On the Degree of Approximation by Manifolds of Finite Pseudo-Dimension

The pseudo-dimension of a real-valued function class is an extension of the VC dimension for set-indicator function classes. A class of finite pseudo-dimension possesses a useful statistical smoothness property. In [10] we introduced a nonlinear approximation width = which measures the worst-case approximation error over all functions by the best manifold of pseudo-dimension n . In this paper we obtain tight upper and lower bounds on ρ _n (W ^r,d _p , L _q ) , both being a constant factor of n ^-r/d , for a Sobolev class W ^r,d _p , . As this is also the estimate of the classical Alexandrov nonlinear n -width, our result proves that approximation of W ^r,d _p by the family of manifolds of pseudo-dimension n is as powerful as approximation by the family of all nonlinear manifolds with continuous selection operators. March 12, 1997. Dates revised: August 26, 1997, October 24, 1997, March 16, 1998, June 15, 1998. Date accepted: June 25, 1998.     

On the subgroup distance problem

We investigate the computational complexity of finding an element of a permutation group H⊆S_n with minimal distance to a given π∈S_n, for different metrics on S_n. We assume that H is given by a set of generators. In particular, the size of H might be exponential in the input size, so that in general the problem cannot be solved in polynomial time by exhaustive enumeration. For the case of the Cayley Distance, this problem has been shown to be NP-hard, even if H is abelian of exponent two [R.G.E. Pinch, The distance of a permutation from a subgroup of S_n, in: G. Brightwell, I. Leader, A. Scott, A. Thomason (Eds.), Combinatorics and Probability, Cambridge University Press, 2007, pp. 473-479]. We present a much simpler proof for this result, which also works for the Hamming Distance, the l_p distance, Lee’s Distance, Kendall’s tau, and Ulam’s Distance. Moreover, we give an NP-hardness proof for the l_∞ distance using a different reduction idea. Finally, we discuss the complexity of the corresponding fixed-parameter and maximization problems.     

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

Asymptotic Distribution of Nodes for Near-Optimal Polynomial Interpolation on Certain Curves in R 2

Let E\subset \Bbb R ^s be compact and let d _n ^E denote the dimension of the space of polynomials of degree at most n in s variables restricted to E . We introduce the notion of an asymptotic interpolation measure (AIM). Such a measure, if it exists , describes the asymptotic behavior of any scheme τ _n ={ \bf x _k,n } _k=1 ^dnE , n=1,2,\ldots , of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with n . We demonstrate the existence of AIMs for the finite union of compact subsets of certain algebraic curves in R ² . It turns out that the theory of logarithmic potentials with external fields plays a useful role in the investigation. Furthermore, for the sets mentioned above, we give a computationally simple construction for ``good' interpolation schemes. November 9, 2000. Date revised: August 4, 2001. Date accepted: September 14, 2001.     

p-Adic dynamical systems of Chebyshev polynomials

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ? _p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T _p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ? _p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T _p in the field ? _p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T _n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.     

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.     

Two finiteness theorems for periodic tilings of d-dimensional euclidean space

Consider the d -dimensional euclidean space E ^d . Two main results are presented: First, for any N∈ N, the number of types of periodic equivariant tilings that have precisely N orbits of (2,4,6, . . . ) -flags with respect to the symmetry group Γ , is finite. Second, for any N∈ N, the number of types of convex, periodic equivariant tilings that have precisely N orbits of tiles with respect to the symmetry group Γ , is finite. The former result (and some generalizations) is proved combinatorially, using Delaney symbols, whereas the proof of the latter result is based on both geometric arguments and Delaney symbols. <lsiheader> <onlinepub>7 August, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>20n2p143.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received September 5, 1996, and in revised form January 6, 1997.     

On graphs without crowns with regularμ-subgraphs

Anm-crown is the complete tripartite graphK _{1, 1,m} with parts of order 1, 1,m, and anm-claw is the complete bipartite graphK _1,m with parts of order 1,m, wherem ≥ 3. A vertexa of a graph Γ is calledweakly reduced iff the subgraph {x є Γ ‖a ^⊥ =x ^⊥} consists of one vertex. A graph Γ is calledweakly reduced iff all its vertices are weakly reduced. In the present paper we classify all connected weakly reduced graphs without 3-crowns, all of whose μ-subgraphs are regular graphs of constant nonzero valency. In particular, we generalize the characterization of Grassman and Johnson graphs due to Numata, and the characterization of connected reduced graphs without 3-claws due to Makhnev. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 874–881, June, 2000. This research was supported by the Russian Foundation for Basic Research under grant No. 99-01-00462.     

On Levels in Arrangements of Lines, Segments, Planes, and Triangles%

We consider the problem of bounding the complexity of the k th level in an arrangement of n curves or surfaces, a problem dual to, and an extension of, the well-known k-set problem. Among other results, we prove a new bound, O(nk ^5/3 ) , on the complexity of the k th level in an arrangement of n planes in R ³ , or on the number of k -sets in a set of n points in three dimensions, and we show that the complexity of the k th level in an arrangement of n line segments in the plane is , and that the complexity of the k th level in an arrangement of n triangles in 3-space is O(n ² k ^5/6 α(n/k)) . <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p315.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received February 7, 1997, and in revised form May 15, 1997, and August 30, 1997.     

On the invertibility of the operator <Emphasis Type="Italic">d/dt</Emphasis> + <Emphasis Type="Italic">A</Emphasis> in certain functional spaces

We prove that the operator d/dt + A constructed on the basis of a sectorial operator A with spectrum in the right half-plane of ℂ is continuously invertible in the Sobolev spaces W _p ¹ (ℝ, D _α), α ≥ 0. Here, D _α is the domain of definition of the operator A ^α and the norm in D _α is the norm of the graph of A ^α. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1020–1025, August, 2007.     

Edge-Ramsey theory

LetK be a configuration, a set of points in some finite-dimensional Euclidean space. Letn andk be positive integers. The notationR(K, n, r) is an abbreviation for the following statement: For everyr-coloring of the points of then-dimensional Euclidean space,R ⁿ , a monochromatic configurationL which is congruent toK exists. A configurationK is Ramsey if the following holds: For every positive integerr, a positive integern=n(K, r) exists such that, for allm≥n, R(K, m, r) holds. A configuration is spherical if it can be embedded in the surface of a sphere inn-space, providedn is sufficiently large. It is relatively easy to show that if a configuration is Ramsey, it must be spherical. Accordingly, a good fraction of the research efforts in Euclidean Ramsey theory is devoted to determining which spherical configurations are Ramsey. It is known that then-dimensional measure polytopes (the higher-dimensional analogs of a cube), then-dimensional simplex, and the regular polyhedra inR ² andR ³ are Ramsey. Now letE denote a set of edges in a configurationK. The pair (K, E) is called an edge-configuration, andR _e (K, E, n, r) is used as an abbreviation for the following statement: For anyr-coloring of the edges ofR ⁿ , there is an edge configuration (L, F) congruent to (K, E) so that all edges inF are assigned the same color. An edge-configuration isedge-Ramsey if, for allr≥1, a positive integern=n(K, E, r) exists so that ifm≥n, the statementR _e (K, E, m, r) holds. IfK is a regular polytope, it is saidK isedge-Ramsey when the configuration determined by the set of edges of minimum length is edge-Ramsey. It is known that then-dimensional simplex is edge-Ramsey and that the nodes of any edge-Ramsey configuration can be partitioned into two spherical sets. Furthermore, the edges of any edge-Ramsey configuration must all have the same length. It is conjectured that the unit square is edge-Ramsey, and it is natural to ask the more general question: Which regular polytopes are edge-Ramsey? In this article it is shown that then-dimensional measure polytope and then-dimensional cross polytope are edge-Ramsey. It is also shown that these two infinite families and then-dimensional simplexes are the only regular edge-Ramsey polytopes, with the possible exceptions of the hexagon and the 24-cell.     

On the interpolation of some generalized function spaces of different anisotropy

In the paper, the interpolation properties of the spacesH _p ^s (v; ℝ _n ) of Sobolev-Liouville type and the spacesB _{p, q} ^s (μ ℝ _n ) of Nikol'skii-Besov type generated by functions of polynomial growth that are infinitely differentiable outside of the origin are studied. Interpolation formulas for the pairs {H(v _o ),H(v ₁)} and {B(μ₀),B(μ₁)} of spaces of the above types for which the anisotropies of the interpolated spaces do not depend on each other are proved. The investigated spaces, for certain specification of the generating functions, coincide with the classical (isotropic and anisotropic) Sobolev-Liouville and Nikol'skii-Besov spaces. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 666–672, November, 1997. Translated by A. I. Shtern     

On the Stabilization of a Solution of the Cauchy Problem for One Class of Integro-Differential Equations

We consider a solution of the Cauchy problem u(t, x), t > 0, x ∈ R ², for one class of integro-differential equations. These equations have the following specific feature: the matrix of the coefficients of higher derivatives is degenerate for all x. We establish conditions for the existence of the limit lim _t→∞ u(t, x) = v(x) and represent the solution of the Cauchy problem in explicit form in terms of the coefficients of the equation.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1699 – 1706, December, 2004.     

New types of 3-D systems of quadratic differential equations with chaotic dynamics based on Ricker discrete population model

The wide class of 3-D autonomous systems of quadratic differential equations, in each of which either there is a couple of coexisting limit cycles or there is a couple of coexisting chaotic attractors, is found. In the second case the couple consists of either Lorentz-type attractor and another attractor of a new type or two Lorentz-type attractors. It is shown that the chaotic behavior of any system of the indicated class can be described by the Ricker discrete population model: z_i+1 = z_i exp(r − z_i), r > 0, z_i > 0, i = 0, 1, … . The values of parameters, at which in the 3-D system appears either the couple of limit cycles or the couple of chaotic attractors, or only one limit cycle, or only one sphere-shaped chaotic attractor, are indicated. Examples are given.     

Momentopes, the Complexity of Vector Partitioning, and Davenport—Schinzel Sequences

The computational complexity of the partition problem , which concerns the partitioning of a set of n vectors in d -space into p parts so as to maximize an objective function which is convex on the sum of vectors in each part, is determined by the number of vertices of the corresponding p-partition polytope defined to be the convex hull in (d\times p) -space of all solutions. In this article, introducing and using the class of Momentopes , we establish the lower bound v _p,d (n)=Ω(n^ \lfloor (d-1)/2 \rfloor p ) on the maximum number of vertices of any p -partition polytope of a set of n points in d -space, which is quite compatible with the recent upper bound v _p,d (n)=O(n ^d(p-1)-1 ) , implying the same bound on the complexity of the partition problem. We also discuss related problems on the realizability of Davenport—Schinzel sequences and describe some further properties of Momentopes. Received April 4, 2001, and in revised form October 3, 2001. Online publication February 28, 2002.     

Some intersection theorems

LetL(A) be the set of submatrices of anm×n matrixA. ThenL(A) is a ranked poset with respect to the inclusion, and the poset rank of a submatrix is the sum of the number of rows and columns minus 1, the rank of the empty matrix is zero. We attack the question: What is the maximum number of submatrices such that any two of them have intersection of rank at leastt? We have a solution fort=1,2 using the followoing theorem of independent interest. Letm(n,i,j,k) = max(|F|;|G|), where maximum is taken for all possible pairs of families of subsets of ann-element set such thatF isi-intersecting,G isj-intersecting andF ansd,G are cross-k-intersecting. Then fori≤j≤k, m(n,i,j,k) is attained ifF is a maximali-intersecting family containing subsets of size at leastn/2, andG is a maximal2k?i-intersecting family. Furthermore, we discuss and Erd?s-Ko-Rado-type question forL(A), as well.     

On a class of equilibrium problems in the real axis

In a series of seminal papers, Thomas J. Stieltjes (1856-1894) gave an elegant electrostatic interpretation for the zeros of classical families of orthogonal polynomials, such as Jacobi, Hermite and Laguerre polynomials. More generally, he extended this approach to the zeros of polynomial solutions of certain second-order linear differential equations (Lamé equations), the so-called Heine-Stieltjes polynomials.In this paper, a class of electrostatic equilibrium problems in R, where the free unit charges x₁,…,x_n∈R are in presence of a finite family of “attractors” (i.e., negative charges) z₁,…,z_m∈C?R, is considered and its connection with certain class of Lamé-type equations is shown. In addition, we study the situation when both n→∞ and m→∞, by analyzing the corresponding (continuous) equilibrium problem in presence of a certain class of external fields.     

On Minimum Stars and Maximum Matchings

We discuss worst-case bounds on the ratio of maximum matching and minimum median values for finite point sets. In particular, we consider ``minimum stars,' which are defined by a center chosen from the given point set, such that the total geometric distance L <sub>S</sub> to all the points in the set is minimized. If the center point is not required to be an element of the set (i.e., the center may be a Steiner point), we get a ``minimum Steiner star' of total length L _SS . As a consequence of triangle inequality, the total length L _M of a maximum matching is a lower bound for the length L _SS of a minimum Steiner star, which makes the worst-case value ρ(SS,M) of the value L _SS /L _M interesting in the context of optimal communication networks. The ratio also appears as the duality gap in an integer programming formulation of a location problem by Tamir and Mitchell. In this paper we show that for a finite set that consists of an even number of points in the plane and Euclidean distances, the worst-case ratio ρ(S,M) cannot exceed . This proves a conjecture of Suri, who gave an example where this bound is achieved. For the case of Euclidean distances in two and three dimensions, we also prove upper and lower bounds for the worst-case value ρ(S,SS) of the ratio L _S /L _SS , and for the worst-case value ρ(S,M) of the ratio L _S /L _M . We give tight upper bounds for the case where distances are measured according to the Manhattan metric: we show that in three-dimensional space, ρ(SS,M) is bounded by 3/2, while in two-dimensional space L _SS =L _M , extending some independent observations by Tamir and Mitchell. Finally, we show that ρ(S,SS) is 3/2 in the two-dimensional case, and 5/3 in the three-dimensional case. Received January 1, 1999, and in revised form July 15, 1999.     

On linear widths of classesH ω

We obtain new results related to the estimation of the linear widths λ _N and λ ^N in the spacesC andL _p for the classesH ^ω (in particular, forH ^α, 0<α<1). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1255–1264, September, 1996.     

Méthode de mesure de la chaleur spécifique par température croissante ou décroissante

Résumé L'obtention des valeurs expérimentales de l'entropie, à l'équilibre, à partir des grandeurs thermiques, est rendue compliquée pour un supraconducteur se trouvant dans une phase mélangée, mixte ou intermédiaire, par le comportement irréversible observé en général dans les échantillons réels. Il est possible de réduire, voire supprimer l'incertitude dans l'interprétation des résultats si l'on détermine les variations d'entropie non seulement pour des températures et champs magnétiques croissants, mais également pour des températures et champs décroissants. On décrit une méthode de mesure de la chaleur spécifique, valable pourT<10°K, inspirée de la méthode d'impulsion d'une part, en ce sens qu'elle a l'avantage de travailler sur des états de quasi équilibre en température du système, et de la méthode de refroidissement continu d'autre part, puisqu'elle utilise une perte thermique permettant la mesure par température décroissante. Le modèle théorique correspondant au dispositif expérimental est traité complètement du point de vue mathématique. Les erreurs systématiques propres à la méthode peuvent être réduites à moins de 0.5% et des variations de température de quelques millidegrés peuvent être encore mesurées avec une précision de l'ordre du pourcent. Le procédé convient donc bien à l'étude détaillée des transitions de phase supraconductrices.

---

Summary It is difficult to obtain equilibrium values of entropy from measured thermal quantities, for superconductors in a mixed phase, because of irreversible behavior generally observed in real samples. It is possible to reduce, even suppress, the uncertainty arising from interpretation of results, if entropy variations are determined not only for increasing magnetic field and temperature, but also for decreasing field and temperature. A method to measure the specific heat is described, which is valid forT<10°K. As in the case of the heat burst method, the advantage is that the determination occurs from quasi-equilibrium states in temperature of the system. In addition, a heat leak allows us also to make measurements with decreasing temperature, as in the cooling curve method. The theoretical model corresponding to the experimental arrangement is thoroughly analysed from a mathematical point of view. Systematic errors, arising from the method itself, may be reduced to less than 0,5% and temperature variations of a few milli-degrees can still be measured with a precision of the order of 1%. This procedure is then well adapted to the detailed study of superconducting phase transitions.