Nonzero Complex Number Research Articles

Representation-reduced stated skein modules and algebras

For any marked three manifold (M,N) and any quantum parameter q12 (a nonzero complex number), we use Sq1/2(M,N) to denote the stated skein module of (M,N). When q12 is a root of unity of odd order, the commutative algebra S1(M,N) acts on Sq1/2(M,N). For any maximal ideal ρ of S1(M,N), define Sq1/2(M,N)ρ=Sq1/2(M,N)⊗S1(M,N)(S1(M,N)/ρ).We prove the splitting map for Sq1/2(M,N) respects the S1(M,N)-module structure, so it reduces to the splitting map for Sq1/2(M,N)ρ. We prove the splitting map for Sq1/2(M,N)ρ is injective if there exists at least one component of N such that this component and the boundary of the splitting disk belong to the same component of ∂M. We also prove the representation-reduced stated skein module of the marked handlebody is an irreducible Azumaya representation of the stated skein algebra of its boundary.Let M be an oriented connected closed three manifold. For any positive integer k, we use Mk to denote the marked three manifold obtained from M by removing k open three dimensional balls and adding one marking to each newly created sphere boundary component. We prove dimCSq1/2(Mk)ρ=1 for any maximal ideal ρ of S1(Mk).

The distribution of polynomials in monotone-independent elements

Building on the work of Arizmendi and Celestino (2021), we derive the ∗-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers [Formula: see text] and [Formula: see text], we derive explicitly the ∗-distribution of [Formula: see text] whenever [Formula: see text] and [Formula: see text] are monotone or infinitesimally monotone independent elements. This encompasses both cases of the commutator and anti-commutator. This approach can be pushed to study more general polynomials. As applications, we derive the limiting distribution with respect to the partial trace of polynomials in a certain class of random matrices.

Semi-c-periodicity, c-uniform recurrence and almost automorphy in the complex plane

This paper is devoted to develop the concepts of semi-c-periodicity, c-uniform recurrence and almost automorphy for functions defined on vertical strips in the complex plane, where c is a non-zero complex number. As an extension of the study performed for functions defined on the real axis, this work aims to investigate the main properties of these classes of functions and establish their connections with the more known class of c-almost periodic functions defined on vertical strips. In fact, we resolve an open problem which was raised in 2020 for the real case. Additionally, we also use this approach to introduce an asymptotic version of these new classes of functions.

Completeness of the systems of Bessel functions of index $-5/2$

Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)\rightarrow\mathbb C$, satisfying $\int_{0}^1 t^4 |f(t)|^2\, dt<+\infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $\big\{\rho_k^2\sqrt{x\rho_k}J_{-5/2}(x\rho_k):k\in\mathbb N\big\}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $\sigma\le 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.

ROUGH LACUNARY STATISTICAL ANALYSIS OF ORDER OF η OF TRIPLE DIFFERENCE SEQUENCE SPACES

We generalized the concepts in the probability of rough lacunary statistical analysis of order η by introducing the difference operator Δ_γ^α of fractional order, where α is a proper fraction and γ=(γ_mnk ) is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving lacunary sequence θ and arbitrary sequence p=(p_rst ) of strictly positive real numbers and investigate the topological structures of related with triple difference sequence spaces. The main focus of the present paper is to generalize rough lacunary statistical analysis of order η of triple difference sequence spaces and give the relations between statistical analysis and lacunary statistical analysis of order η.

The Stability Region for Schur Stable Trinomials with General Complex Coefficients

AbstractIn this paper, we characterize the stability region for trinomials of the form $$f(\zeta ):=a\zeta ^n + b\zeta ^m +c$$ f ( ζ ) : = a ζ n + b ζ m + c , $$\zeta \in \mathbb {C}$$ ζ ∈ C , where a, b and c are non-zero complex numbers and $$n,m\in \mathbb {N}$$ n , m ∈ N with $$n>m$$ n > m . More precisely, we provide necessary and sufficient conditions on the coefficients a, b and c in order that all the roots of the trinomial f belongs to the open unit disc in the complex plane. The proof is based on Bohl’s Theorem (Bohl in Math Ann 65(4):556–566, 1908) introduced in 1908.

On a result of Bao-Qin Li concerning Dirichlet series and shared values

Li (2011) proved that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) satisfy the same functional equation with \(a(1)=1\) and \(L_1^{-1}(c_{j})=L_2^{-1}(c_{j})\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_{1}\) and \(c_{2}\), then \(L_{1}=L_{2}\). Later on, Gonek--Haan--Ki (2014) proved that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal {S}}^{\sharp}\) have the positive degrees and \(L_1^{-1}(c)=L_2^{-1}(c)\) for a finite non-zero complex number \(c\), then \(L_1=L_2\). This implies that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal {S}}^{\sharp}\) have the positive degrees and \(L_1^{-1}(c_j)=L_2^{-1}(c_j)\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_1\) and \(c_2\), then \(L_1=L_2\). In this paper, we prove that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) have the zero degrees and satisfy \(L_1^{-1}(c_{j})=L_2^{-1}(c_{j})\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_{1}\) and \(c_{2}\), and if \(a_1(1)=a_2(1)\) or \(\lim_{r\rightarrow+\infty}\frac{T(r,L_2)}{T(r,L_1)}=1\), then \(L_{1}= L_{2}\). The main results obtained in this paper improve Theorem 1 from Li (2011) when the L-functions in the extended Selberg class \({\mathcal {S}}^{\sharp}\) have the zero degrees. Some examples are provided to show that the results obtained in this paper, in a sense, are best possible.

Analytical study of the pantograph equation using Jacobi theta functions

The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y′(x)=ay(qx)+by(x), where a and b are two non-zero real or complex numbers. When 0<q<1 and y(0)=1, the associated Cauchy problem admits a unique power series solution, ∑n≥0(−a/b;q)nn!(bx)n, that converges in the whole complex x-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a by-product, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.

Twisted regular representations for vertex operator algebras

This paper is to study what we call twisted regular representations for vertex operator algebras. Let V be a vertex operator algebra, let σ1,σ2 be commuting finite-order automorphisms of V and let σ=(σ1σ2)−1. Among the main results, for any σ-twisted V-module W and any nonzero complex number z, we construct a weak σ1⊗σ2-twisted V⊗V-module Dσ1,σ2(z)(W) inside W⁎. Let W1,W2 be σ1-twisted, σ2-twisted V-modules, respectively. We show that P(z)-intertwining maps from W1⊗W2 to W⁎ are the same as homomorphisms of weak σ1⊗σ2-twisted V⊗V-modules from W1⊗W2 into Dσ1,σ2(z)(W). We also show that a P(z)-intertwining map from W1⊗W2 to W⁎ is equivalent to an intertwining operator of type (W′W1W2), which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each τ-twisted V-module M with τ any finite-order automorphism of V, the coefficients of the q-graded trace function lie in Dτ,τ−1(−1)(V) and generate a τ⊗τ−1-twisted V⊗V-submodule isomorphic to M⊗M′.

Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $\Phi$. A subset $D$ of the set $\Phi^+$ of positive roots is called a rook placement if it consists of roots with pairwise non-positive scalar products. To each rook placement $D$ and each map $\xi$ from $D$ to the set $\mathbb{C}^{\times}$ of nonzero complex numbers one can naturally assign the coadjoint orbit $\Omega_{D,\xi}$ in the dual space $\mathfrak{n}^*$. By definition, $\Omega_{D,\xi}$ is the orbit of $f_{D,\xi}$, where $f_{D,\xi}$ is the sum of root covectors $e_{\alpha}^*$ multiplied by $\xi(\alpha)$, $\alpha\in D$. (In fact, almost all coadjoint orbits studied at the moment have such a form for certain $D$ and $\xi$.) It follows from the results of Andr\`e that if $\xi_1$ and $\xi_2$ are distinct maps from $D$ to $\mathbb{C}^{\times}$ then $\Omega_{D,\xi_1}$ and $\Omega_{D,\xi_2}$ do not coincide for classical root systems $\Phi$. We prove that this is true if $\Phi$ is of type $G_2$, or if $\Phi$ is of type $F_4$ and $D$ is orthogonal.

On geometric circulant matrices with geometric sequence

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is ( g , gq , g q 2 , … , g q n − 1 ) , where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples.

Equivariant spectral triple for the quantum group Uq(2) for complex deformation parameters

Let q=|q|eiπθ be a nonzero complex number such that |q|≠1 and consider the compact quantum group Uq(2). For θ∉Q∖{0,1}, we obtain the K-theory of the underlying C⁎-algebra C(Uq(2)). We construct a spectral triple on Uq(2) which is equivariant under its own comultiplication action. The spectral triple obtained here is even, 4+-summable, non-degenerate, and the Dirac operator acts on two copies of the L2-space of Uq(2). The K-homology class of the associated Fredholm module is shown to be nontrivial.

BESSEL POLYNOMIALS AND SOME CONNECTION FORMULAS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS

In this paper, we introduce the concept of the \(\mathbb{B}_{\alpha}\)-classical orthogonal polynomials, where \(\mathbb{B}_{\alpha}\) is the raising operator \(\mathbb{B}_{\alpha}:=x^2 \cdot {d}/{dx}+\big(2(\alpha-1)x+1\big)\mathbb{I}\), with nonzero complex number \(\alpha\) and \(\mathbb{I}\) representing the identity operator. We show that the Bessel polynomials \(B^{(\alpha)}_n(x),\ n\geq0\), where \(\alpha\neq-{m}/{2}, \ m\geq -2, \ m\in \mathbb{Z}\), are the only \(\mathbb{B}_{\alpha}\)-classical orthogonal polynomials. As an application, we present some new formulas for polynomial solution.

An Identity Theorem for the Fourier–Laplace Transform of Polytopes on Nonzero Complex Multiples of Rationally Parameterizable Hypersurfaces

A set mathcal {S} of points in mathbb {R}^n is called a rationally parameterizable hypersurface if there is vector function varvec{sigma }:mathbb {R}^{n-1}rightarrow mathbb {R}^n having as components rational functions defined on some common domain D such that mathcal {S}={varvec{sigma }(textbf{t}):textbf{t}in D}. A generalized n-dimensional polytope in mathbb {R}^n is a union of a finite number of convex n-dimensional polytopes in mathbb {R}^n. The Fourier–Laplace transform of such a generalized polytope mathcal {P} in mathbb {R}^n is defined by F_{mathcal {P}}(textbf{z})=int _{mathcal {P}}e^{textbf{z}cdot textbf{x}},textbf{dx}. Let gamma be a fixed nonzero complex number. We prove that F_{mathcal {P}_1}(gamma varvec{sigma }(textbf{t}))=F_{mathcal {P}_2}(gamma varvec{sigma }(textbf{t})) for all textbf{t} in O implies mathcal {P}_1=mathcal {P}_2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of mathcal {P}. Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres.

Classification of pivotal tensor categories with fusion rules related to SO(4)

In this paper we classify all semisimple pivotal tensor categories with the same fusion rules as Rep(SO(4)), or one of the associated truncations. We show that such categories are explicitly classified by two non-zero complex numbers. Furthermore we show these tensor categories are always braided, and aside from a small number of degenerate cases, there exist exactly 8 braidings.

Nonlinear mappings preserving Jordan-type η-*-products

Let η be a nonzero complex number. Let 𝒜 and ℬ be two von Neumann algebras, one of which has no central abelian projections, and let Φ:𝒜→ℬ be not necessarily a linear bijection. For arbitrary elements A,B∈𝒜, one can define their Jordan η-∗-product in the sense of A◇ηB=AB+ηBA∗. Let pn(X1,X2,…,Xn) be the polynomial defined by n indeterminates X1,…,Xn and their Jordan multiple η-∗-product. In this article, it is shown that Φ satisfies the condition Φ(pn(A1,A2,…,An))=pn(Φ(A1),Φ(A2),…,Φ(An)) for all A1,A2,…,An∈𝒜 if and only if one of the following statements holds true: (1)η∈ℝ and there exists a central projection E∈𝒜 such that Φ(E) is a central projection in ℬ, Φ|𝒜E:𝒜E→ℬΦ(E) is a linear ∗-isomorphism and Φ|𝒜(I−E):𝒜(I−E)→ℬ(I−Φ(E)) is a conjugate linear ∗-isomorphism, (2)η∉ℝ and Φ is a linear ∗-isomorphism.

On the number of roots for harmonic trinomials

In this manuscript we study the counting problem for harmonic trinomials of the form aζn+bζ‾m+c, where n,m∈N, n>m, and a, b and c are non-zero complex numbers. As a consequence, we obtain the Fundamental Theorem of Algebra and the Wilmshurst conjecture for harmonic trinomials. The proof of the counting problem relies on the Bohl method introduced in [1].

Hyers–Ulam Stability Results for a Functional Inequality of s , t -Type in Banach Spaces

We introduce an additive s , t -functional inequality where s and t are nonzero complex numbers with 2 s + t &lt; 1 . Using the direct method and the fixed point method, we give the Hyers–Ulam stability of such functional inequality in Banach spaces.

Properties of trinomials of height at least 2

We consider trinomials of the form zn+azm+b, where 0<m<n are relatively prime integers and a and b are nonzero complex numbers. Typically (but not exclusively), a will be an integer with |a|≥2, while b=±1. Our main results cover the irreducibility, Mahler measure and house of such trinomials.