Multidimensional Generalizations Research Articles

Research on the convergence of some types of functional branched continued fractions

An analysis of research on the problem of convergence of various types of functional branched continued fractions has been carried out. Branched continued fractions with $N$ branching branches and branched continued fractions with independent variables are considered. The definition and, in our opinion, characteristic criteria of convergence of multidimensional generalizations of C-, S-, g-, J-fractions are given, both for branched continued fractions of the general form with $N$ branching branches and branched continued fractions with independent variables. Such multidimensional generalizations of continued fractions arise, in particular, in the development of various classes of hypergeometric functions of several variables, in particular, the functions of Appel, Lauricella, Horn, etc.

Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras

Bethe Ansatz was discovered in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and d-simplex equations]. Here we describe a universal method to solve these equations using Clifford algebras. The Yang-Baxter equation (d=2), Zamolodchikov's tetrahedron equation (d=3) and the Bazhanov-Stroganov equation (d=4) are special cases. Our solutions form a linear space. This helps us to include spectral parameters. Potential applications are discussed.

The periodic N breather anomalous wave solution of the Davey–Stewartson equations; first appearance, recurrence, and blow up properties

The integrable focusing Davey–Stewartson (DS) equations, multidimensional generalizations of the focusing cubic nonlinear Schrödinger equation, provide ideal mathematical models for describing analytically the dynamics of dimensional anomalous (rogue) waves (AWs). In this paper (i) we construct the N-breather AW solution of Akhmediev type of the DS1 and DS2 equations, describing the nonlinear interaction of N unstable modes over the constant background solution. (ii) For the simplest multidimensional solution of DS2 we construct its limiting subcases, and we identify the constraint on its arbitrary parameters giving rise to blow up at finite time. (iii) We use matched asymptotic expansions to describe the relevance of the constructed AW solutions in the spatially doubly periodic Cauchy problem of DS2 for small initial perturbations of the background, in the case of one and two unstable modes. We also show, in the case of two unstable modes, that (i) no blow up takes place generically, although the AW amplitude can be arbitrarily large; (ii) the excellent agreement of our formulas, expressed in terms of elementary functions of the initial data, with numerical experiments.

There is no APTAS for 2-dimensional vector bin packing: Revisited

We study the Vector Bin Packing and the Vector Bin Covering problems, multidimensional generalizations of the Bin Packing and the Bin Covering problems, respectively. In the Vector Bin Packing, we are given a set of d-dimensional vectors from [0,1]d and the aim is to partition the set into the minimum number of bins such that for each bin B, each component of the sum of the vectors in B is at most 1. Woeginger [17] claimed that the problem has no APTAS for dimensions greater than or equal to 2. We note that there was a slight oversight in the original proof. In this work, we give a revised proof using some additional ideas from [3,8]. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 600599.An instance of Vector Bin Packing is called δ-skewed if every item has at most one dimension greater than δ. As a natural extension of our general d-Dimensional Vector Bin Packing result we show that for ε∈(0,12500) it is NP-hard to obtain a (1+ε)-approximation for δ-Skewed Vector Bin Packing if δ>20ε.In the Vector Bin Covering problem given a set of d-dimensional vectors from [0,1]d, the aim is to obtain a family of disjoint subsets (called bins) with the maximum cardinality such that for each bin B, each component of the sum of the vectors in B is at least 1. Using ideas similar to our Vector Bin Packing result, we show that for Vector Bin Covering there is no APTAS for dimensions greater than or equal to 2. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 998997.

Entanglement of inhomogeneous free fermions on hyperplane lattices

We introduce an inhomogeneous model of free fermions on a (D−1)-dimensional lattice with D(D−1)/2 continuous parameters that control the hopping strength between adjacent sites. We solve this model exactly, and find that the eigenfunctions are given by multidimensional generalizations of Krawtchouk polynomials. We construct a Heun operator that commutes with the chopped correlation matrix, and compute the entanglement entropy numerically for D=2,3,4, for a wide range of parameters. For D=2, we observe oscillations in the sub-leading contribution to the entanglement entropy, for which we conjecture an exact expression. For D>2, we find logarithmic violations of the area law for the entanglement entropy with nontrivial dependence on the parameters.

Invariant quadratic operators associated with linear canonical transformations and their eigenstates

The main purpose of this work is to identify invariant quadratic operators associated with Linear Canonical Transformations (LCTs) which could play important roles in physics. In quantum physics, LCTs are the linear transformations which keep invariant the Canonical Commutation Relations (CCRs). In this work, LCTs corresponding to a general pseudo-Euclidian space are considered and related to a phase space representation of quantum theory. Explicit calculations are firstly performed for the monodimensional case to identify the corresponding LCT-invariant quadratic operators then multidimensional generalizations of the obtained results are deduced. The eigenstates of these operators are also identified. A first kind of LCT-invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of the coordinates and momenta operators themselves. It is shown that these statistical variances-covariances can be related with thermodynamic variables. In this context, new quantum corrections to the ideal gas state equation are deduced from correction to the Hamiltonian operator of non-relativistic free quantum particles that is suggested by LCT-covariance. Two other LCT-invariant quadratic operators, which can be considered as the number operators of some quasiparticles, are also identified: the first one is a number operator of bosonic type quasiparticles and the second one corresponds to fermionic type. This fermionic LCT-invariant quadratic operator is directly related to a spin representation of LCTs. It is shown explicitly, in the case of a relativistic pentadimensional theory, that the eigenstates of this operator can be considered as basic quantum states of elementary fermions. A classification of the fundamental fermions, compatible with the Standard Model of particle physics, is established from a classification of these states.

Pentagram maps and refactorization in Poisson-Lie groups

The pentagram map was introduced by R. Schwartz in 1992 and is now one of the most renowned discrete integrable systems. In the present paper we prove that this map, as well as all its known integrable multidimensional generalizations, can be seen as refactorization-type mappings in the Poisson-Lie group of pseudo-difference operators. This brings the pentagram map into the rich framework of Poisson-Lie groups, both describing new structures and simplifying and revealing the origin of its known properties. In particular, for multidimensional pentagram maps the Poisson-Lie group setting provides new Lax forms with a spectral parameter and, more importantly, invariant Poisson structures in all dimensions, the existence of which has been an open problem since the introduction of those maps. Furthermore, for the classical pentagram map our approach naturally yields its combinatorial description in terms of weighted directed networks and cluster algebras.

Proper cyclic symmetries of multidimensional continued fractions

We show that palindromic continued fractions exist in an arbitrary dimension. For dimension $n=4$ we also prove a criterion for an algebraic continued fraction to have a proper cyclic palindromic symmetry. Klein polyhedra are considered as multidimensional generalizations of continued fractions. Bibliography: 11 titles.

Generalized Hypergeometric Function 3F2 Ratios and Branched Continued Fraction Expansions

The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function 3F2 to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function 3F2. To illustrate this, we have given a few numerical experiments at the end.

Dynamics of optical solitons in higher-order Sasa–Satsuma equation

Higher order and multidimensional generalizations of the nonlinear Schrödinger equation are useful in a variety of applications. A generalized (3+1)-dimensional nonlinear Sasa–Satsuma equation is studied via the improved generalized Riccati equation mapping method and its Bäcklund transformation and the extended hyperbolic function method. This nonlinear model can be use to describe the physical processes of wave propagation in optical fibers. We explicitly retrieved varieties of optical solitons like bright, dark, singular, combined bright–dark and combined singular-bright to this multidimensional generalized nonlinear model. Other solutions, such as periodic, rational, and exponential solutions of different structures, are also produced as a result of the evolution of this improved technique. Furthermore, the graphs in two and three dimensions are provided to visualize the structure of the solutions acquired by choosing certain parameter values.

Nonlinear spectral decompositions by gradient flows of one-homogeneous functionals

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by works for the total variation, where interesting results on the eigenvalue problem and the relation to the total variation flow have been proven previously, and by recent results on finite-dimensional polyhedral semi-norms, where gradient flows can yield spectral decompositions into eigenvectors. We provide a geometric characterization of eigenvectors via a dual unit ball and prove them to be subgradients of minimal norm. This establishes the connection to gradient flows, whose time evolution is a decomposition of the initial condition into subgradients of minimal norm. If these are eigenvectors, this implies an interesting orthogonality relation and the equivalence of the gradient flow to a variational regularization method and an inverse scale space flow. Indeed we verify that all scenarios where these equivalences were known before by other arguments - such as one-dimensional total variation, multidimensional generalizations to vector fields, or certain polyhedral semi-norms - yield spectral decompositions, and we provide further examples. We also investigate extinction times and extinction profiles, which we characterize as eigenvectors in a very general setting, generalizing several results from literature.

MAXIMAL REGULARITY ESTIMATE FOR A DIFFERENTIAL EQUATION WITH OSCILLATING COEFFICIENTS

MAXIMAL REGULARITY ESTIMATE FOR A DIFFERENTIAL EQUATION WITH OSCILLATING COEFFICIENTS

The generalized centrally extended Lie algebraic structures and related integrable heavenly type equations

There are studied Lie-algebraic structures of a wide class of heavenly type non-linear integrable equations, related with coadjoint flows on the adjoint space to a loop vector field Lie algebra on the torus. These flows are generated by the loop Lie algebras of vector fields on a torus and their coadjoint orbits and give rise to the compatible Lax-Sato type vector field relationships. The related infinite hierarchy of conservations laws is analysed and its analytical structure, connected with the Casimir invariants, is discussed. We present the typical examples of such equations and demonstrate in details their integrability within the scheme developed. As examples, we found and described new multidimensional generalizations of the Mikhalev-Pavlov and Alonso-Shabat type integrable dispersionless equation, whose seed elements possess a special factorized structure, allowing to extend them to the multidimensional case of arbitrary dimension.

Generalized Compressible Flows and Solutions of the $$H(\mathrm {div})$$ Geodesic Problem

We study the geodesic problem on the group of diffeomorphism of a domain M$\subset$Rd, equipped with the H(div) metric. The geodesic equations coincide with the Camassa-Holm equation when d=1, and represent one of its possible multi-dimensional generalizations when d>1. We propose a relaxation {\`a} la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over M. We use this relaxation to prove that smooth H(div) geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure field associated to solutions of our relaxation. Finally, we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized Camassa-Holm and incompressible Euler solutions.

Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Pleba\'nski heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

Multidimensional Analogs of the Cauchy–Riemann System and Representations of Their Solutions via Harmonic Functions

At present, there are numerous multidimensional generalizations of holomorphic vectors. The most general of these is the four-dimensional generalization of the Cauchy–Riemann system. In the present work, by introducing two quaternion functions and the notion of quaternion differentiation, we obtain, for the first time, a five-dimensional generalization of holomorphic vectors. By using the representation of holomorphic vectors via the quaternion harmonic function and its derivatives, we consider the Riemann–Hilbert problem and one problem in a layer. A new solution of the Riemann–Hilbert problem in the five-dimensional half space is obtained.

Spin chains and Gustafson’s integrals

Gustafson’s integrals are multidimensional generalizations of the classical Mellin–Barnes integrals. We show that some of these integrals arise from relations between matrix elements in Sklyanin’s representation of separated variables in spin chain models. We also present several new integrals.

Approximating permanents and hafnians

Approximating permanents and hafnians, Discrete Analysis 2017:2, 34 pp. The _permanent_ per$(A)$ of an $n\times n$ matrix $A$ is like the determinant but without the sign changes: that is, it is the sum $\sum_{\pi\in S_n}\prod_{i=1}^nA_{i\pi(i)}$. If $A$ is the bipartite adjacency matrix of a bipartite graph $G$, then per$(A)$ is the number of perfect matchings in $G$. Whereas row-operations allow one to compute the determinant of a matrix efficiently, computing the permanent is known, by a famous result of Valiant, to be hard (in the sense that if you could compute it efficiently, then you would be able to give efficient solutions to a wide variety of enumeration problems, and would in particular have shown that P=NP). However, another famous result, due to Jerrum, Sinclair and Vigoda, which built on earlier work of Jerrum and Sinclair, shows that there are probabilistic algorithms that run in polynomial time and approximate the permanent to within a factor $1+\epsilon$ with high probability. This paper concerns the question of how well the permanent can be approximated using a _deterministic_ algorithm. The main result of this paper is that if $\delta,\epsilon>0$ then for each $n\in\mathbb N$ there is a polynomial $p$ in $n^2$ variables of degree $O_\delta(\log n+\log(1/\epsilon))$ such that if $A$ is any $n\times n$ matrix with all its entries between $\delta$ and 1, then the permanent of $A$ can be approximated to within a factor $1+\epsilon$ by $\exp(p(A))$. Here, of course, $p(A)$ is $p$ applied to the entries of $A$. Furthermore, $p(A)$ can be computed in time $n^{O_\delta(\log n+\log(1/\epsilon))}$. As mentioned above, the permanent of a 01-matrix counts the number of perfect matchings in a bipartite graph. If instead one is interested in the number of perfect matchings in a graph with $2n$ vertices (that is, the number of sets of $n$ disjoint edges), then the corresponding quantity is called the _hafnian_. For a general $2n\times 2n$ matrix $A$, the hafnian haf$(A)$ is the sum over all partitions of $\{1,\dots,2n\}$ into $n$ sets $\{i_r,j_r\}$ of the product $\prod_{r=1}^nA_{i_rj_r}$. The author proves the same result for hafnians that he obtains for permanents. This is interesting, because it is not known how to approximate hafnians even with a randomized algorithm. The permanent and hafnian are polynomials of degree $n$ in the matrix entries, so these results say that on the class of matrices being considered (that is, real matrices with all entries between $\delta$ and 1), they can be approximated by polynomials of much lower degree. The proofs depend on results that say that the two polynomials cannot be zero on a complex matrix that is close to this class. Similar results are proved for complex matrices with all their entries close to 1. In this case the methods carry over to permanents of multidimensional tensors, but multidimensional generalizations appear to be harder to obtain for the results mentioned above, for reasons explained in the paper, and are not currently known.

Discrete taut strings and real interpolation

Classical taut strings and their multidimensional generalizations appear in a broad range of applications. In this paper we suggest a general approach based on the K-functional of real interpolatio ...

Iterative residual-based vector methods to accelerate fixed point iterations

Fixed point iterations are still the most common approach to dealing with a variety of numerical problems such as coupled problems (multi-physics, domain decomposition, …) or nonlinear problems (electronic structure, heat transfer, nonlinear mechanics, …). Methods to accelerate fixed point iteration convergence or more generally sequence convergence have been extensively studied since the 1960’s. For scalar sequences, the most popular and efficient acceleration method remains the Δ2 of Aitken. Various vector acceleration algorithms are available in the literature, which often aim at being multi-dimensional generalizations of the Δ2 method.In this paper, we propose and analyse a generic residual-based formulation for accelerating vector sequences. The question of the dynamic use of this residual-based transformation during the fixed point iterations for obtaining a new accelerated fixed point method is then raised. We show that two main classes of such iterative algorithms can be derived and that this approach is generic in that various existing acceleration algorithms for vector sequences are thereby recovered.In order to illustrate the interest of such algorithms, we apply them in the field of nonlinear mechanics on a simplified “point-wise” solver used to perform mechanical behaviour unit testings. The proposed test cases clearly demonstrate that accelerated fixed point iterations based on the elastic operator (quasi-Newton method) are very useful when the mechanical behaviour does not provide the so-called consistent tangent operator. Moreover, such accelerated algorithms also prove to be competitive with respect to the standard Newton–Raphson algorithm when available.