A NON-LINEAR ℓ 1 PROBLEM

Abstract

In this note we consider a non-linear problem posed to the author in a private communication with Per Enflo: If a normed space X contains a non-linear ? ₁ ⁽ⁿ⁾-cube (where n ≥ 2 is a natural number) does it necessarily contain a linear isometric copy of ? ₁ ⁽ⁿ⁾?

We exhibit a strong regularity property of non-linear ? ₁ ⁽ⁿ⁾ cubes and apply it to obtain an affirmative answer to Enflo's problem in the setting X = l _∞ ^(m) that, moreover, coincides precisely with well known linear theory.     

On a cyclic disconjugate operator associated to linear differential equations

In this paper the disconjugate linear differential operator of n-th order D^1/(n) given by $$D_1^{(n)} (x)(t) = \frac{1}{{a_n (t)}}\frac{d}{{dt}}\frac{1}{{a_{n - 1} (t)}} \ldots \frac{1}{{a_1 (t)}}\frac{d}{{dt}}x(t)$$ is considered together with other n?1 operators, which are obtained from D ₁ ⁽ⁿ⁾ by an ordered cyclic permutation of the functions a_i. Such operators play an important role in the study of oscillation of the associated linear differential equation (*) $$D_1^{(n)} (x)(t) \pm x(t) = 0.$$ Some properties of these operators suggest the new idea of «isomorphism of oscillation». The existence of an isomorphism of oscillation allows to describe the oscillatory or nonoscillatory behavior of solutions of (*) by the oscillatory or nonoscillatory behavior of solutions of other n ?1 suitable linear differential equations. From this fact one can easily obtain new results about oscillation or nonoscillation of (*) that might be hard to prove directly. Several interesting consequences concerning the classification of solutions of (*) are also presented together with some new applications to the structure of the set of nonoscillatory solutions of (*).     

Nearly Linear Time Algorithms for Permutation Groups: An Interplay Between Theory and Practice

We survey polynomial time algorithms (both deterministic and random) for computations with permutation groups. Particular emphasis is given to algorithms with running time of the form O(n log ^c |G|), where G is a permutation group of degree n. In the case of small-base groups, i.e., when log |G| is polylogarithmic as a function of n, such algorithms run in nearly linear, O(n log^c' n) time. Important classes of groups, including all permutation representations of simple groups except the alternating ones, as well as most primitive groups, belong to this category. For large n, the majority of practical computations is carried out on small-base groups.In the last section, we present some new nearly linear time algorithms, culminating in the computation of the upper central series in nilpotent groups.     

On Tyler's M-Functional of Scatter in High Dimension

Let y¹, y²,..., yⁿ R^q be independent, identically distributed random vectors with nonsingular covariance matrix , and let S = S(y¹,..., yⁿ) be an estimator for . A quantity of particular interest is the condition number of ^-1 S. If the yⁱ are Gaussian and S is the sample covariance matrix, the condition number of ^-1 S, i.e. the ratio of its extreme eigenvalues, equals 1 + O^p((q/n)^1/2) as q and q/n 0. The present paper shows that the same result can be achieved with two estimators based on Tyler's (1987, Ann. Statist., 15, 234-251) M-functional of scatter, assuming only elliptical symmetry of (yⁱ) or less. The main tool is a linear expansion for this M-functional which holds uniformly in the dimension q. As a by-product we obtain continuous Fréchet-differentiability with respect to weak convergence.     

On spherically convex sets and Q-matrices

Let K be a closed spherically convex subset of S^n?1 that is contained in a hemisphere, and x?(K) the radial projection onto S^n?1 of the centroid of K. Then p^Tx?(K)>0 for all p ? K. A specialization of this result to spherical simplices is used to derive a necessary condition for Q-matrices, i.e., matrices for which every corresponding linear complementarity problem has at least one solution.     

Non linear eigenvalues problems with a small parameter

In this work we study non linear eigenvalues problems like: [–² d²/dt²+(t²–)²+1]u=0 where. More precisely we study the spectrum of the operator: Q(;)=–²d²/dt²+(t²–)²+1 when 0, >0. Our method of proof consists in replacing our problem by a linear eigenvalue problem about a non self adjoint system.     

Far field velocity potential induced by a rapidly decaying vorticity distribution

We study the velocity field induced by a vorticity distribution decaying rapidly in the distancer from the origin. In the far field, the vector potential for the velocity field can be represented by a series A ⁽ⁿ⁾, withA ⁽ⁿ⁾ proportional tor ^–n–1, forn=1, 2, .... We show thatA ⁽ⁿ⁾ can be expressed as a linear combination ofM _n linearly independent vector functions. The numberM _n is equal to 3 forn=1 and 4n forn2 and the coefficient of a vector function is defined by a linear combination of nth moments of vorticity. We then show that only 2n+1 linear combinations of thoseM _n vector functions contribute to the far field velocity which is irrotational. The corresponding scalar potential ⁽ⁿ⁾ is then represented by a linear combination of 2n+1 spherical harmonics ofnth order whose coefficients are again linear combinations ofnth moments of vorticity.

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Zusammenfassung Die vorliegende Arbeit beschreibt das Geschwindigkeitsfeld fernab einer Wirbelverteilung, welche mit dem Abstandr vom Ursprung eines geeigneten Bezugssystems hinreichend schnell abklingt. Die Geschwindigkeit besitzt ein Vektorpotential, dessen Fernfeldverhalten einer Reihenentwicklung A ⁽ⁿ⁾, genügt. Dabei istA ⁽ⁿ⁾ proportional zur ^–n–1 fürn=1, 2, .... Wir entwickeln eine explizite Darstellung vonA ⁽ⁿ⁾ als Linearkombination vonM _n linear unabhängigen Vektorfunktionen. Die auftretenden Koeffizienten sind ihrerseits Kombinationenn-ter Momente der Wirbelverteilung. Die ZahlM ₁ ist gleich 3 und es istM _n=4n fürn2, während die Gesamtzahl dernten Momente beträgt. Weiterhin zeigen wir, da\ nur 2n+1 dieser Vektorfunktionen auch zum drehungsfreien Fernfeld der Geschwindigkeitn-ter Ordnung beitragen können und identifizieren die zugehörigen Kombinationen von Wirbelmomenten. Dieselben Kombinationen liefern dann auch die Koeffizienten in einer Entwicklung desskalaren Fernfeldpotentials nach Kugelfunktionen.

     

典型群的增广商群

设G 为有限域K 上的一般线性群(特殊线性群、酉群、辛群及正交群), 记整群环ZG 的n 次增广理想为△ⁿ(G). 本文着重研究有限域上的典型群的增广商群Q_n(G) = △ⁿ(G)/△ⁿ⁺¹(G), 并刻画了这些连续商群的结构.     

Solving superelliptic Diophantine equations by Baker's method

We describe a method for complete solution of the superelliptic Diophantine equation ay^p=f(x). The method is based on Baker's theory of linear forms in the logarithms. The characteristic feature of our approach (as compared with the classical method) is that we reduce the equation directly to the linear forms in logarithms, without intermediate use of Thue and linear unit equations. We show that the reduction method of Baker and Davenport [3] is applicable for superelliptic equations, and develop a very efficient method for enumerating the solutions below the reduced bound. The method requires computing the algebraic data in number fields of degree pn(n-1)/2 at most; in many cases this number can be reduced. Two examples with p=3 and n=4 are given.     

Integration of differential forms on manifolds with locally-finite variations

It is well known that one can integrate any compactly supported, continuous, differential n-form over n-dimensional C¹-manifolds in ℝ^m (m ≥ n). For n = 1, the integral may be defined over any locally rectifiable curve. Another generalization is the theory of currents (linear functionals on the space of compactly supported C^∞-differential forms). The theme of the article is integration of measurable differential n-forms over n-dimensional n C⁰-manifolds in ℝ^m with locally-finite n-dimensional variations (a generalization of locally rectifiable curves to dimension n > 1). The main result states that any such manifold generates an n-dimensional current in ℝ^m (i.e., any compactly supported C^∞ n-form can be integrated over a manifold with the properties mentioned above). Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 168–206.     

On the limit of square-Cesàro means of contractions on Hilbert spaces

By combining tools from functional analysis and algebraic number theory, we investigate the qualitative properties of the square-Cesàro means ${{\frac{1}{N}}\sum_{n=1}^N T^{n^2}}By combining tools from functional analysis and algebraic number theory, we investigate the qualitative properties of the square-Cesàro means \frac1N?_n=1^N Tⁿ²{{\frac{1}{N}}\sum_{n=1}^N T^{n^2}}, where T is a contraction on a Hilbert space. The existence of their limit in the strong operator topology as N → ∞ is known even for general polynomial sequences, but the limit itself has not yet been studied apart from the linear case solved by the von Neumann Ergodic Theorem. The case of the sequence n ² is the next step beyond this linear case, and we show that even though the limit operator is generally not a projection, it still has a relatively simple and interesting structure.     

Linear spaces withn + n + 2lines

It is shown that every non-degenerate linear space withn² + n + 2lines,n ≥ 6, hasv n² + 1 − epoints, whereeis the unique positive real number withn =½e(e + 1). For values ofnfor whicheis an integer, it is shown that the linear spaces withn² + 1 − epoints andn² + n + 2lines are related to symmetric divisible designs.     

Scattering of small solutions to generalized benjamin-bona-mahony equation in several space dimensions

We show that the supremum norm of solutions with small initial data of the generalized Benjamin-Bona-Mahony equation u_t-△u_t=(b,▽u)+u^p(a,▽u)in x?Rⁿ,n≥2, with integer p≥3 , decays to zero like t^-2/3 if n=2 and like t^-1+6, for any δ0, if n≥3, when t tends to infinity. The proofs of these results are based on an analysis of the linear equation u_t-△=(b,▽u)) and the associated oscillatory integral which may have nonisolated stationary points of the phase function.     

Natural maps on the iterated jet prolongation of a fibred manifold

Summary Using an analytic procedure by the first author [5, 6], we first determine all natural transformations of the second iterated jet prolongation J¹J¹Y of a fibred manifold YX into itself depending on a linear connection on the base manifold X. We obtain two 3-parameter families and we interpret them geometrically. Our results clarify the distinguished role of the involution on J¹ J¹ Y depending on introduced by the second author [11]. Then we discuss the role of our transformations in the theory of the natural operators transforming a connection on a fibred manifold YX and a linear connection on X into a connection on the first jet prolongation J¹YX of Y. In the final remark we determine all natural transformations of the second sesquiholonomic and holomonic jet prolongations of Y into themselves. Our attention to second order jet spaces is due to the role they play in fundamental geometric and physical theories (cf. curvature of connections and lagrangian theories).     

Linear Equations and Regularity Conditions on Semigroups

R. Croisot in 1953, stated a very interesting problem of classification of all types of the regularity of semigroups defined by equations of the form a = a^mxaⁿ, with m,n \ge 0, m + n \ge 2. He proved that any of these equations determines either the ordinary regularity, left, right or complete regularity (see also the book by A. H. Clifford and G. B. Preston, Section 4.1). A similar problem, concerning all types of the regularity of semigroups and their elements defined by equations of the form a = a^pxa^qya^r, with p,q,r \ge 0, was treated by S. Lajos and G. Szász in 1975. The purpose of this paper is to generalize the results by Croisot, Lajos and Szász considering all types of the regularity of semigroups and their elements determined by more general equations called linear. We determine all types of the regularity of elements defined by linear equations, and prove that there are exactly 14 types of the regularity of semigroups defined by such equations. We also give implication diagrams for linear equations and regularity conditions.     

A convolution formula for time‐invariant linear communication systems

For every linear and time‐invariant time‐discrete (communication) system T : l^∞ → l^∞, formally, the following convolution formula can be derived: \input amssym.def $$(Tf)(n)=\sum_{k \in {\Bbb Z}} h(n‐k) f(k),\quad n \in {\Bbb Z}, f \in l^{\infty},$$ where h = Tδ is the delta impulse response. This paper is concerned with the question under which assumptions linear and time‐invariant time‐discrete systems T : l^∞ → l^∞ can be characterized by this formula. For this purpose we derive a convolution formula in a more general situation which also leads to a well‐known convolution formula in the time‐continuous case. Copyright © 1999 John Wiley & Sons, Ltd.     

Maximal Points of Convex Sets in Locally Convex Topological Vector Spaces: Generalization of the Arrow–Barankin–Blackwell Theorem

In 1953, Arrow, Barankin, and Blackwell proved that, if C is a nonempty compact convex set in Rⁿ with its standard ordering, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. In this paper, we present a generalization of this result. We show that that, if C is a compact convex set in a locally convex topological space X and if K is an ordering cone on X such that the quasi-interiors of K and the dual cone K* are nonempty, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. For example, our work shows that, under the appropriate conditions, the density results hold in the spaces Rⁿ, L^p(, ), 1p, l^p, 1p, and C (), a compact Hausdorff space, when they are partially ordered with their natural ordering cones.     

A family of 2-weight codes related to BCH-codes

For every prime-power q and every pair of natural numbers m ≤ n′, we construct a q-ary linear code of length q^m (q^n′ − 1)(q^n′ − q^n′−m + 1)/(q − 1) and dimension 3n′, whose only nonzero weights are and . These code parameters and those of the corresponding family of strongly regular graphs are new in odd characteristic. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 391–396, 1997     

Hierarchical Decompositions and Circular Ray Shooting in Simple Polygons

A hierarchical decomposition of a simple polygon is introduced. The hierarchy has logarithmic depth, linear size, and its regions have at most three neighbors. Using this hierarchy, circular ray shooting queries in a simple polygon on n vertices can be answered in O(log² n) query time and O(n log n) space. If the radius of the circle is fixed, the query time can be improved to O(log n) and the space to O(n).     

An Alternate Characterization of the Bentness of Binary Functions, with Uniqueness

In a previous paper, we have obtained a characterization of the binary bent functions on (GF(2))ⁿ (n even) as linear combinations modulo , with integral coefficients, of characteristic functions (indicators) of -dimensional vector-subspaces of (GF(2))ⁿ. There is no uniqueness of the representation of a given bent function related to this characterization. We obtain now a new characterization for which there is uniqueness of the representation.