Semidefinite Research Articles

Unfinished history and paradoxes of quantum potential. I. Non-relativistic origin, history and paradoxes

This is the first of two related papers analyzing and explaining the origin, manifestations and parodoxical features of the quantum potential (QP) from the non-relativistic and relativistic points of view. The QP arises in the quantum Hamiltonian under various procedures of quantization of natural systems, i.e., those whose Hamilton functions are positive-definite quadratic forms in momenta with coefficients depending on the coordinates in (n-dimensional) configurational space V n thus endowed with a Riemannian structure. The result of quantization may be considered as quantum mechanics (QM) of a particle in V n in the normal Gaussian coordinate system in the globally static space-time V 1,n . Contradiction of the QP to the General Covariance and Equivalence principles is discussed.It is found that actually the historically first Hilbert space-based quantization by E. Schrödinger (1926), after revision in the modern framework of QM, also leads to a QP in the form that B. DeWitt found 26 years later. Efforts to avoid the QP or to reduce its drawbacks are discussed. The general conclusion is that some form of QP and a violation of the principles of general relativity which it induces are inevitable in the non-relativistic quantum Hamiltonian. It is also shown that Feynman (path-integral) quantization of natural systems singles out two versions of the QP, which both determine two bi-scalar (independent of a choice of coordinates) propagators fixing two different algorithms of path integral calculation.The accompanying paper under the same general title and the subtitle “The Relativistic Point of View” (published in the same issue of the journal and referred to as Paper II) considers a relation of the nonrelativistic QP to the quantum theory of a scalar field non-minimally coupled to the curved space-time metric.

On the Hermitian structures of the solution to a pair of matrix equations

Let 𝒮 be a given set consisting of some Hermitian matrices with the same sizes. We say that a matrix A ∈ S is minimal (maximal) if A − W is negative (positive) semidefinite for every matrix W ∈ 𝒮. In this article, we construct the new quasi-quadratic Hermitian structures for a system of matrix equations. i.e., where P = P*, Q = Q* and X is a solution to We first consider the extremal inertias and ranks of (1) and (2). As applications, we derive the necessary and sufficient conditions for (1) and (2) to be positive (negative), positive (negative) semidefinite, nonsingular and the system to be consistent, respectively. Some special cases such as the unitary solvability and the contraction solvability to (3) are also considered. In addition, we present the necessary and sufficient conditions for the existence of the left and the right minimal solutions to (3). The explicit expressions of the left and the right minimal solutions are given when the conditions are met.

Correlations of representation functions of binary quadratic forms

This paper extends previous work on linear correlations of representation functions of positive definite binary quadratic forms to allow indefinite forms.

Note to the Paper ``A Positive Definite Binary Quadratic Form as a Sum of Five Squares of Linear Forms (Completion of Mordell's Proof)'' (Bull. Polish Acad. Sci. Math. 61 (2013), 23–26)

Note to the Paper ``A Positive Definite Binary Quadratic Form as a Sum of Five Squares of Linear Forms (Completion of Mordell's Proof)'' (Bull. Polish Acad. Sci. Math. 61 (2013), 23–26)

Toroidal Algorithms for Mesh Geometries of Root Orbits of the Dynkin Diagram $\mathbb{D}_4$

By applying symbolic and numerical computation and the spectral Coxeter analysis technique of matrix morsifications introduced in our previous paper [Fund. Inform. 1242013], we present a complete algorithmic classification of the rational morsifications and their mesh geometries of root orbits for the Dynkin diagram $\mathbb{D}_4$. The structure of the isotropy group $Gl4, \mathbb{Z}_{\mathbb{D}_4}$ of $\mathbb{D}_4$ is also studied. As a byproduct of our technique we show that, given a connected loop-free positive edge-bipartite graph Δ, with n ≥ 4 vertices in the sense of our paper [SIAM J. Discrete Math. 272013] and the positive definite Gram unit form $q_\Delta : \mathbb{Z}^n \rightarrow \mathbb{Z}$, any positive integer d ≥ 1 can be presented as d = qΔv, with $v \in \mathbb{Z}^n$. In case n = 3, a positive integer d ≥ 1 can be presented as d = qΔv, with $v \in \mathbb{Z}^n$, if and only if d is not of the form 4a16 · b + 14, where a and b are non-negative integers.

A matrix analytic approach to conjugate diameters of an ellipse

Herein, problems pertaining to the conjugate diameters of a nondegenerate ellipse are studied within a matrix analytic framework. Pride of place is reserved for the construction of the matrix associated with the positive definite quadratic form defining a nondegenerate ellipse from a pair of its conjugate diameters. Mathematics Subject Classification: 15A18, 15A23, 15A24, 51N20

On inverse problems for left-definite discrete Sturm-Liouville equations

We establish an expansion theorem and investigate inverse spectral and inverse scattering problems for the discrete Sturm-Liouville problem −u′′(n−1)+q(n)u(n) = λw(n)u(n) where q is nonnegative and w may change sign. If w is positive, the 2 -space weighted by w is a Hilbert space and it is customary to use that space for setting the problem. In the present situation the right-hand-side of the equation does not give rise to a positive-definite quadratic form and we use instead the left-hand side to define such a form and hence a Hilbert space (such problems are called left-definite). The difference equation gives rise to a linear relation which, upon proper restrictions, generates a self-adjoint operator. For this operator we define a Fourier transform and investigate the relationship between two operators with the same transform (the inverse spectral problem). If q− q0 and w− 1 are summable one may define the scattering process and we solve the inverse scattering problem. For coefficients decaying sufficiently fast to q0 and 1 , respectively, the concept of a resonance is introduced as a generalization of the notion of an eigenvalue and the set of iso-resonant operators, i.e., operators having the same eigenvalues and resonances, is described. Mathematics subject classification (2010): 39A70, 34L25.

The emergence domain of an exact ground state in a non-integrable system: the case of the polyphenylene type of chains

A technique based on the transformation in positive semidefinite form of the Hamiltonian (Ĥ) is developed to allow the study of the enlargement possibilities of the emergence domain of a given state. For this reason the calculation of exact ground states for the polyphenylene type of non-integrable chains is performed. Using different exact transcriptions of Ĥ in positive semidefinite form and different solutions of the matching equations, the same type of ground state is deduced in different regions of the parameter space, hence a more global view of its emergence possibilities is obtained. With this procedure an enlarged view of the appearance in the phase diagram of different studied phases can be achieved.

Local deformations of positive-definite quadratic forms

We give a complete description of real numbers that are P-limit numbers for integer-valued positive-definite quadratic forms with unit coefficients of the squares. It is shown that each of these P-limit numbers is realized in the Tits quadratic form of a certain Dynkin diagram.

ASYMPTOTIC FORMULAS FOR CLASS NUMBER SUMS OF INDEFINITE BINARY QUADRATIC FORMS ON ARITHMETIC PROGRESSIONS

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants on arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the remainder terms are sharper.

Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures

We give upper bounds for the deviation of the norm of a perturbed error functional from the norm of the original error of a higher-dimensional spherical cubature formula. The deviation arises as a result of the combined influence on the computation of small variations of the weights of the cubature formula and rounding for the subsequent calculation of the cubature sum in the given standards of approximation to real numbers. We estimate the practical error of the cubature formula for its action on an arbitrary function in the unit ball of the normed space of integrands. The resulting estimates are applied to studying the practical error of spherical cubature formulas in the case of integrands in Sobolev-type spaces on the higher-dimensional unit sphere. We represent the norm of the error functional in the dual space of the Sobolev class as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for spherical cubature formulas, each of which is constructed as the direct product of Gauss’s quadrature formula along the meridian of the sphere and of the rectangle quadrature formula along the equator. The weights of this direct product with 2m2 nodes are positive. The formula itself is exact at all spherical harmonics up to order 2m − 1.

The triangular theorem of eight and representation by quadratic polynomials

We investigate here the representability of integers as sums of triangular numbers, where the n n -th triangular number is given by T n = T_n= n ( n + 1 ) / 2 n(n+1)/2 . In particular, we show that f ( x 1 , x 2 , … , x k ) = b 1 T x 1 + ⋯ + b k T x k f(x_1, x_2, \ldots , x_k)=b_1T_{x_1}+\cdots +b_kT_{x_k} , for fixed positive integers b 1 , b 2 , … , b k b_1, b_2, \ldots , b_k , represents every nonnegative integer if and only if it represents 1 1 , 2 2 , 4 4 , 5 5 , and 8 8 . Moreover, if ‘cross-terms’ are allowed in f f , we show that no finite set of positive integers can play an analogous role, in turn showing that there is no overarching finiteness theorem which generalizes the statement from positive definite quadratic forms to totally positive quadratic polynomials.

Solving the Combined Heat and Power Dispatch Problem: A Semi-definite Programming Approach

This article presents a new solution using the semi-definite programming approach to solve the combined heat and power economic dispatch problem. The proposed method involves reformulating the combined heat and power dispatch problem into a semi-definite programming form. The resulting semi-definite programming problem is solved using the primal-dual solver. The effectiveness of the proposed method is illustrated through some test problems, and the results are compared with those in literature.

Combinatorics of the tropical Torelli map

This paper is a combinatorial and computational study of the moduli space of tropical curves of genus g, the moduli space of principally polarized tropical abelian varieties, and the tropical Torelli map. These objects were introduced recently by Brannetti, Melo, and Viviani. Here, we give a new definition of the category of stacky fans, of which the aforementioned moduli spaces are objects and the Torelli map is a morphism. We compute the poset of cells of tropical M_g and of the tropical Schottky locus for genus at most 5. We show that tropical A_g is Hausdorff, and we also construct a finite-index cover for A_3 which satisfies a tropical-type balancing condition. Many different combinatorial objects, including regular matroids, positive semidefinite forms, and metric graphs, play a role.

On the zeros of Epstein zeta functions

Abstract We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.

Another Proof of the Faithfulness of the Lawrence-Krammer Representation of the Braid Group

The Lawrence-Krammer representation of the braid group was proved to be faithful for by Bigelow and Krammer. In our paper, we give a new proof in the case by using matrix computations. First, we prove that the representation of the braid group is unitary relative to a positive definite Hermitian form. Then we show the faithfulness of the representation by specializing the indeterminates q and t to complex numbers on the unit circle rather than specializing them to real numbers as what was done by Krammer.

Integral forms in vertex operator algebras which are invariant under finite groups

For certain pairs consisting of a vertex operator algebra (e.g., lattice type) and a finite subgroup of its automorphism group, we prove existence of a positive definite integral form invariant under the group. Applications include an integral form in the Moonshine VOA which is invariant under the Monster, and examples in other lattice type VOAs.

The extreme rays of the [formula omitted] copositive cone

We give an explicit characterization of all extreme rays of the cone C5 of 5×5 copositive matrices. We show that the extreme rays which are not positive semi-definite or nonnegative form a 10-dimensional variety, which can be parameterized in a semi-trigonometric way.

On a question of Bhatia and Kittaneh

We settle in the affirmative a question of Bhatia and Kittaneh. For P and Q positive semidefinite n×n matrices, the inequality σr(PQ)⩽12λr(P+Q) holds for r=1,2,…,n.

Sum of squares methods for minimizing polynomial forms over spheres and hypersurfaces

This paper studies the problem of minimizing a homogeneous polynomial (form) f(x) over the unit sphere \(\mathbb{S}^{n - 1} = \left\{ {x \in \mathbb{R}^n :\left\| x \right\|_2 = 1} \right\}\). The problem is NP-hard when f(x) has degree 3 or higher. Denote by fmin (resp. fmax) the minimum (resp. maximum) value of f(x) on \(\mathbb{S}^{n - 1}\). First, when f(x) is an even form of degree 2d, we study the standard sum of squares (SOS) relaxation for finding a lower bound of the minimum fmin: $$\max \gamma s.t. f\left( x \right) - \gamma \cdot \left\| x \right\|_2^{2d} is SOS.$$ Let fsos be the above optimal value. Then we show that for all n ⩾ 2d, $$1 \leqslant \frac{{f_{\max } - f_{sos} }} {{f_{\max } - f_{\min } }} \leqslant C(d)\sqrt {\left( {_{2d}^n } \right)} .$$ Here, the constant C(d) is independent of n. Second, when f(x) is a multi-form and \(\mathbb{S}^{n - 1}\) becomes a multi-unit sphere, we generalize the above SOS relaxation and prove a similar bound. Third, when f(x) is sparse, we prove an improved bound depending on its sparsity pattern; when f(x) is odd, we formulate the problem equivalently as minimizing a certain even form, and prove a similar bound. Last, for minimizing f(x) over a hypersurface H(g) = {x ∈ ℝn: g(x) = 1} defined by a positive definite form g(x), we generalize the above SOS relaxation and prove a similar bound.