The Turán Number Of The Fano Plane

  Let PG₂(2) be the Fano plane, i. e., the unique hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. In this paper we prove that for sufficiently large n, the maximum number of edges in a 3-uniform hypergraph on n vertices not containing a Fano plane is

The horospherical duality

We discuss the horospherical duality as a geometrical background of harmonic analysis on semisimple symmetric spaces.     

The γ-borel conjecture

In this paper we prove that it is consistent that every -set is countable while not every strong measure zero set is countable. We also show that it is consistent that every strong -set is countable while not every -set is countable. On the other hand we show that every strong measure zero set is countable iff every set with the Rothberger property is countable.Thanks to Boise State University for support during the time this paper was written and to Alan Dow for some helpful discussions and to Boaz Tsaban for some suggestions to improve an earlier version.     

The Categories of m-semilatices

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Remarks On The Regularity Of Weak Solutions Of The Navier–Stokes Equations

In this paper we extend the results of Foias–Guillopé–Temam on the regularity and a priori estimates for the weak solutions of the Navier–Stokes equations. More specifically, we obtain upperbounds for the temporal averages of the Gevrey class norm for the weak solutions of the equations. The estimates are obtained first by getting integrated version of Foias–Temam's local in time estimate for Gevrey class norms of strong solutions and next by an induction procedure. We also strengthen slightly the local in time Gevrey class regularization of strong solutions.     

The Isotonic Fuzzy Algebraic Systems

We introduce the concept of isotonic fuzzy algebraic systems and prove that a isotonic fuzzy set of a group G is a fuzzy group iff μ_A(xy)=min {μ_A(x), μ_A(y)} iff A is a constant function.     

The Cameron-Erd?s conjecture

A subset A of integers is said to be sum-free if a+b∉A for any a,b∈A. Let s(n) be the number of sum-free sets in interval [1,n] of integers. P. Cameron and P. Erd?s conjectured that s(n)=O(2^n/2). We show that for even n and for odd n, where are absolute constants, thereby proving the conjecture.     

The generalized Rédei-matrix

Let F be a number field with odd class number, and let E be a quadratic extension of F. Our main aim is to prove that the 4-rank of the class group C(E) of E is equal to m − 1 − rank R _E/F , where m is the number of primes of F ramifying in E, R _E/F is the generalized Rédei-matrix of local Hilbert symbols with coefficients in and the rank is the rank over . We determine the generalized Rédei-matrices R _E/F explicitly for biquadratic number fields E. The research is partly supported by NNSF of China (No. 10371054, No. 10771100) and the Morningside Center of Mathematics in Beijing (MCM).     

The Krzyż conjecture revisited

The Krzy? conjecture concerns the largest values of the Taylor coefficients of a non-vanishing analytic function bounded by one in modulus in the unit disk. It has been open since 1968 even though information on the structure of extremal functions is available. The purpose of this paper is to collect various conditions that the coefficients of an extremal function (and various other quantities associated with it) should satisfy if the conjecture is true and to show that each one of these properties is equivalent to the conjecture itself. This may provide several possible starting points for future attempts at solving the problem.     

The spectral theory of commutative C∗-algebras: The constructive Gelfand-Mazur theorem

It is shown, for a commutative C^?-algebra in any Grothendieck topos E, that the locale MFn A of multiplicative linear functionals on A is isomorphic to the locale Max A of maximal ideals of A, extending the classical result that the space of C^?-algebra homomorphisms from A to the field of complex numbers is isomorphic to the maximal ideal space of A, that is, the Gelfand-Mazur theorem, to the constructive context of any Grothendieck topos. The technique is to present Max A, in analogy with our earlier definition of MFn A, by means of a propositional theory which expresses one's natural intuition of the notion involved, and then to establish various properties, leading up to the final result, by formal reasoning within these theories.     

The Euler—Kronecker constant

We consider lower bounds for the Euler—Kronecker constant in the case of number fields and upper and lower bounds in the case of algebraic manifolds over a finite field.     

The ‘final orderrs problem

When the service department of a company selling machines stops producing and supplying spare parts for certain machines, customers are offered an opportunity to place a so-called final order for these spare parts. We focus on one customer with one machine. The customer plans to use this machine up to a fixed horizon. Based on this horizon, and on the failure rates of the components, the prices of spare components and the consequences of the machine failing before the critical lifetime is reached, the size of the final order is determined.     

The wΔ-space problem

We address the following question: “Must every wΔ-space with a G_δ-diagonal be developable?” Consistently, the answer is “no.”Example. Assume CH. There is a zero-dimensional, scattered, locally compact, wΔ-space with a G_δ-diagonal which is not developable.For normal, locally compact spaces (or slightly weaker), the answer is “yes”.Theorem. If X is ω-sCWH, locally Lindelöf, wΔ-space with a G_δ-diagonal, then X is developable.     

The Linear Arboricity of Graphs

Inthispaper,allgraphsarefinite,simpleandundirected.Forarealnumberx,[x]istheleastintegernotlessthanx.LetG＝(V(G),E(G))beagraph.Weuse△(G)andδ(G)todenotethemaximum(vertex)degreeandtheminimum(vertex)degreeofG.Letw(G)＝min{d(u) d(v):uv∈E(G)}.Thegirthg(G)ofGistheminimumlengthofcycles.Thedensitymad(G)ofagraphGisthemaximumvalueof2|E(H)|/|V(H)|takenoverallsubgraphsHofG.Ifv6V(G),N(v)denotesthesetofvenicesadjacenttov,thedegreedG(v)isING(v)landNc(v)={aluEN(v)andd(u)=k}.Avertexofdegreekisc…     

The equivalence on classical metrics

In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, and prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the new metrics are equivalent to the Einstein-Kahler metric. That means that the Yau's conjecture is true on Cartan-Hartogs domains.     

The work of Lu Jia-xi

Lu Jia-xi was a high school teacher at the ninth middle school in Baot'ou China. The only work of his published in western journals were six brilliant papers published in the Journal of Combinatorial Theory-entitled on Lange Scts of Disjoint Triple Systems I,-VI, [LuI-VI]. This problem in one form or another dates back to the middle