Development of the method of averaging in Clifford geometric algebras

We develop the method of averaging in Clifford (geometric) algebras suggested by the author in previous papers. We consider operators constructed using two different sets of anticommuting elements of real or complexified Clifford algebras. These operators generalize Reynolds operators from the representation theory of finite groups. We prove a number of new properties of these operators. Using the generalized Reynolds operators, we give a complete proof of the generalization of Pauli’s theorem to the case of Clifford algebras of arbitrary dimension. The results can be used in geometry, physics, engineering, computer science, and other applications.

Research target: Mathematics

Language: English

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Keywords: Clifford algebra Pauli's theorem Reynolds operator geometric algebra method of averaging

2-Elliptic Periodic Orbits near a Nonsimple Homoclinic Tangency in Four-Dimensional Symplectic Maps

Lerman L. M., Turaev D. V., Regular and Chaotic Dynamics 2026 Vol. 31 No. 3 P. 349–369

We show that bifurcations of four-dimensional symplectic diffeomorphisms with a quadratic homoclinic tangency to a saddle periodic orbit with real multipliers produce 2-elliptic periodic orbits if the tangency is not partially hyperbolic. We show that a normal form for the rescaled first-return maps near such tangency is given by a four-dimensional symplectic H´enonlike map and study bifurcations of the ...

Added: May 15, 2026

Bibliometric Analysis by Network Models

Aleskerov F. T., Yakuba V. I., Khutorskaya O. et al., Springer, 2026.

The book contains new models of bibliometric analysis based on centrality measures in network analysis, pattern analysis and stability analysis. A distinctive feature of these centrality measures is that they account for the parameters of vertices and group influence of vertices to a vertex. This reveals specific groups of publications, authors, terms, journals and affiliations ...

Added: May 15, 2026

Neural-network maps for two-parameter modeling of bistability and codimension-two bifurcations in two-dimensional flow dynamical systems

Kuptsov P., Panyushev A., Stankevich N., Chaos 2026 Vol. 36 No. 5 Article 053138

We develop a machine-learning approach to reproduce the behavior of two versions of the van der Pol oscillator exhibiting a subcritical Andronov–Hopf bifurcation, with or without a codimension-2 Bautin point. We construct a neural-network model that functions as a recur rent map and train it on short segments of oscillator trajectories. The results show that, ...

Added: May 15, 2026

Bifurcations and Structural Stability of Generic PC-HC Families

Dorovskiy A., / Series arXiv "math". 2026.

In this paper the structural stability of generic families of vector fields of the PC-HC class on the two-dimensional sphere is proved. A classification of these families up to moderate equivalence in neighborhoods of their large bifurcation supports is presented, based on such invariants as the configuration and the characteristic set. The realization lemma is proved. ...

Added: May 14, 2026

The Sobolev space W_2^{1/2}: Simultaneous improvement of functions by a homeomorphism of the circle

Lebedev V., Journal of Mathematical Analysis and Applications 2026 Vol. 563 No. 2 Article 130787

It is known that for every continuous real-valued function $f$ on the circle $\mathbb T=\mathbb R/2\pi\mathbb Z$ there exists a change of variable, i.e., a self-homeomorphism $h$ of $\mathbb T$, such that the superposition $f\circ h$ is in the Sobolev space $W_2^{1/2}(\mathbb T)$. We obtain new results on simultaneous improvement of functions by a single change of variable in relation ...

Added: May 14, 2026

Symmetric Cubic Polynomials

Blokh A., Oversteegen L., Selinger N. et al., Arnold Mathematical Journal 2025 Vol. 12 No. 1 P. 1–40

We describe a model for the boundary of the connectedness locus of the parameter space of cubic symmetric polynomials. We show that there exists a monotone continuous function from the connectedness locus to the model which is a homeomorphism if the former is locally connected. ...

Added: May 13, 2026

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Petrov I., Автоматика и телемеханика 2026 № 6 С. 82–118

Системам связанных агентов и сетевому управлению посвящено большое число отечественных и зарубежных исследований. Исторически, наибольший интерес в теории управления возникал к усредняющим системам и, в частности, к задаче консенсуса. Однако сетевое взаимодействие может характеризоваться более специфическими функциями, отражающими зависимость от действий соседей по сети, что особенно явно проявляется в моделях стратегического взаимодействия на сети, которое ...

Added: May 12, 2026

Архимед: научно-методический сборник

М.: ООО «Макс Пресс», 2026.

В настоящем сборнике представлены тезисы докладов участников семинара "Интеграция основного и дополнительного физико-математического образования", проходившего 11 февраля 2026 года в ГБОУ Школа №2007 ФМШ г. москвы, а также другие публикации, посвящённые вопросам дополнительного физико-математического образования. ...

Added: May 11, 2026

A two-point phase recovering from holographic data on a single plane

Novikov R., Sivkin V., Inverse Problems 2026 Vol. 42 No. 4 Article 045009

We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in Rd, d ⩾ 2. In this region, we consider a hyperplane X with sufficiently large distance s from the origin in Rd. We give two-point local formulas for approximate recovering the radiation ...

Added: May 11, 2026

Multivariate Newton interpolation in downward closed spaces reaches the optimal Bernstein–Walsh approximation rate

Hecht M., Hofmann P., Wicaksono D. et al., IMA Journal of Numerical Analysis 2026 Vol. 00 P. 1–30

Recent advances in Bernstein—Walsh theory have extended Bernstein’s Theorem to multiple dimensions, stating that a multivariate function can be approximated with a geometric rate in a downward-closed polynomial space if and only if it is analytic in a generalized Bernstein polyellipse. To compute approximations of this class of functions—which we term Bos–Levenberg–Trefethen–(BLT) functions—we extend the ...

Added: May 11, 2026

Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing

Kelbert M., Kalimulina E. Y., Entropy 2026 Vol. 28 Article 536

We study binary hypothesis testing for i.i.d. observations under a multiplicative context weight. For the optimal weighted total loss, defined as the sum of weighted type-I and typeII losses, we prove the logarithmic asymptotic L∗n = exp{−nDwC (P,Q) + o(n)}, n →∞, where Dw C is the weighted Chernoff information. The single-letter form of the exponent relies on ...

Added: May 7, 2026

Calogero–Sutherland hyperbolic system and Heckman–Opdam $$\mathfrak {gl}_n$$ gl n hypergeometric function

Белоусов Н. М., Черепанов Л. К., Деркачов С. Э. et al., Selecta Mathematica, New Series 2026 Vol. 32 Article 44

We prove equivalence of two integral representations for the wave functions of hyperbolic Calogero–Sutherland system. For this we study two families of Baxter operators related to hyperbolic Calogero–Sutherland and rational Ruijsenaars models; the first one as a limit from hyperbolic Ruijsenaars system, while the second one independently. Besides, computing asymptotics of integral representations and also ...

Added: May 6, 2026

Hodge Laplacian Eigenvalues on Surfaces with Boundary

Муравьев М. Ю., Annales Mathematiques du Quebec 2025

Recently Rohleder proposed a new variational approach to an inequality between the Neumann and Dirichlet eigenvalues in the simply connected planar case using the language of classical vector analysis. Interpreting his approach in terms of differential forms permits to generalize these results to a much broader context. The spectrum of the absolute boundary problem for ...

Added: May 6, 2026

Об изоморфизме задачи Козлова о движении ферромагнетика в магнитном поле и задачи Шоттки о движении четырехмерного твердого тела

Цыганов А. В., Порубов Е. О., Теоретическая и математическая физика 2026 Т. 227 № 2 С. 336–355

Теория тензорных инвариантов обыкновенных дифференциальных уравнений и классификация Картана простых алгебр Ли используется для установления изоморфизма задачи Козлова о движении ферромагнетика в магнитном поле и задачи Шоттки о движении четырехмерного твердого тела. Найдены новые полиномиальные и рациональные бивекторы Пуассона, инвариантные либо относительно пары коммутирующих фазовых потоков, либо относительно одного из пары потоков. ...

Added: May 5, 2026

Моделирование и оценка ресурсных затрат алгоритмов маршрутизации в сетях на кристалле с двумерной циркулянтной топологией

Монахова Э. А., Монахов О. Г., Rzaev E. et al., Прикладная дискретная математика 2026 Т. 71 С. 112–127

В настоящей работе исследовано совместное конструирование топологий семейств оптимальных по диаметру циркулянтных сетей $C(N; \pm 1, \pm s_2)$ и реализуемых для них оптимальных алгоритмов маршрутизации сложности $O(1)$. Предлагаемый алгоритм маршрутизации основан на использовании масштабируемых параметров $L$-образных шаблонов плотной укладки графов на плоскости для семейств оптимальных сетей. Определены аналитические формулы зависимости этих параметров от диаметра графов семейств ...

Added: May 4, 2026

On Undecidability Degree of Theory of Figures in Countable and Uncountable Linear Spaces

Dudakov S., Lobachevskii Journal of Mathematics 2025 Vol. 46 No. 12 P. 6092–6102

We study the additive theory of arbitrary ﬁgures in linear spaces, that is, the theory of addition extended to sets of vectors. Our main result is the following: if a linear space is inﬁnite, then the additive theory of ﬁgures admits interpreting second-order arithmetic and, therefore, it has such or higher degree of undecidability. For countably inﬁnite spaces, ...

Added: May 1, 2026

On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras

Filimoshina E., Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 Article 16

This paper introduces Lie groups in degenerate geometric (Cliﬀord) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that these Lie groups can be equivalently defined using norm functions of multivectors applied in the theory of spin groups. We also study the ...

Added: January 12, 2026

Lorentz Invariance of the Multidimensional Dirac–Hestenes Equation

Sofia Rumyantseva, Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 Article 5

This paper investigates the Lorentz invariance of the multidimensional Dirac–Hestenes equation, that is, whether the equation remains form-invariant under pseudo-orthogonal transformations of the coordinates. We examine two distinct approaches: the tensor formulation and the spinor formulation. We first present a detailed examination of the four-dimensional Dirac–Hestenes equation, comparing both transformation approaches. These results are subsequently ...

Added: December 19, 2025

On Commutative Analogues of Clifford Algebras and Their Decompositions

Sharma H., Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 Article 9

We investigate commutative analogues of Clifford algebras - algebras whose generators square to ±1 but commute, instead of anti-commuting as they do in Clifford algebras. We observe that commutativity allows for elegant results. We note that these algebras generalise multicomplex spaces - we show that a commutative analogue of Clifford algebra is either isomorphic to a multicomplex space ...

Added: December 2, 2025

Equivariant Neural Networks with Geometric Algebras: A New Approach

Filimoshina E., Shirokov D., , in: 2025 International Joint Conference on Neural Networks (IJCNN).: IEEE, 2025. P. 1–8.

This work is devoted to construction and implementation of new equivariant neural networks based on geometric (Clifford) algebras. We propose, implement, test, and compare with competitors a new architecture of equivariant neural networks, which we call Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations. We introduce ...

Added: November 15, 2025

GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras

Filimoshina E., Shirokov D., , in: Volume 267: International Conference on Machine Learning, 13-19 July 2025, Vancouver Convention Center, Vancouver, CanadaVol. 267.: [б.и.], 2025. P. 17153–17188.

We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization ...

Added: October 28, 2025

Determinant, Characteristic Polynomial, and Inverse in Commutative Analogues of Clifford Algebras

Sharma H., Shirokov D., Advances in Applied Clifford Algebras 2025 Vol. 35 Article 44

Commutative analogues of Clifford algebras are algebras defined in the same way as Clifford algebras except that their generators commute with each other, in contrast to Clifford algebras in which the generators anticommute. In this paper, we solve the problem of finding multiplicative inverses in commutative analogues of Clifford algebras by introducing a matrix representation ...

Added: October 2, 2025

On Grade Automorphism in Ternary Clifford Algebras

Shirokov D., , in: Hypercomplex Analysis and Its Applications.Extended Abstracts of the International Conference Celebrating Paula Cerejeiras’ 60th Birthday. ICHAA 2024. Trends in Mathematics (TM, volume 9)Vol. 9.: Birkhäuser, 2025. P. 143–150.

For the first time, we introduce a grade automorphism in ternary Clifford algebras and discuss a number of its properties. This operation is not an involution, but naturally generalizes the grade involution (or the main involution) in ordinary (quadratic) Clifford algebras. The new operation can be used in different applications of ternary Clifford algebras in ...

Added: July 6, 2025

Generalized Degenerate Clifford and Lipschitz Groups in Geometric Algebras

Filimoshina E., Shirokov D., Advances in Applied Clifford Algebras 2025 Vol. 35 Article 29

This paper introduces and studies generalized degenerate Clifford and Lipschitz groups in geometric (Clifford) algebras. These Lie groups preserve the direct sums of the subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations in degenerate geometric algebras. We prove that the generalized degenerate Clifford and Lipschitz groups can be ...

Added: May 29, 2025