Minimax theorems without linear structure

Hadamard's determinant theorem is used to obtain an upper bound for the modulus of the determinant of the sum of two normal matrices in terms of their eigenvalues. This bound is compared with another given by M. E. Miranda.     

一个行列式不等式的进一步研究

本文的日的在于改进已有的两个复矩阵的行列式的上界,以更精细的两个Hermitian正定矩阵和的行列式为基本工具.利用得到的相关一无二次不等式描述的行列式之间的关系,给出了两个复矩阵和的行列式新上界,作为心用可改进华罗庚行列式不等式的上界.     

ESTIMATES OF THE DOUBLE DETERMINANTS OF QUATERNION MATRICES

An estimate of the upper bound is given for the double determinant of the sum of two arbitrary quaternion matrices, and meanwhile the lower bound on the double determinant is established especially for the sum of two quaternion matrices which form an assortive pair. As applications, some known results are obtained as corollaries and a question in the matrix determinant theory is answered completely.     

The lower bound for the number of local minima of integral lattices

A lower bound for the maximal number of local minima of integral lattices with preassigned determinant is derived. This bound coincides with the upper one up to a constant. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 9–14, 2005.     

Determinant of the Schrödinger operator

For the example of the nonrelativistic Schrödinger operator, methods are formulated for calculating the determinant of an elliptic operator on the basis of scattering theory. It is shown that such a determinant is identical to the Jost determinant at zero energy. In the centrally symmetric case, it reduces to ordinary Jost functions and ultimately to the values of the zero-energy wave functions at the origin. The relationship between the determinant of the Schrödinger operator and the characteristics of the scattering resonances and the number of bound states in a field of opposite sign is noted. This makes it possible to find the first terms in the gradient expansion of the determinant as a functional of the potential. The problem of the correlation free energy of a classical plasma serves as a physical illustration.P. N. Lebedev Physics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 3, pp. 373–384, September, 1993.     

Bound and anti‐bound states for coupled‐channel scattering

We discuss bound and anti‐bound states for 2×2 matrix Schrödinger operator. We analyze the Fredholm determinant for Hamiltonians that can be represented in a multi‐channel framework. Our analysis covers the whole and the half‐line problems. We obtain some results on counting anti‐bound states between successive bound states. Copyright © 2017 John Wiley & Sons, Ltd.     

Regulatorabschätzungen für total reelle algebraische Zahlkörper

A lower bound for the regulator of a totally real algebraic number field is determined. The regulator occurs in the determinant of a suitable positive definite quadratic form, and the desired bound is obtained by estimating the minimum of this quadratic form from below. This can be done by solving an extremal value problem with subsidiary conditions from the properties of the units of the field. Numerous examples (Tables I, III) illustrate the advantage of this method over known results.     

Integer matrices related to Liouville’s function

In this note, we construct some integer matrices with determinant equal to certain summation form of Liouville’s function. Hence, it offers a possible alternative way to explore the Prime Number Theorem by means of inequalities related to matrices, provided a better estimate on the relation between the determinant of a matrix and other information such as its eigenvalues is known. Besides, we also provide some comparisons on the estimate of the lower bound of the smallest singular value. Such discussion may be extended to that of Riemann hypothesis.     

Ma-Minda 型双向单叶函数的三阶~Hankel 行列式

In this paper, we investigate the third Hankel determinant H_3(1) for the class H_σ~μ(λ, φ)(λ≥ 1, μ≥ 1) of Ma-Minda bi-univalent functions in the open unit disk D = {z : |z| 1} and obtain the upper bound of the above determinant H_3(1).     

On the first factor of the class number of prime cyclotomic fields

Let H(l) be the first factor of the class number of the field $\text{Q}$(exp 2πi/l), l a prime. The best-known upper and lower bounds on H(l) are improved for small l. The methods would also improve the best-known bounds for large l. It is shown that H(l) is the absolute value of the determinant of an easily written down matrix whose only entries are 0 and 1. The upper bounds obtained on H(l) significantly improve the Hadamard bound on the determinant of this matrix. Results of Lehmer on the factors of H(l) are explained via class field theory.     

On the Ranks and Border Ranks of Symmetric Tensors

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.     

四元数阵重行列式与复阵行列式的关系

指出四元数阵重行列式可用复阵行列式来表示，于是，复阵的伴随矩阵、求逆阵公式、秩的下界等，都可相应地推广到四元数阵．     

Minimal zonotopes containing the crosspolytope

Motivated by the problem to improve Minkowski’s lower bound on the successive minima for the class of zonotopes we determine the minimal volume of a zonotope containing the standard crosspolytope. It turns out that this volume can be expressed via the maximal determinant of a ±1-matrix, and that in each dimension the set of minimal zonotopes contains a parallelepiped. Based on that link to ±1- matrices, we characterize all zonotopes attaining the minimal volume in dimension 3 and present related results in higher dimensions.     

A note on subset selection for matrices

In an earlier paper the authors examined the problem of selecting rows of a matrix so that the resulting matrix is as “non-singular” as possible. However, the proof of the key result in that paper is not constructive. In this note we give a constructive proof for that result. In addition, we examine a case where as non-singular as possible means maximizing a determinant and provide a new bound and a constructive proof for this case also.     

一类解析函数类的Hankel行列式的上界估计

本文对在单位圆盘U={z∈C：l│z│〈1}内解析且满足（1-λ）f（z）/z＋λf^1（z）〈1＋Az/1＋Bz（-1≤B＜A≤1;z∈U的函数f所对应的Hankel行列式│a2a4-a^23│,利用Toeplitz行列式的性质得到了其上界估计。     

两类六角系统的$A\alpha$-特征多项式和$A\alpha$-谱

2017年, Nikiforov首次提出研究图$G$的$A\alpha$-矩阵, 其定义为:$A\alpha(G)=\alpha D(G)+(1-\alpha)A(G) (\alpha\in [0,1])$, 其中$A(G)$和$D(G)$分别为图$G$的邻接矩阵和度对角矩阵. 设$F_n$和$M_n$分别为圈状六角系统和M\"{o}bius带状六角系统图. 根据循环矩阵的行列式和特征值, 本文首先给出图$F_n$和$M_n$的$A\alph$-特征多项式和$A\alpha$-谱, 进一步得到图$F_n$和$M_n$的$A\alpha$-能量的上界.     

四元数体上重行列式的性质及其应用

本文得到了四元数体上重行列式的一些基本不等式，给出了矩阵为正定自共轭阵时，行列式与重行列式的显式关系，同时也给出了文［１］中行列式与文［５］中行列式两种定义的关系。提出了四元数体上广义正定自共轭阵的概念，并获得了这类阵的基本性质．     

Resultant as the determinant of a Koszul complex

The determinant is a very important characteristic of a linear map between vector spaces. Two generalizations of linear maps are intensively used in modern theory: linear complexes (nilpotent chains of linear maps) and nonlinear maps. The determinant of a complex and the resultant are then the corresponding generalizations of the determinant of a linear map. It turns out that these two quantities are related: the resultant of a nonlinear map is the determinant of the corresponding Koszul complex. We give an elementary introduction into these notions and relations, which will definitely play a role in the future development of theoretical physics.     

Ricci Flow and the Determinant of the Laplacian on Non-Compact Surfaces

On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a metric of constant curvature and, for negative Euler characteristic, exhibited a flow from a given metric to a constant curvature metric along which the determinant increases. The aim of this paper is to perform a similar analysis for the determinant of the Laplacian on a non-compact surface whose ends are asymptotic to hyperbolic funnels or cusps. In that context, we show that the Ricci flow converges to a metric of constant curvature and that the determinant increases along this flow.     

Asymptotics of the Fredholm Determinant Associated with the Correlation Functions of the Quantum Nonlinear Schrödinger Equation

The correlation functions of the quantum nonlinear Schrödinger equation can be presented in terms of the Fredholm determinant. An explicit expression for this determinant is found for large time and long distance. Bibliography: 6 titles.