Related papers: The matrix function $e^{tA+B}$ is representable as…
We consider the matrix spherical function related to the compact symmetric pair $(G,K)=(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$. The irreducible $K$ representations $(\pi,V)$ in the ${\rm U}(n)$ part are considered…
Apparently new expressions are given for the exponential of a hermitian matrix,A, in the 2x2,3x3,and 4x4 cases. Replacing A by iA these are explicit formulas for the Fourier transform of exp(iA).
For an ergodic hyperbolic measure $\omega$ of a $C^{1+{\alpha}}$ diffeomorphism, there is an $\omega$ full-measured set $\tilde\Lambda$ such that every nonempty, compact and connected subset $V$ of $\mathbb{M}_{inv}(\tilde\Lambda)$…
We present several applications of matrix-theoretic inequalities to the magnitude of metric spaces. We first resolve an open problem by showing that the magnitude of any finite metric space of negative type is less than or equal to its…
The transfer matrix of the square-lattice eight-vertex model on a strip with $L\geqslant 1$ vertical lines and open boundary conditions is investigated. It is shown that for vertex weights $a,b,c,d$ that obey the relation…
We give a general construction of entire functions in $d$ complex variables that vanish on a lattice of the form $L = A (Z + i Z )^d$ for an invertible complex-valued matrix. As an application we exhibit a class of lattices of density >1…
This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces $\mathbb{R}^m$ where $\mu$ is a positive constant. If the right-hand side $f$ is a rapidly converging series of separable functions, the solution $u$ can be…
We introduce a vector-valued generalization of the Epstein zeta functions associated with the root lattices of ADE-type Lie algebras. The quadratic forms defining these lattices correspond to the Gram matrices of the simple roots. Using the…
The type IIB matrix model is conjectured to describe superstring theory nonperturbatively in terms of ten $N \times N$ bosonic traceless Hermitian matrices $A_\mu$ ($\mu=0, \ldots , 9$), whose eigenvalues correspond to $(9+1)$-dimensional…
In this paper we solve the following problems: (i) find two differential operators P and Q satisfying [P,Q]=P, where P flows according to the KP hierarchy \partial P/\partial t_n = [(P^{n/p})_+,P], with p := \ord P\ge 2; (ii) find a matrix…
In recent work the author presented a formal expansion for lambda_d associated to the dimer problem on a d-dimensional rectangular lattice. Expressed in terms of d it yielded a presumed asymptotic expansion for lambda_d in inverse powers of…
Markov processes on the lattices with arbitrary dimension are omnipresent in statistical mechanics; however their algebraic description is complete only in dimension 1, for which linear algebra provides many tools complementary to the…
Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is type $2$ if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ does not satisfy a zero-one law. This means that we can find a…
We use the Laplace transform and the Gamma function to introduce a new integral transform and name it the Laplace-type transform possessing the property of mapping a function to a functional sequence, which cannot be achieved by the Laplace…
In this paper, we show that if we have a sequence of Hadamard triples $\{(N_n,B_n,L_n)\}$ with $B_n\subset \{0,1,..,N_n-1\}$ for $n=1,2,...$, except an extreme case, then the associated Cantor-Moran measure $$ \begin{aligned} \mu =…
We characterize a holomorphic positive definite function $f$ defined on a horizontal strip of the complex plane as the Fourier-Laplace transform of a unique exponentially finite measure on $\mathbb{R}$. The classical theorems of Bochner on…
For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of…
Based on Stokes' theorem we derive a non-holomorphic functional calculus for matrices, assuming sufficient smoothness near eigenvalues, corresponding to the size of related Jordan blocks. It is then applied to the complex conjugation…
Given a function on diagonal matrices, there is a unique way to extend this to an invariant (by conjugation) function on symmetric matrices. We show that the extension preserves regularity -- that is, if the original function is k times…
For two matrices $A$ and $B$, and large $n$, we show that most products of $n$ factors of $e^{A/n}$ and $n$ factors of $e^{B/n}$ are close to $e^{A + B}$. This extends the Lie-Trotter formula. The elementary proof is based on the relation…