection between these structures with entropic functionals is addressed in Section 5. Particularly, the nonadditive entropy 𝑆<sup>𝑞</sup> is alternatively obtained through a procedure that uses one of the generalized powers defined in Section 3. Section 6 draws our final remarks and points towards new perspectives. Throughout the text, many expressions use symbols designed for compactness. Some of them appear in their explicit forms in the Appendix A.

#### **2. Deformed** 𝒒**-Numbers**

One fundamental mathematical object deserves a special attention within the present context, namely, the very concept of number. This was implicitly advanced within the nonextensive formalism in Ref. [10], through the variable 𝜁<sup>𝑞</sup> = ln e<sup>𝑧</sup> 𝑞 (𝑧 ∈ C) used in the generalization of Euler's formula, that may be read as a complex generalized number [see Equation (22) of [10], Equation (10a) of the present work]. Deformed numerical sets (𝑞-natural N𝑞, 𝑞-integer Z𝑞, 𝑞-rational Q𝑞, 𝑞-real R<sup>𝑞</sup> numbers) were considered following Peano-like axioms and generalized arithmetic operators were consistently defined [27]. Those generalized numbers are a transformation of the so-called 𝑄*-analog of* 𝑛 — 𝑄 standing for quantum, within the context of quantum calculus (we write it with upper case 𝑄 to avoid confusion with the present lower case index 𝑞) [28]:

[𝑛]<sup>𝑄</sup> = 𝑄 <sup>𝑛</sup> <sup>−</sup> <sup>1</sup> 𝑄 − 1 , (7)

from which we borrow the idea of a 𝑞-number. This connection had been previously realized, see [29]. Deformation of reals had also been reported in Ref. [30].

Given a continuous, analytical, monotonous, invertible function 𝑓 (𝑥) generalized through a real parameter 𝑞 that recovers the ordinary case as a limiting procedure (in this context, 𝑞 → 1), we introduce the generalized numbers through four combinations, such as the ordinary case identically recovered:

$$[\mathbf{x}]\_q \qquad = \quad f(f\_q^{-1}(\mathbf{x}))\_\prime \tag{8a}$$

$$f\_q\left[\mathbf{x}\right] \quad \text{ =} \quad f\_q\left(f^{-1}(\mathbf{x})\right), \tag{8b}$$

$$\{\mathbf{x}\}\_q \quad = \quad f^{-1}(f\_q(\mathbf{x}))\_\prime \tag{8c}$$

$$f\_q\{\mathbf{x}\}\qquad =\quad f\_q^{-1}(f(\mathbf{x})).\tag{8d}$$

The adopted notation obeys the following criteria: the square brackets are used when 𝑓 −1 𝑞 (or 𝑓 −1 ) is the argument of 𝑓 (or 𝑓𝑞) and the curly brackets are used when 𝑓<sup>𝑞</sup> (or 𝑓 ) is the argument of 𝑓 −1 (or 𝑓 −1 𝑞 ); the function labeled as 𝑓 is arbitrary. The deformation parameter 𝑞 is used as a subscripted *postfix* if the *inner* function is deformed, referred to as i-number, Equations (8a) and (8c), and as a subscripted *prefix* if the *outer* function is deformed, referred to as o-number, Equations (8b) and (8d) (in analogy with the notation employed for the generalized hypergeometric series — in that case, prefix for the numerator, postfix for the denominator).

The pair of i/o numbers are inverse of each other, and thus

$$\mathbb{I}\_q\left[\left\{\mathbf{x}\right\}\_q\right] = \left[\mathbf{\left[q\right]}\mathbf{x}\right]\Big|\_q = \mathbf{\left[q\right]}\left\{\mathbf{x}\right\}\_q\Big| = \left\{\mathbf{\left[q\right]}\mathbf{x}\right\}\_q = \mathbf{x}.\tag{9}$$

To be more specific to the case we are focusing upon, we define 𝑓 (𝑥) = ln 𝑥, and, consequently, 𝑓 −1 (𝑥) = e 𝑥 . It follows the le-numbers (l stands for logarithm and e stands for exponential, 'le' expresses the order in which the functions are taken)

$$\mathbf{e}\begin{bmatrix} \mathbf{x} \end{bmatrix}\_q = \begin{array}{c} \ln \mathbf{e}\_q^x \quad (\text{ile-number}), \tag{10a} \\ \mathbf{e} \end{array} \tag{10a}$$

$$\mathbf{h}\_q[\mathbf{x}] = \quad \|\mathbf{h}\_q\| \text{ e}^\mathbf{x} \quad (\text{ole-number}), \tag{10b}$$

and the el-numbers

$$\{\mathbf{x}\}\_q = \quad \mathbf{e}^{\ln\_q x} \quad (\text{id-number}), \tag{11a}$$

$$\mathbf{e}\_q \{ \mathbf{x} \} \ = \quad \mathbf{e}\_q^{\text{ln}, \mathbf{x}} \quad \text{(oel-number)}. \tag{11b}$$

Equation (11) is constrained to 𝑥 ∈ R+. This limitation can be overcome, allowing 𝑥 ∈ R, in analogy to what was done in Ref. [31], by ad hoc redefining the el-numbers as

$$\mathbf{e}\_{\{\mathbf{x}\}\_{q}} = \quad \text{sign}(\mathbf{x}) \, \mathbf{e}^{\text{ln}\_{q} \, |\mathbf{x}\vert} \quad (\text{id-number}), \tag{12a}$$

$$\mathbf{e}\_q\{\mathbf{x}\} = \quad \text{sign}(\mathbf{x})\mathbf{e}\_q^{\ln|\mathbf{x}|} \quad \text{(oel-number)}, \tag{12b}$$

with sign(𝑥) = 𝑥/|𝑥| and sign(0) ≡ 0. The present work uses the el-numbers as defined by Equation (12), but expressions are easily rewritten in its simpler form (11) by taking into consideration the restricted domain.

The le-numbers have one fixed point ([𝑥] = 𝑥) at [0]<sup>𝑞</sup> = 0, and <sup>𝑞</sup> [0] = 0 (ile and ole, respectively) for all values of 𝑞 ≠ 1. The iel-numbers have two fixed points ({𝑥} = 𝑥) for 𝑞 < 1, at {±1}<sup>𝑞</sup> = ±1, — zero is not a fixed point for iel-numbers, since {0}<sup>𝑞</sup> (lim𝑥→<sup>0</sup> <sup>−</sup> {𝑥}<sup>𝑞</sup> = −e −1/(1−𝑞) , lim𝑥→<sup>0</sup> <sup>+</sup> {𝑥}<sup>𝑞</sup> = e −1/(1−𝑞) ) —, and three fixed points for 𝑞 ≥ 1, at {0}<sup>𝑞</sup> = 0 and {±1}<sup>𝑞</sup> = ±1. The oel-numbers have three fixed points, at <sup>𝑞</sup>{0} = 0, and <sup>𝑞</sup>{±1} = ±1. Due to the cutoff condition of the 𝑞-exponential, 𝑞<<sup>1</sup> |𝑥| < e 1/(𝑞−1) = 0, and due to the absolute values, the el-numbers are odd, for both i and o deformed numbers ({−𝑥} = −{𝑥}). lenumbers and el-numbers are monotonous crescent with the ordinary numbers, i.e., if 𝑥 > 𝑦, [𝑥] > [𝑦] and {𝑥} > {𝑦} for both i and o deformed numbers. An exception may apply for the oel-numbers: it may happen 𝑥 > 𝑦 but <sup>𝑞</sup>{𝑥} = <sup>𝑞</sup>{𝑦} = 0 for 𝑞 < 1 within the cutoff region, <sup>|</sup>𝑥| ≤ exp − 1/(1 − 𝑞) and <sup>|</sup>𝑦| ≤ exp − 1/(1 − 𝑞) . The inverse relations between ile/ole and iel/oel numbers expressed by Equation (9) are valid outside the cutoff regions. Figure 1 illustrates the four 𝑞-numbers. These deformed numbers also satisfy the identities

$$\left[\ln x\right]\_q = \quad \ln \left(\_q \{x\}\right)\_\prime \tag{13a}$$

$$\ln\_q\left[\ln x\right] \;=\;\ln\left(\{\mathbf{x}\}\_q\right)\;=\;\ln\_q\mathbf{x}\;\tag{13b}$$

$$\left[\ln\_q x\right]\_q = \quad \ln\_q \left(\_q \{x\}\right) \tag{13c} = \quad \ln x \tag{13c}$$

$$\mathbf{n}\_q \begin{bmatrix} \ln\_q x \end{bmatrix}\_r = \begin{bmatrix} \ln\_q \left( \{x\}\_q \right) \end{bmatrix} \tag{13d}$$

$$\left\{ \exp x \right\}\_q = \left. \exp \left( \,\_q \left[ x \right] \right) \right\} \tag{14a}$$

$$\_q\{\exp x\}\quad = \ \_\exp\left(\begin{array}{c} \[x\]\_q \end{array}\right) = \ \_\exp\_q x. \tag{14b}$$

$$\left\{ \exp\_q \mathbf{x} \right\}\_q = \left. \exp\_q \left( \,\_q [\mathbf{x}] \right) \right\} = \left. \exp \mathbf{x} \right\| \tag{14c}$$

$$\mathbb{E}\_q\{\exp\_q x\} \quad = \ \exp\_q(\{x\}\_q). \tag{14d}$$

Whenever convenient and not ambiguous, for the sake of compactness of notation, we henceforth may occasionally use the symbols h𝑥i<sup>𝑞</sup> to denote the i-numbers (either [𝑥]<sup>𝑞</sup> or {𝑥}𝑞), and <sup>𝑞</sup> h𝑥i to denote the o-numbers (either <sup>𝑞</sup> [𝑥] or <sup>𝑞</sup>{𝑥}), and the most general case h𝑥i, without subscripts, to denote any of the four generalized numbers (not to be confound with mean value or the bra-ket symbols). The expressions 'generalized number' and 'generalized variable' are used interchangeably, just as the convenience of the context, without restricting ourselves to the rigorous mathematical distinction these concepts may have.

The following sections explore the connections of these deformed numbers with their corresponding arithmetics and calculus.

**Figure 1.** 𝑞-numbers , illustrated with 𝑞 = −1 (red), 1 (black), 3 (blue). (**a**) ile-number; [𝑥 ≤ −1/(1 − 𝑞)]𝑞<<sup>1</sup> → −∞, illustrated by the vertical red asymptote for 𝑞 = −1; [𝑥 ≥ 1/(𝑞 − 1)]𝑞><sup>1</sup> → ∞, illustrated by the vertical blue asymptote for 𝑞 = 3. (**b**) ole-number; lim𝑥→−∞ 𝑞<<sup>1</sup> [𝑥] = −1/(1 − 𝑞), illustrated by the horizontal red asymptote for 𝑞 = −1: lim𝑥→∞ 𝑞><sup>1</sup> [𝑥] = 1/(𝑞 − 1), illustrated by the horizontal blue asymptote for 𝑞 = 3. (**c**) iel-number; lim𝑥→<sup>0</sup> <sup>±</sup> {𝑥}𝑞<<sup>1</sup> = ±e −1/(1−𝑞) ; illustrated for 𝑞 = −1; lim𝑥→±∞{𝑥}𝑞><sup>1</sup> = ±e 1/(𝑞−1) , illustrated by the horizontal blue asymptotes for 𝑞 = 3; (**d**) oel-number; 𝑞<1 {|𝑥| ≤ e −1/(1−𝑞) } = 0, illustrated for 𝑞 = −1; 𝑞><sup>1</sup> {|𝑥| ≥ e 1/(𝑞−1) } → sign(𝑥)∞, illustrated by the vertical blue asymptotes for 𝑞 = 3.

#### **3. Deformed** 𝒒**-Arithmetics**

Starting from the generalized numbers (10) and (12) we identify four generalized classes of arithmetics. In this paper, the designation 𝑞-addition, 𝑞-multiplication, etc., are ambiguous, and thus we introduce a different notation: the ile-, ole-, iel-, and oel- arithmetic operators. Particularly, and partially anticipating the results of the next subsections, the deformed addition and subtraction of Ref. [3] belong to the ole-algebra (here symbolized by [𝑞]⊕ and [𝑞] ), considered in Section 3.2, and the deformed multiplication and division of Ref. [3] belong to the oel-algebra (here symbolized by {𝑞}⊗ and {𝑞} ), considered in Section 3.3. By 𝑞-arithmetics we generically denote the set of the four arithmetics described in this paper. They can also be referred to as 𝑞-algebras, understood as algebras over the real numbers, or some subset of the reals.

An i-arithmetic operator is defined as the i-number of the ordinary arithmetic operator of the corresponding o-numbers, and, complementary, an o-arithmetic operator is defined as the o-number of the ordinary arithmetic operator of the corresponding i-numbers. The generating rules follow the lines of the 𝜅-arithmetic operators of Kaniadakis [16–18], more generally expressed by Equation (1) of [32] (also in [20]), and are

$$\text{i-arithmetic:} \qquad \text{x } \circlearrowleft\_{\langle q \rangle} \text{ y } = \left\langle \begin{array}{c} \langle \text{q} \rangle \text{ o } \ \text{q} \end{array} \right\rangle\_{q'} \tag{15a}$$

$$\text{to-arithmetic:} \qquad \text{x }\_{\langle q \rangle} \bigcirc \text{y} = \,\_q\Big\langle \langle \mathbf{x} \rangle\_q \circ \, \langle \mathbf{y} \rangle\_q \Big\rangle. \tag{15b}$$

The symbol ◦, a small circle without subscripts, represents any general usual arithmetic operator, ◦ ∈ {+, <sup>−</sup>, <sup>×</sup>, /}; its generalized version is represented by a larger circle #, with bracket subscripts: prefixed/postfixed, square/curly, in consonance with the case. To avoid ambiguity in notation, the generalized *operators* are represented within a circle with their subscripts within brackets. The generalized *numbers* are represented within brackets, with their subscripts without brackets.

Some general relations are valid for all cases (the symbol 𝑁 without subscript generically represents the neutral element of the addition for any of the four arithmetics 𝑁 ∈ {𝑁[+], [+]𝑁, 𝑁{+}, {+}𝑁}; similarly to 𝐼, the neutral element of the multiplication; 𝐴, the absorbing element of the multiplication): the neutral element of the deformed addition 𝑁, such as 𝑥 ⊕ 𝑁 = 𝑥, is the corresponding deformed zero (𝑁[+] = [0]𝑞, [+]𝑁 = <sup>𝑞</sup> [0], 𝑁{+} = {0}𝑞, {+}𝑁 = <sup>𝑞</sup>{0}); the deformed additive opposite of 𝑥, written as 𝑥 ≡ 0 𝑥, such that 𝑥 ⊕ ( 𝑥) = 𝑁, and 𝑥 𝑦 = 𝑥 ⊕ ( 𝑦). Similarly, the neutral element of the deformed multiplication 𝐼, 𝑥 ⊗ 𝐼 = 𝑥, is the corresponding deformed unity (𝐼[×] = [1]𝑞, [×] 𝐼 = <sup>𝑞</sup> [1], 𝐼{×} = {1}𝑞, {×} 𝐼 = <sup>𝑞</sup>{1}). The deformed multiplicative inverse of 𝑥, written as 𝐼 𝑥, is such that 𝑥 ⊗ (𝐼 𝑥) = 𝐼, and 𝑥 𝑦 = 𝑥 ⊗ (𝐼 𝑦). The absorbing element 𝐴 of the deformed multiplication, such that 𝑥 ⊗ 𝐴 = 𝐴, coincides with the neutral element 𝑁 of the corresponding deformed addition (𝐴[×] = 𝑁[+], [×]𝐴 = [+]𝑁, 𝐴{×} = 𝑁{+}, {×}𝐴 = {+}𝑁). The deformed addition and multiplication are commutative (𝑥 ⊕ 𝑦 = 𝑦 ⊕ 𝑥, 𝑥 ⊗ 𝑦 = 𝑦 ⊗ 𝑥), associative [𝑥 ⊕ (𝑦 ⊕ 𝑧) = (𝑥 ⊕ 𝑦) ⊕ 𝑧, 𝑥 ⊗ (𝑦 ⊗ 𝑧) = (𝑥 ⊗ 𝑦) ⊗ 𝑧], and the deformed multiplication is (left and right) distributive with respect to the deformed addition - 𝑥 ⊗ (𝑦 ⊕ 𝑧) = (𝑥 ⊗ 𝑦) ⊕ (𝑥 ⊗ 𝑧), (𝑦 ⊕ 𝑧) ⊗ 𝑥 = (𝑦 ⊗ 𝑥) ⊕ (𝑧 ⊗ 𝑥) [32]. Some constraints may apply to these relations according to the case, to be detailed in the next subsections.

#### *3.1. ile-Arithmetics*

The ile-algebraic operators follow from the generating rule expressed by (15a). The ile-addition is

$$\begin{aligned} \text{tr}\, \oplus\_{\left[q\right]} \text{y} &= \left[ \,\_{q} \left[ \mathbf{x} \right] + \,\_{q} \left[ \mathbf{y} \right] \right]\_{q'} \\ &= \, \ln \exp\_{q} \left( \ln\_{q} \mathbf{e}^{\mathbf{x}} + \ln\_{q} \mathbf{e}^{\mathbf{y}} \right) . \end{aligned} \tag{16}$$

The neutral element of the ile-addition is 𝑁[+] = [0]<sup>𝑞</sup> = 0, and consequently the opposite ile-additive of 𝑦 is

$$\mathbb{1}\ominus\_{\{q\}}\mathbf{y} = \frac{1}{1-q}\ln\left(2 - \mathbf{e}^{(1-q)\mathbf{y}}\right). \tag{17}$$

The ile-difference (15a) with #h𝑞<sup>i</sup> <sup>=</sup> [𝑞] ,

$$\begin{aligned} \text{tr}\,\ominus\_{\{q\}} \text{y} &= \quad \left[ \begin{smallmatrix} q \end{smallmatrix} \begin{smallmatrix} \mathbf{x} \end{smallmatrix} \right] - \,\_{q} \begin{smallmatrix} \mathbf{y} \end{smallmatrix} \right]\_{q'} \\ &= \quad \ln \exp\_q (\ln\_q \,\mathbf{e}^\mathbf{x} - \ln\_q \,\mathbf{e}^\mathbf{y}) , \end{aligned} \tag{18}$$

consistently satisfies 𝑥 [𝑞] 𝑦 = 𝑥 ⊕[𝑞] ( [𝑞] 𝑦 ) for all 𝑞.

The ile-multiplication is

$$\begin{aligned} \mathbf{x} \otimes\_{\left[q\right]} \mathbf{y} &= \quad \left[ \begin{smallmatrix} q \end{smallmatrix} \begin{smallmatrix} x \end{smallmatrix} \right]\_q \left[ \mathbf{y} \right] \Big|\_{q'} \\ &= \quad \ln \exp\_q \left( \ln\_q \mathbf{e}^\mathbf{x} \; \ln\_q \mathbf{e}^\mathbf{y} \right) , \end{aligned} \tag{19}$$

with its neutral ile-multiplicative element 𝐼[×] = [1]<sup>𝑞</sup> = (1−𝑞) −1 ln(2−𝑞) for 𝑞 < 2 ([1]𝑞≠<sup>1</sup> ≠ 1, 𝐼[×] for 𝑞 ≥ 2), and its ile-absorbing element 𝐴[×] = [0]<sup>𝑞</sup> = 0, for all 𝑞. The ile-division is

$$\begin{aligned} \text{tr}\,\otimes\_{\{q\}}\mathbf{y} &= \quad \left[\frac{q\,\mathrm{[x]}}{q\,\mathrm{[y]}}\right]\_{q}^{\prime} \\ &= \quad \ln \exp\_{q}\left(\frac{\mathrm{ln}\_{q}\,\mathrm{e}^{\mathrm{x}}}{\mathrm{ln}\_{q}\,\mathrm{e}^{\mathrm{y}}}\right) \end{aligned} \tag{20}$$

and 𝐼[×] [𝑞] 0.

The ile-power of 𝑥 is defined as the ile-multiplication of 𝑛 identical factors 𝑥,

$$\text{tr}\,\otimes\_{\left[q\right]} n = \prod\_{\left[q\right]}^{n} \left. x = \left[ \left. \left( \left. q \right| \right. \right)^{n} \right]\_{q} . \tag{21}$$

Its analytical extension from 𝑛 ∈ N to 𝑦 ∈ R is written as

$$\propto \otimes\_{\left[q\right]} \mathbf{y} = \ln \exp\_q \left( (\ln\_q \mathbf{e}^x)^y \right), \quad (x > 0), \tag{22}$$

with the particular cases: 𝑥 <sup>∧</sup> [𝑞] 0 = [1]<sup>𝑞</sup> (𝑥 ≠ 0), 𝑥 <sup>∧</sup> [𝑞] 1 = 𝑥 (𝑥 ≠ 0), 1<sup>∧</sup> [𝑞] 𝑦 ≠ 1 (for 𝑞 ≠ 1), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 <sup>∧</sup> [𝑞] 𝑦) = 0 (𝑦 > 0), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 <sup>∧</sup> [𝑞] 𝑦) → ∞ (𝑦 < 0), and the trivial case 𝑥 <sup>∧</sup> [1] 𝑦 = 𝑥 𝑦 . The ile-power is right-distributive with respect to the ile-multiplication: (𝑥 ⊗[𝑞] 𝑦) <sup>∧</sup> [𝑞] 𝑧 = (𝑥 <sup>∧</sup> [𝑞] 𝑧) ⊗[𝑞] (𝑦 <sup>∧</sup> [𝑞] 𝑧).

The repeated generalized addition defines a different generalized multiplication that can be named as dot-multiplication, identified by the symbol , to distinguish it from the previous generalized multiplication (or times-multiplication), symbolized by ⊗ [Equation (19) for the ile class]. The repeated ile-addition is given by

$$\begin{aligned} \text{In } \oslash\_{\{q\}} \text{ y } &=& \sum\_{\{q\}}^{n} \text{y } \\ &=& \left[ \sum\_{q=1}^{n} \, \_{q} \text{[y]} \right]\_{q} \\ &=& \ln \exp\_{q} (n \ln\_{q} \text{e}^{\text{y}}) , \end{aligned} \tag{23}$$

where we have used the generalized summation symbol for the ile class, Í [𝑞] , compatible with the notation adopted in this work. Analytical extension from 𝑛 ∈ N to 𝑥 ∈ R yields the non commutative generalized ile-dot-multiplication:

$$\mathbf{x} \odot\_{\left[q\right]} \mathbf{y} \quad = \quad \frac{1}{1-q} \ln \left( \mathbf{x} \, \mathbf{e}^{(1-q)y} - (\mathbf{x}-1) \right)\_+ \tag{24}$$

The dot-multiplication with the unity has two behaviors, due to its non-commutativity. The trivial case (1  𝑦 = 𝑦) holds for the four classes (for the ile-dot-multiplication of this subsection, as well as for the ole-, iel-, and oel- of the subsections to come). The other case, 𝑥  1, connects the dot-multiplication with the deformed numbers. The ile-dotmultiplication with unity results 𝑥 [𝑞] 1 = - 𝑥 <sup>𝑞</sup> [1] 𝑞 , with <sup>𝑞</sup> [1] = ln<sup>𝑞</sup> 𝑒 = e <sup>1</sup>−<sup>𝑞</sup> <sup>−</sup> <sup>1</sup> /(1 − 𝑞) ≠ 1 for 𝑞 ≠ 1. Repeated ile-dot-multiplication defines ile-dot-power, not explicitly shown here.

The generating rule (15b) defines the ole-algebraic operators. The ole-addition (or ole-sum) is

$$\begin{aligned} \text{tr}\,\, [q] \oplus \text{y} &= \, \, \_q\left[ \, [\text{x}] \_q + \, [\text{y}] \_q \right] \_{\prime} \\ &= \, \, \ln\_q \exp\left( \ln \, \text{e}\_q^{\text{x}} + \ln \, \text{e}\_q^{\text{y}} \right) \_{\prime} \\ &= \, \, \, \text{x} + \text{y} + (1 - q) \text{xy} . \end{aligned} \tag{25}$$

Its neutral ole-additive element is [+]𝑁 = <sup>𝑞</sup> [0] = 0 and the opposite ole-additive element [𝑞] 𝑦 such as 𝑦 [𝑞]⊕ (0 [𝑞] 𝑦) = 0 is

$$\mathbf{y}\_{[q]} \oplus \mathbf{y}\_{\;\;\;\;\;\prime} = \begin{array}{c} -\mathbf{y} \\ \hline \mathbf{1} + (\mathbf{1} - q)\mathbf{y}^{\prime} \end{array} \quad \text{( $\mathbf{y} \neq \mathbf{1}/(q-1)$ )} \,, \tag{26}$$

and, consequently, the ole-subtraction is

$$\begin{aligned} \text{tr}\,\,\, \lbrack q\rbrack \ominus \mathbf{y}\;\, &=\ \lbrack q\rbrack \left[ \lbrack \mathbf{x} \rbrack \_q - \lbrack \mathbf{y} \rbrack \_q \right] \,, \\ &=\ \ln\_q \exp \left( \ln \mathbf{e}\_q^\mathbf{x} - \ln \mathbf{e}\_q^\mathbf{y} \right) \,, \\ &=\ \frac{\mathbf{x} - \mathbf{y}}{1 + (1 - q)\mathbf{y}} \end{aligned} \tag{27}$$

provided 𝑦 ≠ 1/(𝑞 − 1). These are the generalized addition and subtraction of Ref. [3], referred to as 𝑞-sum and 𝑞-difference, respectively (see also Section 3.3.3 of Ref. [2]).

From (15b), the ole-product

$$\begin{aligned} \text{tr}\,\, [q] \otimes \mathbf{y} &= \, \, \_q\Big[ \, [\mathbf{x}]\_q \, [\mathbf{y}]\_q \, ] \, \_\prime \\ &= \, \, \ln\_q \exp\Big( \ln \mathbf{e}\_q^\mathbf{x} \, \ln \mathbf{e}\_q^\mathbf{y} \Big) , \end{aligned} \tag{28}$$

and its neutral ole-multiplicative element [×] 𝐼 = <sup>𝑞</sup> [1] = e <sup>1</sup>−<sup>𝑞</sup> <sup>−</sup> <sup>1</sup> 1 − 𝑞 ≠ 1 for 𝑞 ≠ 1, together with the ole-division,

$$\begin{aligned} \text{tr}\_{\{q\}} \otimes \mathbf{y} &= \ \_q \left[ \frac{\{\mathbf{x}\}\_q}{\{\mathbf{y}\}\_q} \right] \text{'} \\ &= \ \ln\_q \exp \left( \frac{\ln \mathbf{e}\_q^\mathbf{x}}{\ln \mathbf{e}\_q^\mathbf{y}} \right) \text{'} \end{aligned} \tag{29}$$

are coherent with the ole-multiplicative inverse element [×] <sup>𝐼</sup> [𝑞] <sup>𝑦</sup> <sup>=</sup> ln<sup>𝑞</sup> exp (ln e<sup>𝑦</sup> 𝑞) −1 . The ole-absorbing element is [×]𝐴 = <sup>𝑞</sup> [0] = 0. The generalized diamond multiplication defined by Equation (24) of Ref. [27] is related to the ole-multiplication as 𝑥 [𝑞]⊗ 𝑦 = (𝑥 ^<sup>𝑞</sup> 𝑦) [𝑞]⊗ 1, and this expression connects the distributivity property of the diamond multiplication with respect to the ole-addition (Equation (28) of Ref. [27]) and the distributivity of the ole-multiplication with respect to this generalized addition.

The ole-power (the repeated ole-multiplication),

$$\text{tr}\,\,\|q\|\otimes n = \prod\_{\{q\}}^{n}\mathbf{x} = \,\_q\Big[\left(\{\mathbf{x}\}\_q\right)^n\Big],\tag{30}$$

after analytic continuation, becomes

$$\propto \mathbf{r}\_{\{q\}} \otimes \mathbf{y} = \ln\_q \exp\left(\ln \mathbf{e}\_q^x\right)^y, \quad (x > 0), \tag{31}$$

with 𝑥 [𝑞]<sup>∧</sup> 0 = <sup>𝑞</sup> [1] (𝑥 ≠ 0), 𝑥 [𝑞]<sup>∧</sup> 1 = 𝑥 (𝑥 ≠ 0), 1 [𝑞]<sup>∧</sup> 𝑦 ≠ 1 (for 𝑞 ≠ 1), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 [𝑞]<sup>∧</sup> 𝑦) = 0 (𝑦 > 0), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 [𝑞]<sup>∧</sup> 𝑦) → ∞ (𝑦 < 0 and 𝑞 < 1), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 [𝑞]<sup>∧</sup> 𝑦) = 1/(𝑞 − 1) (𝑦 < 0 and 𝑞 > 1), 𝑥 [1]<sup>∧</sup> 𝑦 = 𝑥 𝑦 . The ole-power is right-distributive with respect to the olemultiplication: (𝑥 [𝑞]⊗ 𝑦) [𝑞]<sup>∧</sup> 𝑧 = (𝑥 [𝑞]<sup>∧</sup> 𝑧) [𝑞]⊗ (𝑦 [𝑞]<sup>∧</sup> 𝑧).

The repeated ole-addition has been defined in Ref. [3], and reads

$$\begin{aligned} \text{In } [q] \odot \text{y} &= \sideset{}{}{\text{el}}\sum\_{\text{[q]}}^{n} \text{y}\_{\text{'}}\\ &= \sideset{}{}{\text{e}}\Big[\sum\_{\text{[q]}}^{n} \text{[y]}\_{q} \Big]\_{\text{'}}\\ &= \sideset{}{}{\text{(1} + (\text{1} - q)\text{y})}\_{\text{1} - q}^{n} . \end{aligned} \tag{32}$$

This is identical to Equation (8) of Ref. [9]. Analytical extension into the real domain yields the non commutative ole-dot-multiplication:

$$\propto \begin{array}{c} \propto \,\_{\left[q\right]} \odot \, \mathbf{y} \end{array} = \begin{array}{c} \left( \mathbf{1} + (\mathbf{1} - q)\mathbf{y} \right)^{\mathbf{x}}\_{+} - \mathbf{1} \\ \mathbf{1} - q \end{array} . \tag{33}$$

The ole-dot-multiplication with the unity is expressed by 𝑥 [𝑞] 1 = <sup>𝑞</sup> - 𝑥 [1]<sup>𝑞</sup> , with [1]<sup>𝑞</sup> = ln exp<sup>𝑞</sup> (1) = (1 − 𝑞) −1 ln(2 − 𝑞) ≠ 1 for 𝑞 ≠ 1 and 𝑞 < 2. This relation connects the ole-dot-multiplication and the le deformed numbers with the 𝑄-analog of 𝑛 (7): 𝑛 [𝑞] 1 = (𝑄 <sup>𝑛</sup> <sup>−</sup> <sup>1</sup>)/(<sup>𝑄</sup> <sup>−</sup> <sup>1</sup>), with <sup>𝑄</sup> <sup>=</sup> <sup>2</sup> <sup>−</sup> <sup>𝑞</sup>. The ole-dot power naturally follows from the repeated ole-dot-multiplication, not shown here.

#### *3.2. iel-Arithmetics*

According to the generating rule for i-algebras (15a), the iel-addition is

$$\begin{aligned} \text{fix}\_{\mathsf{gl}} \ y &= \left. \left\{ \begin{matrix} q \ \{\mathbf{x}\} + q \ \{\mathbf{y}\} \end{matrix} \right\} \right|\_{q'} \\ &= \left. \text{sign}(\mathbf{x} + \mathbf{y}) \exp\left( \text{ln}\_q \left| \text{sign}(\mathbf{x}) \, \mathbf{e}\_q^{\text{ln}\,|\mathbf{x}|} + \text{sign}(\mathbf{y}) \, \mathbf{e}\_q^{\text{ln}\,|\mathbf{y}|} \right| \right) . \end{aligned} \tag{34}$$

The cutoff of the 𝑞-exponential (5) imposes restrictions on the domain of (34). Its neutral iel-additive element 𝑁{+} = {0}𝑞, is

$$\begin{aligned} N\_{\{\star\}} &\to 0, \qquad q \ge 1, \\ N\_{\{\star\}} &\le \mathbf{e}^{\frac{-1}{1-q}}, \quad q < 1. \end{aligned} \tag{35}$$

For 𝑞 < 1, there are infinite neutral iel-additive elements, including the zero. The iel-difference reads

$$\begin{aligned} \text{tr}\, \ominus\_{\mathsf{[g]}} \text{y} &= \left. \left\{ \begin{matrix} \mathbf{x} \end{matrix} - \mathbf{q} \begin{pmatrix} \mathbf{y} \end{pmatrix} \right\} \right\}\_{\mathbf{q}'} \\ &= \left. \text{sign}(\mathbf{x} - \mathbf{y}) \exp \Big| \text{ln}\_{\mathbf{q}} \Big| \text{sign}(\mathbf{x}) \, \mathbf{e}\_{\mathbf{q}}^{\ln|\mathbf{x}|} - \text{sign}(\mathbf{y}) \, \mathbf{e}\_{\mathbf{q}}^{\ln|\mathbf{y}|} \Big| \right\}. \end{aligned} \tag{36}$$

The opposite iel-additive element is

$$\Theta\_{\mathsf{[kl]}} \ y = \begin{cases} -\mathsf{y}, & \text{if } |\mathsf{y}| > \exp\left(\frac{-1}{1-q}\right) \\\\ -\mathsf{sign}(\mathsf{y}) \exp\left(\frac{-1}{1-q}\right) & \text{if } |\mathsf{y}| \le \exp\left(\frac{-1}{1-q}\right) \\\\ -\mathsf{y}, & q = 1, \\\\ -\mathsf{y}, & \text{if } |\mathsf{y}| < \exp\left(\frac{1}{q-1}\right) \\\\ -\mathsf{sign}(\mathsf{y}) \exp\left(\frac{1}{q-1}\right) & \text{if } |\mathsf{y}| \ge \exp\left(\frac{1}{q-1}\right) \end{cases} \tag{37}$$

The iel-multiplication and the iel-division are

$$\begin{aligned} \{\mathbf{x} \otimes\_{\mathbf{kl}} \mathbf{y}\} &= \left\{ \begin{array}{rcl} \mathbf{e}\_q(\mathbf{x}) \ \_q(\mathbf{y}) \end{array} \right\}\_{\mathbf{q}'} \\ &= \text{sign}(\mathbf{x} \mathbf{y}) \exp\Big(\text{ln}\_q\Big(\mathbf{e}\_q^{\ln|\mathbf{x}|} \ \mathbf{e}\_q^{\ln|\mathbf{y}|}\Big)\Big)\_{\mathbf{y}'} \end{aligned} \tag{38}$$

$$\begin{aligned} \text{tr}\,\mathcal{O}\_{\mathsf{[q]}}\,\mathbf{y} &= \; \left\{ \frac{q\left\{\mathbf{x}\right\}}{q\left\{\mathbf{y}\right\}} \right\}\_{q} \\ &= \; \text{sign}(\mathbf{x}/\mathbf{y}) \, \exp\left( \text{ln}\_{q} \left( \frac{\mathbf{e}\_{q}^{\ln|\mathbf{x}|}}{\mathbf{e}\_{q}^{\ln|\mathbf{y}|}} \right) \right) . \end{aligned} \tag{39}$$

The neutral element of the iel-multiplication is 𝐼{×} = {1}<sup>𝑞</sup> = 1.

The iel-absorbing element coincides with the neutral iel-additive element, 𝐴{×} = 𝑁{+} (35). The repeated iel-multiplication (iel-power) is given by

$$\mathbf{x} \otimes\_{\|q\|} n = \prod\_{\{q\}}^{n} \mathbf{x} = \left\{ \left( \begin{smallmatrix} q \ \{x\} \end{smallmatrix} \right)^{n} \right\}\_{q'} \tag{40}$$

which is rewritten as (after analytical extension from 𝑛 ∈ N to 𝑦 ∈ R)

$$\text{tr}\,\otimes\_{\text{fd}}\,\mathbf{y} = \exp\left(\ln\_{q}(\mathbf{e}\_{q}^{\ln|\mathbf{x}|})^{\mathbf{y}}\right), \quad (\mathbf{x} > \mathbf{0}), \tag{41}$$

with the particular cases 𝑥 ∧{𝑞} 0 = 1 (𝑥 ≠ 0), 𝑥 ∧{𝑞} 1 = 𝑥 (𝑥 ≠ 0), 1 ∧{𝑞} 𝑦 = 1 (𝑦 ≠ 0), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 ∧{𝑞} <sup>𝑦</sup>) <sup>=</sup> exp − 1/(1 − 𝑞) (𝑦 > 0 and 𝑞 < 1), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 ∧{𝑞} 𝑦) → ∞ (𝑦 < 0 and 𝑞 < 1), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 ∧{𝑞} 𝑦) = 0 (𝑦 > 0 and 𝑞 > 1), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 ∧{𝑞} <sup>𝑦</sup>) <sup>=</sup> exp − 1/(1 − 𝑞) (𝑦 < 0 and 𝑞 > 1), 𝑥 <sup>∧</sup> {1} 𝑦 = 𝑥 𝑦 . The iel-power is right-distributive with respect to the iel-multiplication: (𝑥 ⊗{𝑞} 𝑦) ∧{𝑞} 𝑧 = (𝑥 ∧{𝑞} 𝑧) ⊗{𝑞} (𝑦 ∧{𝑞} 𝑧).

The repeated iel-addition defines the iel-dot-multiplication:

$$\begin{aligned} \text{in } \oslash\_{\text{df}} \text{ y } &= \quad \sum\_{\text{lcf}}^{n} \text{ y } \\ &= \quad \left\{ \sum\_{q}^{n} \, \_{q} \{ \text{y} \} \right\}\_{q} \\ &= \quad \text{sign}(\text{y}) \, \exp(\text{ln}\_{q} n) \, |\text{y}|^{n^{1-q}} .\end{aligned} \tag{42}$$

Analytical extension from 𝑛 ∈ N to 𝑥 ∈ R<sup>+</sup> can be represented by

$$\text{tr}\,\odot\_{\|\mathbf{q}\|}\mathbf{y} = \text{sign}(\mathbf{y})\,\exp(\ln\_{\mathbf{q}}\mathbf{x})\left|\mathbf{y}\right|^{\mathbf{x}^{1-q}}, \quad (\mathbf{x}>\mathbf{0}).\tag{43}$$

The iel-number is connected to the iel-dot-multiplication by 𝑥 {𝑞} 1 = 𝑥 <sup>𝑞</sup>{1} 𝑞 = {𝑥}𝑞, since <sup>𝑞</sup>{1} = 1.

#### *3.3. oel-Arithmetics*

The oel-arithmetic operators derives from (15b):

$$\begin{aligned} \text{tr}\,\,\vert\_{\mathfrak{gl}} \oplus \,\,\mathbf{y}\,\,\,\, &=\,\,\,\,\_{q}\Big\{\{\mathbf{x}\}\_{q} + \{\mathbf{y}\}\_{q}\,\,\Big\}\,,\\ &=\,\,\text{sign}(\mathbf{x}+\mathbf{y})\,\,\exp\_{q}\Big(\text{ln}\Big|\text{sign}(\mathbf{x})\,\,\text{e}^{\text{ln}\_{q}\,\,\|\mathbf{x}\|} + \text{sign}(\mathbf{y})\,\,\text{e}^{\text{ln}\_{q}\,\,\|\mathbf{y}\|}\Big|\,\,\,\end{aligned}\tag{44}$$

$$\begin{aligned} \text{tr}\,\,\vert\_{\mathfrak{g}} \oplus \,\,\mathbf{y}\,\,\,\, &=\,\,\,\_q\Big\{\{\mathbf{x}\}\_q-\{\mathbf{y}\}\_q\,\,\Big\},\\ &=\,\,\text{sign}(\mathbf{x}-\mathbf{y})\,\,\exp\_q\Big(\text{ln}\Big|\text{sign}(\mathbf{x})\,\,\mathbf{e}^{\text{ln}\_q\,\,\|\mathbf{x}\|}-\text{sign}(\mathbf{y})\,\,\mathbf{e}^{\text{ln}\_q\,\,\|\mathbf{y}\|}\Big|\,\,\end{aligned} \tag{45}$$

$$\begin{aligned} \text{tr}\_{\|q\|} \otimes \mathbf{y} &= \text{tr}\_{q\Big\{ \{ \mathbf{x} \}\_q \{ \mathbf{y} \}\_q \} }, \\ &= \text{sign}(\mathbf{x} \mathbf{y}) \, \exp\_q \Big( \text{ln} \Big| \mathbf{e}^{\text{ln}\_q \cdot \| \mathbf{x} \|} \, \mathbf{e}^{\text{ln}\_q \cdot \| \mathbf{y} \|} \Big| \Big) . \end{aligned} \tag{46}$$

$$\begin{aligned} \text{tr}\,\, \_{\mathsf{q}\emptyset}\otimes\,\, ^{\mathsf{q}}\, &=\,\, \_{\mathsf{q}}\Big(\frac{\{\mathbf{x}\}\_{\mathsf{q}}}{\{\mathbf{y}\}\_{\mathsf{q}}}\Big),\\ &=\,\, \_{\mathsf{s}\text{sign}(\mathsf{x}/\mathsf{y})}\,\, \_{\mathsf{q}}\exp\_{q}\Big(\ln\Big|\frac{\mathsf{e}^{\ln\_{q}|\mathsf{x}|}}{\mathsf{e}^{\ln\_{q}|\mathsf{y}|}}\Big|\Big).\end{aligned} \tag{47}$$

Equations (46) and (47) can be rearranged as

$$<\langle \mathbf{x} | \sqrt{\mathbf{g}} | \mathbf{y} \rangle = \left. \text{sign}(\mathbf{x} \mathbf{y}) \left( |\mathbf{x}|^{1-q} + |\mathbf{y}|^{1-q} - 1 \right) \right|\_{+}^{\frac{1}{1-q}} \tag{48}$$

and

$$\text{tr}\,\,\!\_{\left\|\mathbf{y}\right\|} \otimes \mathbf{y} \quad = \; \text{sign}(\mathbf{x}/\mathbf{y}) \left( |\mathbf{x}|^{1-q} - |\mathbf{y}|^{1-q} + 1 \right)\_{+}^{\frac{1}{1-q}}.\tag{49}$$

The oel-product and the oel-ratio were defined in Ref. [3], referred to as 𝑞-product and 𝑞-ratio, respectively (see also Section 3.3.2 of Ref. [2]). The cutoff that appears in (48) defines regions in which the oel-arithmetical operators are ill-defined. Figures 2 and 3 show the regions for which the cutoff applies for the oel-addition and oel-multiplication, respectively. The first column of each (Figures a and c) shows instances for 𝑞 < 1, and the second column (Figures b and d), for 𝑞 > 1. The first line (Figures a and b) exhibits the cutoff regions with a shaded pattern for one typical value of the parameter 𝑞. The second line (Figures c and d) display superimposed curves of the borders of the cutoff regions for various values of 𝑞, without shading them, otherwise they would be confusing; they follow the same pattern of the corresponding Figures a and b, respectively. The cutoff regions are closed for 𝑞 < 1 (illustrated with 𝑞 = −1 by Figures 2a and 3a), and they are open and not connected, lying on the outer side delimited by the bounding curves, for 𝑞 > 1 (illustrated with 𝑞 = 3 by Figures 2b and 3b). The second line of the figures help us to understand the effect of the deforming parameter 𝑞 on the cutoff regions. As 𝑞 approaches unity from below (Figures 2c and 3c), the cutoff regions become smaller and eventually vanish. For the oel-addition, Figure 2c, the borders of the cutoff region approach the second bisector (𝑦 = −𝑥), and, for the oel-multiplication, Figure 3c, they approach the origin (0, 0). As 𝑞 approaches unity from above (Figures 2d and 3d), the cutoff regions move away from the origin. At 𝑞 = 1, no pair of numbers (𝑥, 𝑦) fall within the cutoff regions, and the ordinary arithmetic operators are defined everywhere.

**Figure 2.** Cutoff regions for the oel-addition (44). Left column: 𝑞 < 1, right column: 𝑞 > 1. Top line: the shaded regions correspond to the cutoff regions of the oel-addition. (**a**) 𝑞 = −1. (**b**) 𝑞 = 3. Bottom line: the curves represent the cutoff borders. Regions are not shaded to avoid excessively heavy representation. Their pattern is similar to (**a**,**b**): for 𝑞 < 1, the cutoff regions lie inside the corresponding closed curves, and for 𝑞 > 1, the cutoff regions lie outside the corresponding curves. (**c**) Different values of 𝑞 < 1 (indicated). The cutoff region shrinks and eventually collapses at 𝑦 = −𝑥 as 𝑞 → 1 <sup>−</sup>. (**d**) Different values of 𝑞 > 1 (indicated). As 𝑞 → 1 + , the non connected regions depart from the origin, and there are no cutoff regions.

The distributivity of the oel-multiplication with respect to the oel-addition is valid whenever the cutoff conditions of the l.h.s. and the r.h.s. of 𝑥 {𝑞}⊗ (𝑦 {𝑞}⊕ 𝑧) = (𝑥 {𝑞}⊗ 𝑦) {𝑞}⊕ (𝑥 {𝑞}⊗ 𝑧) are not met. As 𝑞 approaches unity, even from below or from above, the distributivity of the oel-multiplication with respect to the oel-addition is valid for all real values (𝑥, 𝑦,𝑧).

The neutral oel-additive element is {+}𝑁 = <sup>𝑞</sup>{0} = 0 for 𝑞 ≥ 1, and {+}𝑁 | {+}𝑁 {𝑞}⊕ 𝑥 = 𝑥 for 𝑞 < 1. As a consequence, there is no opposite oel-additive element for 𝑞 < 1. For 𝑞 ≥ 1, {𝑞} 𝑦 = −𝑦. The absorbing element {×}𝐴 = <sup>𝑞</sup>{0} = 0 for 𝑞 ≥ 1, and {×}𝐴 | {×}𝐴 {𝑞}⊗ 𝑥 = 0 for 𝑞 < 1 and |𝑥| > 1. If 𝑞 < 1, and |𝑥| < 1 the cutoff of (48) (see (5)) implies that zero is an absorbing element, and, in this case, differently from the other three generalized algebras, {+}𝑁 ≠ {×}𝐴. The neutral multiplicative element of the oel-multiplication is {×} 𝐼 = <sup>𝑞</sup>{1} = 1, for all values of 𝑞. The inverse oel-multiplicative element is

$$\mathbf{1}\_{\|\mathbf{g}\|} \otimes \mathbf{y} \quad = \begin{cases} \displaystyle \text{sign}(\mathbf{y}) \left( 2 - |\mathbf{y}|^{1-q} \right)^{\frac{1}{1-q}}, & \text{if } |\mathbf{y}| < 2^{\frac{1}{1-q}},\\ \mathbf{0}, & \text{otherwise.} \end{cases} \tag{50}$$

This implies the unorthodox property lim𝑦→<sup>0</sup> <sup>+</sup> (1{𝑞} 𝑦) → 2 1/(1−𝑞) , for 𝑞 < 1.

**Figure 3.** Cutoff regions for the oel-multiplication (46). Left column: 𝑞 < 1, right column: 𝑞 > 1. Top line: the shaded regions correspond to the cutoff regions of the oel-multiplication. (**a**) 𝑞 = −1. (**b**) 𝑞 = 3. Bottom line: the curves represent the cutoff borders, |𝑦| = (1 − |𝑥| 1−𝑞 ) 1/(1−𝑞) . Regions are not shaded to avoid excessively heavy representation. Their pattern is similar to the adopted in (**a**) or (**b**): for 𝑞 < 1, the cutoff regions lie inside the corresponding closed curves, and for 𝑞 > 1, the cutoff regions lie outside the corresponding curves. (**c**) Different values of 𝑞 < 1 (indicated). The cutoff region shrinks and eventually collapses at (0, 0) as 𝑞 → 1 −, when the curves coincide with the axes. (**d**) Different values of 𝑞 > 1 (indicated). As 𝑞 → 1 + , the non connected regions depart from the origin, and there are no cutoff regions.

The oel-power, previously defined in Ref. [3] (with different symbols), is written as

$$\propto \llcorner\_{\|q\|} \otimes n = \prod\_{\{q\}}^{n} \ge =\_{q} \{ \left( \{ \mathbf{x} \}\_{q} \right)^{n} \}, \quad (\mathbf{x} > \mathbf{0}).\tag{51}$$

This operator also appears as Equation (8) of Ref. [9]. We make an analytical extension from 𝑛 ∈ N to 𝑦 ∈ R, and the oel-power can also be written as

$$<\downarrow \neq \downarrow \text{y} = \exp\_q(\text{ y } \ln\_q x), \quad (x > 0). \tag{52}$$

Particular cases are 𝑥 {𝑞}<sup>∧</sup> 0 = 1 (𝑥 ≠ 0), 𝑥 {𝑞}<sup>∧</sup> 1 = 𝑥 (𝑥 ≠ 0), 1 {𝑞}<sup>∧</sup> 𝑦 = 1 (𝑦 ≠ 0), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 {𝑞}<sup>∧</sup> 𝑦) = 0 (𝑦 ≥ 1, 𝑞 < 1), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 {𝑞}<sup>∧</sup> 𝑦) = exp<sup>𝑞</sup> − 𝑦/(1 − 𝑞) (𝑦 < 1, 𝑞 < 1), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 {𝑞}<sup>∧</sup> 𝑦) = 0 (𝑦 > 0, 𝑞 > 1), lim𝑥→<sup>0</sup> <sup>+</sup> (𝑥 {𝑞}<sup>∧</sup> 𝑦) → ∞ (𝑦 < 0, 𝑞 > 1), and, as always, 𝑥 {1}<sup>∧</sup> 𝑦 = 𝑥 𝑦 . The oel-power is right-distributive with respect to the oel-multiplication: (𝑥 {𝑞}⊗ 𝑦) {𝑞}<sup>∧</sup> 𝑧 = (𝑥 {𝑞}<sup>∧</sup> 𝑧) {𝑞}⊗ (𝑦 {𝑞}<sup>∧</sup> 𝑧).

The repeated oel-addition is

$$\sum\_{\{q\}} \sum^n \mathbf{y} = \frac{1}{q} \left\{ \sum^n \{\mathbf{y}\}\_q \right\}. \tag{53}$$

Its analytical extension from 𝑛 ∈ N to 𝑥 ∈ R<sup>+</sup> defines the non commutative oel-dotmultiplication:

$$\text{tr}\_{\|\mathbf{g}\|} \odot \mathbf{y} = \text{sign}(\mathbf{y}) \left( (1 - q) \ln \mathbf{x} + |\mathbf{y}|^{1 - q} \right)^{\frac{1}{1 - q}}\_{\star}, \quad (\mathbf{x} > \mathbf{0}). \tag{54}$$

The oel-number is connected to the oel-dot-multiplication by 𝑥 {𝑞} 1 = <sup>𝑞</sup> 𝑥 {1}<sup>𝑞</sup> } = <sup>𝑞</sup>{𝑥}, since {1}<sup>𝑞</sup> = 1.

#### **4. Deformed** 𝒒**-Calculus**

Following the lines of Ref. [3] (see also Sections II.C and II.D of [33]), we connect the deformed algebra with deformed calculus, and define the deformed differentials of ordinary numbers:

$$\mathbf{d}\_{[q]}\boldsymbol{\chi} = \lim\_{\boldsymbol{\chi}' \to \boldsymbol{\chi}} (\boldsymbol{\chi}' \ominus\_{[q]} \boldsymbol{\chi})\_{\prime} \tag{55a}$$

$$\mathfrak{l}\_{[q]}\mathbb{d}\mathfrak{x} = \lim\_{\mathbf{x'} \to \mathbf{x}} \left( \mathbf{x'}\_{[q]} \ominus \mathfrak{x} \right),\tag{55b}$$

$$\mathsf{cl}\_{\mathsf{[q]}}\,\mathsf{x} = \lim\_{\boldsymbol{\chi}' \to \boldsymbol{\chi}} (\mathsf{x}' \ominus\_{\mathsf{[q]}} \mathsf{x})\_{\mathsf{/}} \tag{55c}$$

$$\mathsf{L}\_{\mathsf{gl}}\mathsf{d}.x = \lim\_{\mathbf{x'} \to \mathbf{x}} \left( \mathsf{x'} \nmid\_{\mathsf{gl}} \ominus \mathsf{x} \right). \tag{55d}$$

The definitions of the corresponding deformed differences, Equations (18), (27), (36), and (45), lead to

$$\mathbf{d}\_{[q]}\mathbf{x} = \mathbf{d}\left(\,\_{q}\{\mathbf{x}\}\right),\tag{56a}$$

$$\mathbf{d}\_{[q]}\mathbf{d}x = \mathbf{d}\left(\{x\}\_q\right),\tag{56b}$$

$$\mathbf{d}\_{[q]}\boldsymbol{x} = \mathbf{d}\left(\,\_{q}\{\boldsymbol{x}\}\right),\tag{56c}$$

$$\mathbf{d}\_{[\mathbf{q}]} \mathbf{d}.\mathbf{x} = \mathbf{d} \left( \{ \mathbf{x} \}\_{\mathbf{q}} \right),\tag{56d}$$

i.e., the deformed differential of an ordinary variable (l.h.s. of (56)) is equal to the ordinary differential of the corresponding complementary deformed variable (r.h.s. of (56)): the i-differential of a variable is equal to the ordinary differential of an o-variable, (56a) and (56c), and the o-differential of a variable is equal to the ordinary differential of an i-variable, (56b) and (56d). All the deformed differentials given by (56) can be arranged as the product of the ordinary differential d𝑥 by a deforming function ℎ <sup>𝛿</sup> (𝑥), with 𝛿 ∈ {ile, ole, iel, oel} representing the deformation ( d[𝑞] 𝑥 = ℎile (𝑥) d 𝑥, [𝑞]d 𝑥 = ℎole (𝑥) d 𝑥, d{𝑞} 𝑥 = ℎiel(𝑥) d 𝑥, {𝑞}d𝑥 = ℎoel(𝑥) d 𝑥). Their explicit forms are

$$h\_{\rm life}(\mathbf{x}) = \mathbf{e}^{(1-q)\cdot x} \,, \tag{57a}$$

$$h\_{\text{ole}} = \frac{1}{1 + (1 - q)\chi}, \quad \left(\mathbf{x} \neq \frac{-1}{1 - q}\right), \tag{57b}$$

$$h\_{\rm iel}(\mathbf{x}) = \frac{1}{\chi} (1 + (1 - q) \ln \mathbf{x})^{\frac{q}{1 - q}} \, \quad (\mathbf{x} > \mathbf{0}) , \tag{57c}$$

$$h\_{\rm oel}(\mathbf{x}) = \frac{1}{\chi^q} \exp\left(\frac{\chi^{1-q} - 1}{1 - q}\right), \quad (\mathbf{x} > \mathbf{0}). \tag{57d}$$

A pair of generalized derivatives of a function 𝑓 (𝑥), holding a duality nature between them, stem from each of the deformed differentials, according to which variable the deformed differential applies on: whether on the independent variable 𝑥, — and thus a linear deformed derivative —, generically represented by D<sup>𝛿</sup> 𝑓 (𝑥), or on the dependent variable 𝑓 , — and thus a nonlinear deformed derivative — generically represented by De<sup>𝛿</sup> 𝑓 (𝑥), resulting in eight different cases:

1. ile-Derivatives Linear ile-derivative:

$$\text{D}\_{\text{lie}}f(\mathbf{x}) \equiv \frac{\text{d}f(\mathbf{x})}{\text{d}\_{\text{[}q]} \propto} = \frac{1}{h\_{\text{like}}(\mathbf{x})} \frac{\text{d}f(\mathbf{x})}{\text{d}\mathbf{x}},\tag{58a}$$

Nonlinear ile-derivative:

$$\widetilde{\mathbf{D}}\_{\text{lie}}f(\mathbf{x}) \equiv \frac{\mathbf{d}\_{[q]} \; f(\mathbf{x})}{\mathbf{d}\mathbf{x}} = h\_{\text{lie}}(f(\mathbf{x})) \; \frac{\mathbf{d}f(\mathbf{x})}{\mathbf{d}\mathbf{x}}.\tag{58b}$$

2. ole-Derivatives Linear ole-derivative:

$$\mathrm{D}\_{\mathrm{ole}}f(\boldsymbol{x}) \equiv \frac{\mathrm{d}f(\boldsymbol{x})}{\mathrm{d}\,\mathrm{d}\,\boldsymbol{x}} = \frac{1}{h\_{\mathrm{ole}}(\boldsymbol{x})} \frac{\mathrm{d}f(\boldsymbol{x})}{\mathrm{d}\,\boldsymbol{x}},\tag{59a}$$

Nonlinear ole-derivative:

$$\tilde{\mathbf{D}}\_{\text{ole}}f(\mathbf{x}) \equiv \frac{\,^{\text{q}}\!\!\!\!\!\!^{\text{f}}f(\mathbf{x})}{\text{d}\mathbf{x}} = h\_{\text{ole}}\left(f(\mathbf{x})\right)\frac{\text{d}f(\mathbf{x})}{\text{d}\mathbf{x}}.\tag{59b}$$

3. iel-Derivatives Linear iel-derivative:

$$\mathrm{D}\_{\mathrm{hel}}f(\boldsymbol{x}) \equiv \frac{\mathrm{d}f(\boldsymbol{x})}{\mathrm{d}\_{\{\boldsymbol{q}\}}\,\boldsymbol{x}} = \frac{1}{h\_{\mathrm{hel}}(\boldsymbol{x})} \frac{\mathrm{d}f(\boldsymbol{x})}{\mathrm{d}\boldsymbol{x}} \,\mathrm{d}\boldsymbol{x} \tag{60a}$$

Nonlinear iel-derivative:

$$\widetilde{\mathcal{D}}\_{\text{el}}f(\mathbf{x}) \equiv \frac{\mathbf{d}\_{\text{el}} \mid f(\mathbf{x})}{\mathbf{d}\mathbf{x}} = h\_{\text{el}}(f(\mathbf{x})) \,\frac{\mathbf{d}f(\mathbf{x})}{\mathbf{d}\mathbf{x}}.\tag{60b}$$

4. oel-Derivatives

Linear oel-derivative: linear oel-derivative:

$$\mathrm{D}\_{\mathrm{oel}}f(\boldsymbol{x}) \equiv \frac{\mathrm{d}f(\boldsymbol{x})}{\underset{\{\boldsymbol{\mathsf{q}}\}}{\mathrm{d}}\,\mathrm{x}} = \frac{1}{h\_{\mathrm{oel}}(\boldsymbol{x})} \frac{\mathrm{d}f(\boldsymbol{x})}{\mathrm{d}\boldsymbol{x}},\tag{61a}$$

Nonlinear oel-derivative:

$$\widetilde{\mathbf{D}}\_{\mathrm{oel}}f(\mathbf{x}) \equiv \frac{\upproj}{\mathbf{d}\mathbf{x}} \, f(\mathbf{x}) = h\_{\mathrm{oel}}(f(\mathbf{x})) \, \frac{\mathbf{d}f(\mathbf{x})}{\mathbf{d}\mathbf{x}}.\tag{61b}$$

The duality between the linear and the nonlinear generalized derivatives is expressed by D<sup>𝛿</sup> 𝑓 (𝑥) = De<sup>𝛿</sup> 𝑓 −1 (𝑥). The el-derivatives are defined for 𝑥 > 0. The ole-derivatives has been defined in Ref. [3], then referred to as 𝑞-derivative (the linear deformed derivative) and its dual 𝑞-derivative (the nonlinear deformed derivative). Particularly, the linear olederivative (59a) was used to generalize Fisher's information measure and the Cramer-Rao inequality [34]. The eigenfunction of the linear i/o-deformed derivative is the ordinary exponential of the o/i-deformed variable, which directly follows from (56). They are (written with the symbols h·i representing either [·] or {·})

$$\frac{\text{d}\,\,\exp(\,\_q\langle\alpha\rangle)}{\text{d}\,\,\_{\langle q\rangle}\,\,\text{x}} = \frac{\text{d}\,\,\exp(\,\_q\langle\alpha\rangle)}{\text{d}\,\,\,\_{\langle q\rangle}\langle\alpha\rangle)} = \exp(\langle\alpha\rangle\_q) \tag{62a}$$

and

$$\frac{\mathbf{d}\,\exp(\langle \mathbf{x} \rangle\_q)}{\mathbf{d}\,\mathbf{x}} = \frac{\mathbf{d}\,\exp(\langle \mathbf{x} \rangle\_q)}{\mathbf{d}\,\left(\langle \mathbf{x} \rangle\_q\right)} = \exp(\langle \mathbf{x} \rangle\_q). \tag{62b}$$

Particularly, the 𝑞-exponential (5) is the eigenfunction of the linear ole-derivative, Dolee 𝑥 <sup>𝑞</sup> = e 𝑥 𝑞 (a particular case of (62b) with e 𝑥 <sup>𝑞</sup> = e [𝑥]<sup>𝑞</sup> , see (14b)). Alternatively, its ordinary derivative is d e<sup>𝑥</sup> 𝑞 / d 𝑥 = e 𝑥 𝑞 𝑞 . The nonlinear deformed derivative of which the 𝑞-exponential is eigenfunction was defined in Ref. [11]:

$$
\widetilde{\mathfrak{D}}\_q f(u) = [f(u)]^{1-q} \frac{df(u)}{du} \, , \tag{63}
$$

where we have used the symbol, 𝔇e<sup>𝑞</sup> to distinguish it from the present deformed derivatives.

The integral of the inverse of a variable, <sup>∫</sup> <sup>𝑥</sup> 1 𝑡 <sup>−</sup>1d𝑡, is typically associated to, and frequently taken as the definition of, the logarithm function. The general nonlinear cases are

$$\frac{\mathbf{d}\_{\langle q\rangle}\,\langle \ln x \rangle\_q}{\mathbf{d}\,x} = \frac{\langle q \rangle \,\mathbf{d}\,\,q\langle \ln x \rangle}{\mathbf{d}\,x} = \frac{1}{x}.\tag{64}$$

The particular case of this equation for the nonlinear ole-derivative is (see (13b)): Deole ln<sup>𝑞</sup> 𝑥 = 1/𝑥. Alternatively, the ordinary derivative of the 𝑞-logarithm is d ln<sup>𝑞</sup> 𝑥 / d 𝑥 = 1/𝑥 𝑞 . This expression yields an integral representation of the 𝑞-logarithm function,

$$\int\_{1}^{\infty} t^{-q} \,\mathrm{d}t = \ln\_{q} \,\mathrm{x}.\tag{65}$$

The dual linear deformed derivative of (63), defined by Equation (25) of Ref. [33],

$$\mathfrak{D}\_q f(\mathbf{x}) = \frac{1}{\mathbf{x}^{1-q}} \frac{\mathbf{d}f(\mathbf{x})}{\mathbf{d}\mathbf{x}},\tag{66}$$

operates on the 𝑞-logarithm similarly to the nonlinear ole-derivative: 𝔇<sup>𝑞</sup> ln<sup>𝑞</sup> 𝑥 = 1/𝑥.

Generalized derivatives of a power (for the linear cases), or generalized powers (for the nonlinear cases), of 𝑞-numbers, are

$$\begin{array}{rcl} \mathrm{D\_i}\left(\_{q}\langle\mathbf{x}\rangle^n\right) &=& n \, \_{q}\langle\mathbf{x}\rangle^{n-1},\\ \mathrm{D\_0}\left(\langle\mathbf{x}\rangle\_q^n\right) &=& n \, \nwarrow \langle\mathbf{x}\rangle\_q^{n-1},\end{array} \tag{67}$$

$$\begin{array}{rcl} \widetilde{\mathcal{D}}\_{\mathbb{I}}\left(\langle\mathbf{x}\rangle\_{q}\otimes\_{\langle q\rangle}n\right) & = & \widetilde{\mathcal{D}}\_{\mathbb{I}}\left(\langle\mathbf{x}^{n}\rangle\_{q}\right) & = & n\,\mathbf{x}^{n-1},\\ \widetilde{\mathcal{D}}\_{\mathbb{O}}\left(\,\_{q}\langle\mathbf{x}\rangle\,\_{\langle q\rangle}\otimes n\right) & = & \widetilde{\mathcal{D}}\_{\mathbb{O}}\left(\,\_{q}\langle\mathbf{x}^{n}\rangle\right) & = & n\,\mathbf{x}^{n-1}. \end{array} \tag{68}$$

Second and higher deformed linear derivatives follow the usual rule, D 2 𝛿 𝑓 (𝑥) = D<sup>𝛿</sup> - D<sup>𝛿</sup> 𝑓 (𝑥) and so on, but for the deformed nonlinear cases, second order derivatives (and similarly for higher order derivatives) are defined as

$$
\widetilde{\mathbf{D}}\_{\delta}^{2}f(\mathbf{x}) = h\_{\delta}\left(f(\mathbf{x})\right) \frac{\mathbf{d}}{\mathbf{d}\mathbf{x}} \left[h\_{\delta}\left(f(\mathbf{x})\right) \frac{\mathbf{d}f(\mathbf{x})}{\mathbf{d}\mathbf{x}}\right].\tag{69}
$$

The product rule for the deformed linear derivatives is identical to the usual one, D<sup>𝛿</sup> 𝑓 (𝑥) 𝑔(𝑥) = D<sup>𝛿</sup> 𝑓 (𝑥) 𝑔(𝑥) + 𝑓 (𝑥) D<sup>𝛿</sup> 𝑔(𝑥) . The product rule for the deformed nonlinear derivatives is

$$\frac{1}{h\_{\delta}\left(f(\mathbf{x})\mathbf{g}(\mathbf{x})\right)}\widetilde{\mathbf{D}}\_{\delta}\Big(f(\mathbf{x})\,\mathbf{g}(\mathbf{x})\Big) = \left(\frac{1}{h\_{\delta}\left(f(\mathbf{x})\right)}\widetilde{\mathbf{D}}\_{\delta}\,f(\mathbf{x})\right)\mathbf{g}(\mathbf{x}) + f(\mathbf{x})\left(\frac{1}{h\_{\delta}\left(\mathbf{g}(\mathbf{x})\right)}\widetilde{\mathbf{D}}\_{\delta}\,\mathbf{g}(\mathbf{x})\right). \tag{70}$$

The deformed antiderivatives, or indefinite deformed integrals, associated to the linear deformed derivatives are defined by

$$\int\_{\left(\delta\right)}^{\infty} f(\mathbf{x}') \, \mathbf{d} \mathbf{x}' \; \equiv \quad \int^{\infty} f(\mathbf{x}') \, \mathbf{d}\_{\delta} \mathbf{x}' \,\tag{71}$$

$$=\int^{\infty} f(\mathbf{x'}) \, h\_{\delta}(\mathbf{x'}) \, \mathbf{dx'} \tag{72}$$

(the symbol (𝛿) within parenthesis refers to the deformation, and not a limit of integration), so

$$\mathbf{D}\_{\delta} \int\_{\left(\delta\right)}^{\infty} f(\mathbf{x'}) \, \mathbf{dx'} = f(\mathbf{x}) \tag{73}$$

and

$$\int\_{\left(\delta\right)}^{\infty} \mathbf{D}\_{\delta} f(\mathbf{x}') \, \mathbf{dx}' = f(\mathbf{x}) + C. \tag{74}$$

One possibility for defining the deformed antiderivatives associated to the nonlinear deformed derivatives, particularly following the definition used in [3] for the 𝛿 = ole case, is

$$\widetilde{\int\_{\left(\delta\right)}^{\infty}} f(\mathbf{x'}) \, \mathrm{d}\mathbf{x'} \equiv \int^{\infty} \frac{1}{h\_{\delta}(f(\mathbf{x'}))} f(\mathbf{x'}) \, \mathrm{d}\mathbf{x'},\tag{75}$$

A significant weakness with this option is that the following important properties are not satisfied:

$$\widetilde{\mathbf{D}}\_{\delta} \, \widetilde{\int\_{\left(\delta\right)}^{\infty}} f(\mathbf{x'}) \, \mathbf{dx'} \neq f(\mathbf{x}) \tag{76}$$

and

$$\widetilde{\int\_{\left(\delta\right)}^{\infty}} \widetilde{\mathcal{D}}\_{\delta} f(\mathbf{x}') \, \mathbf{dx}' \neq f(\mathbf{x}) + C. \tag{77}$$

#### **5. Entropy Generator**

The connection between entropies and derivatives was pointed out by Abe [35]. He observed that the Boltzmann-Gibbs entropy can be rewritten as (with 𝑘 = 1)

$$S\_1 = -\left. \frac{\mathbf{d}}{\mathbf{d}\alpha} g(\alpha) \right|\_{\alpha=1} \tag{78}$$

with

$$\log(\alpha) = \sum\_{i}^{W} p^{\alpha}. \tag{79}$$

He realized that 𝑆<sup>𝑞</sup> entropy can be similarly recast through the Jackson's derivative of a function 𝑓 (𝑥) [36]

$$\mathbf{D}\_q^{(0)} f(\mathbf{x}) \equiv \frac{f(q\mathbf{x}) - f(\mathbf{x})}{q\mathbf{x} - \mathbf{x}} \tag{80}$$

(the same deformed derivative of quantum calculus [28]; Newtonian derivative is recovered as the limiting case 𝑞 → 1), so

$$S\_q = -\mathbf{D}\_q^{(\mathbf{J})} g(\alpha) \Big|\_{\alpha=1}. \tag{81}$$

This property had been interpreted as expressing the association between Boltzmann-Gibbs entropy (𝑆1) to infinitesimal translations, and Tsallis entropy to finite dilations [2]. Abe applied this procedure a step further, and used a different derivative operator on 𝑔(𝛼), generating a new symmetric entropic functional 𝑆 𝑆 <sup>𝑞</sup> with 𝑞 ↔ 𝑞 −1 invariance. Following the same line, a two-parameter derivative operator was used to define a two-parameter 𝑆𝑞,<sup>𝑞</sup> 0

entropy, that recovers the previous 𝑆 𝑆 𝑞 , 𝑆<sup>𝑞</sup> and 𝑆<sup>1</sup> with convenient choices of the indices 𝑞 and 𝑞 0 [37].

All the eight deformed derivatives (58)–(61) applied on (79) result in 𝑆<sup>1</sup> entropy with a multiplying function of the parameter 𝑞: −D𝛿𝑔(𝛼)|𝛼=<sup>1</sup> = ℎ 𝜂 𝛿 (1) 𝑆1, where D<sup>𝛿</sup> represents any of the deformed (linear or nonlinear) derivatives (at this point we do not use the tilde for the nonlinear deformed derivatives), ℎ <sup>𝛿</sup> (1) is a particular value of the corresponding Equation (57), 𝜂 = −1 for the linear deformed derivatives, and 𝜂 = +1 for the nonlinear deformed derivatives. This is a consequence of the generalized derivatives being based on infinitesimal deformed translations, and the infinitesimal nature of the translation determines the entropy (except for a multiplicative constant), despite of the deformations.

A non-trivial result is obtained by inverting the procedure. Instead of applying one of the generalized derivatives on the generating function (79), we apply the ordinary Newtonian derivative on a generalized generating function:

$$S\_q^\delta = -\frac{\mathbf{d}}{\mathbf{d}\alpha} \mathbf{g}\_\delta(\alpha; q)\Big|\_{\alpha=1} \,. \tag{82}$$

The generalized generating functions are obtained through the four generalized powers, (22), (31), (41), (52): 𝑔ile (𝛼; 𝑞) = Í𝑊 𝑖 (𝑝<sup>𝑖</sup> <sup>∧</sup> [𝑞] 𝛼), 𝑔ole (𝛼; 𝑞) = Í𝑊 𝑖 (𝑝<sup>𝑖</sup> [𝑞]<sup>∧</sup> 𝛼), 𝑔iel(𝛼; 𝑞) = Í𝑊 𝑖 (𝑝<sup>𝑖</sup> ∧{𝑞} 𝛼), 𝑔oel(𝛼; 𝑞) = Í𝑊 𝑖 (𝑝<sup>𝑖</sup> {𝑞}<sup>∧</sup> 𝛼). The resulting functionals are

$$S\_q^{\rm elle} = \sum\_i q\left[-p\_i\right] \ln\left(\_q\left[p\_i\right]\right)\_\prime \tag{83a}$$

$$S\_q^{\rm ole} = -\sum\_i [p\_i]\_q \ln\left( [p\_i]\_q \right) - (1 - q) \sum\_i p\_i \left[ p\_i \right]\_q \ln\left( [p\_i]\_q \right),\tag{83b}$$

$$S\_q^{\rm el} = -\sum\_i p\_i \ln \left( \,\_q \{ p\_i \} \right) - (1 - q) \sum\_i p\_i \ln p\_i \ln \left( \,\_q \{ p\_i \} \right), \tag{83c}$$

$$S\_q^{\rm oel} = -\sum\_i p\_i^q \ln \left( \{ p\_i \}\_q \right). \tag{83d}$$

The use of the generalized derivatives essentially result in the same, −D𝛿𝑔<sup>𝛿</sup> (𝛼; 𝑞)|𝛼=<sup>1</sup> = ℎ 𝜂 𝛿 (1) 𝑆 𝛿 𝑞 , except for a multiplicative constant for the le cases, since ℎiel(1) = ℎoel(1) = 1. The certainty distribution originates non zero values for the le functionals: 𝑆 ile 𝑞 [𝑝<sup>𝑖</sup> = 1; 𝑝 <sup>𝑗</sup> = 0,∀𝑗 ≠ 𝑖] ≠ 0 for 𝑞 > 1, and, 𝑆 ole 𝑞 [𝑝<sup>𝑖</sup> = 1; 𝑝 <sup>𝑗</sup> = 0,∀𝑗 ≠ 𝑖] ≠ 0 for 𝑞 < 1, since <sup>𝑞</sup> [1] ≠ 1 and [1]<sup>𝑞</sup> ≠ 1. Additionally, the le functionals present negative values: 𝑆 ile <sup>𝑞</sup> presents negative values for 𝑞 < 1, 𝑆 ole <sup>𝑞</sup> presents negative values for 𝑞 > 1. Besides, there are ranges of values of 𝑞 for which neither 𝑆 ile <sup>𝑞</sup> nor 𝑆 ole <sup>𝑞</sup> present a definite concavity (two instances: 𝑞 = 2.4, for ile; 𝑞 = 2.3, for ole). These are severe drawbacks and consequently (83a) and (83b) can not be considered as legitimate entropic forms.

The iel-functional 𝑆 iel 𝑞 fails on the expansibility property for 𝑞 < 1 (adding events of zero probability), since 𝑞<1{0} is not defined. For 𝑞 > 1, it is expansible, non negative and the certainty distribution (𝑝<sup>𝑖</sup> = 1; 𝑝 <sup>𝑗</sup> = 0, ∀𝑗 ≠ 𝑖) implies 𝑆 iel <sup>𝑞</sup> = 0, so, (83c) is admissible as an entropic form for 𝑞 > 1.

The oel-functional (83d) is the nonadditive entropy 𝑆<sup>𝑞</sup> (see Equation (13b)), vastly considered in the literature. This result permits to amend a previous statement: 𝑆<sup>𝑞</sup> entropy, that is associated to finite dilations, can also be associated to infinitesimal translations, but in a deformed space expressed by the oel-power. Figure 4a illustrates the concavity for the two admissible entropic functionals, Equation (83c) with 𝑞 > 1 and Equation (83d), for a two-state system. Figure 4b illustrates el-entropies as monotonically increasing functions of the number of states 𝑊 for the equiprobable distribution, 𝑝<sup>𝑖</sup> = 1/𝑊, ∀𝑖, with the abscissa in logarithm scale, for which the usual case appears as a straight line.

**Figure 4.** (**a**) el-Entropies for a two-state system. 𝑆 iel 𝑞 (83c) for 𝑞 = 2 (red); 𝑆 oel <sup>𝑞</sup> = 𝑆<sup>𝑞</sup> (83d), for 𝑞 = 0.5 (green), 𝑞 = 2 (blue); 𝑆<sup>1</sup> (black). 𝑆𝑞 entropy is convex for 𝑞 < 0, see [1]. (**b**) el-Entropies for equiprobable states as a function of 𝑊. Abscissa in log scale, for which the Boltzmann case is a straight line (black). 𝑆 iel 𝑞 for 𝑞 = 2 (red), 𝑆 oel 𝑞 for 𝑞 = 0.5 (green), 𝑞 = 2 (blue).

#### **6. Final Remarks**

A forerunner of the transformations given by Equation (10) is the relation between Rényi entropy, 𝑆 R <sup>𝑞</sup> = (1 − 𝑞) −1 ln Í<sup>𝑊</sup> 𝑖 𝑝 𝑞 𝑖 , and Tsallis entropy (2) (see Equation (8) of Ref. [1]), 𝑆 R <sup>𝑞</sup> = [𝑆𝑞]𝑞, and, equivalently, 𝑆<sup>𝑞</sup> = <sup>𝑞</sup> [𝑆 R 𝑞 ]. Another instance of the transformation represented by the ile-number (10a) appeared in Equation (22) of Ref. [10] and allowed the generalization of trigonometric functions. The ole-number <sup>𝑞</sup> [𝑥] appeared as Equation (5) of Ref. [38], as the scaling factor of the generalized Kolmogorov-Nagumo average for expressing the Rényi entropy. A former example of connecting deformed numbers with deformed differential operators have appeared in Ref. [39], with the transformation (10a) and the deformed differential (59a), establishing an equivalence between a positiondependent mass system in a usual space and a constant mass within a deformed space. These works have been recently extended to the deformed version of the Fokker-Planck equation for inhomogeneous medium with position-dependent mass [33]. In addition, the use of the iel-number, Equation (11a), to the generalization of the Riemann's zeta function has been recently advanced in [40].

Expressions with operations belonging to one class of 𝑞-algebra may result in operations belonging to a different class. Some instances: the following are generalizations of the logarithm of a product as a sum of logarithms ln(𝑥𝑦) = ln 𝑥 + ln 𝑦 :

ln<sup>𝑞</sup> 𝑥𝑦 = ln<sup>𝑞</sup> 𝑥 [𝑞]⊕ ln<sup>𝑞</sup> 𝑦, (84a)

$$\ln\_q \left( \mathbf{x}\_{\left[q\right]} \otimes \mathbf{y} \right) = \ln\_q \mathbf{x} + \ln\_q \mathbf{y},\tag{84b}$$

$$\ln\left(x\otimes\_{\left[q\right]}\mathbf{y}\right) = \ln x \ll\_{\left[q\right]} \oplus \ln \mathbf{y},\tag{84c}$$

$$
\ln\left(\mathbf{x}\,\,\,\middle|\,\mathbf{y}\,\,\mathbf{y}\,\,\right) = \ln\mathbf{x}\,\,\oplus\_{[q]}\ln\mathbf{y}.\tag{84d}
$$

Generalizations of the logarithm of a power, ln 𝑥 <sup>𝑦</sup> = 𝑦 ln 𝑥, are

$$\ln\_q \left( \mathbf{x}\_{\|\mathbf{g}\|} \otimes \mathbf{y} \right) = \mathbf{y} \ln\_q \mathbf{x}.\tag{85a}$$

$$\ln\left(\mathbf{x}\otimes\_{\mathbf{H}}\mathbf{y}\right) = \mathbf{y}\restriction\_{\left[q\right]}\odot\operatorname{ln}\mathbf{x},\tag{85b}$$

$$
\ln \left( \mathbf{x}\_{\lfloor q \rfloor} \otimes \mathbf{y} \right) = \mathbf{y} \odot\_{\lceil q \rceil} \ln \mathbf{x}.\tag{85c}
$$

The counterpart of these expressions are generalizations of the exponential of a sum as a product of exponentials, e𝑥+<sup>𝑦</sup> = e 𝑥 e 𝑦 :

$$\mathbf{e}\_q^{\mathbf{x}\_{\{q\}} \oplus \mathcal{Y}} = \mathbf{e}\_q^{\mathbf{x}} \mathbf{e}\_q^{\mathbf{y}} \,\tag{86a}$$

$$\mathbf{e}\_q^{\mathbf{x}+\mathbf{y}} = \mathbf{e}\_q^{\mathbf{x}} \llcorner \mathbf{e}\_q^{\mathbf{y}} \llcorner \tag{86b}$$

$$\mathbf{e}^{\mathbf{x}\ (q)\oplus\mathcal{Y}} = \mathbf{e}^{\mathbf{x}} \otimes\_{\left[q\right]} \mathbf{e}^{\mathbf{y}},\tag{86c}$$

$$\mathbf{e}^{\ \mathbf{x}\ \oplus\_{\{q\}}\mathcal{Y}} = \mathbf{e}^{\ \mathbf{x}}\ \ngg\mathbf{e}^{\mathcal{Y}},\tag{86d}$$

and the power of an exponential as the exponential of a product (e 𝑥 ) <sup>𝑦</sup> = e 𝑦 𝑥

$$\mathbf{e}\_q^{\mathbf{x}} \llcorner\_{\|\mathbf{q}\|} \otimes \mathbf{y} = \mathbf{e}\_q^{\mathbf{y}\mathbf{x}} \llcorner\tag{87a}$$

:

$$\mathbf{e}^{\mathsf{x}} \otimes\_{\left[q\right]} \mathbf{y} = \mathbf{e}^{\mathsf{y} \cdot \left[q\right] \odot \left[\mathbf{x}\right]}\tag{87b}$$

$$\mathbf{e}^{\mathbf{x}} \llcorner\_{\|q\|} \otimes \mathbf{y} = \mathbf{e}^{\mathbf{y} \odot\_{\{q\}} \mathbf{x}}.\tag{87c}$$

Relations (84) are also valid for the logarithm, or for the 𝑞-logarithm, of a ratio, simply replacing ordinary or general products by ordinary or general ratios, and ordinary or general sums by ordinary or general differences. Similarly, relations (86) are also valid for the exponential, or for the 𝑞-exponential, of a difference, by replacing the operators accordingly. Sums of 𝑞-logarithm functions, Equation (84b), appeared in the literature prior to the definition of the 𝑞-product [3] —here called the oel-product, Equation (48)— within the context of the generalization of Boltzmann's molecular chaos hypothesis and the 𝐻 Theorem, see Equation (16) of Ref. [41] and Equation (22) of Ref. [42]. The oel-product has shown to be a key ingredient to the generalization of the Fourier transform and the central limit theorem [43,44]. It is allowed to think that the present algebras may be relevant within these contexts.

Equation (85a) is the one referred to in the Introduction, that makes 𝑆<sup>𝑞</sup> extensive: consider a composed system for which its subsystems have 𝑊<sup>𝑖</sup> > 1 available states. If they are independent, the number of available states of the composed system is 𝑊 = Î𝑁 <sup>𝑖</sup> 𝑊<sup>𝑖</sup> , and, besides, if they are identical, 𝑊 = 𝑊 <sup>𝑁</sup> 1 . Correlations between the subsystems lead to a smaller number of available states for the composed system, and particular strong correlations represented by 𝑊 = 𝑊<sup>1</sup> {𝑞}<sup>∧</sup> 𝑁, with 𝑞 < 1, makes 𝑆<sup>𝑞</sup> = 𝑘 ln<sup>𝑞</sup> 𝑊 = 𝑁 𝑘 ln<sup>𝑞</sup> 𝑊1. This is a non trivial case of extensivity.

Different possibilities for generating rules of arithmetic operations, instead of (15), are <sup>𝑞</sup> [𝑥] [𝑞]# <sup>𝑞</sup> [𝑦] <sup>=</sup> <sup>𝑞</sup> [<sup>𝑥</sup> ◦ <sup>𝑦</sup>], [𝑥]<sup>𝑞</sup> #[𝑞] [𝑦]<sup>𝑞</sup> <sup>=</sup> [<sup>𝑥</sup> ◦ <sup>𝑦</sup>]𝑞; These patterns are used in References [27,40].

Weberszpil, Lazo, and Helayël-Neto [45] have shown that the linear ole-derivative (59a) is the first order expansion of the Hausdorff derivative. Whether the other generalized derivatives are also connected to fractal derivatives and fractal metrics remains to be investigated.

Two of the functionals obtained with the recipe of applying the ordinary derivatives to a generalized version of the generating function, (82), result in admissible entropic forms corresponding to the el-class: 𝑆 iel 𝑞 (83c), and 𝑆 oel 𝑞 (83d). The other functionals (83a) and (83b) are not admissible to be considered as entropies, but this does not mean that the le-algebras or le-calculus they are based on are not feasible for other applications.

Extension to the complex domain of the deformed numbers still remains to be explored. Two-parameter generalization are not addressed here, we just advance a few lines. Twoparameter generalizations of numbers in accordance with the present developments are given by

$$\mathbf{n}\_q[\mathbf{x}]\_{q'} \equiv \quad \text{[ $\mathbf{x}$ ]}\_{q,q'} = \text{ln}\_q \exp\_{q'}(\mathbf{x}) , \tag{88a}$$

$$\mathbf{a}\_{q}\{\mathbf{x}\}\_{q'} \equiv \mathbf{ (x)}\_{q,q'} = \exp\_{q}(\mathbf{ln}\_{q'}\mathbf{x}).\tag{88b}$$

The use of the relatively uncommon subscripted prefix to represent the two parameter deformed number may be avoided, since there is no ambiguity with the symbol h𝑥i𝑞,<sup>𝑞</sup> 0. Two-parameter arithmetic operators follow straightforwardly:

$$\text{tr}\, \bigcirc\_{\langle q,q'\rangle} \text{y} = \left\langle \langle \mathbf{x} \rangle\_{q',q} \circ \langle \mathbf{y} \rangle\_{q',q} \right\rangle\_{q,q'} \tag{89}$$

for which, of course, all the previous developments are particular cases. The two-parameter algebra of Ref. [15] is obtained through a different generating rule than (89): it derives from the two-parameter generalized logarithm and exponential functions [14] (Equations (16) and (17) of [15]).

It also comes naturally the two-parameter derivative D𝑞,<sup>𝑞</sup> <sup>0</sup> 𝑓 (𝑥), with deformation on both the independent and dependent variables. A broader generalization of the derivatives can be defined by using not only deformations on the variation of the independent and dependent variable, but also on the ratio among them, with three parameters, in a rather intricate way, say: 𝑞 for the deformed differential of the independent variable, 𝑞 0 for the deformed differential of the dependent variable, and 𝑞 00 for the deformed ratio between them. A particular case with 𝑞 = 𝑞 0 = 𝑞 00 was shown in Ref. [46], and, more recently, in Ref. [47].

Finally, all the present scenario stands on the pair of 𝑞-logarithm/𝑞-exponential functions, inverse of each other. The whole picture may be differently deformed by using different continuous, monotonous, and invertible pair of functions, in agreement with Equation (8).

**Author Contributions:** Conceptualization, E.P.B. and B.G.d.C.; methodology, E.P.B. and B.G.d.C.; software, E.P.B.; writing—original draft preparation, E.P.B.; writing—review and editing, E.P.B. and B.G.d.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This work was partially supported by National Institute of Science and Technology for Complex Systems (INCT-SC). E.P.B. thanks C. Tsallis and Si Hyung Joo, and both authors thank I.S. Gomez for stimulating discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following prefix abbreviations are used in this manuscript:


#### **Appendix A. A Note on Notations—Explicit Expressions**

The peculiar notation adopted in the present work is conceived for compactness, once the explicit forms of some equations may be large or cumbersome. The notation for the generalized numbers is inspired in the 𝑄*-analog of* 𝑛 [28], a generalized number represented within square brackets (7). The four classes of generalized numbers are grouped into two categories, one, the 'le' category, uses the generalized exponential (or its ordinary version) as argument of the ordinary logarithm (or its generalized version), and the other, the 'el' category, the other way around. We have used square brackets for the former, and curly brackets for the latter. Some ambiguity is unfortunately unavoidable, as square and curly brackets are also used with their usual meanings, and the reader must resolve it by the context. We refer to them as 'le' or 'el' concerning the order in which the logarithm/exponential functions appear. Despite of the unusualness, or even possibly strangeness, of the notation, we consider that it may help identify the classes more promptly than something like 'type 1', 'type 2', etc. Differently from the generalized numbers, we use the subscripts enclosed by their corresponding brackets, when dealing with generalized arithmetic operators, so the reader can easily identify the object being generalized, if it is a number or an operator. We have chosen prefix and postfix subscripts, to avoid using superscripts. These pair of subscripts may play a simplifying role if used appropriately, as

illustrated by Equation (9). In the following we present explicit forms of some expressions, for the benefit of the interested reader. The notation [·]<sup>+</sup> ≡ max{0, ·} is used here.

ile-number (Equation (10a))

$$[x]\_q = \frac{1}{1-q} \ln \left( 1 + (1-q)x \right)\_+ \tag{A1}$$

ole-number (Equation (10b))

$$\mathbf{e}\_q[\mathbf{x}] = \frac{\mathbf{e}^{(1-q)\cdot \mathbf{x}} - 1}{1 - q}. \tag{A2}$$

iel-number (Equation (12a))

$$\{\mathbf{x}\}\_q = \text{sign}(\mathbf{x}) \, \exp\left(\frac{|\mathbf{x}|^{1-q} - 1}{1-q}\right). \tag{A3}$$

oel-number (Equation (12b))

$$\mathbf{a}\_q\{\mathbf{x}\} = \text{sign}(\mathbf{x}) \left( \mathbf{1} + (1 - q) \ln|\mathbf{x}| \right)^{1/(1 - q)}\_+. \tag{A4}$$

ile-addition, ile-subtraction (Equations (16) and (18))

$$\text{tr}\, \oplus\_{\{q\}} \text{y} \quad = \, \frac{1}{1-q} \ln \left[ 1 + (1-q) \left( \frac{\mathbf{e}^{(1-q)\times} - 1}{1-q} \pm \frac{\mathbf{e}^{(1-q)y} - 1}{1-q} \right) \right]\_+ \,\tag{A5}$$

$$=\frac{1}{1-q}\ln\left[\mathbf{e}^{(1-q)\cdot\mathbf{x}}\pm\mathbf{e}^{(1-q)\cdot\mathbf{y}}\mp 1\right]\_{+}.\tag{A6}$$

ile-multiplication (Equation (19))

$$\propto \otimes\_{\left[q\right]} \mathbf{y} = \frac{1}{1-q} \ln \left[ 1 + \frac{\left(\mathbf{e}^{(1-q)\cdot\mathbf{x}} - 1\right)\left(\mathbf{e}^{(1-q)\cdot\mathbf{y}} - 1\right)}{1-q} \right]\_+. \tag{A7}$$

ile-division (Equation (20))

$$\text{tr}\,\otimes\_{\{q\}}\mathbf{y} = \frac{1}{1-q}\ln\left[1 + (1-q)\frac{\mathbf{e}^{(1-q)\cdot\mathbf{x}} - 1}{\mathbf{e}^{(1-q)\cdot\mathbf{y}} - 1}\right]\_+.\tag{A8}$$

ile-power (Equation (22))

$$\propto \otimes\_{\left[q\right]} \mathbf{y} = \frac{1}{1-q} \ln \left[ 1 + (1-q) \left( \frac{\mathbf{e}^{(1-q)\cdot x} - 1}{1-q} \right)^{\mathbf{y}} \right]\_+ \quad (\mathbf{x} > \mathbf{0}).\tag{A9}$$

ole-addition (see Equation (25))

ole-subtraction (see Equation (27))

ole-multiplication (Equation (28))

$$\text{var}\_{[q]} \otimes \mathbf{y}\_{\text{-}} = \frac{\exp\left[\frac{\ln[1 + (1 - q)\boldsymbol{x}]\_{+} \ln[1 + (1 - q)\boldsymbol{y}]\_{+}}{1 - q}\right] - 1}{1 - q},\tag{A10}$$

$$=\frac{\left[1+(1-q)\chi\right]\_{+}^{\frac{1}{1-q}\ln\left[1+(1-q)\chi\right]\_{+}}-1}{1-q},\tag{A11}$$

$$=\frac{\left[1+(1-q)\chi\right]\_{+}^{\frac{1}{1-q}\ln\left[1+(1-q)\chi\right]\_{+}}-1}{1-q}.\tag{A12}$$

ole-division (Equation (29))

$$\text{tr}\_{\lfloor q \rfloor} \otimes \mathbf{y} = \frac{\exp\left[ (1 - q) \frac{\ln\left[ 1 + (1 - q) \ge \lfloor \mathbf{x} \rfloor\_{\star} \right]}{\ln\left[ 1 + (1 - q) \ge \lfloor \mathbf{y} \rfloor\_{\star} \right]} \right] - 1}{1 - q}. \tag{A13}$$

¬

ole-power (Equation (31))

$$\text{tr}\,\,\|q\|\otimes\,\mathbf{y} = \frac{\exp\left[ (1-q)^{1-\chi}\ln^{\chi}\left[1+(1-q)\chi\right]\_{+}\right]-1}{1-q}, \quad (\text{x}>0).\tag{A14}$$

iel-addition, iel-subtraction (Equations (34) and (36))

$$\begin{array}{rcl} \mathbf{x} \ominus\_{\mathsf{[d]}} \mathbf{y} &=& \operatorname{sign}(\mathbf{x} \pm \mathbf{y}) \\\\ & \times & \exp\left(\frac{\left|\operatorname{sign}(\mathbf{x}) \left[1 + (1 - q)\ln|\mathbf{x}|\right]\right|\_{\star}^{\frac{1}{1 - q}} \pm \operatorname{sign}(\mathbf{y}) \left[1 + (1 - q)\ln|\mathbf{y}|\right]\_{\star}^{\frac{1}{1 - q}} \right|^{1 - q} - 1}{1 - q} \end{array} \tag{A15}$$

iel-multiplication (Equation (38))

«

$$\text{Max}\_{\|\mathbf{y}\|} \text{ y = sign(xy)} \exp\left( \frac{\left| \left[ 1 + (1 - q) \ln|\mathbf{x}| \right]\_{+} \left[ 1 + (1 - q) \ln|\mathbf{y}| \right]\_{+} \right|}{1 - q} \right). \tag{A16}$$

iel-division (Equation (39))

$$\mathbf{x} \oslash\_{\|\mathbf{d}\|} \mathbf{y} = \text{sign}(\mathbf{x}/\mathbf{y}) \exp\left[ \left( 1 - q \right)^{-1} \left( \left| \left[ \frac{1 + (1 - q) \ln|\mathbf{x}|}{\left[ 1 + (1 - q) \ln|\mathbf{y}| \right]\_{+}} \right]\_{+} \right| - 1 \right) \right]. \tag{A17}$$

iel-power (Equation (41))

$$\text{fix}\,\otimes\_{\left[q\right]}\text{y} = \text{sign}(x)\,\exp\left(\frac{\left|\left[1+(1-q)\ln|x|\right]\_{+}^{y}\right|-1}{1-q}\right)\quad(x>0).\tag{A18}$$

oel-addition, oel-subtraction (Equations (44) and (45))

$$\begin{aligned} \text{x } \sqrt{\oplus} \text{ y } &= \text{ } \text{sign}(\mathbf{x} \pm \mathbf{y})\\ &\times \left[1 + (1 - q) \ln \left| \operatorname{sign}(\mathbf{x}) \exp\left(\frac{|\mathbf{x}|^{1 - q} - 1}{1 - q}\right) \pm \operatorname{sign}(\mathbf{y}) \exp\left(\frac{|\mathbf{y}|^{1 - q} - 1}{1 - q}\right) \right| \right]\_+^{\frac{1}{1 - q}}. \end{aligned} \tag{A19}$$

oel-multiplication (see Equation (48)) oel-division (see Equation (49))

oel-power (Equation (52))

$$\text{If } \mathbf{x}\_{\|\mathbf{f}\|} \otimes \mathbf{y} = \left(\text{sign}(\mathbf{x})\right)^{\mathbf{y}} \left[\mathbf{y} \, \middle|\, \mathbf{x}\big|\, ^{1-q} - (\mathbf{y} - \mathbf{1})\right]\_{+}^{\frac{1}{1-q}}, \quad (\mathbf{x} > \mathbf{0}). \tag{A20}$$

𝑆 ile 𝑞 functional (Equation (83a))

$$S\_q^{\rm ile} = \sum\_{i}^{W} \frac{\mathbf{e}^{-(1-q)p\_i} - 1}{1 - q} \ln \left[ \frac{\mathbf{e}^{(1-q)p\_i} - 1}{1 - q} \right]. \tag{A21}$$

𝑆 ole 𝑞 functional (Equation (83b))

$$\begin{split} S\_q^{\text{ole}} &= -\sum\_{i}^{W} \frac{\ln\left[1 + (1 - q)p\_i\right]}{1 - q} \ln\left[\frac{\ln\left[1 + (1 - q)p\_i\right]}{1 - q}\right] \\ &- \sum\_{i}^{W} p\_i \ln\left[1 + (1 - q)p\_i\right] \ln\left[\frac{\ln\left[1 + (1 - q)p\_i\right]}{1 - q}\right] \end{split} \tag{A22}$$

𝑆 iel 𝑞 functional (Equation (83c))

$$S\_q^{\rm el} = -\sum\_{i}^{W} \frac{p\_i}{1 - q} \ln\left[1 + (1 - q)\ln p\_i\right] - \sum\_{i}^{W} p\_i \ln p\_i \ln\left[1 + (1 - q)\right] \ln p\_i\right\} \tag{A23}$$

𝑆 oel 𝑞 functional (Equation (83d), see also Equation (2))

#### **References**


### *Article* **Hot Spots in the Weak Detonation Problem and Special Relativity**

**Satyanad Kichenassamy**

Laboratoire de Mathématiques de Reims (CNRS, UMR9008), Université de Reims Champagne-Ardenne, Moulin de la Housse, B.P. 1039, CEDEX 2, F-51687 Reims, France; satyanad.kichenassamy@univ-reims.fr

**Abstract:** *Problem statement:* The initiation of a detonation in an explosive gaseous mixture in the high activation energy regime, in three space dimensions, typically leads to the formation of a singularity at one point, the "hot spot". It would be suitable to have a description of the physical quantities in a full neighborhood of the hot spot. *Results of this paper:* (1) To achieve this, it is necessary to replace the blow-up time, or time when the hot spot first occurs, by the blow-up surface in four dimensions, which is the set of all hot spots for a class of observers related to one another by a Lorentz transformation. (2) A local general solution of the nonlinear system of PDE modeling fluid flow and chemistry, with a given blow-up surface, is obtained by the method of Fuchsian reduction. *Advantages of this solution:* (i) Earlier approximate solutions are contained in it, but the domain of validity of the present solution is larger; (ii) it provides a signature for this type of ignition mechanism; (iii) quantities that remain bounded at the hot spot may be determined, so that, in principle, this model may be tested against measurements; (iv) solutions with any number of hot spots may be constructed. The impact on numerical computation is also discussed.

**Keywords:** weak detonation; high activation regime; nonlinear PDEs; Fuchsian reduction analysis; Lorentz transformation; blow-up; hot spot; chemically reactive flows

**PACS:** 11.30.Cp; 47.40.Rs; 82.33.Vx

**MSC:** 80A25; 35Q07; 35B44

#### **1. Introduction**

Observations of ignitions in explosive gaseous systems indicate that the process is initiated in localized regions called reaction centers, or "hot spots". This phenomenon is ubiquitous, from the combustion engine to stellar explosions [1–4], including engines using carbon-free fuels such as ammonia or hydrogen [5–7]. A given observer typically sees one of these hot spots to first form at a definite point in space and time. We have pointed out earlier [8] (Section 10.4) that the set of hotspots in the weak detonation problem forms a blow-up pattern in the sense of Fuchsian reduction [9]. Here, we give a more detailed description of the solutions of the relevant system of nonlinear PDEs, indicating quantities that could be measurable. We also show that the notion of blow-up time, namely, the time when a singularity first occurs in a given inertial frame, is physically meaningless. As a consequence, the hot spot in the laboratory frame loses physical significance and must be replaced by the set of all hot spots seen by different observers. They form the blow-up surface in the sense of reduction theory [8,10], namely, the set of points at which a given solution presents a singularity. As we have shown in related contexts, reduction theory not only gives a precise description of singularity formation, it also enables its control by boundary action [11] and accounts for the possible concentration of energy or other quantities at the different blow-up points [12].

We first review the modeling assumptions (Section 2) and earlier results on the hot spot problem showing how our approach improves upon them (Section 3). Then, we perform a reduction analysis of the model, leading to a local representation of the general

**Citation:** Kichenassamy, S. Hot Spots in the Weak Detonation Problem and Special Relativity. *Axioms* **2021**, *10*, 311. https://doi.org/10.3390/ axioms10040311

Academic Editor: Hans J. Haubold

Received: 4 October 2021 Accepted: 15 November 2021 Published: 19 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

solution (Sections 4 and 5). The effect of Lorentz transformations on blow-up patterns is then described in a general set-up, common to all applications (Section 6). Section 7 is devoted to a discussion of the results and outlines perspectives for further work. Section 8 summarizes the conclusions. The derivation of the equations from first principles, and their non-dimensionalization, are given in Appendices A and B respectively.

The main new results are: (i) The hot spot first recorded by a given observer is not the cause of ignition. (ii) The present solution of the relevant equations has a wider domain of validity than earlier ones, that are recovered as special limiting cases. (iii) Our approach provides a set of measurable quantities that may be viewed as a signature for this type of ignition mechanism. (iv) Special relativity is relevant for purely geometric and kinematical reasons, even when the relativistic effects on the chemistry are negligible.

#### **2. The Weak Detonation Problem: Modeling Assumptions**

Let us first review the modeling assumptions that seem to represent the current consensus [13]. For background information, see also references [1,4,14,17–30,32,34].

Since ignition occurs on a very small scale, it is reasonable to assume that dissipative and convective effects may be neglected. In that case, near each hot spot, the behavior of the gas may be assumed to be close to a spatially homogeneous explosion. This leads to the following assumptions:


Assumption (A1) is usually expressed by saying that the detonation is "quasisteady". Since fluid elements have no time to drift appreciably away from the hot spot, it is usually called a "weak detonation". This detonation is similar to the "weak detonation" represented by a nearly vertical line connecting two points on two Hugoniot curves in the *p*–*v* diagram. See [21] (p. 19) for details on terminology. Because of (A2), (A4) and (A5), one focuses on the reactive Euler equations with the Arrhenius reaction rate, recalled in Appendix A. Assumptions (A2)–(A4) suggest a choice of scales, leading to a non-dimensional form of the equations, namely, the system (A6) in Appendix B. It is obtained in two steps. One first introduces non-dimensional variables *t* ∗ and **x** ∗ = (*x* ∗ 1 , *x* ∗ 2 , *x* ∗ 3 ) and dependent variables **u** ∗ , *T* ∗ , *p* ∗ , *ρ* ∗ and *y* ∗ , that represent velocity, temperature, pressure, density and reactant mass fraction, respectively. One also introduces the dimensionless inverse activation energy *θ*, the ratio of specific heats *γ* and the non-dimensional heat release parameter *β*. Second, one expands **u** ∗ , *T* ∗ , *p* ∗ , *ρ* ∗ and *y* ∗ in powers of *θ* −1 in the limit when *θ* is large.

$$\mathbf{u}^\* \quad = \begin{array}{c} \mathbf{u}\_1 \\ \frac{\theta}{\theta} + \mathcal{O}(\frac{1}{\theta^2}) \end{array} \tag{1a}$$

$$T^\* = \left. 1 + \frac{T\_1}{\theta} + \mathcal{O}(\frac{1}{\theta^2}) \right. \tag{1b}$$

$$p^\* = \left. 1 + \gamma \frac{p\_1}{\theta} + \mathcal{O}(\frac{1}{\theta^2}) \right. \tag{1c}$$

$$\rho^\* \quad = \quad 1 + \frac{\rho\_1}{\theta} + \mathcal{O}(\frac{1}{\theta^2}) \tag{1d}$$

$$y^\* = 1 + \frac{y\_1}{\theta} + \mathcal{O}(\frac{1}{\theta^2}).\tag{1e}$$

The factor *γ* was introduced in the expansion of *p* ∗ to make formulae simpler. This expansion reflects assumptions (A1) and (A3); *θ* is large and the variables describe a nearly uniform state. Neglecting the terms in <sup>1</sup> *θ* 2 , it follows that the specific internal energy *ε* and the total specific energy *e* are comparable, because the squared velocity is of order <sup>1</sup> *θ* <sup>2</sup> and

$$e = \varepsilon\_0 + \frac{1}{\theta} (c\_v T\_1 + q y\_1)\_{\prime\prime}$$

where *ε*<sup>0</sup> is the specific internal energy in the reference state.

Inserting (1) into (A6) and keeping contributions of order <sup>1</sup> *θ* , we obtain that the first correction to the uniform state is governed by the following nonlinear system, in which the dependent variables are (*ρ*1, **u**1, *p*1, *T*1, *y*1), that represent the departure of the nondimensionalized density, fluid velocity, pressure, temperature and reactant mass fraction, respectively, from their values in the reference, constant state.

$$
\gamma p\_1 \quad = \quad \rho\_1 + T\_1 \tag{2a}
$$

$$
\partial\_{l^\*} \rho\_1 + \text{div}^\* \mathbf{u}\_1 = \begin{array}{ll} 0 \\ \end{array} \tag{2b}
$$

$$
\partial\_{\mathbf{l}^\*} \mathbf{u}\_{\mathbf{l}} + \nabla^\* p\_{\mathbf{l}} \quad = \quad \mathbf{0} \tag{2c}
$$

$$\partial\_{l^\*} y\_1 \quad = \quad -\frac{1}{\beta} \exp T\_1 \tag{2d}$$

$$
\partial\_{t^\*} T\_1 \ = \ (\gamma - 1)\partial\_{t^\*} \rho\_1 + \gamma \exp T\_1. \tag{2e}
$$

More precisely, (A6a)–(A6c) and (A6e) directly imply (2a)–(2d); on the other hand, Equation (A7) yields, at leading order, the simple relation

$$\partial\_{t^\*}\left[T\_1 - (\gamma - 1)p\_1 + \beta y\_1\right] = 0.\tag{3}$$

Using (2d), Equation (3) is equivalent to (2e).

It suffices to determine *T*1, **u**<sup>1</sup> and *ρ*<sup>1</sup> from (2b)–(2d); one may then obtain *p*<sup>1</sup> from (2a) and *y*<sup>1</sup> from (3).

Limits on the validity of this expansion may be estimated as follows:


The solutions of system (2) blow up in finite time and the final phase of rapid increase in temperature, just before singularity formation, is called *thermal runaway*. The hot spot for a given observer is thus very close to the point in spacetime where the first singularity of the system appears in his/her frame. However, it is merely part of a weak detonation locus, or blow-up set, described in this paper by the equation *t* = *ψ*(*x*, *y*, *z*), where *ψ* depends on space variables. This set represents "a locus of nonuniform ignition times, resulting from the nonuniform initial state, in which each fluid particle released its chemical energy at a different time" [28] (p. 1243). The hot spot in a given system corresponds to a spacetime point (*t*0, **x**0) such that *ψ* becomes minimum at **x**0. We shall see that this point of spacetime is not a Lorentz invariant.

Two aspects of this problem are somewhat unusual and make many standard tools in the study of partial differential equations inappropriate. First, this initial phase of the process does not propagate as a wave, even though the equations are of hyperbolic type. Indeed, the hot spots observed by different observers are not causally related to one another. Ignition leads to the formation of a blow-up pattern in the sense of [9]. The second difficulty is that the temperature does not become infinite; the model ceases to be valid as soon as the variables *ρ*1, etc., become of the order of *θ*, or when the reactant is depleted (*y* = 0). Therefore, information on the limit as *T*<sup>1</sup> goes to infinity is indeed irrelevant; we need expressions that make sense for a large, but finite *T*1. Before we obtain such expressions, let us review earlier approaches.

#### **3. Earlier Results**

Numerical work is made difficult by the blow-up singularity [15,22]. Therefore, perturbative approaches are preferred. Three perturbative methods of solution have been applied to this problem. The first method [16] consists, when there is only one space variable, called *x* ∗ , in introducing a shifted dimensionless time variable ˜*t* = *t* ∗ *<sup>d</sup>* − *t* ∗ , with *t* ∗ *d* constant and a self-similar variable *s* = *x* <sup>∗</sup>/˜*t*, such that the hot spot develops at time ˜*t* = 0, when the space coordinate vanishes. Thus, the fact that ignition at different points occurs at different times is neglected in the vicinity of the first hot spot. This leads to expansions involving ˜*t n* (ln ˜*t*) *<sup>m</sup>*, where *n* and *m* are integers. The expansions break down when *<sup>s</sup>* <sup>=</sup> <sup>O</sup>(˜*<sup>t</sup>* <sup>−</sup>1/2); this self-similar scaling "does not quite span the entire hot spot" [16] (p. 439) and variables of the form *x* <sup>∗</sup>/˜*t <sup>m</sup>*, with appropriate *m*, must be introduced. The second method [14] considers the limit in which 1 (*γ* − 1)/*γ θ* −1 , in one space dimension; in this limit, (2e) decouples from the other equations. The hot spot is again investigated using a self-similar variable, leading to a further restriction on the validity of the expansion. The third approach [28] is to insert a small parameter *µ* in front of the space derivatives, making the spatial derivatives less important than the time derivatives and using *µ* as the expansion parameter. The result is a formal expansion involving *τ* = *ψ*(*x* ∗ , *µ*) − *t* ∗ , where *ψ*(*x* ∗ , *µ*), representing the ignition time at the location *x* ∗ , is itself expanded in powers of *µ*. The method may be extended to three-dimensional situations. In all cases, the spatial gradient of *ψ* must have length less than unity. In addition, one requires *θ* <sup>−</sup><sup>1</sup> *<sup>µ</sup>* <sup>=</sup> *<sup>θ</sup>* 1/3 1, see [28] (p. 1258). The solution is not uniform as *<sup>τ</sup>* <sup>→</sup> 0.

The upshot of the above results is the following: The initiation of a weak detonation appears to be well represented by a solution of system (2), with a logarithmic singularity on a set of the form *t* ∗ = *ψ*(**x** ∗ ). Ignition appears to start first at a spacetime point (*t* ∗ 0 , **x** ∗ 0 ), such that *t* ∗ <sup>0</sup> = *ψ*(**x** ∗ 0 ) and *ψ*(**x** ∗ ) has a minimum for **x** ∗ = **x** ∗ 0 , but only if one restricts one's attention to a neighborhood of **x** ∗ 0 that shrinks as *t* ∗ tends to the blow-up time *t* ∗ <sup>0</sup> = *ψ*(**x** ∗ 0 ). It would be desirable to obtain a solution valid in a full neighborhood of the hot spot. This is the result of the present paper.

#### **4. Strategy and Results**

We obtain expansions describing singular solutions of the basic system (2), uniformly in the vicinity of the hot spot, yielding a domain of validity larger than that obtained via self-similar variables. We construct (Theorem 2) a convergent expansion for threedimensional solutions that contains powers and logarithms of *τ* := *ψ*(**x** ∗ ) − *t* ∗ , multiplied by functions of the space variables, assuming |∇*ψ*| is small, which is appropriate near the minimum of *ψ*. If the hot spot appears for *τ* = 0, at **x** ∗ = 0, the expansion is valid in a set defined by inequalities of the form *τ* < 0 and |**x** ∗ | < *δ* (we write |**x**| for the usual length of a 3-vector **x**). Therefore, it is valid in a full neighborhood of the origin. It contains five freely specifiable functions of three variables that are called singularity data, including *ψ*. This number is the greatest possible, since there are five unknowns (density, temperature and the three components of velocity) that determine pressure and reactant mass fraction. The arbitrary functions may be interpreted in terms of the asymptotics of *T*1, **u**<sup>1</sup> and *ρ*1, because, even though they may become very large at the hot spot, there are combinations of these variables that have well-defined limits as *t* <sup>∗</sup> → *ψ*(**x** ∗ ) and these limits have simple expressions in terms of the arbitrary functions—more precisely, the five functions (two scalar functions *ψ*, *σ*<sup>0</sup> and the three components of a 3-vector **w**0). They are related to the asymptotics of the non-dimensionalized variables via

$$\rho\_1/T\_1 \to \frac{|\nabla \psi|^2}{\gamma - |\nabla \psi|^2},$$

$$\mathbf{u}\_1 - \frac{\nabla \psi}{\gamma - |\nabla \psi|^2} T\_1 \to \mathbf{w}\_0 - \frac{\nabla \psi}{\gamma - |\nabla \psi|^2} \ln \frac{1 - |\nabla \psi|^2}{\gamma - |\nabla \psi|^2}.$$

and

$$
\rho\_1 - \mathbf{u}\_1 \cdot \nabla \psi \to \sigma\_0 - \mathbf{w}\_0 \cdot \nabla \psi.
$$

The method leading to these results consists in integrating the system starting from the singularity. Thus, the solution is determined from the arbitrary functions in its singular expansion, i.e., by its singularity data on the blow-up surface, rather than by Cauchy data on some hypersurface away from the singularity (for the relation between Cauchy data and singularity data in a typical case, see [31]). While, in the Cauchy problem, the series solution is determined by its first few terms and contains only integral powers of the time variable, in singular problems such as this, the expansion involves logarithms and the arbitrary coefficients are not the first few coefficients of the series. More complicated functions, such as fractional powers, are required in some cases, but not here. There are general rules to perform the reduction and to predict the form of the solution [8]. In fact, a major advantage of the reduction technique is that it enables one to predict the form of the expansion to any order, without having to compute it, since this task is often unwieldy.

Simple applications of these results include the following.

(1) *Recovering earlier results:* Hot spots correspond to the minima of the function *ψ*, and the large activation energy regime is appropriate in a small neighborhood of these minima. This makes it easy to compare our solution to the three asymptotic approaches in Section 3. The results of the first method may be recovered by expanding our solution after introducing self-similar variables. Expanding our solution in powers of *γ* leads to the second method. Inserting the parameter *µ*, both in the expansion and in *ψ*, leads to the third method. Therefore, each of these is recovered as a particular limit of our solution. As already noted, the introduction of self-similar variables leads to a restriction in the domain of validity of the solution.

(2) *Better approximation for large γ:* The second application of our expansions is the determination of the limits of validity of large activation energy asymptotics. System (2) was obtained from the reactive Euler equations by assuming the activation energy parameter *θ* to be large. This implies that one replaces the material derivative *∂<sup>t</sup>* + **u** · ∇ by *∂<sup>t</sup>* . Indeed, **u** ≈ **u**1/*θ*, so that the spatial derivative term is of higher order in an expansion in powers of *θ*. Our results show that the neglected terms are smaller than the ones that have been kept if

the gradient of *ψ* is small compared with *γ*,

where the detonation locus is given by *t* ∗ = *ψ*(**x** ∗ ) in non-dimensional variables and *γ* is the ratio of specific heats. This suggests that this approximation could be better in cases where *γ* is large.

(3) *Signature of detonation:* The expansions obtained in this paper also enable one to compute quantities that remain finite at blow-up. They can be used as a characteristic signature of this ignition mechanism. This may also be used to monitor the quality of numerical schemes. More generally, the expansions may be used as a substitute for the numerical solutions precisely where the solution is large and mesh refinement may become unwieldy. More applications and perspectives are described in Section 7.

#### **5. Reduction Analysis**

We construct solutions of system (2) that become infinite when *t* reaches a value *ψ*(**x**) that depends on space. This reflects the expectation that ignition does not occur simultaneously everywhere. We first introduce new variables, identify the leading form of the expansion of the solution and prove that solutions with this behavior exist. Furthermore, we show that they are uniquely determined by some of the lower-order terms in the expansion and that these terms have a simple interpretation.

#### *5.1. Introduction of the Detonation Locus*

Let us first introduce new space and time variables:

$$
\tau = \psi(\mathbf{x}^\*) - t^\*; \quad \mathfrak{f} = (\mathfrak{f}\_1 \mathfrak{f}\_2 \mathfrak{f}\_3) = (\mathfrak{x}\_1^\*, \mathfrak{x}\_2^\*, \mathfrak{x}\_3^\*).
$$

We assume that |∇*ψ*| < 1. The derivation operators transform as follows (we let ∇*<sup>ξ</sup>* = (*∂*/*∂ξ*1, *∂*/*∂ξ*2, *∂*/*∂ξ*3)):

$$\begin{aligned} \nabla^\* &= \nabla\_{\tilde{\xi}} + (\nabla^\* \psi) \partial\_{\tau} \\ \partial\_t^\* &= -\partial\_{\tau} \\ \text{div}^\* \mathbf{u}\_1 &= -\text{div}\_{\tilde{\xi}} \mathbf{u}\_1 + (\nabla^\* \psi) \cdot \partial\_{\tau} \mathbf{u}\_1. \end{aligned}$$

In particular, ∇∗*ψ* = ∇*ξψ*. For any expression *F*(*ξ*) that does not depend on *τ*, we write ∇*F* for ∇*ξF*. The set of spacetime points where *τ* = 0 represents the locus where the temperature becomes infinite. The actual detonation front is the locus where *T*<sup>1</sup> = *ηθ*, where *η* is a constant of order unity.

System (2), expressed in the new variables, can be written as

*γp*<sup>1</sup> = *ρ*<sup>1</sup> + *T*<sup>1</sup> (4a)

$$
\partial\_{\mathbf{7}} [\rho\_1 - \mathbf{u}\_1 \cdot \nabla \boldsymbol{\psi}] \quad = \quad \text{div}\_{\tilde{\mathbb{S}}} \mathbf{u}\_1 \tag{4b}
$$

$$
\partial\_{\mathsf{T}} [\mathbf{u}\_1 - p\_1 \nabla \boldsymbol{\psi}] \quad = \quad \nabla\_{\mathsf{J}} p\_1 \tag{4c}
$$

$$
\partial\_{\mathsf{T}} T\_1 \ = \ \ (\gamma - 1) \partial\_{\mathsf{T}} p\_1 - \exp T\_1 \tag{4d}
$$

$$
\partial\_{\overline{\tau}} y\_1 \quad = \quad \frac{1}{\beta} \exp T\_1. \tag{4e}
$$

We now transform this system into an equivalent form that is easier to analyze.

**Theorem 1.** *System (4) is equivalent to the system*

$$
\gamma p\_1 \quad = \quad \rho\_1 + T\_1 \tag{5a}
$$

$$
\partial\_{\mathsf{T}} \rho\_1 \quad = \quad \frac{1}{B} [A - (1 - \mathsf{B}) \exp T\_1] \tag{5b}
$$

$$
\partial\_{\mathsf{T}} \mathbf{u}\_{1} = \left( A - \exp T\_{1} \right) \frac{\nabla \boldsymbol{\psi}}{B} + \nabla\_{\mathsf{S}} p\_{1} \tag{5c}
$$

$$
\partial\_{\tau} T\_1 \quad = \quad (\gamma - 1) \\
\frac{A}{B} - \mathfrak{a} \exp T\_1 \tag{5d}
$$

$$
\partial\_{\mathsf{T}} y\_1 \quad = \quad \frac{1}{\beta} \exp T\_{1\prime} \tag{5e}
$$

*where*

$$\mathfrak{a} \quad = \begin{array}{c} \gamma - |\nabla \psi|^2 \\ \hline 1 - |\nabla \psi|^2 \end{array} \tag{6a}$$

$$A\_- = -\operatorname{div}\_{\xi} \mathbf{u}\_1 + (\nabla \psi) \cdot \nabla\_{\xi} p\_1 \tag{6b}$$

$$B\_- = \|1 - |\nabla \psi|^2. \tag{6c}$$

**Proof.** Equations (5a) and (5e) are identical with (4a) and (4e). Using (6), Equation (4c) yields (5c). The other equations are transformed as follows. Replace (4b) by the linear combination (4b)+(∇*ψ*)·(4c); this yields

$$\partial\_{\tau} \left[ \rho\_1 - p\_1 |\nabla \psi|^2 \right] = A\_{\nu}$$

where *A* is defined in (6). Using (4a), this relation is equivalent to

$$\partial\_{\tau} \left[ (\gamma - |\nabla \psi|^2) \rho\_1 - |\nabla \psi|^2 T\_1 \right] = \gamma A. \tag{7}$$

Using (4a), Equation (4d) may be written as

$$
\partial\_{\tau} [T\_1 - (\gamma - 1)\rho\_1] = -\gamma \exp T\_1.
$$

Replace Equations (7) and (4e) by their linear combinations with coefficients (*γ* − 1, *γ* − |∇*ψ*| 2 ) and (1, |∇*ψ*| 2 ), respectively; this yields (5b) and (5d). Retracing one's steps, one may conversely derive (4) from (5). This completes the proof.

**Remark 1.** *System* (6) *does not give the derivative of p*<sup>1</sup> *directly. However, this derivative may be obtained by adding (5b) and (5c) and using (5a). We obtain*

$$
\partial\_{\tau} p\_1 = (A - \exp T\_1) / B. \tag{8}
$$

#### *5.2. Removing the Leading Singularity*

In the right-hand side of Equation (5d) for *T*1, the most important term should be the exponential, since the process is driven by the reaction. Let us use this observation to obtain some heuristic information on the appropriate behavior of the variables near the singularity, before we set out to construct solutions with this behavior. If the exponential is dominant in the right-hand side of (5d), then *∂τT*<sup>1</sup> ≈ −*α* exp *T*<sup>1</sup> and we expect *T*<sup>1</sup> ≈ ln(1/*ατ*). In that case, we have *∂τρ*<sup>1</sup> ≈ (*B* − 1)*B* −1 exp *T*<sup>1</sup> ≈ (*B* − 1)/(*αBτ*), hence, we expect *ρ*<sup>1</sup> ≈ *k* ln(1/*τ*), with

$$k = \frac{1 - B}{aB} = \frac{|\nabla \psi|^2}{\gamma - |\nabla \psi|^2} = \frac{|\nabla \psi|^2}{aB} = \frac{\gamma}{aB} - 1. \tag{9}$$

Finally, *∂τ***u**<sup>1</sup> ≈ −*B* −1 *e <sup>T</sup>*<sup>1</sup>∇*ψ*; hence, the expected behavior **<sup>u</sup>**<sup>1</sup> <sup>≈</sup> (*αB*) <sup>−</sup>1∇*<sup>ψ</sup>* ln(1/*τ*). Now, the term ∇*<sup>ξ</sup> p*<sup>1</sup> = *γ* −1 (*ρ*<sup>1</sup> + *T*1) contains terms involving ln *τ*; hence, **u**<sup>1</sup> involves terms in *τ* ln *τ*. These considerations suggest the introduction of renormalized variables (Φ, *R*, **U**) by letting

$$T\_1 \quad = \quad \ln \frac{1}{\alpha \tau} + \varphi\_1(\tilde{\xi})\tau \ln \tau + \tau \Phi(\tilde{\xi}, \tau) \tag{10a}$$

$$\rho\_1 = \underbrace{|\nabla \psi|^2}\_{\underline{\mathbf{a}} \mathbf{B}} \ln \frac{1}{\underline{\tau}} + \sigma\_0(\underline{\xi}) + \sigma\_1(\underline{\xi}) \tau \ln \tau + \tau \mathcal{R}(\underline{\xi}, \underline{\tau}) \tag{10b}$$

$$\mathbf{u}\_{1} = -\frac{\nabla \boldsymbol{\psi}}{aB} \ln \frac{1}{\tau} + \mathbf{w}\_{0}(\boldsymbol{\xi}) + \mathbf{w}\_{1}(\boldsymbol{\xi})\tau \ln \tau + \tau \mathbf{U}(\boldsymbol{\xi}, \tau). \tag{10c}$$

These new dependent variables are renormalized in the sense that the "infinite part" of the solution was subtracted off *and* the remainder has been divided by an appropriate power of *τ*. This analysis is an application of the general procedure described in [8] (§ 1.4). Using (2a), we now have

$$p\_1 = (\rho\_1 + T\_1) / \gamma = \frac{1}{aB} \ln \frac{1}{\tau} + (\sigma\_0 - \ln a) / \gamma + (\sigma\_1 + \varphi\_1) / \gamma + \tau (\Phi + R) / \gamma. \tag{11}$$

The main result of this section states that, once *ψ*, *σ*<sup>0</sup> and **w**<sup>0</sup> are given, with |∇*ψ*| < 1, the solution is completely and uniquely determined; *σ*1, *ϕ*<sup>1</sup> and **w**<sup>1</sup> may be found in closed form and the renormalized variables have expansions that may be computed inductively to any order, while the corresponding series converges if the data are analytic, or represent a very smooth function if the data are themselves very smooth.

**Theorem 2.** *System (5) admits, near the origin, a family of solutions given by power series in τ, τ* ln *τ and τ*(ln *τ*) 2 *, with coefficients depending on ξ, provided that* |∇*ψ*| ≤ 1 *and*

$$
\sigma\_1 \quad = \quad -\frac{\tilde{A}}{aB^2} [\gamma - 1 + (\gamma + 1)B] \tag{12a}
$$

$$
\varphi\_1 \quad = \quad -\frac{\gamma - 1}{2B} \tilde{A} \tag{12b}
$$

$$\mathbf{w}\_{1} = -\frac{\tilde{A}}{2\alpha B^{2}} \nabla \psi[\gamma - 1 + 2B] - \frac{1}{\gamma} \nabla k\_{\prime} \tag{12c}$$

*where k is given by* (9) *and*

$$\tilde{A} = \frac{(\gamma - |\nabla \psi|^2) \Delta \psi + 4 \sum\_{i,j} \psi\_i \psi\_j \psi\_{ij}}{(\gamma - |\nabla \psi|^2)^2}. \tag{13}$$

*The functions σ*<sup>0</sup> *and* **w**<sup>0</sup> *and the function ψ may be chosen arbitrarily; they determine all the other terms in the expansion.*

**Proof.** For any quantity *X*, Taylor's expansion gives an analytic function *G*<sup>2</sup> such that *e <sup>X</sup>* = 1 + *X* + *X* <sup>2</sup>*G*1(*X*). Therefore,

$$\begin{aligned} e^{T\_1} &= \frac{1}{\alpha \tau} \exp[\varrho\_1 \tau \ln \tau + \tau \Phi] \\ &= \frac{1}{\alpha \tau} + \frac{\varrho\_1}{\alpha} \ln \tau + \frac{\Phi + G\_2}{\alpha} \end{aligned}$$

where

$$\mathcal{G}\_2 = \mathcal{G}\_2(\not\zeta, \tau, \tau \ln \tau, \tau (\ln \tau)^2, \spadesuit) = \frac{\tau}{\mathfrak{a}} (\not\varphi\_1 \ln \tau + \spadesuit)^2 \mathcal{G}\_1(\not\varphi\_1 \tau \ln \tau + \tau \spadesuit). \tag{14}$$

Additionally, (6), (10c) and (24) yield

$$A = \tilde{A}\ln\frac{1}{\pi} + A\_0 + A\_1\tau\ln\pi + \text{tr}\mathcal{A} \tag{15}$$

where

$$\begin{split} \tilde{A} &=& \text{div}\_{\xi} \Big( \frac{\nabla \psi}{\gamma - |\nabla \psi|^{2}} \Big) + \nabla \psi \cdot \nabla\_{\xi} \frac{1}{\gamma - |\nabla \psi|^{2}} \\ &=& \frac{\Delta \psi}{\gamma - |\nabla \psi|^{2}} + 2 \sum\_{j} \psi\_{j} \partial\_{j} \frac{1}{\gamma - |\nabla \psi|^{2}} \\ &=& (\gamma - |\nabla \psi|^{2})^{-2} \Big[ (\gamma - |\nabla \psi|^{2}) \Delta \psi + 4 \sum\_{j,k} \psi\_{i} \psi\_{k} \psi\_{jk} \Big]. \end{split}$$

and

$$A\_0 \quad = \operatorname{div}\_{\xi} \mathbf{w}\_0 + \nabla \psi \cdot \nabla\_{\psi} (\sigma\_0 - \ln a) / \gamma \tag{16a}$$

$$A\_1 \quad = \operatorname{div}\_{\tilde{\xi}} \mathbf{w}\_1 + \nabla \psi \cdot \nabla\_{\tilde{\xi}} (\sigma\_1 + \varrho\_1) / \gamma \tag{16b}$$

$$\mathcal{A}^{\prime} \quad = \quad \text{div}\_{\tilde{\xi}} \mathbf{U} + \nabla \boldsymbol{\psi} \cdot \nabla\_{\tilde{\xi}} (\Phi + \mathcal{R}) / \gamma. \tag{16c}$$

Note that

$$
\gamma \tilde{A} = (k+1) \Delta \psi + 2 \nabla \psi \cdot \nabla k.
$$

We may now determine *ϕ*1, *σ*<sup>1</sup> and **w**1, and obtain the reduced equations for *R*, Φ and **U**. We let

$$
\mathcal{Y} = \tau \partial\_{\tau}.
$$

We substitute (10) into each of the Equations (5b)–(5d). The following calculations have a common pattern. In each case, the choice of the leading terms in (10) ensures that the leading order terms in the resulting equation vanish. The vanishing of the next terms (in *τ* ln *τ*) determines *ϕ*1, *σ*<sup>1</sup> and **u**1. Finally, dividing by *τ*, one obtains a singular system for the renormalized unknowns, to which solutions are given by an existence result for singular initial-value problems. We now carry out this program.

From Equation (5b), we obtain

$$\mathcal{Q}\rho\_1 = A\tau/B - |\nabla\psi|^2 \frac{\pi}{B} \exp(T\_1).$$

hence, using (10b) and (9),

$$-k + \sigma\_1 \tau (1 + \ln \tau) + \tau (\mathcal{O} + 1)\mathcal{R} = \frac{\tau}{B} A - k [1 + \varphi\_1 \tau \ln \tau + \tau (\Phi + \mathcal{G}\_2)].\tag{17}$$

The first term on the right cancels with one of the terms on the left. Equating the coefficients of *τ* ln *τ*, we obtain

$$\begin{aligned} \sigma\_1 &= -\frac{\tilde{A}}{B} - k\varphi\_1 \\ &= \frac{\tilde{A}}{2B} [k(\gamma - 1) - 2] \\ &= \frac{\tilde{A}}{2B} \frac{(\gamma - 1)|\nabla\psi|^2 - 2(\gamma - |\nabla\psi|^2)}{\gamma - |\nabla\psi|^2} \\ &= \frac{\tilde{A}}{2B} \frac{(\gamma + 1)|\nabla\psi|^2 - 2\gamma}{aB} = -\frac{\tilde{A}}{aB^2} [\gamma - 1 + (\gamma + 1)B]. \end{aligned}$$

This proves (12a).

Dividing (17) by *τ* and using (9), the reduced equation for *R* is obtained:

$$(\mathcal{O} + 1)\mathcal{R} + k\Phi = -\sigma\_1 + \frac{1}{B}[A\_0 + A\_1\tau\ln\tau + \tau\omega'] - kG\_2.\tag{18}$$

From (5d), we obtain

$$\mathcal{Q}T\_1 = (\gamma - 1)\frac{\tau A}{B} - \tau \alpha e^{T\_1} \nu$$

hence,

$$-1 + \varrho\_1 \tau (1 + \ln \tau) + \tau (\mathcal{O} + 1)\Phi = \frac{\gamma - 1}{B} (\tau A) - 1 - \varrho\_1 \tau \ln \tau - \tau (\Phi + G\_2). \tag{19}$$

Equating the coefficients of *τ* ln *τ* and using (13), we obtain *ϕ*<sup>1</sup> = − *γ*−1 *<sup>B</sup> <sup>A</sup>*˜ <sup>−</sup> *<sup>ϕ</sup>*1, or

$$
\varphi\_1 = -\frac{\gamma - 1}{2B} \vec{A}.
$$

This proves (12b).

Dividing (19) by *τ*, the reduced equation for Φ is obtained:

$$(\mathcal{O} + 2)\Phi = -\varrho\_1 + \frac{\gamma - 1}{B} [A\_0 + A\_1 \tau \ln \tau + \tau \mathcal{O}] - \mathcal{G}\_2. \tag{20}$$

From Equation (5c), we obtain

$$\mathcal{Q}\mathbf{u}\_1 = \tau (A - e^{T\_1}) \frac{\nabla \psi}{B} + \tau \nabla\_{\xi} p\_1.$$

or

$$\begin{split} & -\frac{\nabla\psi}{aB} + \tau\mathbf{w}\_{1}(1+\ln\tau) + \tau(\mathcal{O}+1)\mathbf{U} \\ &= \quad \frac{\nabla\psi}{aB} [\tau aA - 1 - \varrho\_{1}\tau\ln\tau - \tau(\Phi + \mathcal{G}\_{2})] \\ &+ \nabla\_{\xi} \left[ \frac{\tau\ln(1/\tau)}{\gamma - |\nabla\psi|^{2}} + (\sigma\_{0} - \ln a)\frac{\tau}{\gamma} + \frac{\varrho\_{1} + \sigma\_{1}}{\gamma}\tau^{2}\ln\tau + \frac{\tau^{2}}{\gamma}(\Phi + \mathcal{R}) \right]. \end{split} \tag{21}$$

Equating the coefficients of *τ* ln *τ*, we obtain, since (*αB*) <sup>−</sup><sup>1</sup> = (*k* + 1)/*γ*,

$$\begin{split} \mathbf{w}\_{1} &= \begin{aligned} -\frac{\nabla\psi}{B}\tilde{A} - \frac{\nabla\psi}{\alpha B}\varphi\_{1} - \nabla\_{\tilde{\xi}}\frac{1}{\alpha B} \\ &= \begin{aligned} -\frac{\nabla\psi}{B}\tilde{A} + \frac{\nabla\psi}{\alpha B}(\gamma - 1)\frac{\tilde{A}}{2B} - \frac{1}{\gamma}\nabla\_{\tilde{\xi}}k \\ &= \frac{\tilde{A}}{2B}\nabla\psi\left[\frac{\gamma - 1}{\gamma - |\nabla\psi|^{2}} - 2\right] - \frac{1}{\gamma}\nabla\_{\tilde{\xi}}k \\ &= \frac{\tilde{A}}{2\alpha B^{2}}\nabla\psi\left[2|\nabla\psi|^{2} - (\gamma + 1)\right] - \frac{1}{\gamma}\nabla\_{\tilde{\xi}}k. \end{aligned} \end{aligned}$$

This proves (12c).

Dividing (21) by *τ*, the reduced equation for **U** is obtained:

$$\begin{split} \mathbf{U} \left( \begin{aligned} \boldsymbol{\Psi} + \mathbf{1} \right) \mathbf{U} + \frac{\nabla \boldsymbol{\Psi}}{\alpha \mathbf{B}} \boldsymbol{\Phi} \\ = \mathbf{0} &= \mathbf{w}\_{1} + \frac{\nabla \boldsymbol{\Psi}}{\mathbf{B}} [A\_{0} + A\_{1} \boldsymbol{\tau} \ln \boldsymbol{\tau} + \boldsymbol{\tau} \boldsymbol{\omega} \boldsymbol{\mathcal{I}} - \mathbf{G}\_{2} / \alpha] \\ &+ \nabla\_{\boldsymbol{\xi}} \left[ (\sigma\_{0} - \ln \boldsymbol{\alpha}) / \gamma + \frac{\boldsymbol{\varrho}\_{1} + \sigma\_{1}}{\gamma} \boldsymbol{\tau} \ln \boldsymbol{\tau} + \frac{\boldsymbol{\tau}}{\gamma} (\boldsymbol{\Phi} + \boldsymbol{\mathcal{R}}) \right]. \end{split} \end{split} \tag{22}$$

It remains to prove that the renormalized unknowns admit the desired expansion. We start from the reduced system formed by (17), (20) and (22), namely

$$\begin{aligned} (\mathcal{O}+1)\mathcal{R}+k\Phi &=& -\sigma\_1 + \frac{1}{B}[A\_0 + A\_1\tau\ln\tau + \tau\varkappa\mathcal{I}] - k\mathcal{G}\_2\\ (\mathcal{O}+2)\Phi &=& -\varphi\_1 + \frac{\gamma-1}{B}[A\_0 + A\_1\tau\ln\tau + \tau\varkappa\mathcal{I}] - \mathcal{G}\_2\\ (\mathcal{O}+1)\mathcal{U} + \frac{\nabla\psi}{\varkappa B}\Phi &=& -\mathbf{w}\_1 + \frac{\nabla\psi}{B}[A\_0 + A\_1\tau\ln\tau + \tau\varkappa\mathcal{I} - \mathcal{G}\_2/\varkappa] \\ &+ \nabla\_{\tilde{\mathbb{S}}}\left[(\sigma\_0 - \ln\alpha)/\gamma + \frac{\varrho\_1 + \sigma\_1}{\gamma}\tau\ln\tau + \frac{\tau}{\gamma}(\Phi + \mathbb{R})\right]. \end{aligned}$$

Letting X = (*R*, Φ, **U**) *T* , this system has the general form

$$(\mathcal{Q} + A)\mathcal{K} = \mathbf{F}(\mathbf{r}, \mathbf{r} \ln \mathbf{r}, \mathbf{r} (\ln \mathbf{r})^2, \mathcal{K}, \mathbf{r} \nabla\_{\xi} \mathcal{K}),\tag{23}$$

where

$$A = \begin{pmatrix} 1 & k & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & \left(\partial\psi/\partial\xi\_1\right)/\alpha B & 1 & 0 & 0 \\ 0 & \left(\partial\psi/\partial\xi\_2\right)/\alpha B & 0 & 1 & 0 \\ 0 & \left(\partial\psi/\partial\xi\_3\right)/\alpha B & 0 & 0 & 1 \end{pmatrix}.$$

The essential features of this system are the presence of a factor of *τ* in front of every space derivative of a component of X and the fact that all the eigenvalues of *A* are positive. This system admits a formal series in powers of *τ*, *τ*(ln *τ*) and *τ*(ln *τ*) 2 (using [8] (Theorem 2.4)), with coefficients depending only on *ξ*—they may be expressed explicitly in terms of the data characterizing the singularity, namely, *ψ*, *σ*<sup>0</sup> and **w**0. By [8] (Theorem 4.5), this series converges for a small *τ*; it is real for *τ* real and positive, because, by induction, all the terms of its expansion are.

**Remark 2.** *We record the expression for p*<sup>1</sup> *that follows from (10) and (5a):*

$$p\_1 = (\rho\_1 + T\_1) / \gamma = \frac{1}{aB} \ln \frac{1}{\tau} + \frac{\sigma\_0 - \ln a}{\gamma} + \frac{\varphi\_1 + \sigma\_1}{\gamma} \tau \ln \tau + \frac{\tau}{\gamma} (\Phi + R). \tag{24}$$

**Remark 3.** *The singularity data* (*σ*0, **w**0, *ψ*) *have a simple meaning at a point where* ∇*ψ* = 0*. It is always possible to achieve this by performing a Lorentz transformation, since* |∇*ψ*| *is small in the situation considered here. In that case, the leading terms in the expansions of ρ*<sup>1</sup> *and* **u**<sup>1</sup> *vanish. Therefore the arbitrary functions σ*<sup>0</sup> *and* **w**<sup>0</sup> *represent the density and Eulerian velocity at the first point of the hot spot. Further, we find that ϕ*<sup>1</sup> *and* **w**<sup>1</sup> *vanish at this point and that σ*<sup>1</sup> *is proportional to* ∆*ψ.*

#### **6. Lorentz Transformation and Blow-Up Patterns**

We now show that the hot spot, defined as the first spacetime point where a singularity forms, is not a Lorentz invariant; therefore, it is not an intrinsic feature of the ignition process. Since this is a purely kinematical phenomenon, of general applicability, we explain our result in general terms. To apply it to the ignition problem, it suffices to set *f* = *ψ* in what follows.

Let us consider a physical phenomenon that exhibits a singularity that is observed to take place along a singular locus Σ described by an equation *t* = *f*(*x*, *y*, *z*) with *f* smooth, in a given inertial system (S), by an observer who labels events in his/her local Minkowski space with coordinates (*x*, *y*, *z*, *t*). In this section, we establish that the first spacetime singularity for (S), corresponding to the minimum of *f* , is not the first spacetime singularity for another inertial observer. To see this, let us perform a Lorentz transformation. By translation of variables, we may assume that *f* admits a minimum for *x* = *y* = *z* = 0. Adding a constant to *f* , we may also assume that *f*(0, 0, 0) = 0. Since *f* is minimum at the origin, there, we have *∂<sup>x</sup> f* = *∂<sup>y</sup> f* = *∂<sup>z</sup> f* = 0. Since performing spatial rotations on coordinates does not change the value of the minimum of *f* , it suffices to consider a special Lorentz transformation in the *x*-direction:

$$\mathbf{x}' = \gamma(\mathbf{x} - vt); \quad t' = \gamma(t - v\mathbf{x}/c^2); \quad y' = y, \; z' = z, \qquad \text{with } \gamma = (1 - v^2/c^2)^{-1/2}.$$

The inverse transformation is given by

$$\mathbf{x} = \gamma(\mathbf{x'} + vt'); \quad t = \gamma(t' + v\mathbf{x'}/c^2); \quad y = y', z = z'.$$

Therefore, the equation of the singular set, namely, *t* − *f*(*x*, *y*, *z*) = 0, becomes

$$F(t', \mathbf{x'}, y', z', v) := \gamma(t' + v\mathbf{x'}/c^2) - f(\gamma(\mathbf{x'} + vt'), y', z') = 0. \tag{25}$$

Equation *F* = 0 is an implicit equation for the singularity locus as viewed in (S'). Since *∂F*/*∂t* <sup>0</sup> = *γ*(1 − *v fx*(*γ*(*x* 0 + *vt*0 ), *y* 0 , *z* 0 )) reduces to *γ* at the origin (because *f<sup>x</sup>* = 0 there), the implicit function theorem enables one to locally solve equation *F* = 0 in the form *t* 0 = *g*(*x* 0 , *y* 0 , *z* 0 , *v*). Differentiating Equation (25) with respect to the primed variables, we obtain the following:

$$
\partial\_{\mathbf{x}'} \mathbf{g} = \frac{\partial\_{\mathbf{x}} f - v/c^2}{1 - v \partial\_{\mathbf{x}} f}, \\
\partial\_{\mathbf{y}'} \mathbf{g} = \frac{\gamma^{-1} \partial\_{\mathbf{y}} f}{1 - v \partial\_{\mathbf{x}} f}, \\
\partial\_{\mathbf{z}'} \mathbf{g} = \frac{\gamma^{-1} \partial\_{\mathbf{z}} f}{1 - v \partial\_{\mathbf{x}} f}.
$$

where *∂<sup>x</sup> f* = *∂ f ∂x* (*x*, *y*, *z*) = *<sup>∂</sup> <sup>f</sup> ∂x* (*γ*(*x* 0 + *vt*0 ), *y* 0 , *z* 0 ), *∂<sup>x</sup>* <sup>0</sup> *g* = *∂g ∂x* <sup>0</sup>(*x* 0 , *y* 0 , *z* 0 , *v*), and similarly for *∂<sup>y</sup> f* , *∂<sup>z</sup> f* , *∂<sup>y</sup>* <sup>0</sup> *g*, *∂<sup>z</sup>* 0 *g*.

Now, the places where *f* exhibits an extremum (*∂<sup>x</sup> f* = *∂<sup>y</sup> f* = *∂<sup>z</sup> f* = 0, point *D* on Figure 1) do not coincide with those where *g* does (*∂<sup>x</sup>* <sup>0</sup> *g* = *∂<sup>y</sup>* <sup>0</sup> *g* = *∂<sup>z</sup>* <sup>0</sup> *g* = 0, point *E* on Figure 1). The location of the first singularity in the second system (S') is obtained by solving the equation *∂<sup>x</sup>* <sup>0</sup> *g* = 0 in (S'); the same spacetime point *E* would be obtained in (S) by solving *∂<sup>x</sup> f* = *v*/*c* 2 in (S). By contrast, in (S), the first singularity *D* satisfies *∂<sup>x</sup> f* = 0. The change in the spacetime point where the first singularity is observed may be seen geometrically in one space dimension (see Figure 1). Therefore, the first singularity in (S') does not correspond to the same spacetime point as the first singularity in (S). The first hot spot in a given inertial system is not the cause of ignition and has no intrinsic

physical significance. By contrast, the blow-up set is a well-defined geometric object and its geometric characteristics are physically meaningful.

**Figure 1.** *Illustration of the transformation of the first hot spot under a Lorentz transformation.* The singular set Σ (in blue) has equation *t* = *f*(*x*) in a two-dimensional Minkowski space, for an observer with rectangular coordinates (*t*, *x*). The time of the first singularity corresponds, for this observer, to the spacetime point *D*. For another inertial observer, his/her time and space axes are slanted lines, as indicated, and the first singularity is observed at a different spacetime point *E*, that may be constructed by finding the tangent to Σ (in red) parallel to the *x* 0 -axis.

#### **7. Discussion and Perspectives**

#### *7.1. Detonation Signature*

The preceding results lead to the identification of combinations of physical quantities that admit limits on the detonation front. Indeed, Theorem 2 shows that the following combinations of the unknowns tend to a finite limit at the singularity (as *τ* → 0):

$$
\rho\_1 - \mathbf{u}\_1 \cdot \nabla \psi \quad \rightarrow \quad \sigma\_0 - \mathbf{w}\_0 \cdot \nabla \psi \tag{26a}
$$

$$
\rho\_1/T\_1 \quad \rightarrow \quad k \tag{26b}
$$

$$\mathbf{u}\_1 - \frac{\nabla \boldsymbol{\psi}}{\alpha \boldsymbol{B}} T\_1 \quad \rightarrow \quad \mathbf{w}\_0 + \frac{\nabla \boldsymbol{\psi}}{\alpha \boldsymbol{B}} \ln \boldsymbol{a} \tag{26c}$$

$$\begin{aligned} \vert \nabla \psi \vert^2 T\_1 - (\gamma - |\nabla \psi|^2) \rho\_1 &\to \vert - \vert \nabla \psi \vert^2 \ln a - (\gamma - |\nabla \psi|^2) \sigma\_0 \end{aligned} \tag{26d}$$

$$(\mathcal{O} - 1) \exp T\_1 \quad \to \quad \mathcal{O}\_1/\alpha. \tag{26e}$$

1 where the quantities *α*, *B* and *ϕ*<sup>1</sup> are determined in terms of the arbitrary functions (see Theorems 1 and 2). In particular, when ∇*ψ* = 0 — that is, on the hot spot in the inertial frame at hand —*the leading order infinities vanish* in the expressions for the density and Eulerian velocities and we obtain the simple result

$$
\rho\_1 \quad \rightarrow \quad \sigma\_0 \tag{27a}
$$

$$\begin{array}{ccccc}\mathbf{u}\_{1} & \rightarrow & \mathbf{w}\_{0} \\ \end{array} \tag{27b}$$

$$(\mathcal{O} - 1) \exp T\_1 \quad \rightarrow \quad \varphi\_1/\mathfrak{a}. \tag{27c}$$

These limiting behaviors, (26) and (27), give a characteristic signature of this detonation mechanism, that might be tested against measurements.

The fact that the first hot spot is not the cause of ignition is reflected in the fact that *ψ* is not a constant. The fact that the blow-up pattern is curved in general is reflected in the dependence of the coefficient *σ*<sup>1</sup> (in the expansion of the density) on the second-order derivatives of *ψ*, see (12a) and (13).

A particularly simple limiting behavior is *ρ*1/*T*<sup>1</sup> → *k*, where, we recall (see (9) and (6)), *k* only depends on the ratio of specific heats and the length of the gradient of *ψ*, that is, on

the normal velocity in (S) of the detonation front. An overview of measurement techniques may be found in the second chapter of [32]; see also recent papers, such as [6,34].

#### *7.2. Other Asymptotics*

In this paper, we assume that the three spatial variables are scaled using the same characteristic length `0. It would be interesting to introduce different scalings, since some measurements are performed on thin domains, that are essentially two-dimensional [33,34]. The gap width plays the role of a second small parameter, in addition to the inverse activation energy. The detonation signature could exhibit a marked dependence on gap width, which would be consistent with the results of [33,34].

An important application of much current interest is the possibility of introducing ammonia rather than methane in fuel composition, since the former does not contain carbon. The literature is extensive [5–7]. This would have obvious advantages from the environmental point of view. It would be of interest to determine the non-dimensional parameters for different fuel compositions.

#### *7.3. Relativistic Effects*

The fact that the first hot spot for one observer may not be the first for another is a consequence of the kinematics of special relativity, irrespective of the importance of relativistic effects on the chemistry. This being said, there are two ways to introduce further relativistic effects in this problem. The first is to consider situations in which relativistic effects are significant, as in astrophysics. The second would be to repeat measurements such as those of [33,34] in a moving frame. For instance, one could set the cell used in these measurements in rapid rotation and perform measurements in the (stationary) laboratory frame. One could similarly consider a circular shock tube. Such measurements are perhaps feasible; their interpretation would depend on the importance of inertial effects.

#### **8. Conclusions**

(1) When a singularity is formed along a smooth hypersurface of Minkowski spacetime, with an equation of the form *τ* := *t* − *ψ*(*x*, *y*, *z*) = 0, the spacetime location of the first hot spot is not a Lorentz invariant. This is a consequence of the Lorentz transformation between observers in special relativity and is independent of the size of the relativistic effects in the modeling of the physical situation that led to singularity-formation. However, the set of all singularities seen by all observers is a well-defined geometric object—the blow-up set.

(2) These ideas apply to the weak detonation problem. We have solved the appropriate system of PDEs, in the limit of high activation energy, by integrating them from singularity data given on the blow-up set, or detonation locus, and obtained a general solution of the equations. It contains the maximum number of arbitrary functions, namely, five. This solution improves earlier results in three respects: (i) It provides a description of the solution that is valid when it is large, but not infinite; in the weak detonation problem, the temperature never becomes actually infinite. (ii) It identifies which combinations of physical quantities remain finite at blow-up. (iii) It takes into account the kinematics of special relativity.

(3) In addition, the arbitrary functions (*σ*0, **w**0, *ψ*) in the general solution admit a physical interpretation in terms of the behavior of density, velocity and temperature at the singularity. This provides a signature for this type of ignition mechanism.

(4) Perspectives include the following: (i) a similar asymptotic study for nearly twodimensional situations used in some measurements; (ii) the measurement of a detonation signature on the basis of the limiting behavior of the physical quantities, including the curvature of the blow-up pattern; (iii) the inclusion of the relativistic effects in the chemistry; (iv) the impact of fuel composition, especially the inclusion of hydrogen or ammonia in the mix, on the non-dimensional parameters, therefore on the domain of validity of this ignition mechanism.

**Funding:** This research was partially funded by CNRS through its research group UMR9008 in Reims. The APC was waived by MDPI.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The author thanks Leila Zhang at MDPI for encouraging him to prepare and submit this paper, and the referees for their kind comments.

**Conflicts of Interest:** The author declares no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Appendix A. Derivation of the Basic System**

Let us consider a reactive fluid, with reactant mass fraction *y* (i.e., one gram of fluid contains *y* grams of reactant and 1 − *y* grams of reaction products). The overall reaction is modeled by a one-step irreversible reaction in the form *A*<sup>1</sup> −→ *A*2, where the reactant *A*<sup>1</sup> and the product *A*<sup>2</sup> have the same specific heats *c<sup>p</sup>* and *cv*, as well as the same molar mass *µ*. This makes it possible to consider *A*<sup>1</sup> and *A*<sup>2</sup> as forming a single perfect fluid, with density *ρ*, specific volume *v* = 1/*ρ*, Eulerian velocity **u** = (*u*1, *u*2, *u*3), pressure *p* and temperature *T*, that all vary with in space and time. Therefore, the equation of state is

$$\frac{p}{\rho} = pv = \frac{R}{\mu}T\_{\prime}$$

where *R* is the perfect gas constant, *γ* = *cp*/*cv*, and

$$\frac{R}{\mu} = c\_p - c\_v = (\gamma - 1)c\_v.$$

The specific internal energy is

$$\varepsilon = c\_{\upsilon}T + qy = \frac{p/\rho}{\gamma - 1} + qy\_{\prime}$$

where the constant *q* is the heat release rate. The total specific energy is

$$e = \varepsilon + \frac{1}{2}\mu^2\nu$$

where *u* <sup>2</sup> <sup>=</sup> **<sup>u</sup>** · **<sup>u</sup>**. The reaction rate is given by the Arrhenius law

$$\frac{Dy}{Dt} = r(y, t) := -Ay \exp\left(-\frac{E}{RT}\right). \tag{A1}$$

.

where *A* and *E* are constants.

To write the conservation laws, let us introduce the material derivative operator

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + \sum\_{i=1}^{3} \mu\_i \frac{\partial}{\partial x\_i}$$

Mass, momentum and energy conservation read

$$\frac{D\rho}{Dt} + \rho \operatorname{div} \mathbf{u} \; \mathbf{u} \; \; \; \;=\;\; \; \mathbf{0} \tag{A2a}$$

$$\frac{D\mathbf{u}}{Dt} + \frac{1}{\rho} \nabla p \; \; \; \;= \; \; \; 0 \tag{A2b}$$

$$
\rho \frac{De}{Dt} + \text{div}\,(p\mathbf{u}) \quad = \quad 0. \tag{A2c}
$$

By (A2a), *ρ<sup>t</sup>* + div(*ρ***u**) = 0. Therefore,

$$\begin{aligned} \text{div}(p\mathbf{u}) &= \begin{aligned} &p \operatorname{div} \mathbf{u} + \mathbf{u} \cdot \nabla p \\ &= \quad -\frac{p}{\rho} \frac{D\rho}{Dt} - \rho \mathbf{u} \frac{D\mathbf{u}}{Dt} \\ &= \quad \rho \frac{D}{Dt} \left[ \frac{p}{\rho} - \frac{1}{2} u^2 \right] - \frac{Dp}{Dt} . \end{aligned} \end{aligned}$$

In addition, *p*/*ρ* = (*c<sup>p</sup>* − *cv*)*T*. On the other hand,

$$
\rho \frac{De}{Dt} = \rho \frac{D}{Dt} \left[ c\_\upsilon T + qy + \frac{1}{2}u^2 \right].
$$

Therefore, Equation (A2c) takes the form

$$
\rho \frac{De}{Dt} + \text{div}(p\mathbf{u}) = \rho \frac{D}{Dt} \left\{ [c\_v T + qy + \frac{1}{2}u^2] + [(c\_p - c\_v)T - \frac{1}{2}u^2] \right\} - \frac{Dp}{Dt}.
$$

Using the Arrhenius law,

$$
\rho c\_p \frac{DT}{Dt} - \frac{Dp}{Dt} = -\rho q \frac{Dy}{Dt} = -\rho qr = A\rho qy \exp(-\frac{E}{RT}).\tag{A3}
$$

To sum up, we have to solve the following system:

$$p \quad = \ (\gamma - 1)c\_{\overline{\nu}}\rho T \tag{A4a}$$

$$\frac{D\rho}{Dt} + \rho \operatorname{div} \mathbf{u} \quad = \begin{array}{c} 0 \\ \end{array} \tag{A4b}$$

*D***u** *Dt* <sup>+</sup> 1 *ρ* ∇*p* = 0 (A4c)

$$
\rho c\_p \frac{DT}{Dt} - \frac{Dp}{Dt} \quad = \quad A \rho q y \exp(-\frac{E}{RT}) \tag{A4d}
$$

$$\frac{Dy}{Dt} \quad = \quad -Ay \exp(-\frac{E}{RT}).\tag{A4e}$$

The objective is to solve this system in the limit when the activation energy *E* is large.

#### **Appendix B. Non-Dimensionalization**

*Appendix B.1. Non-Dimensional Variables*

Let us introduce a reference state characterized by the values *p*0, *ρ*0, *T*<sup>0</sup> and *y*0, a reference length `<sup>0</sup> and reference time *t*0. They determine *u*<sup>0</sup> = `0/*t*0. We introduce scaled variables by

$$\mathbf{x}^\* = \frac{\mathbf{x}}{\ell\_0}; \quad t^\* = \frac{t}{t\_0}; \quad p^\* = \frac{p}{p\_0}; \quad \mathbf{u}^\* = \frac{\mathbf{u}}{u\_0}; \quad \rho^\* = \frac{\rho}{\rho\_0}; \quad y^\* = \frac{y}{y\_0}; \quad T^\* = \frac{T}{T\_0}.$$

We take *y*<sup>0</sup> = 1 and assume the equation of state holds for the reference state *p*<sup>0</sup> = (*γ* − 1)*cvρ*0*T*0. The velocity *c*<sup>0</sup> = p *γp*0/*ρ*<sup>0</sup> determines the acoustic time *t<sup>a</sup>* = `0 *c*0 . The main non-dimensional parameters for the present analysis are

*θ* = *E*/*RT*<sup>0</sup> (non-dimensional activation parameter) *t<sup>r</sup>* = *A* −1 exp *θ* (initial reaction time) *c*<sup>0</sup> = p *γp*0/*ρ*<sup>0</sup> *M* = *ta t*0 = *u*0 *c*0 (Mach number) *β* = *qy*<sup>0</sup> *cpT*<sup>0</sup> (heat release parameter).

Substituting these relations into (A4), we obtain

$$p^\* \quad = \quad \rho^\* T^\* \tag{A5a}$$

$$\frac{D\rho^\*}{Dt^\*} + \rho^\* \operatorname{div}^\* \mathbf{u}^\* = \begin{array}{c} \mathbf{0} \\ \end{array} \tag{A5b}$$

$$
\rho^\* \frac{D\mathbf{u}^\*}{Dt^\*} + \frac{1}{\gamma M^2} \nabla^\* p^\* \quad = \quad 0 \tag{A5c}
$$

$$
\rho^\* \frac{D T^\*}{D t^\*} - \frac{\gamma - 1}{\gamma} \frac{D p^\*}{D t^\*} \quad = \quad \frac{t\_0}{t\_r} \theta \rho^\* y^\* \exp[\theta - \frac{\theta}{T^\*}] \tag{A5d}
$$

$$\frac{Dy^\*}{Dt^\*} \quad = \quad -\frac{t\_0}{t\_r} y^\* \exp[\theta - \frac{\theta}{T^\*}].\tag{A5e}$$

*Appendix B.2. Choice of Time and Length Scales*

The modeling assumptions (A2–A4) translate into the following choices:


$$M = \frac{t\_a}{t\_0} = \frac{t\_a}{t\_r} \beta \theta = \frac{t\_a}{A^{-1} \exp\theta} \frac{q y\_0}{c\_p T\_0} \theta.$$

Therefore, since *t<sup>a</sup>* = *t*0,

The assumptions *M* = 1 and *θ* 1 are compatible if *Aβ* is large; this is consistent with the assumption that the reaction is strongly exothermic. Additionally, since *t*<sup>0</sup> *t<sup>r</sup>* for *βθ* 1, the reaction proceeds slowly with respect to the reference time scale *t*0.

−*θ* .

*M* = 1 = *Aβθtae*

With these assumptions, the dimensionless equations become

$$p^\* = \\_\rho^\* T^\* \tag{A6a}$$

$$\frac{D\rho^\*}{Dt^\*} + \rho^\* \operatorname{div}^\* \mathbf{u}^\* = \begin{array}{c} \mathbf{0} \\ \end{array} \tag{A6b}$$

$$
\rho^\* \frac{D\mathbf{u}^\*}{Dt^\*} + \frac{1}{\gamma} \nabla^\* p^\* \quad = \quad \mathbf{0} \tag{A6c}
$$

$$\rho^\* \frac{D T^\*}{D t^\*} - \frac{\gamma - 1}{\gamma} \frac{D p^\*}{D t^\*} \quad = \frac{1}{\theta} \rho^\* y^\* \exp[\theta - \frac{\theta}{T^\*}] \tag{A6d}$$

$$\frac{Dy^\*}{Dt^\*} \quad = \quad -\frac{1}{\beta\theta} y^\* \exp[\theta - \frac{\theta}{T^\*}].\tag{A6e}$$

Multiplying (A6e) by *βρ*∗ and adding the result to (A6d) yields

$$(\rho^\*-1)\frac{D}{Dt^\*}[T^\*+\beta y^\*]+\frac{D}{Dt^\*}\left[T^\*-\frac{\gamma-1}{\gamma}p^\*+\beta y^\*\right]=0.\tag{A7}$$

#### **References**

