Related papers: Hyperdeterminantal Total Positivity
We establish a positive product formula for the solutions of the Sturm-Liouville equation $\ell(u) = \lambda u$, where $\ell$ belongs to a general class which includes singular and degenerate Sturm-Liouville operators. Our technique relies…
The heat kernel on the symmetric space of positive definite Hermitian matrices is used to endow the spaces of Bergman metrics of degree k on a Riemann surface M with a family of probability measures depending on a choice of the background…
Let k be an arbitrary field (of arbitrary characteristic) and let X = [x_{i,j}] be a generic m x n matrix of variables. Denote by I_2(X) the ideal in k[X] = k[x_{i,j}: i = 1, ..., m; j = 1, ..., n] generated by the 2 x 2 minors of X. We…
These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…
In this paper we provide an identity between determinant and generalized matrix function. Also, a criterion of positive semi-definite matrices affirming the permanent dominant conjecture is given. As a consequence, infinitely many infinite…
Berndtsson's famous theorem asserts that, for a compact K\"ahler fibration $p:X\to Y$, the direct image bundle $p_*(K_{X/Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation…
Characterizing in a constructive way the set of real functions whose Fourier transforms are positive appears to be yet an open problem. Some sufficient conditions are known but they are far from being exhaustive. We propose two constructive…
Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem} asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9…
Relaxation rates are key characteristics of quantum processes, as they determine how quickly a quantum system thermalizes, equilibrates, decoheres, and dissipates. While they play a crucial role in theoretical analyses, relaxation rates are…
Fully commutative elements in types $B$ and $D$ are completely characterized and counted by Stembridge. Recently, Feinberg-Kim-Lee-Oh have extended the study of fully commutative elements from Coxeter groups to the complex setting, giving…
A very general KSGNS type dilation theorem in the context of right (not necessarily Hilbert) modules over $C^*$-algebras is presented. The proof uses Kolmogorov type decompositions for positive-definite kernels with values in spaces of…
We introduce a new deal of kernel density estimation using an exponentiated form of kernel density estimators. The density estimator has two hyperparameters flexibly controlling the smoothness of the resulting density. We tune them in a…
It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…
Interpolation and approximation of functionals with conditionally positive definite kernels is considered on sets of centers that are not determining for polynomials. It is shown that polynomial consistency is sufficient in order to define…
In this paper we extend a gradient estimate of R. Hamilton for positive solutions to the heat equation on closed manifolds to bounded positive solutions on complete, non-compact manifolds with $Rc \geq -Kg$. We accomplish this extension via…
Conner and Hurrelbrink established a method of determining the structure of the 2-Sylow subgroup of the tame kernel $K_2(\mathcal{O})$ for certain quadratic number fields. Specifically, the 4-rank for these fields was characterized in terms…
In this paper, we study partitions of totally positive integral elements $\alpha$ in a real quadratic field $K$. We prove that for a fixed integer $m \geq 1$, an element with $m$ partition exists in almost all $K$. We also obtain an upper…
The classical as well as non commutative Korovkin-type theorems deal with convergence of positive linear maps with respect to modes of convergences such as norm convergence and weak operator convergence. In this article, Korovkin-type…
The paper contains two main parts: in the first part, we analyze the general case of $p\geq 2$ matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain…
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on…