Search Results (64)

According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups (classical Galois theory), algebraic groups (Picard–Vessiot theory) and algebraic pseudogroups (Drach–Vessiot theory). The corresponding automorphic differential extensions are such that $di{m}_K\left(L\right)=\mid L/K\mid <\infty$, the transcendence degree $trd(L/K)<\infty$ and $trd(L/K)=\infty$ with $difftrd(L/K)<\infty$, respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry to revisit DGT, pointing out the deep confusion between prime differential ideals (defined by J.-F. Ritt in 1930) and maximal ideals that has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we utilize Hopf algebras to investigate the structure of the algebraic Lie pseudogroups involved, specifically those defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time to illustrate these results, particularly through the study of the Hamilton–Jacobi equation in analytical mechanics. This paper also pays tribute to Prof. A. Bialynicki-Birula (BB) on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group G acting on itself is the simplest example of a PHS. If G is connected and defined over a field K, we may introduce the algebraic extension $L=K\left(G\right)$; then, there is a Galois correspondence between the intermediate fields $K\subset K^{\prime }\subset L$ and the subgroups $e\subset G^{\prime }\subset G$, provided that $K^{\prime }$ is stable under a Lie algebra $\Delta$ of invariant derivations of $L/K$. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without using group parameters in any way. To the best of the author’s knowledge, algebraic Lie pseudogroups have never been introduced by people dealing with DGT in the spirit of Kolchin; that is, they have only been considered with systems of ordinary differential (OD) equations, but never with systems of partial differential (PD) equations. Full article

We investigate left-invariant generalized cross-curvature solitons on simply connected three-dimensional Lorentzian Lie groups. Working with the assumption that the contravariant tensor $P_{ij}$ (defined from the Ricci tensor and scalar curvature) is invertible, we derive the algebraic soliton equations for left-invariant metrics and classify all left-invariant generalized cross-curvature solitons (for the generalized equation ${\mathcal{L}}_{X}g+\lambda g=2h+2\rho Rg$) on the standard 3D Lorentzian Lie algebra types (unimodular Types Ia, Ib, II, and III and non-unimodular Types IV.1, IV.2, and IV.3). For each Lie algebra type, we state the necessary and sufficient algebraic conditions on the structure constants, provide explicit formulas for the soliton vector fields X (when they exist), and compute the soliton parameter $\lambda$ in terms of the structure constants and the parameter $\rho$. Our results include several existence families, explicit nonexistence results (notably for Type Ib and Type IV.3), and consequences linking the existence of left-invariant solitons with local conformal flatness in certain cases. The classification yields new explicit homogeneous generalized cross-curvature solitons in the Lorentzian setting and clarifies how the parameter $\rho$ modifies the algebraic constraints. Examples and brief geometric remarks are provided. Full article

The modeling of the enthalpy of mixing in binary alloys is essential to thermodynamic assessments and computational alloy design, particularly in data-scarce systems where experimental measurements are limited or incomplete. In this work, we propose a machine learning framework for the prediction of mixing enthalpy in binary alloys under conditions of limited data availability. The method integrates symmetry-augmented embeddings, which enforce physical invariances such as element permutation and compositional mirroring, ensuring consistency across chemically equivalent representations and capturing chemically meaningful similarities between elements, thereby supporting knowledge transfer across alloy systems. To account for data uncertainty and improve trust in predictions, we incorporate Bayesian neural networks, enabling the estimation of predictive confidence, especially in composition ranges lacking experimental data. The model is trained jointly across multiple binary alloy systems, allowing it to share structural insights and improve prediction quality in data-limited concentration intervals. The method achieves a reduction in mean absolute error by more than a factor of eight compared with the classical Miedema model (0.53 kJ·mol⁻¹ vs. 4.27 kJ·mol⁻¹) while maintaining consistent accuracy even when trained on only 25% of the experimental measurements, confirming its robustness thanks to cross-alloy knowledge transfer and symmetry-based data augmentation. We evaluate the method on a benchmark dataset containing both fully and partially characterized binary alloy systems and demonstrate its effectiveness in interpolating and extrapolating enthalpy values while providing reliable uncertainty estimates. The results highlight the value of incorporating domain-specific symmetries and uncertainty-aware learning in data-driven material modeling and suggest that this approach can support predictive thermodynamic assessments even in under-sampled systems. Full article

A graph with $q(a+t)$ vertices is known as a q-broom-like graph $K_q\sqcap B(a;t)$, which is produced by the hierarchical product of the complete graph $K_q$ by the rooted broom $B(a;t)$, where $q\ge 3,a\ge 1$ and $t\ge 1$. A numerical quantity associated with graph structure is called a topological index. The inverse sum indeg index (shortened to $ISI$ index) is a topological index defined as $ISI\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}{\displaystyle \frac{d_{v_i}{d}_{v_j}}{d_{v_i}+d_{v_j}}}$, where $d_{v_i}$ is the degree of the vertex $v_i$. In this paper, we take into consideration the $ISI$ index for q-broom-like graphs and perform a thorough analysis of it. We find the $ISI$ spectrum of q-broom-like graphs and derive the closed formulas for their $ISI$ index and $ISI$ energy. We also characterize extremal graphs and arrange them according to their $ISI$ index and $ISI$ energy, respectively. Further, a quantitative structure–property relationship is used to predict six physicochemical properties of sixteen alkaloid structures using $ISI$ index and $ISI$ energy. Both graph invariants have significant correlation values, indicating the accuracy and utility of the findings. The conclusions made in this article can help chemists and pharmacists research alkaloids’ structures for applications in industry, pharmacy, agriculture, and daily life. Full article

Here all rings have identities. Let R be a ring and let R-mod denote the additive category of left finitely generated R-modules. Note that if R is a noetherian ring, then R-mod is an abelian category and every R-module is a finite direct sum of indecomposable R-modules. Finite Group Modular Representation Theory concerns the study of left finitely generated $\mathcal{O}G$-modules where G is a finite group and $\mathcal{O}$ is a complete discrete valuation ring with $\mathcal{O}/J\left(\mathcal{O}\right)$ a field of prime characteristic p. Thus $\mathcal{O}G$ is a noetherian $\mathcal{O}$-algebra. The Green Theory in this area yields for each isomorphism type of finitely generated indecomposable (and hence for each isomorphism type of finitely generated simple $\mathcal{O}G$-module) a theory of vertices and sources invariants. The vertices are derived from the set of p-subgroups of G. As suggested by the above, in Basic Definition and Main Results for Rings Section, let $\mathrm{\Sigma }$ be a fixed subset of subrings of the ring R and we develop a theory of $\mathrm{\Sigma }$-vertices and sources for finitely generated R-modules. We conclude Basic Definition and Main Results for Rings Section with examples and show that our results are compatible with a ring isomorphic to R. For Idempotent Morita Equivalence and Virtual Vertex-Source Pairs of Modules of a Ring Section, let e be an idempotent of R such that $R=ReR$. Set $B=eRe$ so that B is a subring of R with identity e. Then, the functions $eR{\otimes }_R\ast :R-\mathrm{mod}\to B-\mathrm{mod}$ and $Re{\otimes }_B\ast :B-\mathrm{mod}\to R-\mathrm{mod}$ form a Morita Categorical Equivalence. We show, in this Section, that such a categorical equivalence is compatible with our vertex-source theory. In Two Applications with Idemptent Morita Equivalence Section, we show such compatibility for source algebras in Finite Group Block Theory and for naturally Morita Equivalent Algebras. Full article

Tropical trees represent an important potential archive of climate and ecological information, but their dendrochronology based on conventional techniques has been challenging. We conducted a pilot study of the wood anatomy and dendroclimatological potential of Juglans neotropica Diels (Juglandaceae), an IUCN Red List species, using 225 radii sampled from 57 trees in Piura (4°55′ S, 79° 56′ W), northern Peru. A total of 112 radii from 40 trees passed quality control and are included in the tree-ring width chronology for this species. J. neotropica has demonstrably annual rings, and results are consistent with reports that the species has a dormant period during the dry season, which locally is approximately June–November. Local precipitation is correlated (p = 0.10, 1-tailed test) with tree-ring growth, lagged by one year, consistent with other studies of tropical tree species. The age distribution of the sample collection of J. neotropica is young and invariant, probably because of selective cutting by local villagers. To supplement ring-width analysis, we conducted the first oxygen isotopic (δ¹⁸O) and radiocarbon (∆¹⁴C) analysis for this species on radii from two individuals; results are preliminary given sample size limitations, but consistent with dendrochronological dating, within uncertainties, in all three chronometric analyses. A two-sample composite annually-averaged δ¹⁸O anomaly data series is correlated significantly with gridded regional growing season (December–May) precipitation (1973/74–2005/06). Qualitatively consistent with simulation of ring width and δ18O, responses to El Niño events are manifested in positive ring-growth anomalies and negative isotopic anomalies following known event years. The combination of tree-ring, radiocarbon, stable isotopic analyses, and the application of sensor and chronological modeling provides a degree of confidence in the results that would not have been possible by relying on any single approach and indicates the potential for further investigation of this and other tropical tree species with uncertain ring boundaries. Full article

Recently, the exponential arithmetic–geometric index ($EAG$) was introduced. The exponential arithmetic–geometric index ($EAG$) of a graph G is defined as $EAG\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}{e}^{{\displaystyle \frac{d_i+d_j}{2\sqrt{d_i{d}_j}}}}$, where $d_i$ represents the degree of the vertex $v_i$ in G. The characterization of extreme structures in relation to graph invariants from the class of unicyclic graphs is an important problem in discrete mathematics. Cruz et al., 2022 proposed a unified method for finding extremal unicyclic graphs for exponential degree-based graph invariants. However, in the case of $EAG$, this method is insufficient for generating the maximal unicyclic graph. Consequently, the same article presented an open problem for the investigation of the maximal unicyclic graph with respect to this invariant. This article completely characterizes the maximal unicyclic graph in relation to $EAG$. Full article

Basic principles of ferroelectric activity induced by the noncollinear alignment of spins are reviewed. There is a fundamental reason why the inversion symmetry can be broken by certain magnetic order. This situation occurs when the magnetic order simultaneously involves ferromagnetic ($\mathit{F}$) and antiferromagnetic ($\mathit{A}$) counterparts, transforming under the spatial inversion $\mathcal{I}$ and time reversal $\mathcal{T}$ as $\mathcal{I}\mathit{F}=\mathit{F}$ and $\mathcal{IT}\mathit{A}=\mathit{A}$, respectively. The incompatibility of these two conditions results in breaking the inversion symmetry, which manifests itself in the electric polarization $\overrightarrow{P}$. The noncollinear alignment of spins is one of examples of such coexistence of $\mathit{F}$ and $\mathit{A}$. This coexistence principle imposes a constraint on possible dependencies of $\overrightarrow{P}$ on the directions of spins, which can include only “antisymmetric coupling” in the bond, ${\overrightarrow{\mathcal{P}}}_{ij}·[{\mathit{e}}_i\times {\mathit{e}}_j]$, and “single-ion anisotropy”, ${\mathit{e}}_i·$ $\overrightarrow{\mathbb{\Pi }}$ ${\mathit{e}}_i$. Microscopically, ${\overrightarrow{\mathcal{P}}}_{ij}$ can be evaluated in the framework of superexchange theory. For the single Kramers doublet, this theory yields ${\overrightarrow{\mathcal{P}}}_{ij}\sim {\overrightarrow{\mathit{r}}}_{ij}^{\phantom{0}}$, where ${\overrightarrow{\mathit{r}}}_{ij}^{\phantom{0}}$ is the spin-dependent part of the position operator induced by the relativistic spin-orbit coupling. ${\overrightarrow{\mathit{r}}}_{ij}^{\phantom{0}}$ remains invariant under spatial inversion, providing the microscopic reason why noncollinear alignment of spins can induce $\overrightarrow{P}$ even in centrosymmetric crystals. The symmetry properties of ${\overrightarrow{\mathit{r}}}_{ij}^{\phantom{0}}$ can be rationalized from the viewpoint of symmetry of Kramers states. Particularly, the commonly used Katsura–Nagaosa–Balatsky (KNB) rule $\overrightarrow{P}\propto {\overrightarrow{ϵ}}_{ji}\times [{\mathit{e}}_i\times {\mathit{e}}_j]$ (${\overrightarrow{ϵ}}_{ji}$ being the direction of the bond $ij$) can be justified only for relatively high symmetry of the bonds. The single-ion anisotropy vanishes for the spin ${\displaystyle \frac{1}{2}}$ or if magnetic ions are located in inversion centers, thus severely restricting the applicability of this microscopic mechanism. The properties of multiferroic materials are reconsidered from the viewpoint of these principles. A particular attention is paid to complications caused by possible deviations from the KNB rule. Full article

An inverse-square probability mass function (PMF) is at the Newcomb–Benford law (NBL)’s root and ultimately at the origin of positional notation and conformality. $Pr\left(Z\right)={\left(2Z\right)}^{-2}$, where $Z\in {\mathbb{Z}}^{+}$. Under its tail, we find information as harmonic likelihood $\mathcal{L}\left(\left[s,t\right)\right)=H_{t-1}-H_{s-1}$, where $H_n$ is the nth harmonic number. The global $\mathbb{Q}$-NBL is $Pr\left(b,q\right)={\displaystyle \raisebox{1ex}{$\mathcal{L}\left(\left[q,q+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\mathcal{L}\left(\left[1,b\right)\right)$}\right.}={\left(q{H}_{b-1}\right)}^{-1}$, where b is the base and q is a quantum ($1\le q<b$). Under its tail, we find information as logarithmic likelihood $\ell \left(\left[i,j\right)\right)=ln{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The fiducial $\mathbb{R}$-NBL is $Pr\left(r,d\right)={\displaystyle \raisebox{1ex}{$\ell \left(\left[d,d+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\ell \left(\left[1,r\right)\right)$}\right.}={log}_r\left(1+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}\right)$, where $r\ll b$ is the radix of a local complex system. The global Bayesian rule multiplies the correlation between two numbers, s and t, by a likelihood ratio that is the NBL probability of bucket $\left[s,t\right)$ relative to b’s support. To encode the odds of quantum j against i locally, we multiply the prior odds ${\displaystyle \raisebox{1ex}{$Pr\left(b,j\right)$}\!\left/ \!\raisebox{-1ex}{$Pr\left(b,i\right)$}\right.}$ by a likelihood ratio, which is the NBL probability of bin $\left[i,j\right)$ relative to r’s support; the local Bayesian coding rule is $\tilde{o}\left(j:i|r\right)={\displaystyle \raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$j$}\right.}{log}_r{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The Bayesian rule to recode local data is $\tilde{o}\left(j:i|r^{\prime }\right)=\tilde{o}\left(j:i|r\right){\displaystyle \raisebox{1ex}{$lnr$}\!\left/ \!\raisebox{-1ex}{$ln{r}^{\prime }$}\right.}$. Global and local Bayesian data are elements of the algebraic field of “gap ratios”, ${\displaystyle \frac{A-B}{C-D}}$. The cross-ratio, the central tool in conformal geometry, is a subclass of gap ratio. A one-dimensional coding source reflects the global Bayesian data of the harmonic external world, the annulus $\left\{x\in \mathbb{Q}|1\le \left|x\right|<b\right\}$, into the local Bayesian data of its logarithmic coding space, the ball $\left\{x\in \mathbb{Q}|\left|x\right|<1-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}\right\}$. The source’s conformal encoding function is $y={log}_r\left(2x-1\right)$, where x is the observed Euclidean distance to an object’s position. The conformal decoding function is $x={\displaystyle \frac{1}{2}}\left(1+r^y\right)$. Both functions, unique under basic requirements, enable information- and granularity-invariant recursion to model the multiscale reality. Full article

We introduce a special class of cubic curves whose defining parameter satisfies a linear or quadratic equation provided by the values of a cross ratio. There are only seven such cubics and several properties of the real cubics in this class (some of them being elliptic curves) are discussed. Using the Möbius transformation, we extend this self-duality and obtain new families of remarkable complex cubics. In addition, we study (from the differential geometric viewpoint) the surface parameterized by all real cubic curves and we derive its curvature functions. As a by-product, we find a new classification of real Möbius transformations and some estimates for the number of vertices of real cubic curves. Full article

We propose a Ramsey-theory-based approach for the analysis of the behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by a Ramsey-theory-based, bi-colored, complete graph. Vectors of momenta of the particles ${\overrightarrow{p}}_i$ serve as the vertices of the graph. We start from the graph representing the system in the frame of the center of masses, where the momenta of the particles in this system are ${\overrightarrow{p}}_{cmi}.$ If ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))\ge 0$ is true, the vectors of momenta of the particles numbered i and j are connected with a red link; if ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))<0$ takes place, the vectors of momenta are connected with a green link. Thus, the complete, bi-colored graph emerges. Considering an isolated system built of six interacting particles, according to the Ramsey theorem, the graph inevitably comprises at least one monochromatic triangle. The coloring procedure is invariant relative to the rotations/translations of frames; thus, the graph representing the system contains at least one monochromatic triangle in any of the frames emerging from the rotation/translation of the original frame. This gives rise to a novel kind of mechanical invariant. Similar coloring is introduced for the angular momenta of the particles. However, the coloring procedure is sensitive to Galilean/Lorenz transformations. Extensions of the suggested approach are discussed. Full article

Suppose R is a local ring with invariants $p,n,r,m,k$ and $mr=4,$ that is R of order $p^4.$ Then, $R=R_0+u{R}_0+v{R}_0+w{R}_0$ has maximal ideal $J=u{R}_0+v{R}_0+w{R}_0$ of order $p^{(m-1)r}$ and a residue field F of order $p^r,$ where $R_0=GR(p^n,r)$ is the coefficient subring of $R.$ In this article, the goal is to improve the understanding of linear codes over small-order non-chain rings. In particular, we produce the MacWilliams formulas and generator matrices for linear codes of length N over $R.$ In order to accomplish that, we first list all such rings up to isomorphism for different values of $p,n,r,m,k.$ Furthermore, we fully describe the lattice of ideals in R and their orders. Next, for linear codes C over R, we compute the generator matrices and MacWilliams identities, as shown by numerical examples. Given that non-chain rings are not principal ideals rings, it is crucial to acknowledge the difficulties that arise while studying linear codes over them. Full article

Miller (Arch. Rational Mech. Anal., 2020) posed the question of whether it is possible to prove the Navier–Stokes regularity criterion using only one entry of the strain tensor $S_{ij}$. Although this paper does not fully address this question, we do establish several scaling-invariant Serrin-type criteria based on one row of the strain tensor. Full article

Understanding hydrological nonstationarity under climate change is important for runoff prediction and it enables more robust decisions. Regarding the multiple structural hypotheses, this study aims to identify and interpret hydrological structural nonstationarity using the Bayesian Model Averaging (BMA) method by (i) constructing a nonstationary model through the Bayesian weighted averaging of two lumped conceptual rainfall–runoff (RR) models (the Xinanjiang and GR4J model) with time-varying weights; and (ii) detecting the temporal variation in the optimized Bayesian weights under climate change conditions. By combining the BMA method with period partition and time sliding windows, the efficacy of adopting time-varying model structures is investigated over three basins located in the U.S. and Australia. The results show that (i) the nonstationary ensemble-averaged model with time-varying weights surpasses both individual models and the ensemble-averaged model with time-invariant weights, improving $NSE[\sqrt{Q}]$ from 0.04 to 0.15; (ii) the optimized weights of Xinanjiang model increase and that of GR4J declines with larger precipitation, and vice versa; (iii) the change in the optimized weights is proportional to that of precipitation under monotonic climate change, as otherwise the mechanism changes significantly. Overall, it is recommended to adopt nonstationary structures in hydrological modeling. Full article

This work proposes a matching data science approach for the laser ablation quality, r_eb, the study of Si₃N₄ film based on supervised machine learning classifiers in the CMOS-MEMS process. The study demonstrates that there exists an energy threshold, E_th, for laser ablation. If the laser energy surpasses this threshold, increasing the interval time will not contribute significantly to the recovery of pulse laser energy. Thus, r_eb enhancement is limited. When the energy is greater than 0.258 mJ, there exists a critical value of interval time at which the r_eb value is relatively low for each energy level, respectively. In addition, the variation of r_eb, Δr_eb, is independent of the interval time at the invariant point of energy between 0.32 mJ and 0.36 mJ. Energy and interval time exhibit a Pearson correlation of 0.82 and 0.53 with r_eb, respectively. To maintain Δr_eb below 0.15, green laser ablation of Si₃N₄ at operating energies of 0.258–0.378 mJ can adopt a baseline interval time of the initial baseline multiplied by 1/∜2. Additionally, for operating energies of 0.288–0.378 mJ during Si₃N₄ laser ablation, Δr_eb can be kept below 0.1. With the forced partition methods, namely, the k-means method and percentile method, the XGBoost (v 2.0.3) classifier maintains a competitive accuracy across test sizes of 0.20–0.40, outperforming the machine learning algorithms Random Forest and Logistic Regression, with the highest accuracy of 0.78 at a test size of 0.20. Full article