Transposition generation of alternating permutations

A permutation _{1 2}... _n is alternating if ₁< ₂> ₃< ₄.... Alternating permutations are counted by the Euler numbers. Here we show that alternating permutations can be listed so that successive permutations differ by a transposition, ifn is odd. Extensions and open problems are mentioned.Research supported by the Natural Sciences and Engineering Research Council of Canada under grant A3379.     

On translations of double rays in graphs

Let be an infinite graph, let be a double ray in , and letd andd denote the distance functions in and in , respectively. One calls anaxis ifd(x,y)=d (x,y) and aquasi-axis if lim infd(x,y)/d (x,y)>0 asx, y range over the vertex set of andd (x,y). The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if is a translation whose directionD() is a thin end, then there exists an axis inD() andD(^–1) invariant under ^r for somer not exceeding the maximum number of disjoint rays inD().The thinness ofD() is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.     

Compact and weakly compact weighted composition operators on weighted spaces of continuous functions

In this note we characterize the compact and weakly compact weighted composition operatorsW _, on certain weighted locally convex spacesCV _o(X, E) of vector-valued continuous functions induced by self maps ofX and the operator-valued mappings XB(E).The work of this author was supported in part by CSIR Grant 9/100/92-EMR-IThe work of this author was supported in part by UGC Grant F.8-7/91 (RBB-II)     

The steinitz lemma inl inf∞ sup2

Let (a, b) be a pair of non-negative numbers such that (1)a, b1 and (2)a+b3. Letu ₁,...,u _n be a sequence of vectors from the set {(x, y)R ²: |x|, |y|1}, withu ₁+...+u _n =0. It is shown that there is a permutation of indices such that all partial sumsu ₍₁₎+...+u _(k) lie in the rectangle |x|a, |y|b. Conditions (1) and (2) are also necessary.     

227-01227-01227-01-property and normal subgroups

Summary D-property (=set of primes) in finite groups is not in general inherited by subgroups. In this paper, as evidence in favor of the following conjecture (F. Gross): (o) If a finite group G satisfies D then its normal subgroups satisfy D-property as well. the Author shows that if the D and the D-properties (=set of the primes not in ) hold together in a finite group G, then both are inherited by the normal subgroups of G. As a corollary, the characterization of the groups satisfying both the properties D and D is given in terms of the composition factors.     

On the properties of ε(≥0) optimal policies in discounted unbounded return model

This paper investigates the properties of (0) optimal policies in the model of [2]. It is shown that, if * = ( ₀ ^* , ₁ ^* ,..., _n ^* , _n ^+1/* , ...) is a-discounted optimal policy, then ( ₀ ^* , ₁ ^* , ..., _n ^* ) for alln0 is also a-discounted optimal policy. Under some condition we prove that stochastic stationary policy _n ^* corresponding to the decision rule _n ^* is also optimal for the same discounting factor. We have also shown that for each-optimal stochastic stationary policy ₀ ^* , ₀ ^* can be decomposed into several decision rules to which the corresponding stationary policies are also-optimal separately; and conversely, a proper convex combination of these decision rules is identified with the former ₀ ^* . We have further proved that for any (,)-optimal policy, say *=( ₀ ^* , ₁ ^* , ..., _n ^* , _n ^+1/* , ...), _n–1 ^* ) is ((1– ⁿ )^–1 e, ) optimal forn>0. At the end of this paper we mention that the results about convex combinations and decompositions of optimal policies of § 4 in [1] can be extended to our case.Project supported by the Science Fund of the Chinese Academy of Sciences.     

K-interpolation in problems of uniform approximation of functions

Given an arbitrary metric compactum Q, we compute the K-functional of a pair (C(Q), C(Q)) and derive two-sided bounds for (C_[–,], C² _[–,]). We prove interpolation theorems and study applications to problems of the theory of approximation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 523–533, April, 1992.     

1-formes fermées singulières et groupe fondamental

Summary We study the influence of the ₁ of a closed manifoldM ⁿ (n3) on the foliations ofM defined by closed differential 1-forms with Morse singularities (of index 0,n). Every nonexact form is cohomologous to a weakly complete one, that is one whose leaf space is of the same type as that of a nonsingular form. Generically, a form has compact leaves or is weakly complete. If ₁ M has no quotient isomorphic to *, then every nonexact form onM is weakly complete. We also say a form is complete if every path inM is homotopic to either a path transverse to or a path contained in a leaf of . Completeness of depends only on its de R ham cohomology class. The set of complete cohomology classes depends only on ₁ M and is related to finitely generated normal subgroups of ₁ M with quotient . If ₁ M is nilpotent (or even polycyclic), every nonexact form onM is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.     

Range of the posterior probability of an interval for priors with unimodality preserving contaminations

Range of the posterior probability of an interval over the -contamination class ={=(1–)₀+q:qQ} is derived. Here, ₀ is the elicited prior which is assumed unimodal, is the amount of uncertainty in ₀, andQ is the set of all probability densitiesq for which =(1–)₀+q is unimodal with the same mode as that of ₀. We show that the sup (resp. inf) of the posterior probability of an interval is attained by a prior which is equal to (1–)₀ except in one interval (resp. two disjoint intervals) where it is constant.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

The tree number of a graph with a given girth

A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted by(G). It is known that ifG is connected then(G) |G|/2. In this paper we show that ifG is connected and has girthg 5 then(G) |G|/g + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this case(G) |G|/4 + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequality(G) (|G| – 1)/3 + 1.     

An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates

We consider the sine-Gordon equation in laboratory coordinates with both x and t in [0, ). We assume that u(x, 0), u_t(x, 0), u(0, t) are given, and that they satisfy u(x, 0)2q, u_t(x, 0)0, for large x, u(0, t)2p for large t, where q, p are integers. We also assume that u_x(x, 0), u_t(x, 0), u_t(0, t), u(0, t)-2p, u(x, 0)-2q L₂. We show that the solution of this initial-boundary value problem can be reduced to solving a linear integral equation which is always solvable. The asymptotic analysis of this integral equation for large t shows how the boundary conditions can generate solitons.The authors dedicate this paper to the memory of M. C. PolivanovDepartment of Mathematics and Computer Science; Institute for Nonlinear Studies, Clarkson University, Postdam, New York. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 387–403, September, 1992.     

Nonlinear instability of superposed magnetic fluids in the presence of an oblique magnetic field

The nonlinear evolution of interfacial waves separating two magnetic fluids subjected to an oblique magnetic field is studied in two dimensions, with the use of the method of multiple scales. It is shown that the evolution of the envelope is governed by two partial differential equations. These equations can be combined to yield two alternate Schrödinger equations with cubic nonlinearity; one of them leads to the determination of the cutoff wave number separating stable from unstable deformations while the other Schrödinger equation is used to analyze the stability of the system. The stability of the system is discussed both theoretically and computationally, and the stability diagrams are obtained. It is found in the linear theory that the oblique magnetic field has a stabilizing influence if 0 ₁ + ₂ < /2, or 3/2 < ₁ + ₂ 2 and a destabilizing influence if /2 < ₁ + ₂ < 3/2, where 0 _j , (j=1, 2) and , is the angle between the field and the horizontal axis.In the nonlinear theory, the stability analysis reveals that there exist different regions of stability and instability. It is reported that the oblique magnetic field plays a dual role in the stability criterion and the angles ₁ and ₂ play a distinctive role in this analysis besides the effect of the variation of the magnetic permeabilities.     

m-Systems and Partial m-Systemsof Polar Spaces

LetP be a finite classical polar space of rankr, withr 2. A partialm-systemM ofP, with 0 m r - 1, is any set (₁), ₂,..., _k ofk ( 0) totally singularm-spaces ofP such that no maximal totally singular space containing _i has a point in common with (_{1 2} ... _k) — _i,i = 1, 2,...,k. In a previous paper an upper bound for ¦M¦ was obtained (Theorem 1). If ¦M¦ = , thenM is called anm-system ofP. Form = 0 them-systems are the ovoids ofP; form =r - 1 them-systems are the spreads ofP. In this paper we improve in many cases the upper bound for the number of elements of a partialm-system, thus proving the nonexistence of several classes ofm-systems.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Real analytic curves in Fréchet spaces and their duals

The following results are presented: 1) a characterization through the Liouville property of those Stein manifoldsU such that every germ of holomorphic functions on xU can be developed locally as a vector-valued Taylor series in the first variable with values inH(U); 2) ifT is a surjective convolution operator on the space of scalar-valued real analytic functions, one can find a solutionu of the equationT u=f which depends holomorphically on the parameterz wheneverf depends in the same manner. These results are obtained as an application of a thorough study of vector-valued real analytic maps by means of the modern functional analytic tools. In particular, we give a tensor product representation and a characterization of those Fréchet spaces or LB-spacesE for whichE-valued real analytic functions defined via composition with functionals and via suitably convergent Taylor series are the same.     

Lifting-Probleme für Vektorfunktionen und ⊗-Sequenzen

In this article the topologically exact sequences of locally convex spaces are characterized for which for every locally convex space F the map id : FE F Q is a homomorphism, or equivalently, the map id L : FK F E is a topological injection. This is motivated by the problem of lifting Q-valued functions with certain given properties to E-valued functions with the same or slightly weaker properties, which may also be considered as the investigation of parameter dependences of solutions of linear (differential) equations. Applications to partial differential equations and to Fredholm functions are given.     

L 2-approximation by the translates of a function and related attenuation factors

Summary Let be thek-dimensional subspace spanned by the translates (·–2j/k),j=0, 1, ...,k–1, of a continuous, piecewise smooth, complexvalued, 2-periodic function . For a given functionfL ²(–, ), its least squares approximantS _kf from can be expressed in terms of an orthonormal basis. Iff is continuous,S _kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of include the space of trigonometric polynomials where is the de la Valleé Poussin kernel, algebraic polynomial splines where is the periodic B-spline, and trigonometric polynomial splines where is the trigonometric B-spline.     

A problem arising in finite population sampling theory

A probability sampling design is a probability functionp(s) on subsetss of [1, ..., i, ..., N. Let _ij denote the joint inclusion probability fori andj. The problem is to determine conditions under which a fixed size (n) sampling designp exists so that _ij (x _i –x _j )² for a vector of real numbersx=(x ₁, ...,x _N ), or equivalently, so that for some order of the coefficients _ik = _ij + _jk . Some necessary conditions for the proportionality to hold are obtained, and it is conjectured that it is satisfied only in special circumstances.     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

Divergent Fourier series with respect to orthonormalized systems of smooth functions

In this paper it is proved that for any functionf L ² [–; ], f>0, there exists a complete orthonormalized system of uniformly bounded trigonometric polynomials with respect to which the Fourier series of this function is divergent almost everywhere in the interval [–; ].Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 69–78, July, 1976.The authoress expresses her gratitude to A. M.