The incidence structure of subspaces with well-scaled frames

In this paper, the incidence structure of classes of subspaces that generalize the regular (unimodular) subspaces of rational coordinate spaces is studied. Let F the a field and S - F β {0}. A subspace, V, of a coordinate space over F is S-regular if every elementary vector of V can be scaled by an element of F β {0} so that all of its non-zero entries are elements of S. A subspace that is {−1, +1 }-regular over the rational field is regular.Associated with a subspace, V, over an arbitrary (respectively, ordered) field is a matroid (oriented matroid) having as circuits (signed circuits) the set of supports (signed supports) of elementary vectors of V. Fundamental representation properties are established for the matroids that arise from certain classes of subspaces. Matroids that are (minor) minimally non-representable by various classes of subspaces are identified. A unique representability results is established for the oriented matroids of subspaces that are dyadic (i.e., {±2⁰, ±2¹, ±2², …}-regular) over the rationals. A self-dual characterization is established for the matroids of S-regular subspaces which generalizes Minty's characterization of regular spaces as digraphoids.     

Application of generalized differential transform method to multi-order fractional differential equations

In a recent paper [Odibat Z, Momani S, Erturk VS. Generalized differential transform method: application to differential equations of fractional order, Appl Math Comput. submitted for publication] the authors presented a new generalization of the differential transform method that would extended the application of the method to differential equations of fractional order. In this paper, an application of the new technique is applied to solve fractional differential equations of the form y^(μ)(t)=f(t,y(t),y^(β₁)(t),y^(β₂)(t),…,y^(β_n)(t)) with μ>β_n>β_n-1>…>β₁>0, combined with suitable initial conditions. The fractional derivatives are understood in the Caputo sense. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the new generalization.     

On piecewise-polynomial approximation of functions with a bounded fractional derivative in an Lp-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε [0, 1), and ƒ^{(r + α)} is the classical fractional derivative, i.e., ƒ^{(r + α)}(y) = ∝₀¹, ^α d(ƒ^(r)(t)). We prove that, for any such function ƒ, there exists a piecewise-polynomial of degree s that interpolates ƒ at n equally spaced points and that approximates ƒ with an error (in sup-norm) ƒ^{(r + α)}_p O(n^{−(r+α−1/p}). We also prove that no algorithm based on n function and/or derivative values of ƒ has the error equal ƒ^{(r + α)}_p O(n^{−(r+α−1/p}) for any ƒ. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s r. Hence, even without knowing the actual regularity (r, α, and p) of ƒ, we can approximate the function ƒ with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.     

On the stability of fractional neutron point kinetics (FNPK)

The aim of this work is investigate the stability of fractional neutron point kinetics (FNPK). The method applied in this work considers the stability of FNPK as a linear fractional differential equation by transforming the s − plane to the W − plane. The FNPK equations is an approximation of the dynamics of the reactor that includes three new terms related to fractional derivatives, which are explored in this work with an aim to understand their effect in the system stability. Theoretical study of reactor dynamical systems plays a significant role in understanding the behavior of neutron density, which is important in the analysis of reactor safety. The fractional relaxation time (τ^α) for values of fractional-order derivative (α) were analyzed, and the minimum absolute phase was obtained in order to establish the stability of the system. The results show that nuclear reactor stability with FNPK is a function of the fractional relaxation time.     

A bipartite analogue of Dilworth’s theorem for multiple partial orders

Let r be a fixed positive integer. It is shown that, given any partial orders <₁, …, <_r on the same n-element set P, there exist disjoint subsets A,BP, each with at least n^1−o(1) elements, such that one of the following two conditions is satisfied: (1) there is an such that every element of A is larger than every element of B in the partial order <_i, or (2) no element of A is comparable with any element of B in any of the partial orders <₁, …, <_r. As a corollary, we obtain that any family C of n convex compact sets in the plane has two disjoint subfamilies A,BC, each with at least n^1−o(1) members, such that either every member of A intersects all members of B, or no member of A intersects any member of B.     

The Complexity of Indefinite Elliptic Problems with Noisy Data

We study the complexity of second-order indefinite elliptic problems −div(au) +bu=f(with homogeneous Dirichlet boundary conditions) over ad-dimensional domain Ω, the error being measured in theH¹(Ω)-norm. The problem elementsfbelong to the unit ball ofW^{r, p}, (Ω), wherep [2, ∞] andr>d/p. Information consists of (possibly adaptive) noisy evaluations off,a, orb(or their derivatives). The absolute error in each noisy evaluation is at most δ. We find that thenth minimal radius for this problem is proportional ton^−r/d+ δ and that a noisy finite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be efficiently implemented using multigrid techniques. Using these results, we find tight bounds on the -complexity (minimal cost of calculating an -approximation) for this problem, said bounds depending on the costc(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation isc(δ) = δ^−s(fors> 0), then the complexity is proportional to (1/)^{d/r + s}.     

The Complexity of Two-Point Boundary-Value Problems with Piecewise Analytic Data

Previous work on the ε-complexity of elliptic boundary-value problems Lu = f assumed that the class F of problem elements f was the unit ball of a Sobolev space. In a recent paper, we considered the case of a model two-point boundary-value problem, with F being a class of analytic functions. In this paper, we ask what happens if F is a class of piecewise analytic functions. We find that the complexity depends strongly on how much a priori information we have about the breakpoints. If the location of the breakpoints is known, then the ε-complexity is proportional to ln (ε⁻¹), and there is a finite element p-method (in the sense of Babu ka) whose cost is optimal to within a constant factor. If we know neither the location nor the number of breakpoints, then the problem is unsolvable for ε < √2. If we know only that there are b ≥ 2 breakpoints, but we de not know their location, then the ε-complexity is proportional to bε⁻¹, and a finite element h-method is nearly optimal. In short, knowing the location of the breakpoints is as good as knowing that the problem elements are analytic, whereas only knowing the number of breakpoints is no better than knowing that the problem elements have a bounded derivative in the L₂ sense.     

Signed 2-independence in graphs

A function f : V→{−1,1} defined on the vertices of a graph G=(V,E) is a signed 2-independence function if the sum of its function values over any closed neighbourhood is at most one. That is, for every vV, f(N[v])1, where N[v] consists of v and every vertex adjacent to v. The weight of a signed 2-independence function is f(V)=∑f(v), over all vertices vV. The signed 2-independence number of a graph G, denoted α_s²(G), equals the maximum weight of a signed 2-independence function of G. In this paper, we establish upper bounds for α_s²(G) in terms of the order and size of the graph, and we characterize the graphs attaining these bounds. For a tree T, upper and lower bounds for α_s²(T) are established and the extremal graphs characterized. It is shown that α_s²(G) can be arbitrarily large negative even for a cubic graph G.     

Triply-Logarithmic Parallel Upper and Lower Bounds for Minimum and Range Minima over Small Domains

We consider the problem of computing the minimum ofnvalues, and several well-known generalizations [prefix minima, range minima, and all nearest smaller values (ANSV)] for input elements drawn from the integer domain [1···s], wheres ≥ n. In this article we give simple and efficient algorithms for all of the preceding problems. These algorithms all takeO(log log log s) time using an optimal number of processors andO(ns^ε) space (for constant ε < 1) on the COMMON CRCW PRAM. The best known upper bounds for the range minima and ANSV problems were previouslyO(log log n) (using algorithms for unbounded domains). For the prefix minima and for the minimum problems, the improvement is with regard to the model of computation. We also prove a lower bound of Ω(log log n) for domain sizes = 2^{Ω(log n log log n)}. Since, forsat the lower end of this range, log log n = Ω(log log log s), this demonstrates that any algorithm running ino(log log log s) time must restrict the range ofson which it works.     

Lp Markov–Bernstein Inequalities on All Arcs of the Circle

Let 0<p<∞ and 0α<β2π. We prove that for n1 and trigonometric polynomials s_n of degree n, we have

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cn^p ∫^β_α |s_n(θ)|^p dθ, where c is independent of α, β, n, s_n. The essential feature is the uniformity in [α,β] of the estimate and the fact that as [α,β] approaches [0,2π], we recover the L_p Markov inequality. The result may be viewed as the complete L_p form of Videnskii's inequalities, improving earlier work of the second author.     

Equiangular lines in C (part II)

A subset S of a complex projective space is F-regular provided each two points of S have the same non-zero distance and each subset of three points of S has the same shape invariant. The aim of this paper is the determination for any odd integer r, of the largest integer n(r) such tht CP^r−1 contains an F-regular subset of n(r) points.It is established that n(r) ≤ 2r − 2 for any odd integer r and n(1 + 2^s) = 2^s+1 for any integer s.     

Infinite p-groups containing exactly p2 solutions of the equation xP=1

We study arbitrary infinite 2-groups with three involutions and infinite locally finite p-groups (p2), containing p²–1 elements of order p. For odd p the groupG=a, where A is a direct product of two quasicyclic 3-groups ¦b¦=9, b³A, and subgroup A is generated by the elements of the commutator ladder of element b, is a unique infinite non-Abelian locally finite p-group whose equation xP=1 has p² solutions.Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 11–18, July, 1976.     

Coupled oscillators with power-law interaction and their fractional dynamics analogues

A one-dimensional chain of coupled oscillators with the long-range power-law interactions is considered. Equations of motion in the infrared limit are mapped onto the continuum equation with the Riesz fractional derivative of order α, when 0 < α < 2. The evolution of soliton-like and breather-like structures is obtained numerically and compared for two types of simulations: using the chain of oscillators and using the continuous medium equation with the fractional derivative.     

Smoothness of stopping timesof diffusion processes

For an elliptic diffusion process, we prove that the exit time from an open set is in the fractional Sobolev spaces E^p_α (or D^p_α) provided that pα<1. The result is almost optimal.     

Existence and Regularity of a Class of Weak Solutions to the Navier–Stokes Equations

We construct a class of weak solutions to the Navier–Stokes equations, which have second order spatial derivatives and one order time derivatives, ofppower summability for 1 < p ≤ 5/4. Meanwhile, we show thatu L^s(0, T; W^{2, r}(Ω)) with 1/s + 3/2r = 2 for 1 < r ≤ 5/4.rcan be relaxed not to exceed 3/2 if we consider only in the interior of Ω. In the end, we extend the classical regularity theorem. Our results show thatuis a regular solution if u L^s(0, T; L^r(Ω)) with 1/s + 3/2r = 1 for Ω satisfying (1.3), with 1/s + 1/r = 5/6 for arbitrary domain inR³and 1 < s ≤ 2. For Ω = Rⁿwithn ≥ 3, this result was previously obtained byH. Beirão da Veiga (Chinese Ann. Math. Ser. B16, 1995, 407–412).     

Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians

The Pfaffian method enumerating perfect matchings of plane graphs was discovered by Kasteleyn. We use this method to enumerate perfect matchings in a type of graphs with reflective symmetry which is different from the symmetric graphs considered in [J. Combin. Theory Ser. A 77 (1997) 67, MATCH—Commun. Math. Comput. Chem. 48 (2003) 117]. Here are some of our results: (1) If G is a reflective symmetric plane graph without vertices on the symmetry axis, then the number of perfect matchings of G can be expressed by a determinant of order |G|/2, where |G| denotes the number of vertices of G. (2) If G contains no subgraph which is, after the contraction of at most one cycle of odd length, an even subdivision of K_2,3, then the number of perfect matchings of G×K₂ can be expressed by a determinant of order |G|. (3) Let G be a bipartite graph without cycles of length 4s, s{1,2,…}. Then the number of perfect matchings of G×K₂ equals ∏(1+θ²)^m_θ, where the product ranges over all non-negative eigenvalues θ of G and m_θ is the multiplicity of eigenvalue θ. Particularly, if T is a tree then the number of perfect matchings of T×K₂ equals ∏(1+θ²)^m_θ, where the product ranges over all non-negative eigenvalues θ of T and m_θ is the multiplicity of eigenvalue θ.     

Where does smoothness count the most for Fredholm equations of the second kind with noisy information?

We study the complexity of Fredholm problems (I−T_k)u=f of the second kind on I^d=[0,1]^d, where T_k is an integral operator with kernel k. Previous work on the complexity of this problem has assumed either that we had complete information about k or that k and f had the same smoothness. In addition, most of this work has assumed that the information about k and f was exact. In this paper, we assume that k and f have different smoothness; more precisely, we assume that fW^r,p(I^d) with r>d/p and that kW^s,∞(I^2d) with s>0. In addition, we assume that our information about k and f is contaminated by noise. We find that the nth minimal error is Θ(n^−μ+δ), where μ=min{r/d,s/(2d)} and δ is a bound on the noise. We prove that a noisy modified finite element method has nearly minimal error. This algorithm can be efficiently implemented using multigrid techniques. We thus find tight bounds on the -complexity for this problem. These bounds depend on the cost c(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation is proportional to δ^−t, then the -complexity is roughly (1/)^t+1/μ.     

Numerical differentiation inspired by a formula of R.P. Boas

First, we briefly discuss three classes of numerical differentiation formulae, namely finite difference methods, the method of contour integration, and sampling methods. Then we turn to an interpolation formula of R.P. Boas for the first derivative of an entire function of exponential type bounded on the real line. This formula may be classified as a sampling method. We improve it in two ways by incorporating a Gaussian multiplier for speeding up convergence and by extending it to higher derivatives. For derivatives of order s, we arrive at a differentiation formula with N^′ nodes that applies to all entire functions of exponential type without any additional restriction on their growth on the real line. It has an error bound that converges to zero like e^-αN/N^m as N→∞, where α>0 and N^′=2N, m=3/2 for odd s while N^′=2N+1, m=5/2 for even s. Comparable known formulae have stronger hypotheses and, for the same α, they have m=1/2 only. We also deduce a direct (error-free) generalization of Boas’ formula (Corollary 5). Furthermore, we give a modification of the main result for functions analytic in a domain and consider an extension to non-analytic functions as well. Finally, we illustrate the power of the method by examples.     

Tests for the mean direction of the Langevin distribution with large concentration parameter

In this paper we study the asymptotic behaviors of the likelihood ratio criterion (T_L^(s)), Watson statistic (T_W^(s)) and Rao statistic (T_R^(s)) for testing H_0s: μ (a given subspace) against H_1s: μ , based on a sample of size n from a p-variate Langevin distribution M_p(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ⁻¹. For the case when κ is unknown, it is shown that T_R^(s) T_L^(s) T_W^(s) in their powers up to the order κ⁻¹.     

Discrete time parametric models with long memory and infinite variance

We study a large class of infinite variance time series that display long memory. They can be represented as linear processes (infinite order moving averages) with coefficients that decay slowly to zero and with innovations that are in the domain of attraction of a stable distribution with index 1 < α < 2 (stable fractional ARIMA is a particular example). Assume that the coefficients of the linear process depend on an unknown parameter vector β which is to be estimated from a series of length n. We show that a Whittle-type estimator β_n for β is consistent (β_n converges to the true value β₀ in probability as n → ∞), and, under some additional conditions, we characterize the limiting distribution of the rescaled differences (n/logn)^1/ga(β_n − β₀).