On a generalization of Schur theorem concerning resultants

Let K be a field and put $${\mathcal {A}}:=\{(i,j,k,m)\in \mathbb {N}^{4}:\;i\le j\;\text{ and }\;m\le k\}$$ A : = { ( i , j , k , m ) ∈ N 4 : i ≤ j and m ≤ k } . For any given $$A\in {\mathcal {A}}$$ A ∈ A we consider the sequence of polynomials $$(r_{A,n}(x))_{n\in \mathbb {N}}$$ ( r A , n ( x ) ) n ∈ N defined by the recurrence $$\begin{aligned} r_{A,n}(x)=f_{n}(x)r_{A,n-1}(x)-v_{n}x^{m}r_{A,n-2}(x),\;n\ge 2, \end{aligned}$$ r A , n ( x ) = f n ( x ) r A , n - 1 ( x ) - v n x m r A , n - 2 ( x ) , n ≥ 2 , where the initial polynomials $$r_{A,0}, r_{A,1}\in K[x]$$ r A , 0 , r A , 1 ∈ K [ x ] are of degree i, j respectively and $$f_{n}\in K[x], n\ge 2$$ f n ∈ K [ x ] , n ≥ 2 , is of degree k with variable coefficients. The aim of the paper is to prove the formula for the resultant $${\text {Res}}(r_{A,n}(x),r_{A,n-1}(x))$$ Res ( r A , n ( x ) , r A , n - 1 ( x ) ) . Our result is an extension of the classical Schur formula which is obtained for $$A=(0,1,1,0)$$ A = ( 0 , 1 , 1 , 0 ) . As an application we get the formula for the resultant $${\text {Res}}(r_{A,n},r_{A,n-2})$$ Res ( r A , n , r A , n - 2 ) , where the sequence $$(r_{A,n})_{n\in \mathbb {N}}$$ ( r A , n ) n ∈ N is the sequence of orthogonal polynomials corresponding to a moment functional which is symmetric.

Study of matrix transformation of uniformly almost surely convergent complex uncertain sequences

In this paper, we introduce the concept of convergence of complex uncertain series. We initiate matrix transformation of complex uncertain sequence and extend the study via linearity and boundedness. In this context, we prove Silverman-Toeplitz theorem and Kojima-Schur theorem considering complex uncertain sequences. Finally, we establish some results on co-regular matrices .

The ranks of central factor and commutator groups

The Schur Theorem says that if G is a group whose center Z(G) has finite index n, then the order of the derived group G′ is finite and bounded by a number depending only on n. In this paper we show that if G is a finite group such that G/Z(G) has rank r, then the rank of G′ is r-bounded. We also show that a similar result holds for a large class of infinite groups.

A LINEAR CONNECTION FOR BOTH SUB-RIEMANNIAN GEOMETRY AND NONHOLONOMIC MECHANICS (I)

We study the geometry of a sub-Riemannian manifold (M, HM, VM, g), where HM and VM are the horizontal and vertical distribution respectively, and g is a Riemannian extension of the Riemannian metric on HM. First, without the assumption that HM and VM are orthogonal, we construct a sub- Riemannian connection ▽ on HM and prove some Bianchi identities for ▽. Then, we introduce the horizontal sectional curvature, prove a Schur theorem for sub-Riemannian geometry and find a class of sub-Riemannian manifolds of constant horizontal curvature. Finally, we define the horizontal Ricci tensor and scalar curvature, and some sub-Riemannian differential operators (gradient, divergence, Laplacian), extending some results from geometry to the sub-Riemannian setting.