Continuous embeddings in harmonic mixed norm spaces on the unit ball in ? n

In this paper continuous embeddings in spaces of harmonic functions with mixed norm on the unit ball in ? ⁿ are established, generalizing some Hardy-Littlewood embeddings for similar spaces of holomorphic functions in the unit disc. Differences in indices between the spaces of harmonic and holomorphic spaces are revealed. As a consequence an analogue of classical Fejér-Riesz inequality is obtained. Embeddings in the special case of Riesz systems are also established.     

Lower estimate of Tandori’s norm

The conventional monotonicity hypothesis on the coefficients given in the sharpest lower estimate of the celebrated Tandori-norm is weakened to locally almost monotonicity assumption.     

Can remembering history from predecessor promote cooperation in the next generation?

People usually think that helping the next generation to remember history can promote cooperation in dilemma games. We show that is not always the case when agents have memory. Agents play with each neighbor by game history and strategies (such as TFT and WSLS), and the next generation inherits good strategies from the predecessor. We analyze the system’s cooperation ratio by comparing the 2 sources of history at the beginning of each generation: (a) inherited from the predecessor; (b) randomly initialized with different cooperation ratio. We find that with unconditional imitation update rule, agents who remember history get lower cooperation ratio than those who randomly initialize the history; while with replicator rule, higher initial cooperation ratio promotes higher final cooperation. We also do additional experiments to investigate the R, ST, P reciprocity and strategies distribution of the systems.     

Does polynomial parallel volume imply convexity?

For a non-empty compact set A^d, d2, and r0, let Ar denote the set of points whose distance from A is r at the most. It is well-known that the volume, V_d(A_r), of A_r is a polynomial of degree d in the parameter r if A is convex. We pursue the reverse question and ask whether A is necessarily convex if V_d(A_r) is a polynomial in r. An affirmative answer is given in dimension d=2, counterexamples are provided for d3. A positive resolution of the question in all dimensions is obtained if the assumption of a polynomial parallel volume is strengthened to the validity of a (polynomial) local Steiner formula. Mathematics Subject Classification (2000):52A38, 28A75, 52A22, 53C65     

Does Boris-SDC conserve phase space volume?

In this paper we investigate the conservation of phase space volume of the Boris-SDC algorithm. This method provides a generic way to extend the standard, second-order accurate Lorentz force integrator commonly used for charged particles in an electric and magnetic field to a high-order method using spectral deferred corrections. For a single particle in a Penning trap and different frequencies of the electric and magnetic fields, we assess the conservation properties of the method by computing the update matrix of one step of Boris-SDC as well as its determinant. We compare the results to the convergence regions and relate them to energy conservation properties of the method. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

How Many Comparisons Does Quicksort Use?

We examine the probability distribution of the number of comparisons needed by the Quicksort sorting algorithm, where probability arises due to the items being in random order. We develop a general class of distributions for the permutation of the items to be sorted which includes the uniform distribution on permutations as a special case. For this general class, the distribution of the number of comparisons is given by the solution of a simple recurrence relation. We provide an exact solution of the recurrence for very small n. We show that the solution can be approximated asymptotically by the solution of a "quadratic" operator equation. We exhibit three numerical solutions to the operator equation for two different distributions on permutations from the general class. We also exhibit numerical solutions for the case in which the algorithm is modified so that arrays are partitioned by the median-of-three selected items rather than by a single selected item.     

Circular operators on minimal norm ideals of ℬ(ℋ)

In this article the notion of circular operator is extended to the Banach space setting. In particular, this property is considered for elementary operators of lengths one and two acting on minimal norm ideals of ?(?). Necessary and sufficient conditions for the circularity of generalized derivations and Lüders operators are also obtained.     

The Bernstein–Orlicz norm and deviation inequalities

We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case.     

Does negative type characterize the round sphere?

We discuss the measure-theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio must be a round sphere was put forward by the author in 2004. We resolve this conjecture in the class of Riemannian symmetric spaces by showing that a Riemannian manifold with symmetry ratio must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres.

     

Does the loss of regularity really appear?

In the theory of weakly hyperbolic equations we have the effect of loss of regularity. The present paper devotes to the study of two problems. On the one hand we describe families of weakly hyperbolic Cauchy problems for which we have no loss of regularity. On the other hand we discuss the question if the loss of derivatives really appears. Copyright © 2008 John Wiley & Sons, Ltd.     

Does inventory pooling improve customer service levels?

We study the risk pooling problem in a two-location inventory system, and analytically demonstrate that pooling improves the Type-I service levels at both locations. We also present an example showing that pooling may reduce the Type-II service levels at both locations.     

Does science also prefer a ternary pattern?

Nothing is as difficult to see as the obvious (B. Malinowski)

Through the importance of the number three in our culture and the strange preference for a ternary pattern of our nature one can perceive how and why number theory degraded to numerology. The strong preference of our minds for simple patterns can be read as the key to understanding not only the development of numerology, but also why scientists were and are prone to give their discoveries or creations a structure that is based on simple patterns and, among them, especially the ternary pattern. If some numbers seem to appear quite often throughout the fabric of reality, like zero, one, two, and three, reality has, nevertheless, been preferentially analysed in a way to enhance its ternary character. The present excursion through some scientific domains where the ternary pattern has been used tries to offer an answer to the fascination that some numbers, and patterns based on them, exerted and exerts, even today, on human minds. Throughout the overall models created by the human mind, be they natural or spiritual, the models based on a ‘unary’ and on a ‘dualistic’ pattern are surely important, but the model based on a ternary pattern ended up assuming an archetypical character, which is quite intriguing, and deserves to be deepened. The paper is a useful resource for projects for students and background material for teachers of mathematics touching, as it does, on biology, chemistry and physics.     

Zeroth order H ∞ norm approximation of multivariable systems

In this paper the H _∞ norm approximation of a given stable, proper, rational transfer function by a constant matrix is considered (Zeroth order H _∞ norm approximation problem). The solution method is based on the observation that the H _∞ norm approximation problem can be put into an allpass imbedding problem.     

Incorporating minimum Frobenius norm models in direct search

The goal of this paper is to show that the use of minimum Frobenius norm quadratic models can improve the performance of direct-search methods. The approach taken here is to maintain the structure of directional direct-search methods, organized around a search and a poll step, and to use the set of previously evaluated points generated during a direct-search run to build the models. The minimization of the models within a trust region provides an enhanced search step. Our numerical results show that such a procedure can lead to a significant improvement of direct search for smooth, piecewise smooth, and noisy problems.     

The norm of Bernstein 21—Adjusted interpolation polynomial operators

In this paper we estimated the norm of Bernstein 2l-ad justed interpolation polynomial operators, and proved that the estimation is sharp.