Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality $\|p'\|_{[-1,1]}\leq\frac12\|p\|_{[-1,1]}$ for a constrained polynomial $p$ of degree at most $n$, initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval $(-1,1)$ and establish a new asymptotically sharp inequality., Lai-Yi Zhu, Da-Peng Zhou., and Obsahuje bibliografii
We examine the q-Pell sequences and their applications to weighted partition theorems and values of L-functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous authors noted in the introduction. Showing that simple combinatorial manipulations give rise to an identity published by the present author, a weighted form of a Lebesgue partition theorem is given as the main application to partitions. The conclusion of the paper summarizes some directions for further research, pointing out that certain conditions on the q-polynomial would be desired, and also possibly looking at the operator equation in the present paper from the position of using modular forms., Alexander E. Patkowski., and Obsahuje bibliografii
We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P. M. Sołtan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite noncommutative (NC) space. As special cases three classes of NC objects are introduced: quantum group of gauge transformations, Pontryagin dual of a quantum group, and Galois-Hopf-algebra of an algebra extension., Maysam Maysami Sadr., and Obsahuje bibliografii
In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators M+ and M−. More precisely, we prove that M+ and M− map W¹,p(R) → W1,p(R) with 1 < p < nekonečno, boundedly and continuously. In addition, we show that the discrete versions M+ and M− map BV(ℤ) → BV(ℤ) boundedly and map l¹(ℤ) → BV(ℤ) continuously. Specially, we obtain the sharp variation inequalities of M+ and M−, that is Var(M+(f))<Var(f) and Var(M−(f))<Var(f) if f ∈ BV(ℤ), where Var(f) is the total variation of f on ℤ and BV(ℤ) is the set of all functions f: ℤ → R satisfying Var(f) < nekonečno., Feng Liu, Suzhen Mao., and Obsahuje bibliografii
Let K be a field and S = K[x1, ..., xm, y1,..., yn] be the standard bigraded polynomial ring over K. In this paper, we explicitly describe the structure of finitely generated bigraded “sequentially Cohen-Macaulay” S-modules with respect to Q = (y1, ..., yn). Next, we give a characterization of sequentially Cohen-Macaulay modules with respect to Q in terms of local cohomology modules. Cohen-Macaulay modules that are sequentially Cohen-Macaulay with respect to Q are considered., Leila Parsaei Majd, Ahad Rahimi., and Obsahuje seznam literatury
For any two positive integers n and k\geqslant 2, let G(n, k) be a digraph whose set of vertices is {0, 1, ..., n − 1} and such that there is a directed edge from a vertex a to a vertex b if ak ≡ b (mod n). Let n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} be the prime factorization of n. Let P be the set of all primes dividing n and let P_{1},P_{2} \subseteq P be such that P_{1\cup P_{2}}= P and P_{1\cup P_{2}}=\emptyset . A fundamental constituent of G(n, k), denoted by G_{{P_2}}^*(n,k), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \prod\nolimits_{{p_i} \in {P_2}} {{p_i}} and are relatively prime to all primes q\in P_{1}. L. Somer and M. Křižek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic., Amplify Sawkmie, Madan Mohan Singh., and Obsahuje seznam literatury
Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when it is homotopy equivalent to a sphere., Nela Milošević, Zoran Z. Petrović., and Obsahuje seznam literatury
The packing constant is an important and interesting geometric parameter of Banach spaces. Inspired by the packing constant for Orlicz sequence spaces, the main purpose of this paper is calculating the Kottman constant and the packing constant of the Cesàro-Orlicz sequence spaces ({\text{ce}}{{\text{s}}_\varphi }) defined by an Orlicz function φ equipped with the Luxemburg norm. In order to compute the constants, the paper gives two formulas. On the base of these formulas one can easily obtain the packing constant for the Cesàro sequence space cesp and some other sequence spaces. Finally, a new constant \widetilde D (X), which seems to be relevant to the packing constant, is given., Zhen-Hua Ma, Li-Ning Jiang, Qiao-Ling Xin., and Obsahuje seznam literatury
In this paper we study parallel and totally geodesic hypersurfaces of two-step homogeneous nilmanifolds of dimension five. We give the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces. Moreover, we prove that two-step homogeneous nilmanifolds of dimension five which have one-dimensional centre never admit parallel hypersurfaces. Also we prove that the only two-step homogeneous nilmanifolds of dimension five which admit totally geodesic hypersurfaces have three-dimensional centre., Mehri Nasehi., and Obsahuje seznam literatury
Let f: X → X be a continuous map with the specification property on a compact metric space X. We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller and locally maximal hyperbolic set., Jinjun Li, Min Wu., and Obsahuje seznam literatury