Nonlinear separable equations in linear spaces and commutative Leibniz algebras

It is well-known that sums and products of holomorphic functions are holomorphic, and the holomorphic functions on a complex manifold form a commutative algebra over C . The study of complex manifolds using algebras of holomorphic functions upon them is called complex algebraic geometry. The purpose of this paper is to develop an analogue of complex algebraic geometry, in which the complex numbers C are replaced by the quaternions H . The natural quaternionic analogue of a complex manifold is called a hypercomplex manifold. A class of H -valued functions on hypercomplex manifolds will be defined, called q-holomorphic functions, that are analogues of holomorphic functions on complex manifolds. Now, the set of holomorphic functions on a complex manifold is a commutative algebra over C . Therefore one asks: does the set of q-holomorphic functions on a hypercomplex manifold have an analogous algebraic structure, and if so, what is it? We shall show that the q-holomorphic functions on a (noncompact) hypercomplex manifold do indeed possess a rich algebraic structure. To describe it, we shall introduce a theory of quaternionic algebra, which is a quaternionic analogue of real linear algebra. This theory is built on three building blocks: AH-modules, the analogues of vector spaces, AH-morphisms, the analogues of linear maps, and the quaternionic tensor product, the analogue of tensor product of real vector spaces. As far as the author can tell, these ideas seem to be new. They enable us to construct algebraic structures over H as though H were a commutative field. Quaternionic algebra describes the algebraic structure of hypercomplex manifolds in a remarkable way, and it seems to be the natural language of hypercomplex algebraic geometry, the algebraic geometry of hypercomplex manifolds. We believe that quaternionic algebra is worth studying for its own sake. It has many similarities with linear algebra over R or C , which is why the analogies between complex and quaternionic theories work so well, but there are also deep differences, which give quaternionic algebra a flavour all of its own. Quillen [12] has given a sheaf-theoretic interpretation of the ideas of quaternionic algebra, based on a previous version of this paper. He finds a contravariant equivalence between a class of AH-modules and regular sheaves on a real form of CP. Regular sheaves on CP are equivalent to representations of the Kronecker quiver, and out of such a representation Quillen constructs an AH-module. Under Quillen’s equivalence stable AH-modules correspond to regular vector bundles over the real form of CP.

Boolean matrices, i.e., matrices over a Boolean algebra, do not satisfy many of the fundamental properties of matrices over the real or complex field. Algebraic aspects of the Boolean pseudoinverse matrix concept are investigated. Maps operating on Boolean vector spaces by analogy with those on linear spaces are defined. Some properties of them are obtained. Most of these properties remain the same in the case of linear spaces and they are essential for solving linear systems. In particular orthogonal projections in linear spaces receive their algebraic analogue in Boolean vector spaces. The maps defined are useful for solving equations in the Boolean algebra, if no pseudoinverse matrix to a given matrix exists. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra. Some examples for computations with binary relations and Boolean matrices using computer algebra systems are given.

We present a new package ZpL for the mathematical software system SageMath. It implements a sharp tracking of precision on p-adic numbers, following the theory of ultrametric precision introduced in a previous paper by the same authors. The underlying algorithms are mostly based on automatic differentiation techniques. We introduce them, study their complexity and discuss our design choices. We illustrate the benefits of our package (in comparison with previous implementations) with a large sample of examples coming from linear algebra, commutative algebra and differential equations.

In this paper, the Cauchy problem of the nonlinear Helmholtz-type equation is discussed. This problem is well known to be severely ill-posed. Compared with linear equations, nonlinear equations are more difficult to deal with because of their lack of linearity. In order to obtain the approximate solution of this nonlinear problem, a combination of the quasi-boundary value method and the quasi-reversibility method is proposed. Using the variable separation method, the approximate solution is equivalent to solving a class of integral equations. The well-posedness of the approximation problem is proved by the Banach fixed-point theorem. Convergence analysis and error estimation are discussed. Since the convergence of error estimates cannot be obtained by the traditional a-priori bound, we introduce a new a-priori bound to obtain the convergence of error estimates, and the Hölder-type error estimate is achieved. Finally, some numerical experiments are given to corroborate the qualitative analysis and show the regularization method works well.

The analysis of the transfer matrices associated with the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method, which generalizes to these integrable quantum models the method first introduced by Sklyanin. For representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras, for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of the transfer matrix spectrum (eigenvalues and eigenstates) and its non-degeneracy. Moreover, we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states, which particularly apply to transfer matrix eigenstates. A comparison of our analysis of the 8-vertex reflection algebra with that of (Niccoli G 2012 J. Stat. Mech. P10025, Faldella S et al 2014 J. Stat. Mech. P01011) for the 6-vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products, which exists for these two different realizations of the reflection algebra once they are described by the SOV method. As will be shown in a future publication, this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8-vertex and 6-vertex type.

The Probabilistic Solution of the Plate with Simple-Supported and Stretched Boundary and Uniform Load Being Gaussian White Noise

Linear Algebra in Dioids: A Survey of Recent Results

Modeling and simulation of water infiltration from the ground surface into unsaturated–saturated soil is very important in the study of the hydrological cycle process. Transient fluid flow through unsaturated–saturated soil is generally described by Richards’ equation. The equation is a nonlinear parabolic partial differential equation for which many numerical and limited analytical solutions exist. In this paper, exponential models for water content and hydraulic conductivity are selected so the nonlinear equation becomes linear and then separation of variables and Fourier series expansion techniques are used to derive analytical solutions to 2D water infiltration into unsaturated–semi-saturated soils with different distributions of water content on a part and/or parts of the top boundary. 2D water infiltrations are investigated in a rectangular soil domain, where high water content zone is considered as semi-saturated zone and residual water content zone is considered as unsaturated zone at initial time. A total of three cases are investigated in this paper. In case 1, a uniform distribution of water content on a part at the top boundary and no-flow boundaries at side edges are considered. Also, horizontal water table in unsaturated–semi-saturated domain is assumed as initial condition. In case 2, an inclined water table initial condition and a uniform distribution of water content on a part at the top boundary are investigated. Finally, in case 3, an analytical solution for ramp distributions of water content on two parts at the top boundary and an inclined water table initial condition is obtained. To illustrate the use of the derived equations, water content values are obtained from a numerical solution and compared to those from analytical solutions for two cases, showing less than 2% errors. These analytical solutions may be used as a benchmark for verification and efficiency assessment of numerical approaches.

Introduction.Given a field k and a fc-algebra A, Gerstenhaber has introduced the concept of a deformation of A [8] which is a [[/]]-algebra structure on the module A [[/]].One of the main problems of the theory of deformations is to determine those algebras for which the only deformation (modulo an equivalence relation) is the usual A-[[r]]-algebra structure on .4[[*]].Such algebras are said to be rigid.If A is a commutative /c-algebra, we may restrict our attention to those deformations of A which are commutative.This leads us to consider the commutative cohomology groups of A, Hn(A, M)s, defined by Harrison [11].The principal purpose of this paper is to investigate the following conditions on a commutative algebra A over a field k:(1) H2(A, M)s = 0 for all ^-modules M.(2) H\A,f=0.(3) A is a rigid /r-algebra in the commutative deformation theory.(4) The module of differentials of A over k is a projective /4-module.(5) A is a finite product of (not necessarily algebraic) separable extension fields of k.We may show that (1) implies (4) [9, Theorem (20.4.9)] and that (2) implies (3) [8, p. 65].Condition (1) may be expressed by saying that A is a formally smooth fc-algebra [9, 19.4.4].When A is a field, Gerstenhaber has shown that (1), (2), and ( 5) are equivalent and he conjectured that these conditions were equivalent to (3).We show that (1), ( 2), (3), and (5) are equivalent even for A semisimple (Theorem 6.5).Moreover, if A is artinian, then (1) and ( 5) are equivalent (Proposition 5.9).In another direction, if A is a geometric complete intersection, then (1), (2), and (3) are equivalent (Theorem 6.8a).If in addition A is an integral domain with a separable quotient field, then (1), ( 2), (3), and (4) are equivalent (Theorem 6.8c).In the final section, we investigate the above conditions for the group algebra B[G] of an abelian group G which is a direct sum of cyclic groups with B a commutative algebra over a field k of positive characteristic p.In addition, we consider the following condition on G and k:(6) G has no element of order p = char (k).

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

This advanced textbook on linear algebra and geometry covers a wide range of classical and modern topics. Differing from existing textbooks in approach, the work illustrates the many-sided applications and connections of linear algebra with functional analysis, quantum mechanics and algebraic and differential geometry. The subjects covered in some

This paper describes a class of exact solutions of the equations of motion for an unsteady, incompressible non-Newtonian second grade fluid in the Cartesian co-ordinates. The exact solutions of non-linear equations governing the flow of second grade fluids are obtained through generalized separation variables method by assuming certain form of stream functions. The expressions for velocity profile and vorticity stream function are constructed. Key words: Exact solutions, vorticity functions, second grade fluids, separation of variables.

Variable separation method for a nonlinear time fractional partial differential equation with forcing term

We develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations. Using it we obtain a number of new two-dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions.

Linear Algebra Linear algebra studies linear transformations and vector spaces, or in another language, matrix multiplication and the vector space R n . You should know how to translate between the language of abstract vector spaces and the language of matrices. In particular, given a basis for a vector space, you should know how to represent any linear transformation as a matrix. Further, given two matrices, you should know how to determine if these matrices actually represent the same linear transformation, but under different choices of bases. The key theorem of linear algebra is a statement that gives many equivalent descriptions for when a matrix is invertible. These equivalences should be known cold. You should also know why eigenvectors and eigenvalues occur naturally in linear algebra. Real Analysis The basic definitions of a limit, continuity, differentiation and integration should be known and understood in terms of ∈'s and δ's. Using this ∈ and δ language, you should be comfortable with the idea of uniform convergence of functions. Differentiating Vector-Valued Functions The goal of the Inverse Function Theorem is to show that a differentiable function f : R n → R n is locally invertible if and only if the determinant of its derivative (the Jacobian) is non-zero. You should be comfortable with what it means for a vector-valued function to be differentiable, why its derivative must be a linear map (and hence representable as a matrix, the Jacobian) and how to compute the Jacobian.