Dubins’ problem on surfaces. I. nonnegative curvature

Let M be a complete, connected, two-dimensional Riemannian manifold. Consider the following question: Given any (p₁,v₁) and (p₂, v₂) in T M, is it possible to connect p₁ to P₂ by a curve y in M with arbitrary small geodesic curvature such that, for i = 1, 2, y is equal to v_i at p_i? In this article, we bring a positive answer to the question if M verifies one of the following three conditions: (a) M is compact, (b) M is asymptotically flat, and (c) M has bounded nonnegative curvature outside a compact subset.     

The tangent approximation to one-sided Brownian exit densities

Summary Let p(t) be the density of the first-exit time of a Brownian motion over a one-sided moving boundary, and let p ₁(t) be the density at t of the first-exit time over the tangent to the boundary at t. When is p ₁(t) a good approximation to p(t)? We investigate this question by means of a new integral equation for p(t) which makes possible explicit estimates for the error of the approximation.Work supported by the Deutsche Forschungsgemeinschaft at the Sonderforschungsbereich 123, Universität Heidelberg     

One condition for solution uniqueness and robustness of both l1-synthesis and l1-analysis minimizations

The ? ₁-synthesis model and the ? ₁-analysis model recover structured signals from their undersampled measurements. The solution of the former is a sparse sum of dictionary atoms, and that of the latter makes sparse correlations with dictionary atoms. This paper addresses the question: when can we trust these models to recover specific signals? We answer the question with a condition that is both necessary and sufficient to guarantee the recovery to be unique and exact and, in the presence of measurement noise, to be robust. The condition is one–for–all in the sense that it applies to both the ? ₁-synthesis and ? ₁-analysis models, to both constrained and unconstrained formulations, and to both the exact recovery and robust recovery cases. Furthermore, a convex infinity–norm optimization problem is introduced for numerically verifying the condition. A comprehensive comparison with related existing conditions is included.     

Topologically Complete Representations of Inverse Semigroups

A representation of an inverse semigroup by means of partial open homeomorphisms of a topological T ₀ -space is called topologically complete if the domains of these partial homeomorphisms form a base of the topology. It is shown how to construct topologically complete representations on the base of a ternary relation satisfying some elementary axioms. This result makes it possible to obtain a pseudo-elementary axiomatization for inverse semigroups that have faithful topologically complete representations in T ₁ ,T ₂ and T ₃ -spaces. A topology is introduced on any antigroup; this topology is a concomitant of the algebraic structure and every topologically complete representation is continuous with respect to this topology.     

Topologically complete representations of inverse semigroups

A representation of an inverse semigroup by means of partial open homeomorphisms of a topological T₀-space is called topologically complete if the domains of these partial homeomorphisms form a base of the topology. It is shown how to construct topologically complete representations on the base of a ternary relation satisfying some elementary axioms. This result makes it possible to obtain a pseudo-elementary axiomatization for inverse semigroups that have faithful topologically complete representations in T₁, T₂ and T₃-spaces. A topology is introduced on any antigroup; this topology is a concomitant of the algebraic structure and every topologically complete representation is continuous with respect to this topology.     

Saturation, Suslin trees and meager sets

We show, using a variation of Woodin’s partial order ℙ_max, that it is possible to destroy the saturation of the nonstationary ideal on ω₁ by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ_max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ₁, saturation of the nonstationary ideal, and no weakly Lusin sequences, answering a question of Shelah and Zapletal.Supported by the Japan Society for the Promotion of Science, the Mittag-Leffler Institute and the São Paulo State Research Support Foundation (FAPESP, Grant # 02/11551-3).     

The mixed Lp affine surface areas for functions

In this paper, we propose a definition of a general mixed L_p Affine surface area, ?n ≠ p ∈ ?, for multiple functions. Our definition is di?erent from and is “dual” to the one in [11] by Caglar and Ye. In particular, our definition makes it possible to establish an integral formula for the general mixed L_p Affine surface area of multiple functions (see Theorem 3.1 for more precise statements). Properties of the newly introduced functional are proved such as affine invariance, and related affine isoperimetric inequalities are proved.     

Perfect pre-images of ω1 and the PFA

Assuming the Proper Forcing Axiom, I give a widely applicable condition under which a pre-image of ω₁ includes a homeomorphic image of ω₁. This makes it possible to strengthen a number of results of Z. Balogh and P.J. Nyikos on non-metrizable manifolds and related spaces.     

RETRACTED ARTICLE: Discrete optimization method based on grassmannian parameterization in multidimensional dichotomic data structuring

A solution of the large computational time problem arising in multidimensional data structuring is addressed by employing algebraic properties of finite geometries. A vector parameterization of the Grassmannian Gr₂(k, n) is proposed which makes it possible to minimize the amount of memory and reduce the number of operations required to solve the problem. An algorithm based on this parameterization and the Gray codes is constructed; the algorithm is suitable for parallel computation, which further reduces computation time.     

Russian doll search for the Steiner triple covering problem

Russian doll search is applied to finding maximum independent sets in hypergraphs, focusing on a particular subproblem of the hitting set problem, the Steiner triple covering problem. An instance denoted A ₁₃₅ is solved considerably faster with Russian doll search than with integer linear programming and a state-of-the-art optimization tool (using otherwise a similar established approach to split the problem into subproblems). In addition, the improvement in speed makes it possible to carry out a search proving that all optimal solutions for A ₁₃₅ are isomorphic.     

Multiplicative inequalities for the L 1 norm: Applications in analysis and number theory

The paper is devoted to multiplicative lower estimates for the L ₁ norm and their applications in analysis and number theory. Multiplicative inequalities of the following three types are considered: martingale (for the Haar system), complex trigonometric (for exponential sums), and real trigonometric. A new method for obtaining sharp bounds for the integral norm of trigonometric and power series is proposed; this method uses the number-theoretic and combinatorial characteristics of the spectrum. Applications of the method (both in H ¹ and L ₁) to an important class of power density spectra, including [n ^α] with 1 ≤ α < ∞, are developed. A new combinatorial theorem is proved that makes it possible to estimate the arithmetic characteristics of spectra under fairly general assumptions.     

Some remarks on metaplectic cusp forms and the correspondences of Shimura and Waldspurger

The simple relation between representations of the covering groups of SL₂ and GL₂ makes it possible to fuse and extend the recent metaplectic results of Shimura, Waldspurger, Flicker, and ourselves. By giving a new (purely local andL-function theoretic) treatment of the Waldspurger-Shintani correspondence, we also simplify some of Waldspurger’s original results.     

On complex matrices that are unitarily similar to real matrices

There are well-known conditions ensuring that a complex n × n matrix A can be converted by a similarity transformation into a real matrix. Is it possible to realize this conversion via unitary similarity rather than a general one? The following answer to this question is given in this paper: A matrix A ∈ M _n (?) can be made real by a unitary similarity transformation if and only if A and ā are unitarily similar and the matrix P transforming A into ā can be chosen unitary and symmetric at the same time. Effective ways for verifying this criterion are discussed.     

The superstar packing problem

Hell and Kirkpatrick proved that in an undirected graph, a maximum size packing by a set of non-singleton stars can be found in polynomial time if this star-set is of the form {S ₁, S ₂, ..., S _k } for some k∈ℤ₊ (S _i is the star with i leaves), and it is NP-hard otherwise. This may raise the question whether it is possible to enlarge a set of stars not of the form {S ₁, S ₂, ..., S _k } by other non-star graphs to get a polynomially solvable graph packing problem. This paper shows such families of depth 2 trees. We show two approaches to this problem, a polynomial alternating forest algorithm, which implies a Berge-Tutte type min-max theorem, and a reduction to the degree constrained subgraph problem of Lovász. Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No. 504438.     

Non-isochronicity of the center in polynomial Hamiltonian systems

In 2002 Jarque and Villadelprat proved that planar polynomial Hamiltonian systems of degree 4 have no isochronous centers and raised an open question for general planar polynomial Hamiltonian systems of even degree. Recently, it was proved that a planar polynomial Hamiltonian system is non-isochronous if a quantity, denoted by M_2m−2, can be computed such that M_2m−2≤0. As a corollary of this criterion, the open question was answered for those systems with only even degree nonlinearities. In this paper we consider the case of M_2m−2>0 and give a new criterion for non-isochronicity. Applying the new criterion, we also answer the open question for some cases in which some terms of odd degree are included.     

Eight octads suffice

The following question is answered: What is the minimum number of octads required to define a Steiner system $\text{l}$(5, 8, 24)? A set of octads is said to define a particular Steiner system if it is a subset of the special octads of that system, but of no other. As the title suggests, it is possible to produce a set of eight octads which defines the system, whereas any collection of seven octads is a subset of no Steiner system or of more than one. Note that, given an element of S₂₄, one is now able to decide whether it is in M₂₄ by simply applying it to a defining set of eight octads. If the images are in the system the element must be in M₂₄.     

Graded Nilpotent Algebras and Eggert's Conjecture

A group G is (l,m,n)-generated if it is a quotient group of the triangle group T(l,m,n) = (x,y,z|x ^l= y ^m= z ⁿ= xyz= 1). In [8] the problem is posed to find all possible (l,m,n)-generations for the non-abelian finite simple groups. In this paper we partially answer this question for the Janko group J ₃. We find all (2, 3, t)-generations as well as (2, 2,2,p)-generations, p a prime, for J ₃     

Method of fast expansions for solving nonlinear differential equations

A method is proposed for constructing fast converging Fourier series with the help of a special boundary function M _q . The convergence rate of the series is determined by the order q of M _q , which makes it possible to use a small number of series terms. The general theory of constructing fast expansions is described, the error of the partial sum of a series is estimated, and an example of a non- linear integrodifferential problem is considered. Due to its remarkable properties, the fast expansion method can be effectively used in applications.     

A Class of Dust-Like Self-Similar Solutions of the Massless Einstein�CVlasov System

In this paper the existence of a class of self-similar solutions of the Einstein–Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein–Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein–Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. This solution starts at a particular point P ₀ and converges to a stationary solution P ₁ as the independent variable tends to infinity. The existence proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes. The spacetimes constructed do not constitute a counterexample to cosmic censorship since they are not asymptotically flat. They should be seen as the first step on a possible path towards constructing solutions of importance for understanding the issue of the formation of naked singularities in general relativity.