Related papers: Linear quantum addition rules

This paper deals with strong invariance principles (known also as strong approximation theorems) for sums of the form $\sum_{n=1}^{[Nt]}F\big(X(n),X(2n),...,X(kn), X(q_{k+1}(n)),X(q_{k+2}(n)),..., X(q_\ell(n))\big)$

Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and…

We are dealing in this work with such formal and conceptual extensions of nonrelativistic quantum mechanics (QM) which contain QM with its standard formalism and interpretation as a subtheory. QM is here primarily equivalently reformulated…

Representations of the $s\ell_q(2)$ algebra are constructed in the space of polynomials of real (complex) variable for $q^N=1$. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and…

We investigate the conditions on an integer sequence f(n), n 2 N, with f(1) = 0, such that the sequence q(n), computed recursively via q(n) = q(n - q(n - 1)) + f(n), with q(1) = 1, exists. We prove that f(n + 1) - f(n) in {0,1}, n > 0, is a…

In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f^2 = I. We describe several generalisations of well-known results in the theory of boolean functions, including…

Research on quantum computing has recently gained significant momentum since first physical devices became available. Many quantum algorithms make use of so-called oracles that implement Boolean functions and are queried with highly…

Let $k\leq n$ be two positive integers and $q$ a prime power. The basic question in minimal linear codes is to determine if there exists an $[n,k]_q$ minimal linear code. The first objective of this paper is to present a new sufficient and…

Let $(U_n)_{n=0}^\infty$ and $(V_m)_{m=0}^\infty$ be two linear recurrence sequences. For fixed positive integers $k$ and $\ell$, fixed $k$-tuple $(a_1,\dots,a_k)\in \mathbb{Z}^k$ and fixed $\ell$-tuple $(b_1,\dots,b_\ell)\in…

For a rational number $q$, a rational $D(q)$-$n$-tuple is a set of $n$ distinct nonzero rationals $\{a_1, a_2, \dots, a_n\}$ such that $a_ia_j+q$ is a rational square for all $1 \leqslant i < j \leqslant n$. For every $q$ we find all…

Let $s_{ij}$ represent a tranposition in $S_n$. A polynomial $P$ in $\mathbb{Q}[X_n]$ is said to be $m$-quasiinvariant with respect to $S_n$ if $(x_i-x_j)^{2m+1}$ divides $(1-s_{ij})P$ for all $1 \leq i, j \leq n$. We call the ring…

The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further…

For a two parameter family of Askey-Wilson polynomials, that can be regarded as basic analogues of the Legendre polynomials, an addition formula is derived. The addition formula is a two-parameter extension of Koornwinder's addition formula…

We construct representations of the quantum algebras ~$U_{q{\bf q}}(gl(n))$ and ~$U_{q{\bf q}}(sl(n))$~ which are in duality with the multiparameter quantum groups ~$GL_{q{\bf q}}(n)$, ~$SL_{q{\bf q}}(n)$,~ respectively. These objects…

We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in \cite{fanqin}. We apply the formula to construct some $\mathbb{ZP}$-bases…

A quantum Latin square of order $n$ (denoted as QLS$(n)$) is an $n\times n$ array whose entries are unit column vectors from the $n$-dimensional Hilbert space $\mathcal{H}_n$, such that each row and column forms an orthonormal basis. Two…

For a fixed integer N, and fixed numbers b_1,...,b_N, we consider sequences, the nth term (a_n) of which is the sum of the squares of the terms in the expansion of (b_1 + ... + b_N)^n. In the case all b_i=1, we give a formula for a…

A linear quaternionic equation in one quaternionic variable q is of the form $a_1 q b_1+a_2 q b_2+ ... +a_m q b_m = c$, where the $a_i, b_j, c$ are given quaternionic coefficients. If introducing basis elements $\bf i, j, k$ of pure…

Let $\mathbb{F}_q$ be the finite field of $q$ elements. In this paper we obtain bounds on the following counting problem: given a polynomial $f(x)\in \mathbb{F}_q[x]$ of degree $k+m$ and a non-negative integer $r$, count the number of…

The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…