Search Results (20)

The Clifford–Weyl algebra ${𝒜}_q^{\pm }$, as a class of solvable polynomial algebras, combines the anti-commutation relations of Clifford algebras ${𝒜}_q^{+}$ with the differential operator structure of Weyl algebras ${𝒜}_q^{-}$. It exhibits rich algebraic and geometric properties. This paper employs Gröbner–Shirshov basis principles in concert with Poincaré–Birkhoff–Witt (PBW) basis methodology to delineate the iterated skew polynomial structures within ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$. By constructing explicit PBW generators, we analyze the structural properties of both algebras and their modules using constructive methods. Furthermore, we prove that ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$ are Auslander regular, Cohen–Macaulay, and Artin–Schelter regular. These results provide new tools for the representation theory in noncommutative geometry. Full article

In this paper, we generalize and study the concept of Hadamard product of ideals of projective varieties to the case of monomial ideals. We have a research direction similar to the one of the join of monomial ideals contained in a paper of Sturmfels and Sullivant. In the second part of the paper, we give a brief tutorial on the Hadamard.m2 package of Macaulay2. Full article

In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen–Macaulay property. As a result, we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex splittable ideals. Furthermore, we provide new and simpler combinatorial proofs of known Cohen–Macaulay criteria for several families of monomial ideals, such as (vector-spread) strongly stable ideals and (componentwise) polymatroidals. Finally, we characterize the family of bi-Cohen–Macaulay graphs by the novel criterion for the Cohen–Macaulayness of vertex splittable ideals. Full article

We already know that the noncommutative $\mathbb{N}$-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties to noncommutative $\mathbb{N}$-graded Noetherian algebra. By generalizing the conclusions about commutative rings and combining what we already know about noncommutative graded algebras, we identify a class of noncommutative graded algebras with the property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic. We also discuss how the relationship between AS–Gorenstein algebras and AS–Cohen–Macaulay algebras admits a balanced dualizing complex. We show that AS–Gorenstein algebras and AS–Cohen–Macaulay algebras with a balanced dualizing complex belong to this algebra. Full article

Multivariate public-key cryptosystems are potential candidates for post-quantum cryptography. The security of multivariate public-key cryptosystems relies on the hardness of solving a system of multivariate quadratic polynomial equations. Faugère’s F4 algorithm is one of the solution techniques based on the theory of Gröbner bases and selects critical pairs to compose the Macaulay matrix. Reducing the matrix size is essential. Previous research has not fully examined how many critical pairs it takes to reduce to zero when echelonizing the Macaulay matrix in rows. Ito et al. (2021) proposed a new critical-pair selection strategy for solving multivariate quadratic problems associated with encryption schemes. Instead, this paper extends their selection strategy for solving the problems associated with digital signature schemes. Using the OpenF4 library, we compare the software performance between the integrated F4-style algorithm of the proposed methods and the original F4-style algorithm. Our experimental results demonstrate that the proposed methods can reduce the processing time of the F4-style algorithm by up to a factor of about seven under certain specific parameters. Moreover, we compute the minimum number of critical pairs to reduce to zero and propose their extrapolation outside our experimental scope for further research. Full article

In 1974, the author proved that the codimension of the ideal $(g_1,g_2,\dots ,g_d)$ generated in the group algebra $\mathbb{K}\left[{\mathbb{Z}}^d\right]$ over a field $\mathbb{K}$ of characteristic 0 by generic Laurent polynomials having the same Newton polytope $\mathsf{\Gamma }$ is equal to $d!\times Volume\left(\mathsf{\Gamma }\right)$. Assuming that Newtons polytope is simplicial and super-convenient (that is, containing some neighborhood of the origin), the author strengthens the 1974 result by explicitly specifying the set ${\mathbb{B}}^{sh}$ of monomials of cardinality $d!\times Volume\left(\mathsf{\Gamma }\right)$, whose equivalence classes form a basis of the quotient algebra $\mathbb{K}\left[{\mathbb{Z}}^d\right]/(g_1,g_2,\dots ,g_d)$. The set ${\mathbb{B}}^{sh}$ is constructed inductively from any shelling $sh$ of the polytope $\mathsf{\Gamma }$. Using the ${\mathbb{B}}^{sh}$ structure, we prove that the associated graded $\mathbb{K}$ -algebra $g{r}^{\mathsf{\Gamma }}\left(\mathbb{K}\left[{\mathbb{Z}}^d\right]\right)$ constructed from the Arnold–Newton filtration of $\mathbb{K}$ -algebra $\mathbb{K}\left[{\mathbb{Z}}^d\right]$ has the Cohen–Macaulay property. This proof is a generalization of B. Kind and P. Kleinschmitt’s 1979 proof that Stanley–Reisner rings of simplicial complexes admitting shelling are Cohen–Macaulay. Finally, we prove that for generic Laurent polynomials $(f_1,f_2,\dots ,f_d)$ with the same Newton polytope $\mathsf{\Gamma }$, the set ${\mathbb{B}}^{sh}$ defines a monomial basis of the quotient algebra $\mathbb{K}\left[{\mathbb{Z}}^d\right]/(g_1,g_2,\dots ,g_d)$. Full article

In this paper, we discuss a few simple methods for computing the local topological index and its various analogs for vector fields and differential forms given on complex varieties with singularities of different types. They are based on properties of regular meromorphic and logarithmic differential forms, of the dualizing (canonical) module and related constructions. In particular, we show how to compute the index on Cohen–Macaulay, Gorenstein and monomial curves, on normal and non-normal surfaces and some others. In contrast with known traditional approaches, we use neither computers, nor integration, perturbations, deformations, resolution of singularities, spectral sequences or other related standard tools of pure mathematics. Full article

We discuss several known formulas that use the Macaulay duration and convexity of commonly used cash flow streams to approximate their net present value, and compare them with a new approximation formula that involves hyperbolic functions. Our objective is to assess the reliability of each approximation formula under different scenarios. The results in this note should be of interest to actuarial candidates and educators as well as analysts working in all areas of actuarial practice. Full article

The purpose of this publication is to quantify and compare the market risk on the external government debt of Kazakhstan and Bulgaria in the conditions of COVID-19, the emerging energy crisis, and the coup attempt in the first country. In particular, the authors invest the market risk of sovereign bonds issued on global financial markets. Market risk is assessed both as a single issue and at a portfolio level using the Value-at-risk approach. Sixteen samples with historical observations of all issues of Kazakhstan’s and Bulgarian Sovereign Bonds issued on the international financial markets were formed. The duration method was used in the calculation of Delta normal bond VaR and CVaR. It was found that with the same credit rating, similar portfolio duration levels, similar GDP per capita, Debt (% of GDP), and Debt Per Capita, the market risk on their portfolio differed significantly. The comparison of risk levels between the two portfolios was made by six indicators–two indicators measuring linearly the sensitivity of bond prices to changes in market interest rates (Weighted average Macaulay duration and Weighted average modified duration) and four downside indicators (Undiversified VaR, Diversified VaR, Undiversified CVaR, amd Diversified CVaR). The return/risk performance of both portfolios was assessed by the Sharpe ratio in three variants (SR Undiversified VaR, SR Diversified VaR, and SR Diversified CVaR). When evaluating the bond portfolio VaR and CVaR, a practical version of the Duration method was proposed, which allows the use of an unlimited number of assets, taking into account the correlations between yield returns and historical price volatility. Full article

Contemporary Indian philosophy is a distinct genre of philosophy that draws both on classical Indian philosophical sources and on Western materials, old and new. It is comparative philosophy without borders. In this paper, I attempt to show how contemporary Indian philosophy works through five instances from five of its protagonists: Krishnachandra Bhattacharyya (his new interpretation of the old rope-snake parable in his essay “Śaṅkara’s Doctrine of Maya”, 1925); Daya Krishna (I focus on the “moral monadism” that the theory of karma in his reading leads to, drawing on his book Discussion and Debate in Indian Philosophy, 2004); Ramchandra Gandhi (his commentary on the concept of Brahmacharya in correspondence with his grandfather, the Mahatma, in his essay “Brahmacharya”, 1981); Mukund Lath (on identity through—not despite—change, with classical Indian music, Rāga music, as his case-study, in his essay “Identity through Necessary Change”, 2003); and Rajendra Swaroop Bhatnagar (on suffering, in his paper “No Suffering if Human Beings Were Not Sensitive”, 2021). My aim is twofold. First, to introduce five contemporary Indian philosophers; and second, to raise the question of newness and philosophy. Is there anything new in philosophy, or is contemporary philosophy just a footnote—à la Whitehead—to the writings of great thinkers of the past? Is contemporary Indian philosophy, my protagonists included, just a series of footnotes to classical thinkers both in India and Europe? Footnotes to the Upaniṣads, Nāgārjuna, Dharmakīrti and Śaṅkara, as much as (let us not forget colonialism and Macaulay) to Plato, Aristotle, Kant and Hegel? Footnotes can be creative and work almost as a parallel text, interpretive, critical, even subversive. However, my contention is that contemporary Indian philosophy (I leave it to others to plea for contemporary Western philosophy) is not a footnote, it is a text with agency of its own, validity of its own, power of its own. It is wholly and thoroughly a text worth reading. In this paper, I make an attempt to substantiate this claim through the philosophical mosaic I offer, in each instance highlighting both the continuity with classical sources and my protagonists’ courageous transgressions and innovations. Full article

Piezoelectric nanogenerators can convert energy from ambient vibrations into electrical energy. In the future, these nanogenerators could substitute conventional electrochemical batteries to supply electrical energy to consumer electronics. The optimal design of nanogenerators is fundamental in order to achieve their best electromechanical behavior. We present the analytical electromechanical modeling of a vibration-based piezoelectric nanogenerator composed of a double-clamped beam with five multilayered cross-sections. This nanogenerator design has a central seismic mass (910 μm thickness) and substrate (125 μm thickness) of polyethylene terephthalate (PET) as well as a zinc oxide film (100 nm thickness) at the bottom of each end. The zinc oxide (ZnO) films have two aluminum electrodes (100 nm thickness) through which the generated electrical energy is extracted. The analytical electromechanical modeling is based on the Rayleigh method, Euler–Bernoulli beam theory and Macaulay method. In addition, finite element method (FEM) models are developed to estimate the electromechanical behavior of the nanogenerator. These FEM models consider air damping at atmospheric pressure and optimum load resistance. The analytical modeling results agree well with respect to those of FEM models. For applications under accelerations in y-direction of 2.50 m/s² and an optimal load resistance of 32,458 Ω, the maximum output power and output power density of the nanogenerator at resonance (119.9 Hz) are 50.44 μW and 82.36 W/m³, respectively. This nanogenerator could be used to convert the ambient mechanical vibrations into electrical energy and supply low-power consumption devices. Full article

In the requirement engineering phase, the team members work to get the user requirements, comprehend them and specify them for the next process. There are many models for the requirement engineering phase. There is a need to select the best Requirement Engineering model, and integrate it with cloud computing, that can give the best response to the users and software developers and avoid mistakes in the requirement engineering phase. In this study, these models are integrated with the cloud computing domain, and we report on the security considerations of all the selected models. Four requirement engineering process models are selected for this study: the Linear approach, the Macaulay Linear approach, and the Iterative and Spiral models. The focus of this study is to check the security aspects being introduced by the cloud platform and assess the feasibility of these models for the popular cloud environment SaaS. For the classification of the security aspects that affect the performance of these model, a framework is proposed, and we check the results regarding selected security parameters and RE models. By classifying the selected RE models for security aspects based on deep learning techniques, we determine that the Loucopoulos and Karakostas iterative requirements engineering process model performs better than all the other models. Full article

Microelectromechanical system (MEMS)-based piezoelectric energy harvesting (PEH) devices can convert the mechanical vibrations of their surrounding environment into electrical energy for low-power sensors. This electrical energy is amplified when the operation resonant frequency of the PEH device matches with the vibration frequency of its surrounding environment. We present the electromechanical modeling of two MEMS-based PEH devices to transform the mechanical vibrations of domestic washing machines into electrical energy. These devices have resonant structures with a T shape, which are formed by an array of multilayer beams and a ultraviolet (UV)-resin seismic mass. The first layer is a substrate of polyethylene terephthalate (PET), the second and fourth layers are Al and Pt electrodes, and the third layer is piezoelectric material. Two different types of piezoelectric materials (ZnO and PZT-5A) are considered in the designs of PEH devices. The mechanical behavior of each PEH device is obtained using analytical models based on the Rayleigh–Ritz and Macaulay methods, as well as the Euler–Bernoulli beam theory. In addition, finite element method (FEM) models are developed to predict the electromechanical response of the PEH devices. The results of the mechanical behavior of these devices obtained with the analytical models agree well with those of the FEM models. The PEH devices of ZnO and PZT-5A can generate up to 1.97 and 1.35 µW with voltages of 545.32 and 45.10 mV, and load resistances of 151.12 and 1.5 kΩ, respectively. These PEH devices could supply power to internet of things (IoT) sensors of domestic washing machines. Full article

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each $1\le s\le 5$ , an O-sequence $(h_0,h_1,\dots ,h_{2s-1})\in {\mathbb{Z}}_{\ge 0}^{2s}$ satisfying the properties that (i) $h_0\le h_1\le \cdots \le h_{s-1}$ , (ii) $h_{2s-1}=h_0$ , $h_{2s-2}=h_1$ and (iii) $h_{2s-1-i}=h_i+{(-1)}^i$ , $2\le i\le s-1$ , can be the h-vector of a Cohen-Macaulay standard G-domain. Full article