Measuring the Size(1)

How do you measure the size of a one-dimensional figure?The size of a line segment is given by its length.How do you measure the size of two-dimensional figures?Rectangles and other polygons are two-dimensional.There are two common ways of measuring these figures.Perimeter measures the boundary.Area measures the space inside the figure.     

Small representations of the relation algebra ɛn+1(1, 2, 3)

Applying combinatorial methods, we prove that the symmetric relation algebra _n+1(1, 2, 3) ofn+1 atoms is finitely representable for alln 1, on at most (2+o(1))n² elements asn . We explicitly construct a representation of size 4.5n², for every n >1.Presented by B. Jonsson.     

On the reduction (mod 1) of completely monotone functions (0, ∞)

Summary The Euler-Maclaurin summation formula and its harmonic analysis (Poisson) are applied to the case of functions which are completely monotone on an open half-line. What thus results is a curious class of Fourier series, which can be determined explicitly and which represent completely monotone functions on the first half of the period. A by-product is the complete monotony (on the first half-period) of the Bernoulli functions, whether the index is integral or fractional.     

Ramsey Number of K_(2,s 1) vs.K_(1,n)

It is shown that the Ramsey number r(K_(2,s 1),K_(1,n))(?)n sn~(1/2) (s 3)/2 o(1) for large n,and r(K_(2,s 1),K_(1,n))∈{((q-1)~2/s) 1,((q-1)~2/s) 2},where n=((q-1)~2/s)-q 2 and q is a prime power such that s|(q-1).     

From a (G, k, 1) to a (Ck ⊕ G, k, 1) Difference Family

Given a subgroup N of an additive group G, a (G,N,k,1) difference family (DF) is a set D of k-subsets of G such that (d – d | d, d D, d d, D D) = G – N. Generalizing a construction by Genma, Jimbo, and Mishima [4], we give a new condition for realizing a (Ck G, Ck × {0}, k, 1)-DF starting from a (G, {0}, k, 1)-DF. Among the consequences, new cyclic Steiner 2-designs are obtained.     

Sequences in the range of a vector measure and Banach spaces satisfying Π1(X,l 1) = Π2(X,l 1)

If X is a Banach space such that Π₁(X,l ₁) = Π₂ (X, l ₁), we prove the following results: 1) A bounded sequence (x _n ) lies inside the range of some X-valued measure if and only if the operator (α _n ) ? l ₁ → Σ _n α _n x _n ? X is 1-summing, and 2) If A is a bounded subset of X lying in the range of some X ^??-valued measure, then A is necessarily contained in the range of some X-valued measure.     

The Missing Boundaries of the Santaló Diagrams for the Cases (d,ω, R ) and (ω ,R,r )

In this paper we solve two open problems posed by Santaló: to obtain complete systems of inequalities for some triples of measures of a planar convex set. Received February 17, 1999, and in revised form July 15, 1999.     

On the size of the set A(A + 1)

Let F _p be the field of a prime order p. For a subset A ì F_p{A \subset F_p} we consider the product set A(A + 1). This set is an image of A ×  A under the polynomial mapping f(x, y) = xy +  x : F _p ×  F _p → F _p . In the present note we show that if |A| <  p ^1/2, then

Commutator identities of the homotopes of (−1, 1)-algebras

We study the commutator algebras of the homotopes of (?1, 1)-algebras and prove that they are Malcev algebras satisfying the Filippov identity h _a (x, y, z) = 0 in the case of strictly (?1, 1)-algebras. We also proved that every Malcev algebra with the identities xy ³ = 0, xy ² z ² = 0, and h _a (x, y, z) = 0 is nilpotent of index at most 6.     

Explicit solutions of the equation –div z = f under the constraint ∥z ∥∞ = 1

We consider the eigenvalue problem for the 1-Laplace operator and the corresponding variational problem. Since the underlying functionals are not differentiable, the derivation of the Euler-Lagrange equation is highly nontrivial and involves methods of nonsmooth analysis. A remarkable fact is, that there are infinitely many functions f for which the equation –div z = f has to be satisfied. The purpose of the present work is the investigation of this equation. We show that in a square 2-dimensional domain for each right hand side f there are infinitely many vector fields z satisfying the Euler-Lagrange equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unusual interactions of (1/2, 0) ⊕ (0, 1/2) particles

We continue to study the ‘fermion — 4-vector potential’ interactions in the framework of the McLennan-Case construct which is a reformulation of the Majorana theory of the neutrino. This theory is shown after applying Majorana-likeanzatzen to give rise to appearance of unusual terms as σ [A×A^*], which were recently discussed in non-linear optics.     

On the stability of quasidouble step spline approximations for solutions ofy (n) = f(x,y, ⋯, y (n−1) )

In this paper, we continue the study [4] of the stability question of the quasidouble step spline function approximations,s(x) C ^m–2 , to the initial value problemy ⁽ⁿ⁾ = f(x, y, , y ^(n–1) ). It will be shown that the method is unstable and hence divergent form n + 4.     

Discrete Morse theory and a reformulation of the K(π,1)-conjecture

A recent theorem of Dobrinskaya [20 Dobrinskaya, N.È. (2006). Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes. Proc. Steklov Inst. Math. 252(1):30–46.[Crossref] , [Google Scholar]] states that the K(π,1)-conjecture holds for an Artin group G if and only if the canonical map BM→BG is a homotopy equivalence, where M denotes the Artin monoid associated to G. The aim of this paper is to give an alternative proof by means of discrete Morse theory and abstract homotopy theory. Moreover, we exhibit a new model for the classifying space of an Artin monoid, in the spirit of [13 Charney, R., Meier, J., Whittlesey, K. (2004). Bestvina’s normal form complex and the homology of Garside groups. Geom. Dedicata 105:171–188.[Crossref], [Web of Science ®] , [Google Scholar]], and a small chain complex for computing its monoid homology, similar to the one of [44 Squier, C. C. (1994). The homological algebra of Artin groups. Math. Scand. 75(1):5–43.[Crossref], [Web of Science ®] , [Google Scholar]].     

Symbolic dynamics: from the N-centre to the (N + 1)-body problem,a preliminary study

We consider a rotating N-centre problem, with N ≥ 3 and homogeneous potentials of degree ?α < 0, α ∈ [1,2). We prove the existence of infinitely many collision-free periodic solutions with negative and small Jacobi constant and small values of the angular velocity, for any initial configuration of the centres. We will introduce a Maupertuis’ type variational principle in order to apply the broken geodesics technique developed in Soave and Terracini (Discrete Contin Dyn Syst 32:3245–3301, 2012). Major difficulties arise from the fact that, contrary to the classical Jacobi length, the related functional does not come from a Riemaniann structure but from a Finslerian one. Our existence result allows us to characterize the associated dynamical system with a symbolic dynamics, where the symbols are given partitions of the centres in two non-empty sets.     

On the Congruence σ(n)≡1(mod n)

Let k≥2 be an integer,and let σ(n) denote the sum of the positive divisors of an integer n.We call n a quasi-multiperfect number if σ(n)=kn+1.In this paper,we give some necessary properties of them.     

On the chromatic uniqueness of the graphW(n,n − 2, k)

This paper shows that the graphW(n, n – 2, k) is chromatically unique for any even integern 6 and any integerk 1.