Three-Way Tiling Sets in Two Dimensions

In this article we show that there exist measurable sets W⊂ℝ² with finite measure that tile ℝ² in a measurable way under the action of a expansive matrix A, an affine Weyl group [(W)\tilde]\widetilde{W} , and a full rank lattice [\varGamma\tilde] ì \mathbbR²\widetilde{\varGamma}\subset\mathbb{R}^{2} . This note is follow-up research to the earlier article “Coxeter groups and wavelet sets” by the first and second authors, and is also relevant to the earlier article “Coxeter groups, wavelets, multiresolution and sampling” by M. Dobrescu and the third author. After writing these two articles, the three authors participated in a workshop at the Banff Center on “Operator methods in fractal analysis, wavelets and dynamical systems,” December 2–7, 2006, organized by O. Bratteli, P. Jorgensen, D. Kribs, G. ólafsson, and S. Silvestrov, and discussed the interrelationships and differences between the articles, and worked on two open problems posed in the Larson-Massopust article. We solved part of Problem 2, including a surprising positive solution to a conjecture that was raised, and we present our results in this article.     

On the security analysis of an image scrambling encryption of pixel bit and its improved scheme based on self-correlation encryption

An image scrambling encryption scheme for pixel bits was presented by Ye [Ye GD. Image scrambling encryption algorithm of pixel bit based on chaos map. Pattern Recognit Lett 2010;31:347-54], which can be seen as one kind of typical binary image scrambling encryption considering from the bit-plain of size M × (8N). However, recently, some defects existing in the original image encryption scheme, i.e., Ye’s scheme, have been observed by Li and Lo [Li CQ, Lo KT. Optimal quantitative cryptanalysis of permutation-only multimedia ciphers against plaintext attacks. Signal Process 2011;91:949-54]. In the attack proposed by Li and Lo at least 3 + ⌈log₂(MN)⌉ plain images of size M × N are used to reveal the permutation matrix W = [w(i, k)] (i ∈ {1, 2, … , M}; k ∈ {1, 2, … , 8N}) which can be applied to recover the exact plain image. In the current paper, at first, one type of special plain image/cipher image is used to analyze the security weakness of the original image scrambling scheme under study. The final encryption vectors TM and TN or the decryption vectors TM′ and TN′ are revealed completely according to our attack. To demonstrate the performance of our attack, a quantified comparison is drawn between our attack and the attack proposed by Li and Lo. Compared with Li and Lo’s attack, our attack is more efficient in the general conditions. In particular, when the sizes of images satisfy the condition M = N or M ? 8N, the number of the used plain images/cipher images is at most 9, which is sharply less than 3 + ⌈log₂(MN)⌉ when M and N are of large size. To overcome the weaknesses of the original scheme, in this paper, an improved image scrambling encryption scheme is proposed. In the improved scheme, the idea of the “self-correlation” method is used to resist the chosen-plaintext attack/known-plaintext attack. The corresponding simulations and analyses illustrate that the improved encryption method has good cryptographic properties, and can overcome the weakness of the original image encryption scheme. Finally, farther improvement is briefly presented for the future work.     

Enumeration of rectangular arrays of zeros and ones

For r=1,2 the rectangular arrays of zeros and ones with r rows and n columns, with m_i zeros and r_i changes in the ith row, and with s_i vertical changes between the ith and (i+1)st rows, i=1,…,r-1, are enumerated. The number of arrays of zeros and ones with 3 rows and n columns, with r_i changes in the ith row, i=1,2,3, and with s_i vertical changes between the ith and (i+1)st rows, i=1,2, is also determined.     

Empirical wavelet analysis of tail and memory properties of LARCH and FIGARCH models

Using computationally efficient wavelet methods, we study two nonlinear models of financial returns {r _t }: linear ARCH (LARCH) and fractionally integrated GARCH (FIGARCH). We estimate the tail index α and the long memory parameter d of the squared returns X_t = r_t²{X_t= r_t^2} of LARCH, and of the powers X _t = |r _t | ^p of FIGARCH. We find that the X _t have infinite variance and long memory, and show how the estimates of α and d depend on the model parameters. These relationships are determined empirically, as the setting is quite complex, and no suitable theory has been developed so far. In particular, we provide empirical relationships between the estimates [^(d)]{\hat d} and the difference parameters in LARCH and FIGARCH. Our computational work uncovers tail and memory properties of LARCH and FIGARCH for practically relevant parameter ranges, and provides some guidance on modeling returns on speculative assets including FX rates, stocks and market indices.     

m-Fold Hypergeometric Solutions of Linear Recurrence Equations Revisited

We present two algorithms to compute m-fold hypergeometric solutions of linear recurrence equations for the classical shift case and for the q-case, respectively. The first is an m-fold generalization and q-generalization of the algorithm by van Hoeij (Appl Algebra Eng Commun Comput 17:83–115, 2005; J. Pure Appl Algebra 139:109–131, 1998) for recurrence equations. The second is a combination of an improved version of the algorithms by Petkovšek (Discrete Math 180:3–22, 1998; J Symb Comput 14(2–3):243–264, 1992) for recurrence and q-recurrence equations and the m-fold algorithm from Petkovšek and Salvy (ISSAC 1993 Proceedings, pp 27–33, 1993) for recurrence equations. We will refer to the classical algorithms as van Hoeij or Petkovšek respectively. To formulate our ideas, we first need to introduce an adapted version of an m-fold Newton polygon and its characteristic polynomials for the classical case and q-case, and to prove the important properties in this case. Using the data from the Newton polygon, we are able to present efficient m-fold versions of the van Hoeij and Petkovšek algorithms for the classical shift case and for the q-case, respectively. Furthermore, we show how one can use the Newton polygon and our characteristic polynomials to conclude for which m ? \mathbbN{m\in \mathbb{N}} there might be an m-fold hypergeometric solution at all. Again by using the information obtained from the Newton polygon, the presentation of the q-Petkovšek algorithm can be simplified and streamlined. Finally, we give timings for the ‘classical’ q-Petkovšek, our q-van Hoeij and our modified q-Petkovšek algorithm on some classes of problems and we present a Maple implementation of the m-fold algorithms for the q-case.     

Traveling wave solutions of Lotka–Volterra type two predators‐one prey model

In this work, we consider a model with one basal resource and two species of predators feeding by the same resource. There are three non‐trivial boundary equilibria. One is the saturated state E_K of the prey without any predator. Other two equilibria, E₁ and E₂, are the coexistence states of the prey with only one species of predators. Using a high‐dimensional shooting method, the Wazewski' principle, we establish the conditions for the existence of traveling wave solutions from E_K to E₂ and from E₁ to E₂. These results show that the advantageous species v₂ always win in the competition and exclude species v₁ eventually. Finally, some numerical simulations are presented, and biological interpretations are given. Copyright © 2016 John Wiley & Sons, Ltd.     

Counting derangements, involutions and unimodal elements in the wreath product Cr ? Sn

We count derangements, involutions and unimodal elements in the wreath product C _r ≀ S _n by the numbers of excedances, fixed points and 2-cycles. Properties of the generating functions, including combinatorial formulas, recurrence relations and real-rootedness are studied. The results obtained specialize to those on the symmetric group S _n and on the hyperoctahedral group B _n when r = 1, 2, respectively.     

Fonctions de partitions à parité périodique

Let be the set of positive integers and a subset of . For , let denote the number of partitions of n with parts in . In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any and , there is a unique set , such that is even for n>N. Soon after, Ben Saïd and Nicolas (Acta Arith. 106 (2003) 183) considered , and proved that for all k≥0, the sequence is periodic on n. In this paper, we generalise the above works for any formal power series f in with f(0)=1, by constructing a set such that the generating function of is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in with P(0)=Q(0)=1, then for all k≥0 the sequence is periodic on n.     

Pointfree Aspects of the Td Axiom of Classical Topology

Abstract

Abstraction from the condition defining To-spaces leads to the following notion in an arbitrary frame L: a filter F in L is called slicing if it is prime and there exist a, b e L such that a £ F, b e F, and a is covered by 6. This paper deals with various aspects of these slicing filters. As a first step, we present several results about the original td condition. Next, concerning slicing filters, we show they are completely prime and characterize them in various ways. In addition, we prove for the frames £>X of open subsets of a space X that every slicing filter is an open neighbourhood filter U(x) and X is td iff every U(x) is slicing. Further, for Top_D and Prm_D the categories of td spaces and their continuous maps, and all frames and those homomorphisms whose associated spectral maps preserve the completely prime elements, respectively, we show that the usual contravariant functors between Top and Frm induce analogous functors here, providing a dual equivalence between Top_D and the subcategory of Prm_D given by the To-spatial frames (not coinciding with the spatial ones). In addition, we show that Top_D is mono-coreflective in a suitable subcategory of Top. Finally, we provide a comparison between Jo-separation and sobriety showing they may be viewed, in some sense, as mirror images of each other.     

Minimal, rigid foliations by curves on ℂℙ n

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂℙ ⁿ for every dimension n≥2 and every degree d≥2. Precisely, we construct a foliation ℱ which is induced by a homogeneous vector field of degree d, has a finite singular set and all the regular leaves are dense in the whole of ℂℙ ⁿ . Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if ℱ is conjugate to another holomorphic foliation by a homeomorphism sufficiently close to the identity, then these foliations are also conjugate by a projective transformation. Finally, all these properties are persistent for small perturbations of ℱ.?This is done by considering pseudo-groups generated on the unit ball 𝔹 ⁿ ⊆ℂ ⁿ by small perturbations of elements in Diff(ℂ ⁿ ,0). Under open conditions on the generators, we prove the existence of many pseudo-flows in their closure (for the C ⁰-topology) acting transitively on the ball. Dynamical features as minimality, ergodicity, positive entropy and rigidity may easily be derived from this approach. Finally, some of these pseudo-groups are realized in the transverse dynamics of polynomial vector fields in ℂℙ ⁿ . Received March 7, 2002 / final version received November 26, 2002?Published online February 7, 2003 Most of this work has been carried out during a visit of the first author to IMPA/RJ and a visit of the second author to the University of Lille 1. We would like to thank these institutes for hospitality and express our gratitude to CNPq-Brazil and CNRS-France for the financial support which made these visits possible. We are also indebted to Paulo Sad, Marcel Nicolau and the referee whose comments helped us to improve on the preliminary version. Finally, the second author has partially conducted this research for the Clay Mathematics Institute.     

Transmission of sound through a pair of rectangular elastic plates

^** Email: f.leppington{at}ukonline.co.uk Two thin elastic plates, which may have different physical parameters,occupy the parallel regions 0 < x < a, 0 < y < b,z = – d and 0 < x < a, 0 < y < b, z = 0, ina Cartesian coordinate system, where the separation distanced is much less than the dimensions a and b. The plates are simplyconnected to a structure that occupies the rest of the region– d < z < 0. The side walls (x = 0 or a, 0 <y < b, – d < z < 0 and y = 0 or b, 0 < x <a, – d < z < 0) are taken to be acoustically soft;the surfaces z = – d and z = 0, outside the rectangularregions occupied by the plates, are taken to be acousticallyhard. Light compressible fluid (air) is in each of the half-spacesoutside the plates and in the rectangular region between them.A sound field is induced by a high-frequency plane wave. Allowanceis made for a small element of damping in the fluid trappedbetween the plates, as well as damping in the plates. Estimatesare derived for the transmitted power, averaged over a frequencyband and with respect to all incidence directions, in the asymptoticlimits of small fluid loading and high frequency. For the casewhere the two plates have identical physical properties, detailedasymptotic estimates are given for the form of the peaks inthe transmitted power, at a discrete set of operating frequencies,both above and below the plates' coincidence frequency.     

On Suboptimal LCS-alignments for Independent Bernoulli Sequences with Asymmetric Distributions

Let X = X ₁ ... X _n and Y = Y ₁ ... Y _n be two binary sequences with length n. A common subsequence of X and Y is any subsequence of X that at the same time is a subsequence of Y; The common subsequence with maximal length is called the longest common subsequence (LCS) of X and Y. LCS is a common tool for measuring the closeness of X and Y. In this note, we consider the case when X and Y are both i.i.d. Bernoulli sequences with the parameters ϵ and 1 − ϵ, respectively. Hence, typically the sequences consist of large and short blocks of different colors. This gives an idea to the so-called block-by-block alignment, where the short blocks in one sequence are matched to the long blocks of the same color in another sequence. Such and alignment is not necessarily a LCS, but it is computationally easy to obtain and, therefore, of practical interest. We investigate the asymptotical properties of several block-by-block type of alignments. The paper ends with the simulation study, where the of block-by-block type of alignments are compared with the LCS.     

An approximation algorithm for the generalized assignment problem

The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei requires timep _ij and incurs a cost ofc _ij ; each machinei is available forT _i time units, and the objective is to minimize the total cost incurred. Our main result is as follows. There is a polynomial-time algorithm that, given a valueC, either proves that no feasible schedule of costC exists, or else finds a schedule of cost at mostC where each machinei is used for at most 2T _i time units.We also extend this result to a variant of the problem where, instead of a fixed processing timep _ij , there is a range of possible processing times for each machine—job pair, and the cost linearly increases as the processing time decreases. We show that these results imply a polynomial-time 2-approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomial-time algorithm that, given valuesM andT, either proves that no schedule of mean job completion timeM and makespanT exists, or else finds a schedule of mean job completion time at mostM and makespan at most 2T. Research partially supported by an NSF PYI award CCR-89-96272 with matching support from UPS, and Sun Microsystems, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.Research supported in part by a Packard Fellowship, a Sloan Fellowship, an NSF PYI award, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

LM-g splines

As an extension of the notion of an L-g spline, three mathematical structures called LM-g splines of types I, II, and III are introduced. Each is defined in terms of two differential operators the coefficients a_j, J = 0,…, n − 1, and b_i, I = 0,…, m, are sufficiently smooth; and b_m is bounded away from zero on [0, T]. Each of the above types of splines is the solution of an optimization problem more general than the one used in the definition of the L-g spline and hence it is recognized as an entity which is distinct from and more general mathematically than the L-g spline. The LM-g splines introduced here reduce to an L-g spline in the special case in which m = 0 and b₀ = constant ≠ 0. After the existence and uniqueness conditions, characterization, and best approximation properties for the proposed splines are obtained in an appropriate reproducing kernel Hilbert space framework, their usefulness in extending the range of applicability of spline theory to problems in estimation, optimal control, and digital signal processing are indicated. Also, as an extension of recent results in the generalized spline literature, state variable models for the LM-g splines introduced here are exhibited, based on which existing least squares algorithms can be used for the recursive calculation of these splines from the data.     

On Some Euler Sequence Spaces of Nonabsolute Type

In the present paper, we introduce Euler sequence spaces e ₀ ^r and e _c ^r of nonabsolute type that are BK-spaces including the spaces c ₀ and c and prove that the spaces e ₀ ^r and e _c ^r are linearly isomorphic to the spaces c ₀ and c, respectively. Furthermore, some inclusion theorems are presented. Moreover, the α-, β-, γ- and continuous duals of the spaces e ₀ ^r and e _c ^r are computed and their bases are constructed. Finally, necessary and sufficient conditions on an infinite matrix belonging to the classes and are established, and characterizations of some other classes of infinite matrices are also derived by means of a given basic lemma, where 1 ≤ p ≤ ∞.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 3–17, January, 2005.     

Asymptotic bias of some election methods

Consider an election where N seats are distributed among parties with proportions p ₁,…,p _m of the votes. We study, for the common divisor and quota methods, the asymptotic distribution, and in particular the mean, of the seat excess of a party, i.e. the difference between the number of seats given to the party and the (real) number Np _i that yields exact proportionality. Our approach is to keep p ₁,…,p _m fixed and let N→∞, with N random in a suitable way. In particular, we give formulas showing the bias favouring large or small parties for the different election methods.     

Prior influence in linear regression when the number of covariates increases to infinity

It is becoming more typical in regression problems today to have the situation where “p>n”, that is, where the number of covariates is greater than the number of observations. Approaches to this problem include such strategies as model selection and dimension reduction, and, of course, a Bayesian approach. However, the discrepancy between p and n can be so large, especially in genomic data, that examining the limiting case where p→∞ can be a relevant calculation. Here we look at the effect of a prior distribution on the coefficients, and in particular characterize the conditions under which, as p→∞, the prior does not overwhelm the data. Specifically, we find that the prior variance on the growing number of covariates must approach zero at rate 1/p, otherwise the prior will overwhelm the data and the posterior distribution of the regression coefficient will equal the prior distribution.     

Identifiability of the multinormal and other distributions under competing risks model

Let X₁, X₂ ,…, X_p be p random variables with joint distribution function F(x₁ ,…, x_p). Let Z = min(X₁, X₂ ,…, X_p) and I = i if Z = X_i. In this paper the problem of identifying the distribution function F(x₁ ,…, x_p), given the distribution Z or that of the identified minimum (Z, I), has been considered when F is a multivariate normal distribution. For the case p = 2, the problem is completely solved. If p = 3 and the distribution of (Z, I) is given, we get a partial solution allowing us to identify the independent case. These results seem to be highly nontrivial and depend upon Liouville's result that the (univariate) normal distribution function is a nonelementary function. Some other examples are given including the bivariate exponential distribution of Marshall and Olkin, Gumbel, and the absolutely continuous bivariate exponential extension of Block and Basu.     

Growth properties of semigroups generated by fractional powers of certain linear operators

Certain semigroups are generated by powers ?(?A)^a, for closed operators A in Banach space and 0 < a < 1. Properties of extent of the resolvent set and size of the resolvent operator of A correspond to properties relating to the sectors of holomorphy of the semigroups, and their growth near the origin and infinity. In this paper, we deal with semigroups having two different types of growth properties. In the first instance, the semigroup grows near the origin as r^?t, 0 < t < 1. We show that such semigroups are fractional-power semi-groups of operators A, whose resolvents decay as r^?s, 0 < s < 1, in subsectors of the right-hand half-plane. In the second instance, the semigroups are bounded near the origin, and admit special estimates on growth at the periphery of their sectors of definition. We show that for the corresponding A, the resolvent is defined and admits special growth estimates in a region which contains every subsector of the right half-plane; and in these subsectors, the resolvent decays as r^?1.