共查询到20条相似文献,搜索用时 31 毫秒
1.
Paul Sundheim 《Linear and Multilinear Algebra》2013,61(9):1872-1900
Constructions of several multidimensional determinants for matrices in dimensions n = 2m are described. With one exception, the constructions are facilitated by isomorphisms with rings of commutative hypercomplex numbers. Properties shared by the Cayley hyperdeterminant, such as the multiplicative property |AB|?=?|A||B|, are established and the degrees and homogeneity of the determinants are discussed. These considerations emphasize the close relationship between order 2 multidimensional matrices in dimension m and hypercomplex systems in dimension n = 2m. 相似文献
2.
Francesco Catoni 《Advances in Applied Clifford Algebras》2009,19(1):29-42
In a paper published in 1916, the mathematician L. Bianchi has shown that the commutative hypercomplex numbers can be related
with Riemannian spaces with null curvature (Euclidean spaces).
After summarizing this paper we take into account the developments from the time of publication, in particular the formalization
of the theory of functions of a hypercomplex variable that allows us to complement and simplify the solution of the problem.
Afterwards we look for a physical interpretation of the results: beginning with the two-dimensional number systems, we give
to Bianchi’s transformation the physical meaning of a field generated by a point source. In particular the field associated
with hyperbolic numbers can be related with the equivalence principle, one of the Einstein’s starting point for the formulation
of general relativity. 相似文献
3.
E. Hitzer 《Mathematical Methods in the Applied Sciences》2013,36(9):1042-1055
A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g., for quaternions). Already in the case of quaternions, we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
4.
The main goal of this paper is to construct a spatial analog to the Kolosov–Muskhelishvili formulae using the framework of the hypercomplex function theory. We prove a generalization of Goursat's representation theorem for solutions of the biharmonic equation in three dimensions. On the basis of this result, we construct explicitly hypercomplex displacement and stress formulae in terms of two monogenic functions. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
5.
Denis Constales Rolf Sören Kraußhar 《Mathematical Methods in the Applied Sciences》2009,32(16):2050-2070
In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus, the solutions can be expressed as multi‐periodic eigensolutions to the Dirac operator associated with a complex parameter λ∈?. Physically, these can be interpreted as the solutions to the time‐harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green‐type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n‐torus. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
6.
Giampaolo Cicogna 《Mathematical Methods in the Applied Sciences》2014,37(12):1819-1827
This paper describes the notion of σ‐symmetry, which extends the one of λ‐symmetry, and its application to reduction procedures of systems of ordinary differential equations (ODEs) and of dynamical systems (DS) as well. We also consider orbital symmetries, which give rise to a different form of reduction of DS. Finally, we discuss how DS can be transformed into higher order ODEs, and how these symmetry properties of the DS can be transferred into reduction properties of the corresponding ODEs. Many examples illustrate the various situations. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
7.
Henrik Pedersen Yat Sun Poon 《Proceedings of the American Mathematical Society》1999,127(10):3027-3034
Making use of the relations among 3-Sasakian manifolds, hypercomplex manifolds and quaternionic Kähler orbifolds, we prove that complete 3-Sasakian manifolds are rigid.
8.
We study invariant Abelian hypercomplex structures on 8-dimensional nilpotent Lie groups. We prove that a group N admitting such a structure is either Abelian or an Abelian extension of a group of type H. We determine the Poincaré polynomials of the associated nilmanifolds and study the existence of symplectic and quaternionic structures on such spaces. 相似文献
9.
Yilmaz Simsek 《Mathematical Methods in the Applied Sciences》2019,42(18):7030-7046
The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed. 相似文献
10.
We investigate the analytical and numerical solutions of the modified Kortweg de Vries equation by applying the idea of commutative hypercomplex mathematics, He's homotopy perturbation method as a simple particular procedure, and the Runge–Kutta discontinuous Galerkin methods. Moreover, we discuss at great length the convergence conditions for this equation by using the Banach fixed point theory, which could provide a good iteration algorithm. Finally, we compare the homotopy perturbation method with some standard ideas same as the Runge–Kutta discontinuous Galerkin method by some numerical illustrations. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
11.
Jorge Lauret 《Annals of Global Analysis and Geometry》2006,30(2):107-138
Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants.Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory.Communicated by: Nigel Hitchin (Oxford)
Mathematics Subject Classifications (2000): Primary: 53D05, 53D55; Secondary: 22E25, 53D20, 14L24, 53C30. 相似文献
12.
Francesco Catoni Roberto Cannata Enrico Nichelatti Paolo Zampetti 《Advances in Applied Clifford Algebras》2005,15(2):183-212
Systems of hypercomplex numbers, which had been studied and developed at the end of the 19th century, are nowadays quite unknown to the scientific community. It is believed that study of their applications ended just
before one of the fundamental discoveries of the 20th century, Einstein’s equivalence between space and time. Owing to this equivalence, not-defined quadratic forms have got concrete
physical meaning and have been recently recognized to be in strong relationship with a system of bidimensional hypercomplex
numbers. These numbers (called hyperbolic) can be considered as the most suitable mathematic language for describing the bidimensional space-time, in spite of some
unfamiliar algebraic properties common to all the commutative hypercomplex systems with more than two dimensions: they are
decomposable systems and there are non-zero numbers whose product is zero. With respect to the famous Hamilton quaternions,
one can introduce the differential calculus for the hyperbolic numbers and for all the commutative hypercomplex systems; moreover,
one can even introduce functions of hypercomplex variable.
The aim of this work is to study the systems of commutative hypercomplex numbers and the functions of hypercomplex variable
by describing them in terms of a familiar mathematical tool, i.e. matrix algebra. 相似文献
13.
Charles P. Boyer Krzysztof Galicki Benjamin M. Mann 《Annals of Global Analysis and Geometry》1996,14(1):81-105
This paper describes a family of hypercomplex structures {% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFqessaaa!4076!\[\mathcal{I}\]a(p)}a=1,2,3 depending on n real non-zero parameters p = (p
1,...,p
n) on the Stiefel manifold of complex 2-planes in n for all n > 2. Generally, these hypercomplex structures are inhomogenous with the exception of the case when all the p
i's are equal. We also determine the Lie algebra of infinitesimal hypercomplex automorphisms for each structure. Furthermore, we solve the equivalence problem for the hypercomplex structures in the case that the components of p are pairwise commensurable. Finally, some of these examples admit discrete hypercomplex quotients whose topology we also analyze.During the preparation of this work all three authors were supported by NSF grants. 相似文献
14.
Francesco Catoni Roberto Cannata Vincenzo Catoni Paolo Zampetti 《Advances in Applied Clifford Algebras》2005,15(1):1-25
The geometries in N-dimensional Euclidean spaces can be described by Clifford algebras that were introduced as extensions of complex numbers.
These applications are due to the fact that the Euclidean invariant (the distance between two points) is the same as the one
of Clifford numbers.
In this paper we consider the more general extension of complex numbers due to their group properties (hypercomplex systems),
and we introduce the N-dimensional geometries associated with these systems. For N > 2 these geometries are different from the N-dimensional Euclidean geometries; then their investigation could open new applications. Moreover for the commutative systems
the differential calculus does exist and this property allows one to define the functions of hypercomplex variable that can
be used for studying some partial differential equations as well as the non-flat N-dimensional spaces. This last property can be relevant in general relativity and in field theories. 相似文献
15.
Computational bounds on polynomial differential equations 总被引:1,自引:0,他引:1
Daniel S. Graça Jorge Buescu Manuel L. Campagnolo 《Applied mathematics and computation》2009,215(4):1375-1385
In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations.We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines.We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56.Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines. 相似文献
16.
Nelson Faustino 《Mathematical Methods in the Applied Sciences》2019,42(4):1312-1320
In this paper, we exploit the umbral calculus framework to reformulate the so‐called discrete Cauchy‐Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for the underlying solution, but also integral representations for the Chebyshev polynomials of first and second kind by means of its Cauchy principal values. It turns out that the resulting integral representation associated to our toy problem is a space‐time Fourier type inversion formula. Moreover, with the aid of some Laplace transform identities involving the generalized Mittag‐Leffler function, we are able to establish a link with a Cauchy problem of differential‐difference type. 相似文献
17.
Carla Cruz Maria Irene Falcão Helmuth R. Malonek 《Mathematical Methods in the Applied Sciences》2014,37(12):1723-1735
The use of a non‐commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D‐elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan–Sofo type binomial identity is also proved. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
18.
Cristian A. Coclici Gheorghe Moroanu Wolfgang L. Wendland 《Mathematical Methods in the Applied Sciences》2000,23(5):401-440
We consider a one‐dimensional coupled problem for elliptic second‐order ODEs with natural transmission conditions. In one subinterval, the coefficient ϵ>0 of the second derivative tends to zero. Then the equation becomes there hyperbolic and the natural transmission conditions are not fulfilled anymore. The solution of the degenerate coupled problem with a flux transmission condition is corrected by an internal boundary layer term taking into account the viscosity ϵ. By using singular perturbation techniques, we show that the remainders in our first‐order asymptotic expansion converge to zero uniformly. Our analysis provides an a posteriori correction procedure for the numerical treatment of exterior viscous compressible flow problems with coupled Navier–Stokes/Euler models. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
19.
Jihyeok Choi Sunder Sethuraman Shankar C. Venkataramani 《Random Structures and Algorithms》2016,48(4):703-731
We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to ‘non‐tree’ evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C0‐semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 703–731, 2016 相似文献
20.
Radica Bojičić Marko D. Petković Predrag M. Rajković 《Mathematical Methods in the Applied Sciences》2017,40(16):5810-5820
We study Hankel transform of the sequences (u,l,d),t, and the classical Motzkin numbers. Using the method based on orthogonal polynomials, we give closed‐form evaluations of the Hankel transform of the aforementioned sequences, sums of two consecutive, and shifted sequences. We also show that these sequences satisfy some interesting convolutional properties. Finally, we partially consider the Hankel transform evaluation of the sums of two consecutive shifted (u,l,d)‐Motzkin numbers. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献