A note on dual prehomomorphisms from a group into the Margolis–Meakin expansion of a group

Semigroup Forum - We give a category-free order theoretic variant of a key result in Auinger and Szendrei (J Pure Appl Algebra 204(3):493–506, 2006) and illustrate how it might be used to...     

A Krylov–Schur approach to the truncated SVD

Computing a small number of singular values is required in many practical applications and it is therefore desirable to have efficient and robust methods that can generate such truncated singular value decompositions. A method based on the Lanczos bidiagonalization and the Krylov–Schur method is presented. It is shown that deflation strategies can be easily implemented in this method and possible stopping criteria are discussed. Numerical experiments show the efficiency of the Krylov–Schur method.     

A Gamma-convergence approach to the Cahn–Hilliard equation

We study the asymptotic dynamics of the Cahn–Hilliard equation via the “Gamma-convergence” of gradient flows scheme initiated by Sandier and Serfaty. This gives rise to an H ¹-version of a conjecture by De Giorgi, namely, the slope of the Allen–Cahn functional with respect to the H ⁻¹-structure Gamma-converges to a homogeneous Sobolev norm of the scalar mean curvature of the limiting interface. We confirm this conjecture in the case of constant multiplicity of the limiting interface. Finally, under suitable conditions for which the conjecture is true, we prove that the limiting dynamics for the Cahn–Hilliard equation is motion by Mullins–Sekerka law. Partially supported by a Vietnam Education Foundation graduate fellowship.     

A q-series expansion formula and the Askey–Wilson polynomials

Previously, we proved a q-series expansion formula which allows us to recover many important classical results for q-series. Based on this formula, we derive a new q-formula in this paper, which clearly includes infinitely many q-identities. This new formula is used to give a new proof of the orthogonality relation for the Askey–Wilson polynomials. A curious q-transformation formula is proved, and many applications of this transformation to Hecke type series are given. Some Lambert series identities are also derived.     

A probabilistic approach to the Yang–Mills heat equation

We construct a parallel transport U in a vector bundle E, along the paths of a Brownian motion in the underlying manifold, with respect to a time dependent covariant derivative ∇ on E, and consider the covariant derivative ∇₀U of the parallel transport with respect to perturbations of the Brownian motion. We show that the vertical part U⁻¹∇₀U of this covariant derivative has quadratic variation twice the Yang–Mills energy density (i.e., the square norm of the curvature 2-form) integrated along the Brownian motion, and that the drift of such processes vanishes if and only if ∇ solves the Yang–Mills heat equation. A monotonicity property for the quadratic variation of U⁻¹∇₀U is given, both in terms of change of time and in terms of scaling of U⁻¹∇₀U. This allows us to find a priori energy bounds for solutions to the Yang–Mills heat equation, as well as criteria for non-explosion given in terms of this quadratic variation.     

A finite field approach to the Carlitz–Riordan q-Catalan numbers

Journal of Algebraic Combinatorics - Let q be a prime power and $${\mathbb {F}}_q$$ be the finite field with q elements. Suppose that $$n\ge 1$$ and $${{\mathscr {F}}}=\{E_1,\ldots ,E_{2n-1}\}$$ is...     

A new approach to the Assad–Kirk fixed point theorem

In the present paper, considering the Wardowski’s technique, we give a new approach to the Assad–Kirk fixed point theorem on metrically convex metric spaces.We also provide a nontrivial example showing that our result is a proper extension of the Assad–Kirk fixed point theorem.     

A reliable approach to the solution of Navier–Stokes equations

In this paper we will demonstrate an affective approach of solving Navier–Stokes equations by using a very reliable transformation method known as the Cole–Hopf transformation, which reduces the problem from nonlinear into a linear differential equation which, in turn, can be solved effectively.     

A new approach to the stabilization of a fluid–structure interaction model

This article is a continuation of our work on a linear fluid–structure interaction model [Grobbelaar-Van Dalsen, On a fluid–structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech. 10(3) (2008), pp. 388–401; Grobbelaar-Van Dalsen, Strong stability for a fluid––structure model, Math. Methods Appl. Sci., 32(2009) pp. 1452–1466]. The model describes the interaction between a 3-D incompressible fluid and a 2-D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with the inclusion of the shear stress that the fluid exerts on the plate. A dissipative damping mechanism of Kelvin–Voigt type is applied to the interior of the plate. While our earlier work shows that weak solutions in a space of finite energy are strongly asymptotically stable under no-slip transmission conditions at the interface with uniform exponential stability only attainable under an additional domination condition, the present research is directed at achieving uniform exponential stability of weak solutions without imposing the domination condition. Using energy methods we establish uniform exponential decay under a modified transmission condition at the interface. This condition entails that the fluid velocity at the interface is coupled to a linear combination of the plate velocity and displacement.     

A bilinear approach to the pooling problem†

In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refinery planning models. The main difficulty is that pooling causes an inherent nonlinearity in the otherwise linear models. We shall define the problem by formulating an aggregate mathematical model of a refinery, comment on solution methods for pooling problems that have been presented in the literature, and develop a new method based on convex approximations of the bilinear terms. The method is illustrated on numerical examples     

A forward–backward SDE approach to affine models

We consider factor models for interest rates and asset prices where the risk- neutral dynamics of the factors process is modelled by an affine diffusion. We characterize the factors process and bond price in terms of forward–backward stochastic differential equations (FBSDEs), prove an existence and uniqueness theorem which gives the solution explicitly, and characterize the bond price as an exponential affine function of the factors in a new way. Our approach unifies the results, based on stochastic flows, of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001) with the approach, based on the Feynman-Kac formula, of Duffie and Kan (Math Finance 6(4):379–406, 1996), and addresses a mistake in the approach of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). We extend our results on the bond price to consider the futures and forward price of a risky asset or commodity.      

Percival Lagrangian approach to the Aubry–Mather theory

We present some streamlined proofs of some of the basic results in the Aubry–Mather theory (existence of quasi-periodic minimizers, multiplicity results when there are gaps among minimizers) based on the study of hull functions. We present results in arbitrary number of dimensions.     

A Fourier–Mukai approach to the K-theory of compact Lie groups

Let G be a compact, connected, simply-connected Lie group. We use the Fourier–Mukai transform in twisted K-theory to give a new proof of the ring structure of the K-theory of G.     

A conditional least squares approach to PGARCH and PARMA–PGARCH time series estimation

In this Note, a conditional least squares (CLS) estimates for periodic GARCH (PGARCH) models with martingale difference centered squared innovations is developed. The approach is extended to the PARMA–PGARCH models. We establish the strong consistency and the asymptotic normality for our estimate.     

A quadratic programming approach to the Randić index

Let G(k, n) be the set of connected graphs without multiple edges or loops which have n vertices and the minimum degree of vertices is k. The Randi? index χ = χ(G) of a graph G   is defined by χ(G)=_∑(uv)(_δuδv)^-1/2$\chi (G)={\sum }_{(\mathit{uv})}({\delta }_u{\delta }_v{)}^{-1/2}$, where δ_u is the degree of vertex u and the summation extends over all edges (uv) of G. Caporossi et al. [G. Caporossi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs IV: Chemical trees with extremal connectivity index, Computers and Chemistry 23 (1999) 469–477] proposed the use of linear programming as one of the tools for finding the extremal graphs. In this paper we introduce a new approach based on quadratic programming for finding the extremal graphs in G(k, n) for this index. We found the extremal graphs or gave good bounds for this index when the number n_k of vertices of degree k is between n − k and n. We also tried to find the graphs for which the Randi? index attained its minimum value with given k (k ? n/2) and n. We have solved this problem partially, that is, we have showed that the extremal graphs must have the number n_k of vertices of degree k less or equal n − k and the number of vertices of degree n − 1 less or equal k.