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This paper focuses on different reductions of the two-dimensional (2d)-Toda hierarchy. Symmetric and skew-symmetric moment matrices are first considered, resulting in differential relations between symmetric/skew-symmetric tau functions and 2d-Toda’s tau functions, respectively. Furthermore, motivated by the Cauchy two-matrix model and Bures ensemble from random matrix theory, we study the rank-one shift condition in the symmetric case and rank-two shift condition in the skew-symmetric case, from which the C-Toda and B-Toda hierarchies are determined, together with their special Lax matrices and integrable structures. [ABSTRACT FROM AUTHOR]

In this article we study the bivariate truncated moment problem (TMP) of degree 2 k on the union of parallel lines. First we present an alternative proof of Fialkow's solution [28] to the TMP on the union of two parallel lines (TMP–2pl) using the solution of the truncated Hamburger moment problem (THMP). We add a new equivalent solvability condition, which is then used together with the THMP, to solve the TMP on the union of three parallel lines (TMP–3pl), our second main result of the article. Finally, we establish a sufficient condition for the existence of a solution to the TMP on the union of n parallel lines in the pure case, i.e. when the moment matrix M k is of the highest possible rank, or equivalently the only column relations come from the union of n lines. The condition is based on the feasibility of a certain linear matrix inequality, corresponding to the extension of M k by adding rows and columns indexed by some monomials of degree k + 1. The proof is by induction on n , where n ≥ 2 and for the base of induction n = 2 we use the solution of the TMP–2pl. [ABSTRACT FROM AUTHOR]

In this article we solve four special cases of the truncated Hamburger moment problem (THMP) of degree 2k with one or two missing moments in the sequence. As corollaries we obtain via appropriate substitutions, the solutions to bivariate truncated moment problems of degree 2k for the curves y = x 3 (first solved by Fialkow in Trans Am Math Soc 363:3133–3165, 2011), y 2 = x 3 , y = x 4 where a certain moment of degree 2 k + 1 is known and y 3 = x 4 with a certain moment given. The main technique is the completion of the partial positive semidefinite matrix (ppsd) such that the conditions of Curto and Fialkow's solution of the THMP are satisfied. The main tools are the use of the properties of positive semidefinite Hankel matrices and a result on all completions of a ppsd matrix with one unknown entry, proved by the use of the Schur complements for 2 × 2 and 3 × 3 block matrices. [ABSTRACT FROM AUTHOR]

We extend the concept of Pascal matrix to all matrices with the property that entries are the sum of left and above entries. We have obtained an algebraic relation that allows us to obtain a Toeplitz matrix from a Pascal matrix, and vice versa. This procedure can be applied to any Pascal matrices and any Toeplitz matrices, Hermitian matrices or not, moment matrices or not. The relationship with the difference operator and with the shift operator is also studied. [ABSTRACT FROM AUTHOR]

In this paper, a new measurement to compare two large-scale graphs based on the theory of quantum probability is proposed. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. Our proposed distance between two graphs is defined as the distance between the corresponding moment matrices of their spectral distributions. It is shown that the spectral distributions of their adjacency matrices in a vector state includes information not only about their eigenvalues, but also about the corresponding eigenvectors. Moreover, we prove that the proposed distance is graph invariant and sub-structure invariant. Examples with various graphs are given, and distances between graphs with few vertices are checked. Computational results for real large-scale graphs show that its accuracy is better than any existing methods and time cost is extensively cheap. [ABSTRACT FROM AUTHOR]

We propose a new solver for the sparse spikes superresolution problem over the space of Radon measures. A common approach to off-the-grid deconvolution considers semidefinite relaxations of the total variation (the total mass of the absolute value of the measure) minimization problem. The direct resolution of this semidefinite program (SDP) is, however, intractable for large scale settings, since the problem size grows as f2d c, where fc is the cutoff frequency of the filter and d the ambient dimension. Our first contribution is a Fourier approximation scheme of the forward operator, making the TV-minimization problem expressible as an SDP. Our second contribution introduces a penalized formulation of this semidefinite lifting, which we prove to have low-rank solutions. Our last contribution is the FFW algorithm, a Fourier-based Frank--Wolfe scheme with nonconvex updates. FFW leverages both the low-rank and the Fourier structure of the problem, resulting in an O(fd c log fc) complexity per iteration. Numerical simulations are promising and show that the algorithm converges in exactly r steps, r being the number of Diracs composing the solution. [ABSTRACT FROM AUTHOR]

A generalization of the flat extension theorems of Curto and Fialkow and Laurent and Mourrain is obtained by seeing the problem as one of ideal projection. [ABSTRACT FROM AUTHOR]

The (classical) truncated moment problem, extensively studied by Curto and Fialkow, asks to characterize when a finite sequence of real numbers indexes by words in commuting variables can be represented with moments of a positive Borel measure μ on Rn. Burgdorf and Klep (J Oper Theory 68:141-163, 2012) introduced its tracial analog, the truncated tracial moment problem, which replaces commuting variables with non-commuting ones and moments of μ with tracial moments of matrices. In the bivariate quartic case, where indices run over words in two variables of degree at most four, every sequence with a positive definite 7×7 moment matrix M2 can be represented with tracial moments (Burgdorf and Klep in C R Math Acad Sci Paris 348:721-726, 2010, 2012). In this article the case of singular M2 is studied. For M2 of rank at most 5 the problem is solved completely; namely, concrete measures are obtained whenever they exist and the uniqueness question of the minimal measures is answered. For M2 of rank 6 the problem splits into four cases, in two of which it is equivalent to the feasibility problem of certain linear matrix inequalities. Finally, the question of a flat extension of the moment matrix M2 is addressed. While this is the most powerful tool for solving the classical case, it is shown here by examples that, while sufficient, flat extensions are mostly not a necessary condition for the existence of a measure in the tracial case. [ABSTRACT FROM AUTHOR]

Let β ≡ β ( m ) = { β i } i ∈ Z + n , | i | ≤ m , β 0 > 0 , denote a real n -dimensional multisequence of finite degree m . The Truncated Moment Problem concerns the existence of a positive Borel measure μ , supported in R n , such that (0.1) β i = ∫ R n x i d μ ( i ∈ Z + n , | i | ≤ m ) . We associate to β ≡ β ( 2 d ) an algebraic variety in R n called the core variety , V ≡ V ( β ) . The core variety contains the support of each representing measure μ . We show that if V is nonempty, then β ( 2 d − 1 ) has a representing measure. Moreover, if V is a nonempty compact or determining set, then β ( 2 d ) has a representing measure. We also use the core variety to exhibit a sequence β , with positive definite moment matrix and positive Riesz functional, which fails to have a representing measure. [ABSTRACT FROM AUTHOR]

Mixture experiments in which linear constraints are imposed on the components of the mixture are used extensively in practice. The problem of constructing designs which are in some sense optimal for this experimental setting is not straightforward. More specifically, the design space is a polytope embedded in a regular simplex. In the present paper, a new approach to the problem, which builds on the fact that points in the polytope can be represented as convex combinations of the vertices of that polytope, is introduced. Some theory underpinning this idea and rooted in the notion of barycentric coordinates is developed and algorithms for the construction of exact and approximate D - and I -optimal designs for the Scheffé model are delineated. The methodology is illustrated by means of examples involving three- and four-component mixtures. [ABSTRACT FROM AUTHOR]