On non-strong jumping numbers and density structures of hypergraphs

Estimating Turán densities of hypergraphs is believed to be one of the most challenging problems in extremal set theory. The concept of ‘jump’ concerns the distribution of Turán densities. A number α∈[0,1) is a jump for r-uniform graphs if there exists a constant c>0 such that for any family F of r-uniform graphs, if the Turán density of F is greater than α, then the Turán density of F is at least α+c. A fundamental result in extremal graph theory due to Erd?s and Stone implies that every number in [0,1) is a jump for graphs. Erd?s also showed that every number in [0,r!/r^r) is a jump for r-uniform hypergraphs. Furthermore, Frankl and Rödl showed the existence of non-jumps for hypergraphs. Recently, more non-jumps were found in [r!/r^r,1) for r-uniform hypergraphs. But there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we propose a new but related concept-strong-jump and describe several sequences of non-strong-jumps. It might help us to understand the distribution of Turán densities for hypergraphs better by finding more non-strong-jumps.     

Proof of the Loebl–Komlós–Sós conjecture for large, dense graphs

The Loebl–Komlós–Sós conjecture states that for any integers k and n, if a graph G on n vertices has at least n/2 vertices of degree at least k, then G contains as subgraphs all trees on k+1 vertices. We prove this conjecture in the case when k is linear in n, and n is sufficiently large.     

Operators associated with the Jacobi semigroup

The aim of this paper is to introduce some operators induced by the Jacobi differential operator and associated with the Jacobi semigroup, where the Jacobi measure is considered in the multidimensional case.In this context, we introduce potential operators, fractional integrals, fractional derivates, Bessel potentials and give a version of Carleson measures.We establish a version of Meyer’s multiplier theorem and by means of this theorem, we study fractional integrals and fractional derivates.Potential spaces related to Jacobi expansions are introduced and using fractional derivates, we give a characterization of these spaces. A version of Calderon’s Reproduction Formula and a version of Fefferman’s theorem are given.Finally, we present a definition of Triebel–Lizorkin spaces and Besov spaces in the Jacobi setting.     

A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.     

The Bishop–Phelps–Bollobás theorem fails for bilinear forms on

In this paper we show that the Bishop–Phelps–Bollobás theorem fails for bilinear forms on l₁×l₁, while it holds for linear operators from l₁ to l_∞.     

Bidimensional shallow water model with polynomial dependence on depth through vorticity

In this paper, we obtain a bidimensional shallow water model with polynomial dependence on depth. With this aim, we introduce a small non-dimensional parameter ε and we study three-dimensional Euler equations in a domain depending on ε (in such a way that, when ε becomes small, the domain has small depth). Then, we use asymptotic analysis to study what happens when ε approaches to zero. Asymptotic analysis allows us to obtain a new bidimensional shallow water model that not only computes the average velocity (as the classical model does) but also provides the horizontal velocity at different depths. This represents a significant improvement over the classical model. We must also remark that we obtain the model without making assumptions about velocity or pressure behavior (only the usual ansatz in asymptotic analysis). Finally, we present some numerical results showing that the new model is able to approximate the non-constant in depth solutions to Euler equations, whereas the classical model can only obtain the average velocity.     

A fast and accurate algorithm for computing radial transonic flows

An efficient algorithm is described for calculating stationary one-dimensional transonic outflow solutions of the compressible Euler equations with gravity and heat source terms. The stationary equations are solved directly by exploiting their dynamical system form. Transonic expansions are the stable manifolds of saddle-point-type critical points, and can be obtained efficiently and accurately by adaptive integration outward from the critical points. The particular transonic solution and critical point that match the inflow boundary conditions are obtained by a two-by-two Newton iteration which allows the critical point to vary within the manifold of possible critical points. The proposed Newton Critical Point (NCP) method typically converges in a small number of Newton steps, and the adaptively calculated solution trajectories are highly accurate. A sample application area for this method is the calculation of transonic hydrodynamic escape flows from extrasolar planets and the early Earth. The method is also illustrated for an example flow problem that models accretion onto a black hole with a shock.     

Johnson’s rule,composite jobs and the relocation problem

Two-machine flowshop scheduling to minimize makespan is one of the most well-known classical scheduling problems. Johnson’s rule for solving this problem has been widely cited in the literature. We introduce in this paper the concept of composite job, which is an artificially constructed job with processing times such that it will incur the same amount of idle time on the second machine as that incurred by a chain of jobs in a given processing sequence. This concept due to Kurisu first appeared in 1976 to deal with the two-machine flowshop scheduling problem involving precedence constraints among the jobs. We show that this concept can be applied to reduce the computational time to solve some related scheduling problems. We also establish a link between solving the two-machine flowshop makespan minimization problem using Johnson’s rule and the relocation problem introduced by Kaplan. We present an intuitive interpretation of Johnson’s rule in the context of the relocation problem.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

Factoring an infinite abelian group by lacunary cyclic subsets

It was proved in Corrádi and Szabó (Math Pannonica 5:275–280, 1994) that if a finite abelian group of odd order is factored into a direct product of lacunary cyclic subsets, then at least one of the factors must be a subgroup. The paper extends this result for certain infinite torsion groups.