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

Note On The Utility Of A Theorem Of Leindler–Meir–Totik

Our purpose is to generalize and to extend a theorem of S. Sharma and S. K. Varma [15] concerning the order of approximation by Abel means in the Lipschitz norm. The proof is basically based on a simple extension of a general theorem of L. Leindler, A. Meir and V. Totik [6] related to approximation by finite summability methods.     

An Extension Of The Ruzsa-Szemerédi Theorem

  We let G ^(r)(n,m) denote the set of r-uniform hypergraphs with n vertices and m edges, and f ^(r)(n,p,s) is the smallest m such that every member of G ^(r)(n,m) contains amember of G ^(r)(p,s). In this paper we are interested in fixed values r,p and s for which f ^(r)(n,p,s) grows quadratically with n. A probabilistic construction of Brown, Erds and T. Sós ([2]) implies that f ^(r)(n,s(r-2)+2,s)=(n ²). In the other direction the most interesting question they could not settle was whether f ⁽³⁾(n,6, 3) = o(n ²). This was proved by Ruzsa and Szemerédi [11]. Then Erds, Frankl and Rödl [6] extended this result to any r: f ^(r)(n, 3(r-2)+3, 3)=o(n ²), and they conjectured ([4], [6]) that the Brown, Erds and T. Sós bound is best possible in the sense that f ^(r)(n,s(r-2)+3,s)=o(n ²).In this paper by giving an extension of the Erds, Frankl, Rödl Theorem (and thus the Ruzsa–Szemerédi Theorem) we show that indeed the Brown, Erds, T. Sós Theorem is not far from being best possible. Our main result is

The Further Application Of Inductive Reasoning And Deductive Reasoning

Now we have a good understanding of inductive reasoning and deductive reasoning. In order to put them into practice, we should do some exercises. Practice Exercises Which reasoning process is shown in the following example? Explain your answer. 1. We examine the fingerprints of 1000 people. No two individuals in this group of people have identical fingerprints. We conclude that for all people, no two people have identical fingerprints.     

On The Effective Computation Of Łojasiewicz Exponents Via Newton Polyhedra

In this paper we give some conclusions on Newton non-degenerate analytic map germs on Kn (K = ? or ?), using information from their Newton polyhedra. As a consequence, we obtain the exact value of the Lojasiewicz exponent at the origin of Newton non-degenerate analytic map germs. In particular, we establish a connection between Newton non-degenerate ideals and their integral closures, thus leading to a simple proof of a result of Saia. Similar results are also considered to polynomial maps which are Newton non-degenerate at infinity.     

On Inversion Of Bessel Potentials Associated With The Laplace–Bessel Differential Operator

Explicit inversion formulas of Balakrishnan–Rubin type and a characterization of Bessel potentials associated with the Laplace–Bessel differential operator are obtained. As an auxiliary tool the B-metaharmonic semigroup is introduced and some of its properties are investigated.     

The Spectral Pictures of Completely Irreducible Operators and Decomposition Theorem Of Hilbert Space Operators

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Proof Of A Conjecture Of Erdős On Triangles In Set-Systems

A triangle is a family of three sets A,B,C such that A∩B, B∩C, C∩A are each nonempty, and . Let be a family of r-element subsets of an n-element set, containing no triangle. Our main result implies that for r ≥ 3 and n ≥ 3r/2, we have This settles a longstanding conjecture of Erdős [7], by improving on earlier results of Bermond, Chvátal, Frankl, and Füredi. We also show that equality holds if and only if consists of all r-element subsets containing a fixed element. Analogous results are obtained for nonuniform families.     

L p – L q Estimates To The Solution Of The Cauchy Problem For Partial Differential Equations Describing Non-Simple Thermoelastic Materials

Theory of non-simple materials is different from that of simple materials because in it the first strain gradient is taken into consideration as the constitutive variable. The consequence of this fact, from mathematical point of view, is that the equation of motion consists either of higher order derivatives of displacement (four order derivatives) and some material parameters can depend not only on the temperature and the gradient of displacement but also on the second derivative of displacement. We consider the system of partial differential equations describing non-simple thermoelastic materials. This system consists of four scalar equations, three equations of motion and one of energy balance, describing the field of displacement and the temperature in an elastic body. Using the Fourier transform, we found the L ^p – L ^q time decay estimates of the solution of the Cauchy problem for the system of equation describing the non-simple thermoelastic materials, being important for proving the global-in-time solution of this problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The λ- structure of the green ring Of GL(2,F p) in characteristic,IV

In this paper the reader is assumed to have taken notice of [I]. In [III] 1 we described the $lambda;, and s-, structure of the Green ring of GL(2,F _p), and Sl(2,F _p). We shall now construct a subring of the Green ring which is invariant for the $lambda;, and s-, operations. It is generated by all the indecomposables with odd-dimensional composition factors. This sheds another light on the results in the previous sections. We shall also study a certain quotient of the Green ring, which is in fact the Green ring of a certain subgroup of GL(2,F _p) consisting of upper triangular matrices. The multiplication and the λ, s-, structure of this quotient Green ring is described. Moreover it is shown how this λ and s-, structure controls the deviation from being a λ, respectively s-, ring of the Green ring of any finite group with a normal Sylow subgroup of order p. The sequence of Adams operations for these groups is shown to be periodic, and the period reflects the internal p-structure of these groups.     

Hajós Theorem For Colorings Of Edge-Weighted Graphs

Hajós theorem states that every graph with chromatic number at least k can be obtained from the complete graph K _k by a sequence of simple operations such that every intermediate graph also has chromatic number at least k. Here, Hajós theorem is extended in three slightly different ways to colorings and circular colorings of edge-weighted graphs. These extensions shed some new light on the Hajós theorem and show that colorings of edge-weighted graphs are most natural extension of usual graph colorings.* Supported in part by the Ministry of Education, Science and Sport of Slovenia, Research Program P0–0507–0101.     

On A Hypergraph Turán Problem Of Frankl

Let be the 2k-uniform hypergraph obtained by letting P₁, . . .,P_r be pairwise disjoint sets of size k and taking as edges all sets P_i∪P_j with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph K_r: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain can be obtained by partitioning V = V1 ∪V₂ and taking as edges all sets of size 2k that intersect each of V₁ and V₂ in an odd number of elements. Let denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no has at most as many edges as . Sidorenko has given an upper bound of for the Tur′an density of for any r, and a construction establishing a matching lower bound when r is of the form 2^p+1. In this paper we also show that when r=2^p+1, any -free hypergraph of density looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of to , where c(r) is a constant depending only on r. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials. * Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship.     

Stability Analysis Of Block ɵ-Methods For Neutral Multidelay-Differential-Algebraic Equations

This paper is concerned with the stability analysis of Block ɵ -met hods forsolving neutral multidelay-di®erential- algebraic equations. We shown that if the coefficient matrices of neutral multidelay-differential-algebraic equations satisfying somestability conditions and ɵ ϵ [ 1 2; 1], then the numerical solution of Block µ-methods forsolving neutral multidelay-differential-algebraic equations is asymptotically stable.     

Invertibility And Positive Invertibility Of Hille–Tamarkin Integral Operators

Hille–Tamarkin integral operators on the space are considered. Invertibility conditions, estimates for the norm of the inverse operators and positive invertibility conditions are established. In addition, bounds for the spectral radius are suggested. Applications to nonselfadjoint differential operators and integro-differential ones are discussed.     

The horospherical duality

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

Generalizations Of A “folk-Theorem” On Simple Functional Equations In A Single Variable

It is known (“mathematical folklore”) that, to every function defined on [1,2], there exists a solution of f(2x) = 2f(x) on ]0,∞[ of which the given function is a restriction to [1,2]. With a little care in the definition on [1,2], with still a lot of arbitrariness left, the resulting solution will be continuous, even C^∞ on ]0,∞[ (a behaviour markedly different from that of the Cauchy equation f(x + y) = f(x) + f(y), which has f(x) = cx as only continuous solution on ]0,∞[, even though, with y = x, it degenerates into the above equation). If 0 is added to the domain and we choose the “arbitrary function” bounded on [1,2[, then the solution will even be continuous (from the right) at 0. However, if f is supposed to be differentiable at 0 (from the right), then f(x) = cx is the only solution on [0,∞[. p In this paper we present similar and further results concerning general, Cⁿ (n ≤ ∞), analytic, locally monotonie or γ-th order convex solutions of the somewhat more general equation f(kx) = k^γf(x) (k ≠ 1 a positive, γ a real constant), which seems to be of importance in meterology. Some of the results are not quite what one expects.     

On An Extremal Hypergraph Problem Of Brown, Erdős And Sós

Let f_r(n,v,e) denote the maximum number of edges in an r-uniform hypergraph on n vertices, which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and Szemerédi and of Erdős, Frankl and R?dl, we partially resolve a problem raised by Brown, Erdős and Sós in 1973, by showing that for any fixed 2≤k<r, we have