Spline interpolation at knot averages

It is well known that when interpolation points coincide with knots, the knot sequence must obey some restriction in order to guarantee the existence and boundedness of the interpolation projector. But, when the interpolation points are chosen to be the knot averages, the corresponding quadratic or cubic spline interpolation projectors are bounded independently of the knot sequence. Based on this fact, de Boor in 1975 made a conjecture that interpolation by splines of orderk at knot averages is bounded for anyk. In this paper we disprove de Boor's conjecture fork 20.Communicated by Wolfgang Dahmen.     

The Field Descent Method

We obtain a broadly applicable decomposition of group ring elements into a “subfield part” and a “kernel part”. Applications include the verification of Lander’s conjecture for all difference sets whose order is a power of a prime >3 and for all McFarland, Spence and Chen/Davis/Jedwab difference sets. We obtain a new general exponent bound for difference sets. We show that there is no circulant Hadamard matrix of order v with 4<v<548, 964, 900 and no Barker sequence of length l with 13 < l ≤ 10²².     

Topological realizations of ortho-projection graphs

We discuss the production of ortho-projection graphs from alternating knot diagrams, and introduce a more general construction of such graphs from “splittings” of closed, non-orientable surfaces. As our main result, we prove that this new topological construction generates all ortho-projection graphs. We present a minimal example of an ortho-projection graph that does not arise from a knot diagram, and provide a surface-splitting that realizes this graph.     

On deformation of associative algebras and graph homology

Deformation theory of associative algebras and in particular of Poisson algebras is reviewed. The role of an “almost contraction” leading to a canonical solution of the corresponding Maurer–Cartan equation is noted. This role is reminiscent of the Homotopical Perturbation Lemma, with the infinitesimal deformation cocycle as “initiator.”Applied to star-products, we show how Moyal's formula can be obtained using such an almost contraction and conjecture that the “merger operation” provides a canonical solution at least in the case of linear Poisson structures.     

Upper limits of Sinai’s walk in random scenery

We consider Sinai’s walk in i.i.d. random scenery and focus our attention on a conjecture of Révész concerning the upper limits of Sinai’s walk in random scenery when the scenery is bounded from above. A close study of the competition between the concentration property for Sinai’s walk and negative values for the scenery enables us to prove that the conjecture is true if the scenery has “thin” negative tails and is false otherwise.     

On a long-standing conjecture by Pólya–Szegö and related topics

The electrostatic capacity of a convex body is usually not simple to compute. We discuss two possible approximations of it. The first one is related to a long-standing conjecture by Pólya–Szegö. It states that, among all convex bodies, the “worst shape” for the approximation exists and is the planar disk. We prove the first part of this conjecture, and we establish some related results which give further evidence for the validity of the second part. We also suggest some complementary conjectures and open problems. The second approximation we study is based on the use of web functions.Received: September 29, 2003     

Relative Stability for Strictly Stationary Sequences

For a nonnegative strictly stationary random sequence satisfying the “minimal” dependence condition necessary and sufficient conditions for the relative stability are found. As an application the well-known Khinchine stability result for i.i.d. random variables is proved for uniformly strong mixing sequences.     

On a long-standing conjecture by Pólya–Szeg? and related topics

The electrostatic capacity of a convex body is usually not simple to compute. We discuss two possible approximations of it. The first one is related to a long-standing conjecture by Pólya–Szegö. It states that, among all convex bodies, the “worst shape” for the approximation exists and is the planar disk. We prove the first part of this conjecture, and we establish some related results which give further evidence for the validity of the second part. We also suggest some complementary conjectures and open problems. The second approximation we study is based on the use of web functions.     

Two positivity conjectures for Kerov polynomials

Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the “free cumulants” of the associated Young diagram. We present two positivity conjectures for their coefficients. The latter are stronger than the positivity conjecture of Kerov–Biane, recently proved by Féray.     

A version of the volume conjecture

We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special linear group of degree two over complex numbers. We also confirm the conjecture for the figure-eight knot and torus knots. This version is different from S. Gukov's because of a choice of polarization.     

An algorithm for segment approximation

Summary An algorithm for computing a set of knots which is optimal for the segment approximation problem is developed. The method yields a sequence of real numbers which converges to the minimal deviation and a corresponding sequence of knot sets. This sequence splits into at most two subsequences which converge to leveled sets of knots. Such knot sets are optimal. Numerical results concerning piecewise polynomial approximation are given.     

The Kervaire-Laudenbach Conjecture and Presentations of Simple Groups

The statement “no non-Abelian simple group can be obtained from a non-simple one by adding one generator and one relator” first is equivalent to the Kervaire-Laudenbach conjecture, and second, becomes true under the additional assumption that an initial non-simple group is either finite or torsion free.Supported by RFBR grant No. 02-01-00170.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 399–437, July–August, 2005.     

Face Size and the Maximum Genus of a Graph 1. Simple Graphs

This paper shows that a simple graph which can be cellularly embedded on some closed surface in such a way that the size of each face does not exceed 7 is upper embeddable. This settles one of two conjectures posed by Nedela and koviera (1990, in “Topics in Combinatorics and Graph Theory,” pp. 519–529, Physica Verlag, Heidelberg). The other conjecture will be proved in a sequel to this paper.     

Successive approximation and linear stability involving convergent sequences of optimization problems

A certain convergence notion for extended real-valued functions, which has been studied by a number of authors in various applied contexts since the latter 1960s, is examined here in relation to abstract optimization problems in normed linear spaces. The main facts concerning behavior of the optimal values, the optimal solution sets and the -optimal solution sets corresponding to “convergent” sequences of such problems are developed. General linear perturbations are incorporated explicitly into the problems of the sequence, lending a stability-theoretic character to the results. Most of the results apply to nonconvex minimization.     

Reachability realization and stabilizability of switched linear discrete-time systems

In this paper, the reachability realization of a switched linear discrete-time system, which is a collection of linear time-invariant discrete-time systems along with some maps for “switching” among them, is addressed. The main contribution of this paper is to prove that for a switched linear discrete-time system, there exists a basic switching sequence such that the reachable (controllable) state set of this basic switching sequence is equal to the reachable (controllable) state set of the system. Hence, the reachability (controllability) can be realized by using only one switching sequence. We also discuss the stabilizability of switched systems, and obtain a sufficient condition for stabilizability. Two numeric examples are given to illustrate the results.     

The Teichmüller TQFT volume conjecture for twist knots

The Teichmüller TQFT, defined by Andersen and Kashaev, gives rise to a quantum invariant of triangulated hyperbolic knot complements; it has an associated volume conjecture, where the hyperbolic volume of the knot appears as a certain asymptotic coefficient.In this note, we announce a proof of this volume conjecture for all twist knots up to 14 crossings; along the way we explicitly compute the partition function of the Teichmüller TQFT for the whole infinite family of twist knots.Among other tools, we use an algorithm of Thurston to construct a convenient ideal triangulation of a twist knot complement, as well as the saddle point method for computing limits of complex integrals with parameters.     

Random creation and dispersion of mass

Consider evolution of density of a mass or a population, geographically situated in a compact region of space, assuming random creation-annihilation and migration, or dispersion of mass, so the evolution is a random measure. When the creation-annihilation and dispersion are diffusions the situation is described formally by a stochastic partial differential equation; ignoring dispersion make approximations to the initial density by atomic measures and if the corresponding discrete random measures converge “in law” to a unique random measure call it a solution. To account for dispersion Trotter's product formula is applied to semiflows corresponding to dispersion and creation-annihilation. Existence of solutions has been a conjecture for several years despite a claim in ([2], J. Multivariate Anal. 5, 1–52). We show that solutions exist and that non-deterministic solutions are “smeared” continuous-state branching diffusions.     

Genetic algorithms with noisy fitness

The convergence properties of genetic algorithms with noisy fitness information are studied here. In the proposed scheme, hypothesis testing methods are used to compare sample fitness values. The “best” individual of each generation is kept and a greater-than-zero mutation rate is used so that every individual will be generated with positive probability in each generation. The convergence criterion is different from the frequently-used uniform population criterion; instead, the sequence of the “best” individual in each generation is considered, and the algorithm is regarded as convergent if the sequence of the “best” individuals converges with probability one to a point with optimal average fitness.