This paper considers random processes of the form $$X_{n+1}=a_nX_n+b_n\pmod p$$ where p is odd, $$X_0=0$$ , $$(a_0,b_0), (a_1,b_1), (a_2,b_2),\ldots $$ are i.i.d., and $$a_n$$ $$b_n$$ independent with $$P(a_n=2)=P(a_n=(p+1)/2)=1/2$$ $$P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$$ . can be viewed as a multiplicatively symmetrized version process Chung, Diaconis, Graham. shows that order $$(\log p)^2$$ steps...

Abstract Given a random process which undergoes stochastic resetting at constant rate r to position drawn from distribution ${\cal P}(x)$ , we consider sequence of dynamical observables $A_1, \ldots, A_n$ associated the intervals between events. We calculate exactly probabilities various events related this sequence: that last element is larger than all previous ones, monot...

The famous linear diophantine problem of Frobenius is the to determine largest integer (Frobenius number) whose number representations in terms $a_1,\dots,a_k$ at most zero, that not representable. In other words, all integers greater than this can be represented for least one way. One natural generalizations find (generalized a given nonnegative $p$. It easy explicit form case two variables. H...

let $r$ be a commutative ring with identity‎. ‎a proper ideal $p$ of $r$ is a $(n-1,n)$-$phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin r$‎, ‎$a_1cdots a_nin pbackslash p^m$ ($a_1cdots a_nin pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin p$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎in this paper several results concerning $(n-1,n)$-$phi_m$-prime and $(n-1,n)$-...

For fixed non-negative integers $k$, $t$, and $n$, with $t < k$, a $k_t$-Dyck path of length $(k+1)n$ is lattice that starts at $(0, 0)$, ends $((k+1)n, stays weakly above the line $y = -t$, consists steps from step-set $\{(1, 1), (1, -k)\}$. We enumerate family paths by considering number down-steps height $i$ modulo $k$. Given tuple $(a_1, a_2, \ldots, a_k)$ we find an exact enumeration fo...

In this paper, we study the root distributions of Ehrhart polynomials free sums certain reflexive polytopes. We investigate cases where roots $A_d^\vee$'s or $A_d$'s lie on canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ complex plane $\mathbb{C}$, $A_d$ denotes polytope type A dimension $d$ and $A_d^\vee$ its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} ...

We propose a Nekrasov-type formula for the instanton partition functions of four-dimensional N=2 U(2) gauge theories coupled to (A_1,D_{2n}) Argyres-Douglas theories. This is carried out by extending generalized AGT correspondence case group, which requires us define irregular states direct sum Virasoro and Heisenberg algebras. Using our formula, one can evaluate contribution theory at each fix...

Let $$n\ne 0$$ be an integer. A set of m distinct positive integers $$\{a_1,a_2,\ldots ,a_m\}$$ is called a D(n)-m-tuple if $$a_ia_j + n$$ perfect square for all $$1\le i < j \le m$$ . k In this paper, we prove that $$\{k,k+1,c,d\}$$ $$D(-k)$$ -quadruple with $$c>1$$ , then $$d=1$$ The proof relies not only on standard methods in field (Baker’s linear forms logarithms and the hypergeometric met...

This paper is devoted to the multivariable $$H^\infty $$ functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if $$(A_1,\ldots , A_d)$$ such family, $$A_k$$ R-sectorial R-type $$\omega _k\in (0,\pi )$$ $$k=1,\ldots ,d$$ and admits bounded (\Sigma _{\theta _1}\times \cdots \times \Sigma _d})$$ joint for some $$\theta (\omega _k...

Twisted generalized Weyl algebras (TGWAs) are a large family of that includes several interest for ring theory and representation theory, such as algebras, primitive quotients $U(\mathfrak{sl}_2)$, multiparameter quantized algebras. In this work, we study invariants TGWAs under diagonal automorphisms. Under certain conditions, able to show the fixed TGWA by an automorphism is again TGWA. partic...