λφ4model and Higgs mass in standard model calculated by Gaussian effective potential approach with a new regularization-renormalization method

Based on a new regularization-renormalization method, the λφ⁴ model used in standard model (SM) is studied both perturbatively and nonperturbatively by Gaussian effective potential (GEP). The invariant property of two mass scales is stressed and the existence of a (Landau) pole is emphasized. Then after coupling with theSU(2) ×U(1) gauge fields, the Higgs mass in standard model (SM) can be calculated to bem _H≈138 GeV. The critical temperature (T _c ) for restoration of symmetry of Higgs field, the critical energy scale (μ_max, the maximum energy scale under which the lower excitation sector of the GEP is valid) and the maximum energy scale (μ_max, at which the symmetry of the Higgs field is restored) in the SM areT _c ≈476 GeV, μ_c≈0.547 × 10¹⁵ and μ_max≈0.873 × 10¹⁵, respectively. Project supported in part by the National Natural Science Foundation of China.     

New bounds for expected delay in FIFO M/G/c queues

Most bounds for expected delay, E[D], in GI/GI/c queues are modifications of bounds for the GI/GI/1 case. In this paper we exploit a new delay recursion for the GI/GI/c queue to produce bounds of a different sort when the traffic intensity p = λ/μ = E[S]/E[T] is less than the integer portion of the number of servers divided by two. (S AND T denote generic service and interarrival times, respectively.) We derive two different families of new bounds for expected delay, both in terms of moments of S AND T. Our first bound is applicable when E[S₂] < ∞. Our second bound for the first time does not require finite variance of S; it only involves terms of the form E[S^β], where 1 < β < 2. We conclude by comparing our bounds to the best known bound of this type, as well as values obtained from simulation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tail bounds for occupancy and the satisfiability threshold conjecture

The classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and probabilistic analysis. Our motivating application is the following well-known conjecture on threshold phenomenon for the satisfiability problem. Consider random 3-SAT formulas with cn clauses over n variables, where each clause is chosen uniformly and independently from the space of all clauses of size 3. It has been conjectured that there is a sharp threshold for satisfiability at c* ≈? 4.2. We provide a strong upper bound on the value of c*, showing that for c > 4.758 a random 3-SAT formula is unsatisfiable with high probability. This result is based on a structural property, possibly of independent interest, whose proof needs several applications of the occupancy tail bounds.     

Koszul homology and syzygies of Veronese subalgebras

A graded K-algebra R has property N _p if it is generated in degree 1, has relations in degree 2 and the syzygies of order ≤ p on the relations are linear. The Green–Lazarsfeld index of R is the largest p such that it satisfies the property N _p . Our main results assert that (under a mild assumption on the base field) the cth Veronese subring of a polynomial ring has Green–Lazarsfeld index ≥ c + 1. The same conclusion also holds for an arbitrary standard graded algebra, provided c >> 0{c\gg 0}.     

Upper Bounds for the Laplacian Graph Eigenvalues

We first apply non-negative matrix theory to the matrix K = D A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1 (G) of G and the spectral radius p(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extremal graphs which achieve the upper bounds.     

On the average crosscap number II: Bounds for a graph

The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less thanβ(G)-1/2β(G)-1β(G) and not larger thanβ(G). Furthermore, we also describe the structure of the graphs which attain the bounds of the average crosscap number.     

Canonical edge-colourings of locally finite graphs

A variety of results on edge-colourings are proved, the main one being the following: ifG is a graph without loops or multiple edges and with maximum degree Δ=Δ(G), and if ν is a given integer 1≦ν≦Δ(G), thenG can be given a proper edge-colouring with the coloursc ₁, ...,c _Δ+1 with the additional property that any edge colouredc _μ with μ≧ν is on a vertex which has on it edges coloured with at least ν − 1 ofc ₁, ...,c _v .     

Fractional and integral colourings

LetG=(V, E) be an undirected graph andc any vector in ℤ ^V(G) ₊. Denote byχ(G _c) (resp.η(G _c)) the chromatic number (resp. fractional chromatic number) ofG with respect toc. We study graphs for whichχ(G _c)−[η(G _c)]⩽1. We show that for the class of graphs satisfyingχ(G _c)=[η(G _c)] (a class generalizing perfect graphs), an analogue of the Duplication Lemma does not hold. We also describe a 2-vertex cut decomposition procedure related to the integer decomposition property. We use this procedure to show thatχ(G _c)=[η(G _c)] for series-parallel graphs andχ(G _c)⩽[η(G _c)]+1 for graphs that do not have the 4-wheel as a minor. The work of this author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERCC) under grant A9126.     

On the number of views of polyhedral terrains

We show that the number of topologically different orthographic views of a polyhedral terrain withn edges isO(n ^5+ɛ ), and that the number of topologically different perspective views of such a terrain isO(n ^8+ɛ ), for any ɛ>0. Both bounds are almost tight in the worst case. The proofs are simple consequences of the recent almost-tight bounds of [11] on the complexity of lower envelopes in higher dimensions. Pankaj Agarwal has been supported by National Science Foundation Grant CCR-91-06514. Micha Sharir has been supported by National Science Foundation Grant CCR-91-22103, and by grants from the U.S.—Israeli Binational Science Foundation, the G.I.F.—the German Israeli Foundation for Scientific Research and Development- and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics

 We study traveling waves of a discrete system

Universal graphs and induced-universal graphs

We construct graphs that contain all bounded-degree trees on n vertices as induced subgraphs and have only cn edges for some constant c depending only on the maximum degree. In general, we consider the problem of determining the graphs, so-called universal graphs (or induced-universal graphs), with as few vertices and edges as possible having the property that all graphs in a specified family are contained as subgraphs (or induced subgraphs). We obtain bounds for the size of universal and induced-universal graphs for many classes of graphs such as trees and planar graphs. These bounds are obtained by establishing relationships between the universal graphs and the induced-universal graphs.     

On a reduced load equivalence for fluid queues under subexponentiality

We propose a general framework for obtaining asymptotic distributional bounds on the stationary backlog in a buffer fed by a combined fluid process A ₁ + A ₂ and drained at a constant rate c. The fluid process A ₁ is an (independent) on–off source with average and peak rates ρ₁ and r₁ , respectively, and with distribution G for the activity periods. The fluid process A ₂ of average rate ρ₂ is arbitrary but independent of A ₁. These bounds are used to identify subexponential distributions G and fairly general fluid processes A ₂ such that the asymptotic equivalence P[W ^A1+A2,c >ϰ]∼P[W ^A1,c—ρ2>ϰ] (ϰ → ∞) holds under the stability condition ρ₁ + ρ₂ < c and the non-triviality condition c – ρ₂ < r ₁. In these asymptotics the stationary backlog results from feeding source A ₁ into a buffer drained at reduced rate c – ρ₂. This reduced load asymptotic equivalence extends to a larger class of distributions G a result obtained by Jelenkovic and Lazar [19] in the case when G belongs to the class of regular intermediate varying distributions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some Properties of the Injective Tensor Product of Banach Spaces

Let X and Y be Banach spaces such that X has an unconditional basis. Then X Y, the injective tensor product of X and Y, has the Radon-Nikodym property （respectively, the analytic Radon-Nikodym property, the near Radon-Nikodym property, non-containment of a copy of co, weakly sequential completeness） if and only if both X and Y have the same property and each continuous linear operator from the predual of X to Y is compact.     

Hitting Simplices with Points in ℝ3

The so-called first selection lemma states the following: given any set P of n points in ℝ ^d , there exists a point in ℝ ^d contained in at least c _d n ^d+1−O(n ^d ) simplices spanned by P, where the constant c _d depends on d. We present improved bounds on the first selection lemma in ℝ³. In particular, we prove that c ₃≥0.00227, improving the previous best result of c ₃≥0.00162 by Wagner (On k-sets and applications. Ph.D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c ₃≤1/4⁴≈0.00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69–77, 1984) (where the two-dimensional case was settled).     

A note about the cyclotomic polynomial Фpq（x） and some related results

Abstract. The expression of cyclotomic polynomial Фpq （x） is concerned for a long time. A simple and explicit expression of Фpq （x） in Z[x] has been showed. The form of the factors of Фpq （x） over F2 and the upper, lower bounds of their Hamming weight are provided.     

The Upper Bounds of Arbitrary Eigenvalues for Uniformly Elliptic Operators with Higher Orders

Let Ωbelong to R^m （m≥ 2） be a bounded domain with piecewise smooth and Lipschitz boundary δΩ Let t and r be two nonnegative integers with t ≥ r ＋ 1. In this paper, we consider the variable-coefficient eigenvalue problems with uniformly elliptic differential operators on the left-hand side and （-Δ）^T on the right-hand side. Some upper bounds of the arbitrary eigenvalue are obtained, and several known results are generalized.     

Birational models for varieties of Poncelet curves

We consider Poncelet pairs (S,C), whereS is a smooth conic andC is a degree—c plane curve having the Poncelet property with respect toS. We prove that forc>4 the projection (S,C)→C is generically one-to-one and use this to describe a birational model of the variety of Poncelet curves forc odd.     

The asymmetric random cluster model and comparison of Ising and Potts models

We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas, and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with different parameter values; we give, for example, values (β, h) for which the 0‘s configuration in the Potts lattice gas is dominated by the “+” configuration of the (β, h) Ising model. The Potts model, with possibly an external field applied to one of the spins, is a special case of the Potts lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example, we obtain 0.571 ≤ 1 − exp(−β _c ) ≤ 0.600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can also be compared to related models such as the partial FK model, obtained by deleting a fraction of the nonsingleton clusters from a realization of the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the Potts model. Received: 27 August 2000 / Revised version: 31 August 2000 / Published online: 8 May 2001     

The maximum clique and the signless Laplacian eigenvalues

Lower and upper bounds are obtained for the clique number ω(G) and the independence number α(G), in terms of the eigenvalues of the signless Laplacian matrix of a graph G. This work was supported by the National Natural Science Foundation of China (No. 10771080), SRFDP of China (No. 20070574006) and by the Foundation to the Educational Committee of Fujian (No. JB07020).     

Designs in Product Association Schemes

A number of important families of association schemes—such as the Hamming and Johnson schemes—enjoy the property that, in each member of the family, Delsarte t-designs can be characterised combinatorially as designs in a certain partially ordered set attached to the scheme. In this paper, we extend this characterisation to designs in a product association scheme each of whose components admits a characterisation of the above type. As a consequence of our main result, we immediately obtain linear programming bounds for a wide variety of combinatorial objects as well as bounds on the size and degree of such designs analogous to Delsarte's bounds for t-designs in Q-polynomial association schemes.