Related papers: The Hilbert's-Tenth-Problem Operator

For an invertible (bounded) linear operator Q acting in a Hilbert space ${\cal H}$, we consider the consequences of the QT-symmetry of a non-Hermitian Hamiltonian $H:{\cal H}\to{\cal H}$ where T is the time-reversal operator. If H is…

This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers. Hilbert's Tenth Problem (H10) asks about the…

The family of Tremblay-Turbiner-Winternitz (TTW) Hamiltonians $H_k$ on a plane, corresponding to any positive real value of $k$, is shown to admit another ${\cal N} = 2$ supersymmetric extension than that previously introduced by the…

It is observed that the infinite matrix with entries $(\sqrt{mn}\log (mn))^{-1}$ for $m, n\ge 2$ appears as the matrix of the integral operator $\mathbf{H}f(s):=\int_{1/2}^{+\infty}f(w)(\zeta(w+s)-1)dw$ with respect to the basis…

The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $\text{Ber}(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the…

We start with a Riemann-Hilbert Problems (RHP) with canonical normalization whose sewing functions depends on several additional variables. Using Zakharov-Shabat theorem we are able to construct a family of ordinary differential operators…

Wiener-Hopf plus Hankel operators $W(a)+H(b):L^p(\mathbb{R}^+)\to L^p(\mathbb{R}^+)$ with generating functions $a$ and $b$ from a subalgebra of $L^\infty(\mathbb{R})$ containing almost periodic functions and Fourier images of…

Given Hilbert space operators $P,T\in B(\H), P\geq 0$ invertible, $T$ is $(m,P)-$ expansive (resp., $(m,P)-$ isometric) for some positive integer $m$ if…

The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving…

The halting problem for Turing machines is decidable on a set of asymptotic probability one. Specifically, there is a set B of Turing machine programs such that (i) B has asymptotic probability one, so that as the number of states n…

We prove an analogue of Hilbert's Tenth Problem for complex meromorphic functions. More precisely, we prove that the set of integers is positive existentially definable in fields of complex meromorphic functions in several variables over…

We obtain various upper bounds for the numerical radius $w(T)$ of a bounded linear operator $T$ defined on a complex Hilbert space $\mathcal{H}$, by developing the upper bounds for the $\alpha$-norm of $T$, which is defined as…

A deformation of Heisenberg algebra induces among other consequences a loss of Hermiticity of some operators that generate this algebra. Therefore, these operators are not Hermitian, nor is the Hamiltonian operator built from them. In the…

In this chapter, the Hilbert space framework in the mathematical theory of composite materials is introduced for studying the properties of effective operators. The goal is to introduce some of the key concepts and fundamental theorems in…

Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation $[\hat P,\hat M]=1$. In ordinary quantum mechanics $\hat P$ is the derivative and $\hat M$ the coordinate operator. Here we shall realize $\hat P$ as…

The Hilbert transform is essentially the \textit{only} singular operator in one dimension. This undoubtedly makes it one of the the most important linear operators in harmonic analysis. The Hilbert transform has had a profound bearing on…

We investigate a Riemann-Hilbert problem (RHP), whose solution corresponds to a group of $q$-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the…

Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial- time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs)…

Tensor operators associated with a given quantum Lie algebra admit a natural description in the R-matrix language. Here we employ the R-matrix approach to discuss the problem of fusion of tensor operators. The most interesting case is…

Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following…