Set Pn Research Articles

Irregularities of distribution and geometry of planar convex sets

We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When ∂C is C2 regardless of curvature, we prove that for every set PN of N points in T2 we have the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN1/2. When ∂C is only piecewise C2 and is not a polygon we prove the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN2/5. We also give a whole range of intermediate sharp results between N2/5 and N1/2. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of C.

Parking functions, tree depth and factorizations of the full cycle into transpositions

Consider the set Fn of factorizations of the full cycle (0 1 2 · · · n) ∈ S{0,1,...,n} into n transpositions. Write any such factorization (a1 b1) · · · (an bn) with all ai < bi to define its lower and upper sequences (a1, . . . , an) and (b1,...,bn), respectively. Remarkably, any factorization can be uniquely recovered from its lower (or upper) sequence. In fact, Biane (2002) showed that the simple map sending a factorization to its lower sequence is a bijection from Fn to the set Pn of parking functions of length n. Reversing this map to recover the factorization (and, hence, upper sequence) corresponding to a given lower sequence is nontrivial.

Boolean Properties and Bell-Like Inequalities of Numerical Events

Let S be a set of states of a physical system and p(s) be the probability of the occurrence of an event when the system is in state s. A function p from S to [0,1] is called a numerical event or alternatively an S-probability. If a set P:={p(s)|s in S} is ordered by the order of real functions such that certain plausible requirements are fulfilled, P becomes an orthomodular poset in which properties can be described by the addition and comparison of functions. P is then called an algebra of S-probabilities or algebra of numerical events. We first answer the question under which circumstances it is possible to consider sets of empirically found numerical events as members of an algebra of S-probabilities. Then we discuss the problem to decide whether a given small set Pn of S-probabilities can be embedded into a Boolean subalgebra of an algebra P of S-probabilities, in which case we will call Pn Boolean embeddable. If Pn is not Boolean embeddable, then the physical system at hand will most likely be non-classical. In the case of a concrete logic P, that is a quantum logic which can be represented by sets, we derive criteria for the subset Pn of P to be Boolean embeddable which can be checked by very simple procedures, for arbitrary S-probabilities we provide sets of Bell-like inequalities which characterize the Boolean embeddability of Pn. Finally we will show how these Bell-like inequalities fit into a general framework of Bell inequalities by providing a method for generating Bell inequalities for S-probabilities from elementary Bell valuations.

Sets and multisets of range uniqueness for polynomials

A set S⊆R is called a set of range uniqueness (SRU) for the set Pn of real polynomials of degree at most n, if for all f,g∈Pn, f[S]=g[S]⇒f=g. We show that for every natural number n, there are SRUs for Pn of cardinality 2n+1, but there are no such SRUs of size 2n. We also construct SRUs for the set P of all real polynomials.

The minimal [formula omitted]-dispersion of point sets in high dimensions

In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set Pn={x1,…,xn}⊂[0,1]d and k∈{0,1,…,n}, we define the k-dispersion to be the volume of the largest box amidst a point set containing at most k points. The minimal k-dispersion is then given by the infimum over all possible point sets of cardinality n. We provide both upper and lower bounds for the minimal k-dispersion that coincide with the known bounds for the classical minimal dispersion for a surprisingly large range of k’s.

Matrix geometric means based on shuffled inductive sequences

We propose a new deterministic sequence converging to the least squares mean of N symmetric positive definite (SPD) matrices. By “least squares mean”, we refer to the Riemannian barycenter with respect to the natural metric (also known as the trace metric or affine-invariant metric) on the set Pn of n×n SPD matrices. In some papers, this mean is also referred to as the Karcher mean. The sequence we propose belongs to the family of inductive sequences, obtained by letting a point take successive steps towards the data points, with step lengths progressively diminishing to zero. We use the word “inductive” since each point of those sequences can be seen as the inductive mean of a well-chosen set of points containing multiple replications of the data points. We show that visiting the data points in a cyclic order usually results in a slow convergence, and remedy this weakness by reordering repeatedly the data points using a shuffling algorithm. We prove the convergence of the resulting inductive sequence to the least squares mean of the data in the general framework of NPC spaces (complete metric spaces with non-positive curvature), which includes the set Pn. We also illustrate numerically on Pn that some points of this sequence are fairly accurate estimates of the least squares mean, while being considerably cheaper to compute, and perform well as initializers for state-of-the-art algorithms. To reduce further the computation time, we finally exploit inductive sequences to produce a family of parallelizable algorithms for estimating the least squares mean.

A lower bound for the dispersion on the torus

We consider the volume of the largest axis-parallel box in the d-dimensional torus that contains no point of a given point set Pn with n elements. We prove that, for all natural numbers d,n and every point set Pn, this volume is bounded from below by min{1,d/n}. This implies the same lower bound for the discrepancy on the torus.

Several extremal approximation problems for the characteristic function of a spherical layer

We discuss three interrelated extremal problems on the set Pn,m of algebraic polynomials of a given degree n on the unit sphere \(\mathbb{S}^{m - 1}\) of the Euclidean space ℝm of dimension m ≥ 2. (1) Find the norm of the functional \(F\left( \eta \right) = F_h P_n = \int_{\mathbb{G}\left( \eta \right)} {P_n (x)dx}\), which is the integral over the spherical layer \(\mathbb{G}\left( \eta \right) = \left\{ {x = \left( {x_1 , \ldots ,x_m } \right) \in \mathbb{S}^{m - 1} :h' \leqslant x_m \leqslant h''} \right\}\) defined by a pair of real numbers η = (h′, h″), −1 ≤ h′ < h″ ≤ 1, on the set Pn,m with the norm of the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of functions summable on the sphere. (2) Find the best approximation in \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) of the characteristic function χη of the layer \(\mathbb{G}\left( \eta \right)\) by the subspace Pn,m⊥ of functions from \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) that are orthogonal to the space of polynomials Pn,m. (3) Find the best approximation in the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of the function χη by the space of polynomials Pn,m. We present a solution of all three problems for the values h′ and h″ that are neighboring roots of the polynomial in a single variable of degree n + 1 that deviates least from zero in the space L1φ (−1, 1) of functions summable on the interval (−1, 1) with ultraspherical weight φ(t) = (1 − t2)α, α = (m − 3)/2. We study the respective one-dimensional problems in the space of functions summable on (−1, 1) with an arbitrary not necessarily ultraspherical weight.

On the key exchange with new cubical maps based on graphs

Families of edge transitive algebraic graphs Fn(K), over the commutative ring K were used for the graph based cryptographic algorithms. We introduce a key exchange protocol defined in terms of bipartite graph An(K), n i¾? 2 with point set Pn and line set Ln isomorphic to n-dimensional free module Kn. Graphs A(n, K) are not vertex and edge transitive. There is a well defined projective limit lim A(n, K) = A(K), n i¾? ∞ which is an infinite bipatrtite graph with point set P = lim Pn and line set L = limLn. Let K be a commutative ring contain at least 3 regular elements (not zero divisors). For each pair of (n, d), n i¾? 2, n i¾? 1 and sequence of elements α1, α2, ', α2d, such that α1, αi+αi+1, i = 1, 2, ', 2d, i = 1, 2, ' 2d-1 and α2d+α1 are regular elements of the ring K. We define polynomial automorphism hn = hn (d, α1, α2, ', α2d) of variety Ln (or Pn). The existence of projective limit lim An(K) guarantees the existence of projective limit h = h(d, α1, α2, ', α2d) = lim hn, n i¾? ∞ which is cubical automorphism of infinite dimensional varieties L (or P). We state that the order of h is an infinity. There is a constant n0 such that hn, n i¾? n0 is a cubical map. Obviously the order of hn is growing with the growth of n and the degree of polynomial map (hn)k from the Cremona group of all polynomial automorphisms of free module Kn with operation of composition is bounded by 3. Let i¾? be affine automorphism of Kn i.e. the element of Cremona group of degree 1. We suggest symbolic Diffie Hellman key exchange with the use of cyclic subgroup of Cremona group generated by i¾?-1hni¾?. In the case of K = Fp, p is prime, the order of hn is the power of p. So the order is growing with the growth of p. We use computer simulation to evaluate the orders in some cases of K = Zm, where m is a composite integer.

The essentially tame local Langlands correspondence, III: the general case

Let F be a non-Archimedean local field with finite residue field. An irreducible smooth representation of the Weil group 𝒲F of F is called essentially tame if its restriction to wild inertia is a sum of characters. Let 𝒢net (F) denote the set of isomorphism classes of irreducible, essentially tame representations of 𝒲F of dimension n. The Langlands correspondence induces a bijection of 𝒢net (F) with a certain set 𝒜net (F) of irreducible cuspidal representations of GLn(F). We work with the set Pn(F) of isomorphism classes of admissible pairs (E/F, ξ) of degree n. There is an obvious bijection of Pn(F) with 𝒢net (F) and an explicit bijection of Pn(F) with 𝒜net (F). Together, these maps give a canonical bijection of 𝒢net (F) with 𝒜net (F). We showed in an earlier paper that the Langlands correspondence is obtained by composing the map Pn (F) → 𝒜net (F) with a permutation of Pn(F) of the form (E/F, ξ) ↦ (E/F, μξξ), where μξ is a tamely ramified character of E× depending on ξ. In this paper, we show that there is a canonical choice for the character μξ and determine it explicitly.

The Solution Polyhedron of the Dissipative Inequality for Relaxation SISO Systems

maximum multiplicity singular points of the semialgebraic set 1 . The polyhedron convPn provides a good approximation to the set . Moreover, note that the solution polyhedron convPn has a number of properties similar to the geometric properties of the correlation polyhedron COR n and is likely to be combinatorially equivalent to COR n . The geometric properties of the polyhedron COR n and the anely equivalent cut polyhedron were considered in [9]. Possible applications of the correlation and cut polyhedra in discrete mathematics, analysis, mathematical physics, and probability theory were also considered there. In a number of papers on combinatorial optimization (see references in [9]) ,COR n is referred to as the Boolean quadratic polyhedron. The present paper is organized as follows. In the rst section, we give necessary information on the dierential-geometric structure of the set and an explicit description of the set Pn on the basis of the exact solution [7] of algebraic Riccati equations of a special class. We dene two forms of indexing of elements of Pn to be used in the second and third sections for the description of anely independent and anely dependent systems of points in Pn. This description is based on an anely equivalent representation of the set Pn in the vector space spn;2 of real symmetric polynomials of degree n in each of the two variables. Corollaries to results of the second section for the solution and correlation polyhedra are given in the third section. The results were partly announced in [10].

Preconditioned Equation Sets and Their Applications

AbstractWe establish the w‐circulant preconditioned equations set PN[w]Xw = B corresponding to the general Toeplitz equation set ANX = B. By choosing different criteria for constructing preconditioners, we obtain different preconditioned equation sets PN[w]Xw = B and an optimal value of w. Theoretical analysis and practical calculation prove that the inversed results by using this method are much better than those by the traditional Fourier method.

Super magical Partitions of an Integer

Let n≥2 be an integer and Pn={p1,p2,…,pr} be a set of r≥2 partitions of n satisfying the following conditions: (i) in each pi the integers are all distinct, and (ii) any two distinct pi's have at most one part in common. Further, the set Pn is called a supermagical configuration of n and of order p if it satisfies ⋃rj=1pj={1,2,…,p} for some positive integer p. Two special types of supermagical configurations of n called cycles and stars are defined and some necessary conditions for a positive integer n to admit such a configuration are obtained.

Graphs of Non-Crossing Perfect Matchings

Let Pn be a set of n=2m points that are the vertices of a convex polygon, and let ℳm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross, and edges joining two perfect matchings M1 and M2 if M2=M1−(a,b)−(c,d)+(a,d)+(b,c) for some points a,b,c,d of Pn. We prove the following results about ℳm: its diameter is m−1; it is bipartite for every m; the connectivity is equal to m−1; it has no Hamilton path for m odd, m>3; and finally it has a Hamilton cycle for every m even, m≥4.

Powers of the Fundamental Ideal in the Witt Ring

Let K be a field of characteristic different from 2. In the algebraic theory of quadratic forms, one studies the Witt ring W(K) of equivalence classes of non-degenerate quadratic forms. The Witt ring has a filtration given by the powers In(K) of the fundamental ideal I(K) of even-dimensional forms. The ideal In(K) is generated by the set Pn(K) of n-fold Pfister forms, 〈〈a1,…,an〉〉≔⊗ni=1〈1,−ai〉, where ai∈K*=K\\;{0}. Many questions about this filtration and its quotients have arisen in the study of W(K). The spectacular work of Voevodsky, together with his collaborative work with Orlov and Vishik, allows one to answer many old questions. The purpose of this paper is to indicate some of these solutions.

A proof of the Mullineux conjecture

A partition λ of a positive integer n is a sequence λ1 λ2 λm 0 of integers such that ∑λi n. For a positive integer p, a partition λ λ1 λ2 λm (or its Young diagram) is called p-regular if it does not have p or more equal parts, i.e. if there does not exist t m p 1 with λt λt 1 λt p 1. Let F be a field of characteristic p 0. It is well known that irreducible representations of the symmetric group Sn over F are naturally parametrized by pregular partitions of n (cf. for example [9, 12]). If λ is such a partition we denote the corresponding irreducible module by Dλ. Let sgnn be the one-dimensional sign representation of Sn over F; i.e., sgnn F as a vector space and g f sgn g f for any g Sn f F . Here sgn g is just the sign of the permutation g. It is clear that for any irreducible Dλ, the tensor product Dλ sgnn is also irreducible. The problem, usually called the problem of Mullineux, is to find the p-regular partition μ such that Dλ sgnn Dμ. Put Dλ sgnn Dn λ In this way a bijection bn on the set Pn of p-regular partitions of n is defined for each positive integer n, and the problem is:

Parameter estimation for a stationary process on a d-dimensional lattice

SUMMARY We study asymptotic properties of various estimation procedures for a general stationary regular process on a d-dimensional lattice. Differences between d = 1, time series, and d > 2, spatial processes, are pointed out. We suppose that the process is observed on a set PN, with CN points, which tends to infinity with the same speed in all directions. The relative edge effect is of order CN lId, increasing with d: this effect is without statistical importance if d = 1, but is important if d > 2. We give a lCN-consistent, asymptotically normal estimator of the underlying parameter, the procedure being constructed by a modification of Whittle's approximation to the log likelihood. In the Gaussian case, this procedure is asymptotically efficient.

On the classification of balanced binary sequences of period&amp;lt;tex&amp;gt;2^n-1&amp;lt;/tex&amp;gt;(Corresp.)

Let U be the set of all binary sequences of period p=2^{n}-1 containing (p+1)/2 ones and (p-1)/2 zeros per period. There is a lattice of interesting subsets of U , the smallest of which is the set PN (the maximum-length linear shift register sequences of period p ). In between are sets with the run statistics of PN , with the correlation properties of PN , with the n (that every nonzero subseqnonce of length n occurs in each period), and others. Results concerning the interrelationships of these subsets are obtained, examples are given to show that certain intersections of subsets are nonempty, and conjectures are formulated regarding other intersections of subsets. For example, it is conjectured that all span- n sequences with the two-level autocorrelation property are in class PN . Some relationships between run properties and correlation properties of binary sequences are also obtained.

Distribution of the Number of Claims in Motor Insurance according to the Lag of Settlement

Let Sn be the set of motor claims Si, i = 1, 2, …, N occurred during a given year n, namely Sn represents the set of claims relevant to the generation (or cohort) n.If Tn is a subset (even empty) of claims resulting without payement (that is the set of zero-claims), the set Pn = Sn − Tn shall denote the set of claims that should be settled.For every si ∈ Pn, we can define the r.v. Xi which represents the period of time required for its settlement (namely the lag of settlement).It is not sensible to deem that the r.v. Xi are equally distributed: as a matter of fact we know that the larger is the claim, the longer the lag of payement.However, we can assume that in a subset U of Pn, the r.v. Xi(U) have the same distribution function, which will be denoted by FU(x) or in short F(x).As F(x) represents the probability that a claim sj ∈ U is settled within a period o − x, the function 1 − F(x) = l(x) denotes the probability that the claim results unsettled after a lag x, that is the survival function of the claim.In this study we intend to find an analytical expression of the function 1(x) on the basis of particular assumptions about the behaviour of the adjuster with regard to the settlement of claims.The assumptions will be tested by fitting the function to some observed data.

On Some Applications of Graph Theory to Geometry

Let [Pn(k)] be the class of all subsets Pn(k) of the k-dimensional Euclidean space consisting of n distinct points and having diameter 1. Denote by dk(n, r) the maximum number of times a given distance r can occur among points of a set Pn(k).PutIn other words Dk(n) denotes the maximum number of times the same distance can occur between n suitably chosen points in k-dimensional space.