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## Method of averaging in Clifford algebras

Advances in Applied Clifford Algebras. 2017. Vol. 27. No. 1. P. 149-163.

In this paper we consider different operators acting on Clifford algebras. We consider Reynolds operator of Salingaros’ vee group. This operator “average” an action of Salingaros’ vee group on Clifford algebra. We consider conjugate action on Clifford algebra. We present a relation between these operators and projection operators onto fixed subspaces of Clifford algebras. Using method of averaging we present solutions of system of commutator equations.

Shirokov D., / Cornell University. Series math "arxiv.org". 2016.

We consider real and complex Clifford algebras of arbitrary even and odd dimensions and prove generalizations of Pauli's theorem for two sets of Clifford algebra elements that satisfy the main anticommutative conditions. In our proof we use some special operators - generalized Reynolds operators. This method allows us to obtain an algorithm to compute elements ...

Added: October 9, 2016

Shirokov D., Вестник Самарского государственного технического университета. Серия: Физико-математические науки 2015 Т. 19 № 1 С. 117-135

In this paper we consider expressions in real and complex Clifford algebras, which we call contractions or averaging. We consider contractions of arbitrary Clifford algebra element. Each contraction is a sum of several summands with different basis elements of Clifford algebra. We consider even and odd contractions, contractions on ranks and contractions on quaternion types. ...

Added: October 16, 2015

Shirokov D., Advances in Applied Clifford Algebras 2021 Vol. 31 Article 30

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras — subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted ...

Added: May 10, 2021

Covolo T., Journal of Noncommutative Geometry 2015 Vol. 9 No. 2 P. 543-565

We develop the theory of linear algebra over a (Z2)n-commutative algebra (n∈N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduce analogues of the notions of trace and determinant. Our construction ...

Added: September 28, 2015

Yurii Burman, Shapiro B., / Cornell University. Series math "arxiv.org". 2016. No. 06935.

For a point p in a complex projective plane and a triple (g,d,l) of non-negative
integers we define a plane Hurwitz number of the Severi variety
W_{g,d,l} consisting of all reduced irreducible plane curves of
genus g and degree d+l having an l-fold node at p and at
most ordinary nodes as singularities at the other points. In the ...

Added: July 5, 2016

Shirokov D., Marchuk N., Journal of Geometry and Symmetry in Physics 2016 Vol. 42 P. 53-72

In this paper we present some new equations which we call Yang-Mills-Proca equations (or generalized Proca equations). This system of equations is a generalization of Proca equation and Yang-Mills equations and it is not gauge invariant. We present a number of constant solutions of this system of equations in the case of arbitrary Lie algebra. ...

Added: February 14, 2017

Shirokov D., Advances in Applied Clifford Algebras 2015 Vol. 25 No. 3 P. 707-718

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras. ...

Added: March 12, 2015

Shirokov D., Theoretical and Mathematical Physics 2013 Vol. 175 No. 1 P. 454-474

We discuss a generalized Pauli theorem and its possible applications for describing n-dimensional (Dirac, Weyl, Majorana, and Majorana–Weyl) spinors in the Clifford algebra formalism. We give the explicit form of elements that realize generalizations of Dirac, charge, and Majorana conjugations in the case of arbitrary space dimensions and signatures, using the notion of the Clifford ...

Added: March 11, 2015

Shirokov D., Marchuk N., Reports on Mathematical Physics 2016 Vol. 78 No. 3 P. 305-326

We find general solutions of some field equations (systems of equations) in pseudo-Euclidian spaces (so-called primitive field equations). These equations are used in the study of the Dirac equation and Yang-Mills equations. These equations are invariant under orthogonal O(p,q) coordinate transformations and invariant under gauge transformations, which depend on some Lie groups. In this paper ...

Added: September 27, 2016

Shirokov D., Advances in Applied Clifford Algebras 2019 Vol. 29 No. 50 P. 1-12

We present a method of computing elements of spin groups in the case of arbitrary dimension. This method generalizes Hestenes method for the case of dimension 4. We use the method of averaging in Clifford’s geometric algebra previously proposed by the author. We present explicit formulas for elements of spin group that correspond to the ...

Added: July 22, 2019

Shirokov D., Advances in Applied Clifford Algebras 2012 Vol. 22 No. 1 P. 243-256

We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements. ...

Added: June 16, 2015

Yu. Burman, Serge Lvovski, Moscow Mathematical Journal 2015 Vol. 15 No. 1 P. 31-48

Suppose that C ⊂ P^2 is a general enough nodal plane curve
of degree > 2, : \hat C → C is its normalization, and π: C′ → P^1 is a finite
morphism simply ramified over the same set of points as a projection
pr_p ◦ν : \hat C → P1, where p ∈ P^2 ...

Added: January 14, 2015

M.V.Karasev, E.M.Novikova, Russian Journal of Mathematical Physics 2015 Vol. 22 No. 4 P. 463-468

We study dynamics of a charge in the planar Penning trap in which the direction of magnetic ﬁeld does not coincide with the trap axis. Under some combined resonance condition on the deviation angle and magnitudes of magnetic and electric ﬁelds, the trajectories of a charge are near-periodic. We describe the reduction to a top-like ...

Added: October 22, 2015

Shirokov D., Journal of Geometry and Symmetry in Physics 2016 Vol. 42 P. 73-94

In this paper we consider some Lie groups in complexified Clifford algebras. Using relations between operations of conjugation in Clifford algebras and matrix operations we prove isomorphisms between these groups and classical matrix groups (symplectic, orthogonal, linear, unitary) in the cases of arbitrary dimension and arbitrary signature. Also we obtain isomorphisms of corresponding Lie algebras ...

Added: December 14, 2016

Gaydukov R., Сибирский журнал вычислительной математики 2022 Т. 15 № 2 С. 97-109

A viscous liquid flow along a semi-infinite plate with small periodic irregularities on the surface was
considered for large Reynolds numbers. The flow near the plate is described by Prandtl equations with
induced pressure which are non-classical PDE, because they contain a limiting term. The main goal is to
construct a numerical algorithm for solving these equations with ...

Added: June 10, 2020

Shirokov D., Mathematics 2023 Vol. 11 No. 16 Article 3607

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Open AccessFeature PaperArticle
Development of the Method of Averaging in Clifford Geometric Algebras
by
Dmitry Shirokov
1,2
1
HSE University, Myasnitskaya Str. 20, Moscow 101000, Russia
2
Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetny Per. 19, Moscow 127051, Russia
Mathematics 2023, 11(16), 3607; https://doi.org/10.3390/math11163607
Received: 29 June 2023 / Revised: 15 August 2023 / Accepted: 17 August 2023 / Published: 21 August 2023
(This article belongs to the ...

Added: October 5, 2023

Bogachev V., 2020 Vol. 75 No. 3 P. 393-425

Generalizations and refnements are given for results of Kozlov and Treschev on non-uniform averagings in the ergodic theorem in the case of operator semigroups on spaces of integrable functions and semigroups of
measure-preserving transformations. Conditions on the averaging measures are studied under which the averages converge for broad classes of integrable functions. ...

Added: October 23, 2020

Anikin A. Y., Brüning J., Dobrokhotov S. et al., Russian Journal of Mathematical Physics 2019 Vol. 26 No. 3 P. 265-276

In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (x1, x2) with the constant magnetic field normal to this plane and with the potential V having the form of a harmonic oscillator in the direction x1 and periodic with respect to variable x2. Such a potential can ...

Added: September 18, 2019

Shirokov D., Advances in Applied Clifford Algebras 2015 Vol. 25 No. 1 P. 227-244

We formulate generalizations of Pauli’s theorem on the cases of real and complex Clifford algebras of even and odd dimensions. We give analogues of these theorems in matrix formalism. Using these theorems we present an algorithm for computing elements of spin groups that correspond to elements of orthogonal groups as double cover. ...

Added: March 11, 2015

Shirokov D., Marchuk N., Advances in Applied Clifford Algebras 2008 Vol. 18 No. 2 P. 237-254

For the complex Clifford algebra <img /> (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These ...

Added: June 16, 2015

Shirokov D., Advances in Applied Clifford Algebras 2010 Vol. 20 No. 2 P. 411-425

In this paper we present new formulas, which represent commutators and anticommutators of Clifford algebra elements as sums of elements of different ranks. Using these formulas we consider subalgebras of Lie algebras of pseudo-unitary groups. Our main techniques are Clifford algebras. We have found 12 types of subalgebras of Lie algebras of pseudo-unitary groups. ...

Added: June 16, 2015

Danilov V., European Journal of Mechanics - B/Fluids 2019 Vol. 74 P. 152-158

A stratified liquid with two layers separated by a fast oscillating interface in the case of Raleigh--Taylor instability
is considered. The averaged equations are derived, and it is shown that a mushy region of a certain density appears after averaging. The similarity between this fact and the case of unstable jump decay is discussed. ...

Added: October 21, 2018

Shirokov D., Computational and Applied Mathematics 2021 Vol. 40 P. 1-29

In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension $n$. The formulas involve only ...

Added: July 15, 2021

Shirokov D., Наноструктуры. Математическая физика и моделирование 2013 Т. 9 № 1 С. 93-104

В работе доказаны утверждения, которые обобщают так называемую фундаментальную теорему Паули о гамма-матрицах. Рассмотрены алгебры Клиффорда над полем вещественных и комплексных чисел произвольной размерности. Для произвольных двух наборов из четного или нечетного числа элементов, удовлетворяющих определяющим антикоммутационным соотношениям алгебры Клиффорда, доказаны обобщения теоремы Паули. Предъявлены алгоритмы для вычисления элемента, осуществляющего связь между двумя наборами. ...

Added: July 22, 2019