Determinants Of Certain Matrices Research Articles

Stability of Traveling Wave Solutions of Nonlinear Dispersive Equations of NLS Type

In this paper we present a rigorous modulational stability theory for periodic traveling wave solutions to equations of nonlinear Schr\"odinger (NLS) type. We first argue that, for Hamiltonian dispersive equations with a non-singular symplectic form and $d$ conserved quantities (in addition to the Hamiltonian), one expects that generically ${\mathcal L}$, the linearization around a periodic traveling wave, will have a $2d$ dimensional generalized kernel, with a particular Jordan structure: The kernel of ${\mathcal L}$ is expected to be $d$ dimensional, the first generalized kernel is expected to be $d$ dimensional, and there are expected to be no higher generalized kernels. The breakup of this $2d$ dimensional kernel under perturbations arising from a change in boundary conditions dictates the modulational stability or instability of the underlying periodic traveling wave. This general picture is worked out in detail for the case of equations of NLS type. We give explicit genericity conditions that guarantee that the Jordan form is the generic one: these take the form of non-vanishing determinants of certain matrices whose entries can be expressed in terms of a finite number of moments of the traveling wave solution. Assuming that these genericity conditions are met we give a normal form for the small eigenvalues that result from the break-up of the generalized kernel, in the form of the eigenvalues of a quadratic matrix pencil. We compare these results to direct numerical simulation in a number of cases of interest: the cubic and quintic NLS equations, for focusing and defocusing nonlinearities, subject to both longitudinal and transverse perturbations. The longitudinal stability of traveling waves of the cubic NLS has been previously studied using the integrability: in this case our results agree with those in the literature. All of the remaining cases appear to be new.

Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

We show that an asymptotic property of the determinants of certain matrices whose entries are finite sums of cotangents with rational arguments is equivalent to the GRH for odd Dirichlet characters. This is then connected to the existence of certain quantum modular forms related to Maass Eisenstein series.

Bonded contact and general crack problems in generalized materials and their relationships

The term of generalized material is introduced here as the material, whose state is described by the second order homogeneous differential equations with constant coefficients. The generalized point sources are described by homogeneous expressions, containing the first order derivatives with constant coefficients. The Green’s function for a half-space, made of generalized material, subjected to the action of generalized point sources, is derived in the form of a single integral over a circle. Some of the components of the surface Green’s function are presented in finite form, no computation of any integral is needed. The bonded contact problem is described as mixed–mixed boundary value problem for a half-space, with normal and tangential displacements prescribed inside domain S and vanishing tractions outside this domain on the plane $$x_{3}=0$$ . A set of governing integral equations for bonded contact problem was derived, with some of the kernels defined in finite form. The general crack problem is defined as that of a flat crack of arbitrary shape S in the plane $$x_{3}=0$$ , with normal tractions $$\sigma _{33}$$ applied symmetrically to the crack faces and tangential tractions $$\sigma _{31}$$ and $$\sigma _{23}$$ applied anti-symmetrically to the crack faces. A set of 3 governing integral equations is derived, with some of the kernels presented in the finite form. The relationship between the integrands in Fourier transform of the kernels of the sets of the integral equations of both problems is established: they are related in such a way, as if they were the coefficients in the schematic sets of algebraic equations. This relationship can also be presented as a general relationship between matrices of arbitrary rank n, with their components being determinants of certain matrices of rank q, with $$q\ge n$$ .

Dually weighted Stirling-type sequences

We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other sequences such as the p,q-binomial coefficients. Recurrence relations, generating functions, orthogonality relations, convolution formulas, and determinants of certain matrices involving the obtained sequences are derived. We also give combinatorial interpretations of certain cases in terms of colored partitions and permutations.

Bell numbers, determinants and series

In this article, we study Bell numbers and Uppuluri Carpenter numbers. We obtain various expressions and relations between them. These include polynomial recurrences and expressions as determinants of certain matrices of binomial coefficients.

Hyperdeterminantal point processes

As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between 2M points for some integer M. The role of matrices is now played by 2M-dimensional “hypercubic” arrays, and the determinant is replaced by a suitable generalization of it to such arrays—Cayley’s first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization.

The determinants of certain matrices arising from the Boolean lattice

Let B be the Boolean lattice on an n -set with B = ⊔ B i the rank decomposition. Let M ( n , i ) be the incidence matrix between B i and B n − i . We obtain a recursive formula for the determinant of the matrix M ( n , i ) .

Triangular powers of integers from determinants of binomial coefficient matrices

Formulas are obtained for the determinants of certain matrices whose entries are zero and either binomial coefficients or their negatives. A consequence is that, for all integers n ≥ 2 and k ≥ 2, there exists an ( n − 1)( k − 1) × ( n − 1)( k − 1) matrix M ( n , k ) whose entries are the alternating binomial coefficients (−1) j +1 ( j n ) and zeros such that det( M ( n , k )) = ± k t n −1 , where t n −1 is the ( n − l)th triangular number. Further, if we form the infinite matrix whose k th row is ( 0 k ), ( 1 k ), ( 2 k ),…, then each of the above mentioned determinants is, up to sign, the determinant of an n × n submatrix A of obtained by selecting the initial n columns, and some choice of n rows of . The matrices M ( n , k ), and others that we will consider also have the unexpected property that det(| M ( n , k )|) = |det( M ( n , k )|, where | M | denotes the matrix obtained from M by replacing each entry with its absolute value.

Quantifier Elimination for Formulas Constrained by Quadratic Equations via Slope Resultants

An algorithm is given for eliminating the quantifier from a formula: (∃x∈R)[α 2 x 2 +α 1 z+α 0 =0ΛF], where F is a quantifier free formula in x 1 ,..., x r , x, and α 2 , α 1 , α 0 are polynomials in x 1 ,..., x r with real coefficients such that the system {α 2 =0, α 1 =0, α 0 =0} has no solution in R r . The output formulas are made of resultants and their variants, which we call slope resultants. The slope resultants can be, like the resultants, expressed as determinants of certain matrices

Determinantal formulae for the solution set of zero-dimensional ideals

In this paper we study the properties of a forgotten construction introduced by V.W. Habicht in 1948 related with the problem of computing a basis for the ideal generated by n polynomials F 1,…, F n in K[x n,…,x n allowing to determine in an easy way its zeros and its radical for the zero-dimensional case. This construction is achieved generalizing to K[x 1,…,x n the definition of subresultants of indexes 0 and 1 and so, the coefficients of the polynomials in the new basis are defined as the determinants of certain matrices constructed with the coefficients of the polynomials F 1,…, F n . The construction is introduced for the generic case and it is showed that its specialization works for all the zero-dimensional ideals in general position with respect to x n except when the coefficients of the polynomials F 1,…, F n are in some hypersurface in K n whose equation we shall give in an explicit way. A basis with a similar structure to the one here introduced can be computed using Groebner Basis (the Shape Lemma) imposing the radicality hypothesis to the initial ideal.

Solution of some combinatorial problems which arise in calculating correlators in exactly solvable models

In this paper the recursion relations which arise in calculating correlators in exactly solvable models by the method of V. E. Korepin are explicitly solved. The solution is given in the form of determinants of certain matrices.

Corrigendum: “Explicit expressions for the determinants of certain matrices” (Math. Comp. 24 (1970), 151–154) by J. L. Lavoie and R. Michaud

Corrigendum: “Explicit expressions for the determinants of certain matrices” (Math. Comp. 24 (1970), 151–154) by J. L. Lavoie and R. Michaud

On the Evaluation of Certain Determinants

Using the properties of the generalized Hilbert matrix and familiar results from the theory of hypergeometric series, we evaluate the determinants of certain matrices whose general terms are known explicitly. In certain cases it is even possible to find the analytic expression for the general terms of the inverses.

On the evaluation of certain determinants

Using the properties of the generalized Hilbert matrix and familiar results from the theory of hypergeometric series, we evaluate the determinants of certain matrices whose general terms are known explicitly. In certain cases it is even possible to find the analytic expression for the general terms of the inverses.