On a division property of consecutive integers

Pillai and Brauer proved that form≧17 we can find blocksB _m ofm consecutive integers such that no element in the block is pairwise prime with each of the other elements. The following basic generalization is proved: For eachd>1 there is a numberG(d) such that for everym≧G(d) there exist infinitely many blocksB _m ofm consecutive integers, such that for eachr∈B _m there existss∈B _m , (r,s)≧d.     

Maximum a posteriori decision and evaluation of class probabilitiesby Boltzmann perceptron classifiers

It is shown that neural network architectures may offer a valuable alternative to the Bayesian classifier. With neural networks, the a posteriori probabilities are computed with no a priori assumptions about the probability distribution functions (PDFs) that generate the data. Rather than assuming certain types of PDFs for the input data, the neural classifier uses a general type of input-output mapping which is then designed to optimally comply with a given set of examples called the training set. It is demonstrated that the a posteriori class probabilities can be efficiently computed by a deterministic feedforward network which is called the Boltzmann perceptron classifier (BPC). Maximum a posteriori (MAP) classifiers are also constructed as a special case of the BPC. Structural relationships between the BPC and a conventional multilayer perceptron (MLP) are given, and it is demonstrated that rather intricate boundaries between classes can be formed even with a relatively modest number of network units. Simulation results show that the BPC is comparable in performance to a Bayesian classifier     

Volume in General Metric Spaces

A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be “faithfully preserved” by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of \(k\) points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set \(S = \{v_1, \ldots , v_k \}\) of points in a general metric space: the product of the distances from \(v_i\) to \(\{ v_1, \dots , v_{i-1} \}\) , normalized by \(\tfrac{1}{(k-1)!}\) , where the ordering of the points is that given by Prim’s minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of \(S\) into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige’s definition: the use of any other order in the product affects volume \(^{1/(k-1)}\) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embedding.     

Quasinormal Modes for Schwarzschild–AdS Black Holes: Exponential Convergence to the Real Axis

We study quasinormal modes for massive scalar fields in Schwarzschild–anti-de Sitter black holes. When the mass-squared is above the Breitenlohner–Freedman bound, we show that for large angular momenta, ?, there exist quasinormal modes with imaginary parts of size exp(??/C). We provide an asymptotic expansion for the real parts of the modes closest to the real axis and identify the vanishing of certain coefficients depending on the dimension.     

CONTROLLED POLLUTION BUFFER ZONE – AN APPROACH TO WASTE MANAGEMENT

We consider pollution that arises from the leakage of a container into a contiguous water region. It is shown that if one considers this to be an “effective diffusion” process, the creation of a “Controlled Pollution Buffer Zone” significantly decreases the degradation of the water quality in the adjacent region.     

An interior points algorithm for the convex feasibility problem

The problem of finding a point in the intersection of a finite family of convex sets in the Euclidean space R″ is considered here. We present a general algorithmic scheme which employs projections onto separating hyperplanes instead of projections onto the convex sets. This scheme includes the method of successive projections of Gubin et al., USSR Comp. Math. and Math. Phys. 7 (1967), 1–24, as a special case. A different realization proposed here is capable of handling the problem when the sets are solid and an interior point of each set is available. This alternative algorithm may, in certain cases, be more attractive than the method of Gubin et al.     

Automorphism Groups of Trees Acting Locally with Affine Permutations

We consider the structure of groups that act on a p ⁿ -regular tree in a vertex transitive way with the local action (i.e. the action of the vertex stabilizer on the link) isomorphic to the group of affine transformations on a finite affine line.     

Explicit formulas for matrix-valued Padé approximants

Formulas are derived for calculating the [N + m, N] Padé approximants to an operator-valued function from the first 2N + m + 1 expansion coefficients of the perturbation series for this function.     

Analysis of a Static VAR Compensator with Optimal Energy Storage Element

This paper analyzes a single-phase static VAR compensator, based on the current source-forced commutated inverter. Emphasis is given to the subject of energy storage in the circuit.     

Flexible waveguides for Er-YAG laser radiation delivery

Flexible plastic waveguides (FPW) were devised for the delivery of Er-YAG laser radiation. The FPW characteristics were studied under various conditions. In vitro studies were carried out to explore the drilling procedure on extracted teeth and the FPW-tissue mutual effects. The results which were obtained proved that the FPW as a delivery device might be a substitute hand applicator for the pneumatic turbine for drilling in teeth