Turán's extremal problem in random graphs: Forbidding odd cycles

For 0<1 and graphsG andH, we writeGH if any -proportion of the edges ofG span at least one copy ofH inG. As customary, we writeC ^k for a cycle of lengthk. We show that, for every fixed integerl1 and real >0, there exists a real constantC=C(l, ), such that almost every random graphG _{n, p} withp=p(n)Cn ^–1+1/2l satisfiesG _n,p1/2+ C ^2l+1. In particular, for any fixedl1 and >0, this result implies the existence of very sparse graphsG withG _1/2+ C ^2l+1.The first author was partially supported by NSERC. The second author was partially supported by FAPESP (Proc. 93/0603-1) and by CNP_q (Proc. 300334/93-1). The third author was partially sopported by KBN grant 2 1087 91 01.     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

OnK 4-free subgraphs of random graphs

For 0<1 and graphsG andH, writeGH if any -proportion of the edges ofG spans at least one copy ofH inG. As customary, writeK ^r for the complete graph onr vertices. We show that for every fixed real >0 there exists a constantC=C() such that almost every random graphG _n,p withp=p(n)Cn ^–2/5 satisfiesG _{n,p 2/3+} K ⁴. The proof makes use of a variant of Szemerédi's regularity lemma for sparse graphs and is based on a certain superexponential estimate for the number of pseudo-random tripartite graphs whose triangles are not too well distributed. Related results and a general conjecture concerningH-free subgraphs of random graphs in the spirit of the Erds-Stone theorem are discussed.The first author was partially supported by FAPESP (Proc. 93/0603-1) and by CNPq (Proc. 300334/93-1 and ProTeM-CC-II Project ProComb). Part of this work was done while the second author was visiting the University of São Paulo, supported by FAPESP (Proc. 94/4276-8). The third author was partially supported by the NSF grant DMS-9401559.     

On ηα-groups and fields

Let :=. The following are known: two -sets of power are isomorphic. Let >0. Two ordered divisible Abelian groups that are -sets of power are isomorphic, two real closed fields that are -sets of power are isomorphic. The following is shown: (1) there exist 2 nonisomorphic ordered Abelian groups (respectively ordered fields) that are -sets of power ; (2) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) of power all having the same order type; (3) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) that are -sets having the same order type.     

Representations of free lattice-ordered groups

We show for any uncountable cardinal that the free groupG of rank has a linear right ordering on which the natural action of the free lattice-ordered groupF of rank is faithful and pathologically 2-transitive. As a consequence, we obtain results on the root system of prime subgroups ofF . This generalizes previous results of McCleary which required the generalized continuum hypothesis and to be regular.     

Über ein Problem von Mordell in der Geometrie der Zahlen

For a latticeL in ⁿ with determinantd(L), let (L) denote the supremum of the values 2^–2 V(P)/d(L), taken over theL-admissible parallelepidesP, symmetric with respect to the origin and with faces parallel to the coordinate-axes. In 1936, Mordell asked for the constants _n = min (L) over alln-dimensional lattices. In this paper we investigate isolated minima of (L) in all over alln-dimensional lattices. In this paper we (Satz 1) and some examples are given. In particular, forn<=4, the set of lattices with isolated turns out to be dense in the space of lattices. Conversely, the set of (algebraically generated) lattices with non-isolated is dense, at least in the case of a plane (Satz 2).     

Non removable edges in 3-connected cubic graphs

LetG be a cyclicallyk-edge-connected cubic graph withk 3. Lete be an edge ofG. LetG be the cubic graph obtained fromG by deletinge and its end vertices. The edgee is said to bek-removable ifG is also cyclicallyk-edge-connected. Let us denote by S _k (G) the graph induced by thek-removable edges and by N _k (G) the graph induced by the non 3-removable edges ofG. In a previous paper [7], we have proved that N ₃(G) is empty if and only ifG is cyclically 4-edge connected and that if N ₃(G) is not empty then it is a forest containing at least three trees. Andersen, Fleischner and Jackson [1] and, independently, McCuaig [11] studied N ₄(G). Here, we study the structure of N _k (G) fork 5 and we give some constructions of graphs such thatN _k (G) = E(G). We note that the main result of this paper (Theorem 5) has been announced independently by McCuaig [11].

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Résumé SoitG un graphe cubique cyliquementk-arête-connexe, aveck 3. Soite une arête deG et soitG le graphe cubique obtenu à partir deG en supprimante et ses extrémités. L'arêtee est ditek-suppressible siG est aussi cycliquementk-arête-connexe. Désignons par S _k (G) le graphe induit par les arêtesk-suppressibles et par N _k (G) celui induit par les arêtes nonk-suppressibles. Dans un précédent article [7], nous avons montré que N ₃(G) est vide si et seulement siG est cycliquement 4-arête-connexe et que si N ₃(G) n'est pas vide alors c'est une forêt possédant au moins trois arbres. Andersen, Fleischner and Jackson [1] et, indépendemment, McCuaig [11] ont étudié N ₄(G). Ici, nous étudions la structure de N _k (G) pourk 5 et nous donnons des constructions de graphes pour lesquelsN _k (G) = E(G). Nous signalons que le résultat principal de cet article (Théorème 5) a été annoncé indépendamment par McCuaig [11].

     

On the concentration of stresses due to a small elliptic inclusion on the neutral axis of a deep beam under constant bending moment

Zusammenfassung Es wird die Spannungsverteilung untersucht, die sich in einem breiten Balken mit konstanter Höhe unter einem konstanten Biegemoment ausbildet, wenn er eine kleine elliptische Einschliessung mit Zentrum auf der Neutralachse enthält. Insbesondere werden die Fälle eines sehr starren Einschlusses sowie eines elliptischen Loches im Detail diskutiert.

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Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - _x , _y normal stress in Cartesian coordinates - _xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h ₁, 1/h ₂ stretch ratios - e _x + _y dilatation - 2 rotation - M bending moment     

Barrelled spaces with a B-complete completion

Let (E, ) be a barrelled locally convex space. The aim of this paper is to describe the barrelled topologies on E weaker than . When the completion Ê of (E, ) is B-complete, various properties of the barrelled topologies weaker than are proved. Some examples are given to illustrate the possible situations.     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Combined free and forced laminar convection in internally finned square ducts

Analysis is presented for the heat transfer performance of square ducts with internal fins from each wall in the case of combined free and forced convection by fully developed laminar flow. Numerical results are obtained for the Nusselt number and the pressure drop parameter for various values of finlengths and heat source parameter. For various values of Rayleigh numbers, the Nusselt number increases with the increase in finlength and decreases with the increase in heat source parameter.

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Zusammenfassung Es wird eine Analyse für den Wärmeaustausch von quadratischen Rohren mit inneren Rippen an jeder Wand im Falle einer Kombination von freier und erzwungener Konvektion bei voll entwickelter laminarer Strömung gegeben. Numerische Resultate für die Nusselt-Zahl und den Druckabfall-Koeffizienten für verschiedene Rippenbreiten und Parameter der Wärmequelle werden erhalten. Für einige Werte der Rayleighzahl wächst die Nusselt-Zahl mit der Rippenbreite und fällt mit wachsendem Parameter der Wärmequelle.

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Nomenclature A cross sectional area of the duct - B _2k Bernoulli numbers - c _p specific heat at constant pressure - D _h hydraulic diameter of finless duct - E _n complex constants (20) - F heat source parameter,Q/c _p - F _n () defined by Equation (14) - G(, , , ) Green's function (15, 16) - g gravitational acceleration - H() Heaviside function - h() defined by Equation (22) - i imaginary unit,i ²=–1 - ImW imaginary part ofW - K(,t) kernel of the integral equation, defined by (25) - k thermal conductivity - L pressure drop parameter, –D _h ² (p/x+ _w )/ - l fin length of each fin, Figure (1) - N _u Nusselt number, Equation (32) - p pressure - Q heat generation rate - R() defined by Equation (26) - R _A Rayleigh number, _w gc _p D _h ⁴ /k - ReW real part ofW - T dimensionless temperature, (t–t _w )/(c _p D _h ² /k) - T _mx dimensionless mixed mean temperature, Equation (33) - t fluid temperature - t ₀ reference temperature atx=0 - u local axial velocity - mean axial velocity - V u/ - W complex function defined by Equation (6) - w suffix denoting wall conditions - W ₀ defined by Equation (9) - W ₁ W–W ₀, Equation (18) - x axial coordinate along the length of the duct - y, z cross-sectional coordinates - constant temperature gradient, t/x - coefficient of thermal expansion of the fluid - fluid density - _n - dynamic viscosity - () Dirac delta function - ² Laplacian operator, ²/y ²/²/z ² - , y/D _h ,z/D _h      

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type–div(a(x,u,Du))=H(x,u,Du)+f, uW ^1,p ₀()L (),where a(x,,)b(||)|| ^p , b is a continuous monotone decreasing function and |H(x,,)| k()|| ^p , k is a continuous monotone increasing function.     

An Investigation of the Fields of Bounded Formal Power Series by Means of Theory of Cuts

We consider a class K of real closed fields F, |F|=|G|=₁, where G is a group of Archimedean classes of F, and cofinality of each symmetric gap of F is ₁. We will show that this class is exactly a class of all bounded formal power series RG,₁, where G is a divisible Abelian group, card(G)=₁, under CH. A nonstandard real line ^*R, which is ₁-set belongs to this class; we will also consider a construction RG(L,P),₁ of fields from this class, where L is a totally ordered set, P is a totally ordered field, G(L,P) is a group of finite words. It will be describes symmetric gaps of such two fields in K, which are not ₁-set. Mathematics Subject Classifications (2000) 03E04, 12J15, 12J25.The work was supported by grant of Ministry of Education PD02-1.1-386.     

Stability and uniqueness of the solution of the inverse kinematic problem of seismology in higher dimensions

The following inverse kinematic problem of seismology is considered. In the compact domain M of dimension ,2 with the metric, we consider the problem of constructing a new metricdu=nds according to the known formula where ,M and K_, is the geodesic in the metric du, connecting the points , . One proves uniqueness and one obtains a stability estimate, where the refraction indices n₁, n₂ are the solutions of the inverse kinematic problem, constructed relative to the functions ₁, ₂, respectively, is the differential form on M×Mwhere =₂–₁,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 84, pp. 3–6, 1979.     

Galerkin methods for parabolic equations with nonlinear boundary conditions

A variety of Galerkin methods are studied for the parabolic equationu _t =(a(x) u),x ⁿ ,t (O,T], subject to the nonlinear boundary conditionu _v =g(x,t,u),x,t (O,T] and the usual initial condition. Optimal order error estimates are derived both inL ² () andH ¹ () norms for all methods treated, including several that produce linear computational procedures.The authors were partially supported by The National Science Foundation during the preparation of this paper.     

Generalized concentration dependence of the viscosity of concentrated polymer solutions

The construction of a generalized concentration dependence of the viscosity of concentrated solutions is analyzed. It is shown that there should be a single dependence for different polymers in different solvents of the form: (/_o)^1–2k=1+(1–2k)c[], where k is the Huggins-Martin constant, and [] is the intrinsic viscosity. Deviations from this relation may be observed in the presence of structure formation in the solution or when the experimental temperature is close to the glass-transition temperature of the system."Plastpolimer" Okhtinsk Research-Production Association, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 172–175, January–February, 1976.     

Estimation of Spectral Densities with Multiplicative Parameter

A two-step estimation procedure is presented for spectral densities of the form f()=g() with and being unknown parameters. The classes of random fields for which the procedure is applicable are defined by restrictions on spectral densities of second and higher orders. The procedure suggests a minimum contrast estimator for the parameter which is then used to construct the estimator for . The delta method provides the asymptotic normality of our estimator for the parameter .     

On the long term behavior of some finite particle systems

Summary We consider the problem of comparing large finite and infinite systems with locally interacting components, and present a general comparison scheme for the case when the infinite system is nonergodic. We show that this scheme holds for some specific models. One of these is critical branching random walk onZ ^d . Let _t denote this system, and let _t ^N denote a finite version of _t defined on the torus [–N,N] ^d Z ^d . Ford3 we prove that for stationary, shift ergodic initial measures with density , that ifT(N) andT(N)/(2N+1)^d s[0,] asN, then {v }, 0 is the set of extremal invariant measures for the infinite system _t andQ _s is the transition function of Feller's branching diffusion. We prove several extensions and refinements of this result. The other systems we consider are the voter model and the contact process.Work supported in part by the National Science Foundation under Grant DMS-8802055, by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University and by the Deutsche Forschungsgemeinschaft through the SFB 123 at the Universität Heidelberg     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

On Interpolation Sequences of One Class of Functions, Analytic in the Half-Plane

A new criterion of solvability of the interpolation problem f( _n )=b_n in the class of functions f, analytic in the right half-plane and such that there exists c ₁(0;+) such that |f(z)|c ₁exp((c₁|z|)) for all z , where is a positive increasing continuous differentiable function on [0;+), for which (t)+ as t+ and there exists c ₂(0;+) such that