Distance sets relating to orthogonal exponentials

The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. By proving a shifted version of Erdös-Solymosi’s theorem on the distance sets, we give some grounds on the conjecture. The results obtained here extend the corresponding results of Iosevich and Jaming in a simple manner.     

Explicit and implicit multi‐center integrations

We present truncated expansions of multicenter one‐electron nuclear attraction and two‐electron repulsion integrals over localized basis functions in terms of one‐ and two‐center integrals of “Coulomb,” “exchange,” and “hybrid” type. Two variants are discussed: the “Explicit Multi‐center Integrations” and the “Implicit Multi‐Center Integrations” (abbreviated as “EMCI” and “IMCI”, respectively). While EMCI also deals with individual integrals, the IMCI option is the more appealing one: it enables us to evaluate the entire matrix elements of “Restricted Hartree–Fock”‐type in a very effective and chemically meaningful way. Due to the diatomic nature of our expansions, integrations over “Slater‐Type Orbitals” become well‐feasible, too. © 2012 Wiley Periodicals, Inc.     

A constructive solution to a problem of ranking tournaments

A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erd?s and J. W. Moon. By probabilistic methods, the existence of ?? ?? unrankable” tournaments was proved. On the other hand, they also mentioned the problem of explicit constructions. However, there seems to be only a few of explicit constructions of such tournaments. In this note, we give a construction of many such tournaments by using skew Hadamard difference sets which have been investigated in combinatorial design theory.     

Random and stochastic disturbances on the input flow in chemostat models with wall growth

In this paper, we analyze a chemostat model with wall growth where the input flow is perturbed by two different stochastic processes: the well-known standard Wiener process, which leads into several drawbacks from the biological point of view, and a suitable Ornstein-Uhlenbeck process depending on some parameters which allow us to control the noise to be bounded inside some interval that can be fixed previously by practitioners. Thanks to this last approach, which has already proved to be very realistic when modeling other simplest chemostat models, it will be possible to prove the persistence and coexistence of the species in the model without needing the theory of random dynamical systems and pullback attractors needed when dealing with the Wiener process. This is an advantage since the theoretical framework in this paper is much less complicated and provides us much more information than the other.     

Toward the classification of biangular harmonic frames

Equiangular tight frames (ETFs) and biangular tight frames (BTFs) – sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectively – are useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames – vector sets defined in terms of the characters of finite abelian groups – because they are characterized by combinatorial objects called difference sets.This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties – all three of which are generalizations of difference sets that fall under the umbrella term “bidifference set” – then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.     

Projections of random Cantor sets

Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases.     

Sub-nanometric precision displacement sensing using low mass and rough surfaces as movable targets

A novel speckle interferometric method has been developed to measure lengths and displacements over a distance of 100 μm with a resolution less than 1 nm using rough surfaces as movable targets instead of using conventional cube corners or mirrors. The speckle method is able to use Michelson and new configurations to combine the two scattered beams or fields from rough movable and stationary reference targets. The targets can be attached and can be produced directly onto movable actuator type of mechanisms. The phase information is extracted from the randomly modulated signals by employing an apertured disc system or an alternative optical fibre arrangement. This method exploits the concept of using narrow angle scatterers for economical use of laser light, enhanced reliability and modulation criteria. A prototype low-mass interferometric displacement sensor has been constructed to provide a compact and a flexible system. Apart from the measurement of changes in displacements in force and pressure measuring devices, it can also be used for the assessment of small range scanning probes, stylus profilers and in medical field the measurement of elastic properties of ear drums, etc. The sub-nanometre resolution and displacement values are repeatable to within ±1 nm can be demonstrated over distances of up to 2 mm by employing appropriate actuators and translation devices. The performance has been confirmed by comparison work against a conventional interferometric transducer.     

A bulk arrival queueing model with fuzzy parameters and varying batch sizes

This paper develops a nonlinear programming approach to derive the membership functions of the steady-state performance measures in bulk arrival queueing systems with varying batch sizes, in that the arrival rate and service rate are fuzzy numbers. The basic idea is based on Zadeh’s extension principle. Two pairs of mixed integer nonlinear programs (MINLP) with binary variables are formulated to calculate the upper and lower bounds of the system performance measure at possibility level α. From different values of α, the membership function of the system performance measure is constructed. For practice use, the defuzzification of performance measures is also provided via Yager ranking index. To demonstrate the validity of the proposed method, a numerical example is solved successfully.     

A Krasnosel’skii-type result for planar sets starshaped via orthogonally convex paths

A Krasnosel’skii-type theorem for compact sets that are starshaped via staircase paths may be extended to compact sets that are starshaped via orthogonally convex paths: Let S be a nonempty compact planar set having connected complement. If every two points of S are visible via orthogonally convex paths from a common point of S, then S is starshaped via orthogonally convex paths. Moreover, the associated kernel Ker S has the expected property that every two of its points are joined in Ker S by an orthogonally convex path. If S is an arbitrary nonempty planar set that is starshaped via orthogonally convex paths, then for each component C of Ker S, every two of points of C are joined in C by an orthogonally convex path. Communicated by Imre Bárány     

乘子猜想在n=5n_1时的一些结果

用［１］中的方法，我们得到乘子猜想在ｎ＝５ｎ１情形时的一些新结果，对此情形下的ＭａｃＦａｒｌａｎｄ的结论（见［２］）作了较大的改进．