Orbits and Lattices for Linear Random Number Generators with Composite Moduli

In order to analyze certain types of combinations of multiple recursive linear congruential generators (MRGs), we introduce a generalized spectral test. We show how to apply the test in large dimensions by a recursive procedure based on the fact that such combinations are subgenerators of other MRGs with composite moduli. We illustrate this with the well-known RANMAR generator. We also design an algorithm generalizing the procedure to arbitrary random number generators.

     

Birational automorphisms of quartic Hessian surfaces

We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice of rank 26 and signature . The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kondo. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.

     

Graphs of zeros of analytic families

Let be a family of holomorphic functions in a domain depending holomorphically on . We study the distribution of zeros of in a subdomain whose boundary is a closed non-singular analytic curve. As an application, we obtain several results about distributions of zeros of families of generalized exponential polynomials and displacement maps related to certain ODE's.

     

The Whitney embedding theorem for tropical torsion modules: Classification of tropical modules

We prove here a tropical version of the well-known Whitney embedding theorem [32] stating that a smooth connected m-dimensional compact differential manifold can be embedded into R^2m+1.The tropical version of this theorem states that a tropical torsion module with m generators can always be embedded into the free tropical module , where p (equals to 2 for m=2, and otherwise) is the number of rows supporting the torsion, when the generators are given by the (independent) columns of a matrix of size n×m.As a corollary, we get that tropical m-dimensional torsion modules are classified by a (m-1)(m(m-1)-1)-parameter family.     

Lattice computations for random numbers

We improve on a lattice algorithm of Tezuka for the computation of the -distribution of a class of random number generators based on finite fields. We show how this is applied to the problem of constructing, for such generators, an output mapping yielding optimal -distribution.

     

Dynamical approach to some problems in integral geometry

As is well known, the main problem in integral geometry is to reconstruct a function in a given domain , where its integrals over a family of subdomains in are known. Such a problem is interesting not only as an object of pure analysis, but also in connection with various applications in practical disciplines. The most remarkable example of such a connection is the Radon problem and tomography. In this paper we solve one of these problems when is a bounded domain in with a piecewise smooth boundary. Some intermediate results related to dynamical systems with two generators and to some functional-integral equations are new and interesting per se. As an application of the results obtained we briefly study a boundary problem for a general third order hyperbolic partial differential equation in a bounded domain with data on the whole boundary .

     

Well-posedness of non-autonomous linear evolution equations for generators whose commutators are scalar

We prove the well-posedness of non-autonomous linear evolution equations for generators \({A(t): D(A(t)) \subset X \to X}\) whose pairwise commutators are complex scalars, and in addition, we establish an explicit representation formula for the evolution. We also prove well-posedness in the more general case where instead of the onefold commutators only the p-fold commutators of the operators A(t) are complex scalars. All these results are furnished with rather mild stability and regularity assumptions: Indeed, stability in X and strong continuity conditions are sufficient. Additionally, we improve a well-posedness result of Kato for group generators A(t) by showing that the original norm continuity condition can be relaxed to strong continuity. Applications include Segal field operators and Schrödinger operators for particles in external electric fields.     

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.

     

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Let \(A \in \mathbb {R}^{d \times d}\), \(d \ge 1\) be a dilation matrix with integer entries and \(| \det A|=2\). We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is \(2^d+d\) and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case \(d=2\) and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.     

Analysis and synthesis of Ultra-uniform Pseudorandom Number Generators

The Wichmann–Hill algorithm is a high-performance generatorof uniformly distributed pseudorandom numbers, designed foruse on, and portability between, 8-bit of 16-bit machines. Twoanalyses (one number-theoretic, the other probability-theoretic)are presented in order to explain its superb performance. Itis shown that the original Wichmann–Hill configurationcan be regarded as a single linear congruential generator withunrealizably large multiplier and modulus decomposed into threerealizable subgenerators. This provides an obvious insight intothe source of the generator's high quality, but more importantlypermits, for the first time, the application of the extremelystringent Coveyou-MacPherson spectral test—which is passedwith flying colours. The techniques used for analysis have also been applied to designand test a large family of three-component generalized Wichmann–Hill-typegenerators with substantially the same very high performanceas the original. Over one hundred such generators have beenfound. There is no difficulty in extending the design to configurationssuitable for 32-bit machines, with some improvement in the quality.Increasing the number of subgenerators produces a more dramaticenhancement: this is illustrated by means of an example employingfour components.     

Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string

We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on a complex parameter . Corresponding to different values of , the problem describes either vibrations of a finite string or propagation of elastic waves on an infinite string. Our main object of interest is the family of non-selfadjoint operators in the energy space of two-component initial data. These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems. We show that the operators are dissipative, simple, maximal operators, which differ from each other by rank-one perturbations. We also prove that the operator coincides with the generator of the Lax-Phillips semigroup, which plays an important role in the aforementioned scattering problem. The results of this work are applied in our two forthcoming papers both to the proof of the Riesz basis property of the eigenvectors and associated vectors of the operators and to establishing the exact and approximate controllability of the system governed by the damped wave equation.

     

Cayley graph expanders and groups of finite width

We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equal to 2) and our second family is given as Cayley graphs of finite groups with very short presentations with only two generators and four relations. Both families are based on particular finite quotients of a group G of infinite upper triangular matrices over the ring .We present explicit vector space bases for the finite abelian quotients of the lower exponent-2 groups of G by upper triangular subgroups and prove a particular 3-periodicity of these quotients. We also conjecture that the group G has finite width 3 and finite average width 8/3.     

Projective curves with maximal regularity and applications to syzygies and surfaces

We first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves ${{\mathcal C}\subset \mathbb P^r_K}$ of maximal regularity with ${{\rm deg}\, {\mathcal C}\leq 2r -3}$ . In particular we specify the number and degrees of generators of the vanishing ideal of such curves. We apply these results to study surfaces ${X \subset \mathbb P^r_K}$ whose generic hyperplane section is a curve of maximal regularity. We first give a criterion for ??an early descent of the Hartshorne-Rao function?? of such surfaces. We use this criterion to give a lower bound on the degree for a class of these surfaces. Then, we study surfaces ${X \subset\mathbb P^r_K}$ for which ${h^1(\mathbb P^r_K, {\mathcal I}_X(1))}$ takes a value close to the possible maximum deg X ? r +?1. We give a lower bound on the degree of such surfaces. We illustrate our results by a number of examples, computed by means of Singular, which show a rich variety of occuring phenomena.     

The abelian monodromy extension property for families of curves

Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne–Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily–Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties.     

Finding induced trees

Let G=(V(G),E(G)) be a finite connected undirected graph and WV(G) a subset of vertices. We are searching for a subset XV(G) such that WX and the subgraph induced on X is a tree. -completeness results and polynomial time algorithms are given assuming that the cardinality of W is fixed or not. Besides we give complexity results when X has to induce a path or when G is a weighted graph. We also consider problems where the cardinality of X has to be minimized.     

Affine curves with infinitely many integral points

Let be an irreducible affine curve of (geometric) genus 0 defined by a finite family of polynomials having integer coefficients. In this note we give a necessary and sufficient condition for to possess infinitely many integer points, correcting a statement of J. H. Silverman (Theoret. Comput. Sci., 2000).

     

Entire regularizations of strongly continuous groups and products of analytic generators

Using entire regularizations of groups, we give a characterization of their analytic generators which we apply to the study of products of such generators.

     

Properties of subgenerators of regularized semigroups

We introduce two operations , in the set of subgenerators of a given - regularized semigroup and prove that is a complete partially ordered lattice with respect to , and the operator inclusion . Also presented are some other properties and examples for