Theorem 1. The convolution ∗ admits a commutativity constraint making Sph into a rigid tensor category. There exists a faithful, exact tensor “fiber” functor Sat : Sph → Vect inducing an equivalence (modulo a sign in the commutativity constraint) of Sph with Rep(G) as tensor categories, where G is the Langlands dual group of the reductive group G, whose weights are the coweights of G and vice v...

We obtain minimal shape-preserving projections onto the n-th degree algebraic polynomials via a geometric approach.

In this work we consider a given root of family n-degree polynomials as one-variable function that depends only on the independent term. Then prove satisfies several ordinary differential equations (ODE). More concretely, it simple separated variables ODE, first order generalized Abel ODE degree n−1 and an (n−1)-th linear ODE. Although some our results are not new, approach is self-contained. F...

‎we consider two commutativity ratios $pr(g)$ and $f(g)$ in a finite group $g$‎ ‎and examine the properties of $g$ when these ratios are `large'‎. ‎we show that‎ ‎if $pr(g) > frac{7}{24}$‎, ‎then $g$ is metabelian and we give threshold‎ ‎results in the cases where $g$ is insoluble and $g'$ is nilpotent‎. ‎we also‎ ‎show that if $f(g) > frac{1}{2}$‎, ‎then $f(g) = frac{n+1}{2n}$‎, ‎for some‎ ‎na...

Let $G = (V, E)$ be a simple graph. Denote by $D(G)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$  and  $A(G)$ the adjacency matrix of $G$. The  signless Laplacianmatrix of $G$ is $Q(G) = D(G) + A(G)$ and the $k-$th signless Laplacian spectral moment of  graph $G$ is defined as $T_k(G)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...

Let q be a power of an odd prime and let k, n ∈ N such that 1 < k ≤ n. We investigate the existence of self-reciprocal irreducible monic polynomials over Fq, of degree 2n and their k-th coefficient prescribed.

Let be a d-dimensional simplex with vertices v 0 ; ; v d and B n (f;) denote the n th degree Bernstein polynomial of a continuous function f on. Dahmen and Micchelli 4] proved that B n (f;) B n+1 (f;); n 2 N; for any convex function f on , and it is clear that a necessary and suucient condition for the inequality to become an identity for all n 2 N is that f is an aane polynomial. Let m be the ...

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space Pn−1. In this paper, we achieve Serre’s conjecture in the special case of smooth cyclic covers of any degree when n ≥ 10, and surpass it for covers of degree r ≥ 3 when n > 10. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsin...