Three enumeration formulas of standard Young tableaux of truncated shapes

In this paper we consider the enumeration of three kinds of standard Young tableaux (SYT) of truncated shapes by use of the method of multiple integrals. A product formula for the number of truncated shapes of the form (n^m, n ? r)\δ_k–1 is given, which implies that the number of SYT of truncated shape (n², 1)\(1) is the number of level steps in all 2-Motzkin paths. The number of SYT with three rows truncated by some boxes ((n + k)³)\(k) is discussed. Furthermore, the integral representation of the number of SYT of truncated shape (n^m)\(3, 2) is derived, which implies a simple formula of the number of SYT of truncated shape (nⁿ)\(3, 2).     

On Hilbert''s Integral Inequality

In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff, g L²[0, ∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π²is also the best value (cf. [[1], Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x) = 2/π∫^∞₀(c(t²x)/(1 + t²)) dt − c(x), 1 − c(x) + c(y) ≥ 0, andf, g ≥ 0 (cf. [[2]]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [ [2]].     

Proof of a conjecture on fractional Ramsey numbers

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r_f (a₁, a₂, …, a_k) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k=2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010     

Continuous dependence of the scattering amplitude on the surface of an obstacle

Let D_j,j = 1,2, be two bounded domains (obstacles) in ?ⁿ, n ≥ 2, with the boundaries Γ_j. Let A_j be the scattering amplitude corresponding to D_j. The Dirichlet boundary condition is assumed on Γ_j. A formula is derived for A:= A₁ ? A₂. This formula is used for a derivation of the estimate of ∣A₁ ? A₂∣ in terms of the distance d(Γ₁, Γ₂) between Γ₁ and Γ₂. If d(Gamma;₁, Gamma;₂) ? ?, then ∣A∣ ? c?, where c is a positive constant which depends on Γ₁ and Γ₂ provided that one of the boundaries is of C^1,λ class, 0 < λ < 1, and the other one is a polyhedron which approximates the first one. The results are useful, in particular, for boundary elements method of solving scattering problems.     

A relative Kuznietsov trace formula on G2

In this paper, we set up the general formulation to study distinguished residual representations of a reductive group G by the relative trace formula approach. This approach simplifies the argument of [JR], which deals with this type of relative trace formula for a special symmetric pair (GL(2n), Sp(2n)) and also works for non-symmetric, spherical pairs. To illustrate our idea and method, we complete our relative trace formula (both the geometric side identity and the spectral side identity) for the case (G ₂, SL(3)). Received: 6 February 1999     

The scaling window of the 2‐SAT transition

We consider the random 2‐satisfiability (2‐SAT) problem, in which each instance is a formula that is the conjunction of m clauses of the form x∨y, chosen uniformly at random from among all 2‐clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n→α, the problem is known to have a phase transition at α_c=1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite‐size scaling about this transition, namely the scaling of the maximal window W(n, δ)=(α_?(n,δ), α₊(n,δ)) such that the probability of satisfiability is greater than 1?δ for α<α_? and is less than δ for α>α₊. We show that W(n,δ)=(1?Θ(n^?1/3), 1+Θ(n^?1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+ε)n, where ε may depend on n as long as |ε| is sufficiently small and |ε|n^1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(?Θ(nε³)) above the window, and goes to one like 1?Θ(n^?1|ε|^?3 below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2‐SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2‐SAT are identical to those of the random graph. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 201–256 2001     

Numerical differentiation inspired by a formula of R.P. Boas

First, we briefly discuss three classes of numerical differentiation formulae, namely finite difference methods, the method of contour integration, and sampling methods. Then we turn to an interpolation formula of R.P. Boas for the first derivative of an entire function of exponential type bounded on the real line. This formula may be classified as a sampling method. We improve it in two ways by incorporating a Gaussian multiplier for speeding up convergence and by extending it to higher derivatives. For derivatives of order s, we arrive at a differentiation formula with N^′ nodes that applies to all entire functions of exponential type without any additional restriction on their growth on the real line. It has an error bound that converges to zero like e^-αN/N^m as N→∞, where α>0 and N^′=2N, m=3/2 for odd s while N^′=2N+1, m=5/2 for even s. Comparable known formulae have stronger hypotheses and, for the same α, they have m=1/2 only. We also deduce a direct (error-free) generalization of Boas’ formula (Corollary 5). Furthermore, we give a modification of the main result for functions analytic in a domain and consider an extension to non-analytic functions as well. Finally, we illustrate the power of the method by examples.     

Location of the travelling wave for the Kolmogorov equation

Summary Under appropriate initial data, solutions of the Kolmogorov equation converge to travelling waves w ^λ (x), λ≧2^1/2, as t→∞. In the case λ>2^1/2, a general formula for the asymptotic position of the wave is known, as are formulas for certain cases of λ=2^1/2. The general formula for the case λ=2^1/2 given in [4] involves the behavior of the solution at earlier times and is typically not explicit enough for computation. Here, we give an improvement of this formula not requiring such information. The methodology involves use of sample path estimates for Brownian bridge and manipulation of certain formulas from [4]. The research for this paper was done at the University of Wisconsin at Madison and supported in part by the National Science Foundation under Grant No. DMS-83-01080     

Scrambling non-uniform nets

In this paper, we consider Owen’s scrambling of an (m−1, m, d)-net in base b which consists of d copies of a (0, m, 1)-net in base b, and derive an exact formula for the gain coefficients of these nets. This formula leads us to a necessary and sufficient condition for scrambled (m − 1, m, d)-nets to have smaller variance than simple Monte Carlo methods for the class of L ₂ functions on [0, 1] ^d . Secondly, from the viewpoint of the Latin hypercube scrambling, we compare scrambled non-uniform nets with scrambled uniform nets. An important consequence is that in the case of base two, many more gain coefficients are equal to zero in scrambled (m − 1, m, d)-nets than in scrambled Sobol’ points for practical size of samples and dimensions.      

A Fredholm determinant formula for Toeplitz determinants

We prove a formula expressing a generaln byn Toeplitz determinant as a Fredholm determinant of an operator 1 –K acting onl ₂ (n,n+1,...), where the kernelK admits an integral representation in terms of the symbol of the original Toeplitz matrix. The proof is based on the results of one of the authors, see [14], and a formula due to Gessel which expands any Toeplitz determinant into a series of Schur functions. We also consider 3 examples where the kernel involves the Gauss hypergeometric function and its degenerations.     

Multiplicative properties of projectively dual varieties

We present a general formula for the dimension of the projectively dual of the product of two projective varietiesX ₁ andX ₂, in terms of dimensions ofX ₁,X ₂ and their projective duals (Theorem 0.1). The proof is based on the formula due to N. Katz expressing the dimension of the dual variety in terms of the rank of certain Hessian matrix. Some consequences and related results are given, including the “Cayley trick” from [3] and its dual version. Partially supported by the NSF (DMS-9102432) Partially supported by the NSF (DMS-9104867) This article was processed by the author using the Springer-Verlag T_EE mamath macro package 1990.     

Divided differences of multivariate implicit functions

Under general conditions, the equation g(x ¹,…,x ^q ,y)=0 implicitly defines y locally as a function of x ¹,…,x ^q . In this article, we express divided differences of y in terms of divided differences of g, generalizing a recent formula for the case where y is univariate. The formula involves a sum over a combinatorial structure whose elements can be viewed either as polygonal partitions or as plane trees. Through this connection we prove as a corollary a formula for derivatives of y in terms of derivatives of g.     

Integrodifferential systems with infinitely many delays

Summary In this paper we consider the initial-value problems: (P ₁ )X(t)=(AX)(t) for t>0, X(0+)=I, X(t)=0 for t<0 and (P ₂ ) Y(t)=(QY)(t) for t>0, Y(0+)=I, Y(t)=0 for t<0, where A and Q are linear specified operators, I and0 — the identity and null matrices of order n, and X(t), Y(t) are unknown functions whose values are square matrices of order n. Sufficient conditions are established under which the problems (P ₁ ) and (P ₂ ) have the same unique solution, locally summable on the half-axis t ⩾0. Using this fact and some properties of the Laplace transform we find a new proof for the variation of constants formula given in[1, 2]. On the basis of this formula we derive certain results concerning a class of integrodifferential systems with infinite delay. Entrata in Redazione il 2 marzo 1977.     

A multi-dimensional Markov chain and the Meixner ensemble

We show that the transition probability of the Markov chain (G(i,1),...,G(i,n)) _i≥1, where the G(i,j)’s are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for G(m,n). We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.     

Analysis on real hyperbolic spaces

The paper gives (a) an integral formula for eigenfunctions of invariant differential operators on the homogeneous space O(p, q)/O(p, q − 1) and (b) a direct integral decomposition of its L²-space under the regular representation of O(p, q).     

Delaying satisfiability for random 2SAT

Let (C₁,C′_(*)),(C₂,C′_(*)),…,(C _m,C′_(*)) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4 \begin{align*}\binom{n}{2}\end{align*} clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on‐line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on‐line choice algorithm in the above process which results whp in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)^1/4. This contrasts with the well‐known fact that a random m ‐clause formula constructed without the choice of two clauses at each step is unsatisfiable whp whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as “Achlioptas processes.” This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on‐line setting above, we also consider an off‐line version in which all m clause‐pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off‐line setting, we show that the two‐choice satisfiability threshold for k ‐SAT for any fixed k coincides with the standard satisfiability threshold for random 2k ‐SAT.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in C ⁿ given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C ⁿ , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C ⁿ , which are different from the classical ones, when we properly select the vector function W.     

An explicit formula for the square of the Riemann zeta-function in the critical strip

In this article, we prove an explicit formula for |ζ(σ + iT)|², where ζ(s) is the Riemann zeta-function and 1/2 < σ < 1, which is an analogue of Jutila’s formula. Our proof differs from that of Jutila. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 381–398, July–September, 2007.     

On the Fourier Transform of Causal Distributions

We extend the distributional Bochner formula [1, p. 72, Theorem 26] to certain kinds of distributions. Theorem I.1 gives a formula [Eq. (I.1.14)] which makes it possible to obtain easily the Fourier transform of distributions of the form As applications of the formula (I.1.14) we evaluate the Fourier transforms of the distributions G_α(P±i0, m, n) [Eq. (I.4.1)] and H_α(P±i0,n) [Eq. (II.1.1)]. It follows from Theorem II.3 that H_zk(P±i0,n) is, for 2K≠n+2r, r=0,1..., an elementary solution of the n-dimensional ultrahyperbolic operator iterated k times.