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
In this paper, we have studied the problem of minimizing the ratio of two indefinite quadratic functions subject to a strictly convex quadratic constraint. First utilizing the relationship between fractional and parametric programming problems due to Dinkelbach, we reformulate the fractional problem as a univariate equation. To find the root of the univariate equation, the generalized Newton method is utilized that requires solving a nonconvex quadratic optimization problem at each iteration. A key difficulty with this problem is its nonconvexity. Using Lagrange duality, we show that this problem can be solved by solving a convex univariate minimization problem. Attainment of the global optimality conditions is discussed. Our preliminary numerical experiments on several randomly generated test problems show that, the new approach is much faster in finding the global optimal solution than the known semidefinite relaxation approach, especially when solving large scale problems.
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
A multi-server M/M/n-type queueing system with a bounded total volume and finite queue size is considered. An AQM algorithm with the "accepting'' function is being used to control the arrival process of incoming packets. The stationary queue-size distribution and the loss probability are derived. Numerical examples illustrating theoretical results are attached as well.
Let L(H) denote the algebra of operators on a complex infinite dimensional Hilbert space H. For A, B ∈ L(H), the generalized derivation δ A,B and the elementary operator δ A,B are defined by δ A,B(X)=AX-XB and δ A,B}(X)=AXB-X for all X\in L(H). In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of δ A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of Δ A,B with respect to the wider class of unitarily invariant norms on L(H).
A conjecture due to Honda predicts that given any abelian variety over a number field $K$, all of its quadratic twists (or twists of a fixed order in general) have bounded Mordell-Weil rank. About 15 years ago, Rubin and Silverberg obtained an analytic criterion for Honda's conjecture for a family of quadratic twists of an elliptic curve defined over the field of rational numbers. In this paper, we consider this problem over number fields. We will prove that the existence of a uniform upper bound for the ranks of elliptic curves in this family is equivalent to the convergence of a certain infinite series. This result extends the work of Rubin and Silverberg over $\mathbb Q$.
The main objective of this paper is to study the boundedness character, the periodic character, the convergence and the global stability of positive solutions of the difference equation xn+1 = ( A + ∑ k i=0 αixn−i)
⁄ ∑ k i=0 βixn−i , n = 0, 1, 2, . . . where the coefficients A, αi , βi and the initial conditions x−k, x−k+1, . . . , x−1, x0 are positive real numbers, while k is a positive integer number.
The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability oThe main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation xn+1 = α0xn + α1xn−l + α2xn−k ⁄ β0xn + β1xn−l + β2xn−k , n = 0, 1, 2, . . . where the coefficients αi , βi ∈ (0,∞) for i = 0, 1, 2, and l, k are positive integers. The initial conditions x−k, . . . , x−l , . . . , x−1, x0 are arbitrary positive real numbers such that l < k. Some numerical experiments are presented.