ζ-function and heat kernel formulae

We present a systematic study of asymptotic behaviour of (generalised) ζ-functions and heat kernels used in noncommutative geometry and clarify their connections with Dixmier traces. We strengthen and complete a number of results from the recent literature and answer (in the affirmative) the question raised by M. Benameur and T. Fack (2006) [1].     

On simultaneous approximation to (α,α) with α+kα−1=0

We show that the modified Jacobi-Perron algorithm gives the best simultaneous approximation to (α,α²) with α³+kα−1=0. We claim the following facts:

(1)

the limit set of become an ellipse, where (p_n,q_n,r_n) is the nth convergent (p_n/q_n,r_n/q_n) of (α,α²) by the modified Jacobi-Perron algorithm,

(2)

the limit set of belongs to outside of the ellipse in (1).

     

Symmetric duality with (p, r) − ρ − (η, θ)-invexity

In this paper we introduce a new type of generalized invex function, called (p, r) − ρ − (η, θ)-invex function and study symmetric duality results under these assumptions. In our study the nonnegative orthants for the constraints are replaced by closed convex cones and their polars. We establish weak and strong duality theorems under (p, r) − ρ − (η, θ)-invexity assumptions for the symmetric dual problems. We also give many examples to justify our results.     

π and the hypergeometric functions of complex argument

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In this article we derive some new identities concerning π, algebraic radicals and some special occurrences of the Gauss hypergeometric function ₂F₁ in the analytic continuation. All of them have been derived by tackling some elliptic or hyperelliptic known integral, and looking for another representation of it by means of hypergeometric functions like those of Gauss, Appell or Lauricella. In any case we have focused on integrand functions having at least one couple of complex-conjugate roots. Founding upon a special hyperelliptic reduction formula due to Hermite (1876) [6], π is obtained as a ratio of a complete elliptic integral and the four-variable Lauricella function. Furthermore, starting with a certain binomial integral, we succeed in providing as a ratio of a linear combination of complete elliptic integrals of the first and second kinds to the Appell hypergeometric function of two complex-conjugate arguments. Each of the formulae we found theoretically has been satisfactorily tested by means of Mathematica^®.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=rQqtVtAf-RQ.     

Nonsmooth minimax programming under locally Lipschitz (Φ, ρ)-invexity

Minimax programming problems involving locally Lipschitz (Φ, ρ)-invex functions are considered. The parametric and non-parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable (Φ, ρ)-invexity assumption imposed on objective and constraint functions. When the sufficient conditions are utilized, parametric and non-parametric dual problems in the sense of Mond-Weir and Wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem. With the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems.     

η Forms and determinant lines

We construct a canonical zeta function (Quillen) connection on the determinant line bundle for a family of APS elliptic boundary value problems and show that its curvature is the 2-form component of a relative eta form.     

The hyperelliptic integrals and π

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A class of hyperelliptic integrals are expressed through hypergeometric functions, like those of Gauss, Lauricella and Appell, namely multiple power series. Whenever they can on their own be reduced to elliptic integrals through an algebraic transformation, we obtain a two-fold representation of the same mathematical object, and then several completely new π determinations through the above special functions and/or Euler integrals. All our π formulae have been successfully tested by means of convenient Mathematica_®'s packages and enter in a wide historical/sound context of π-formulae quite far from being exhausted. Due to their structure, the formulae's practical value does not lie in computing π, but in allowing, through π, a benchmark for computing the involved special functions, particularly those less elementary.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=UHZIHgAeCUc.     

Quotients and colimits of κ-quantales

Let κQnt be the category of κ-quantales, quantales closed under κ-joins in which the monoid identity is the largest element. (κ is an infinite regular cardinal.) Although the lack of lattice completeness in this setting would seem to mitigate against the techniques which lend themselves so readily to the calculation of frame quotients, we show how to easily compute κQnt quotients by applying generalizations of the frame techniques to suitable extensions of this category.The second major tool in the analysis is the free κ-quantale over a λ-quantale, κ?λ. Surprisingly, these can be characterized intrinsically, and the generating sub-κ-quantale can even be identified. The result that the λ-free κ-quantales coincide with the λ-coherent κ-quantales directly generalizes Madden?s corresponding result for κ-frames.These tools permit a direct and intuitive construction of κQnt colimits. We provide two applications: an intrinsic characterization of κQnt colimits, and of free (over sets) κ-quantales. The latter is a direct generalization of Whitman?s condition for distributive lattices.     

On Tauberian theorems for (A)(C, α) summability method

In this paper we introduce some Tauberian conditions for the (A)(C, α) summability method. These results extend and generalize some of the classical Tauberian theorems for the Abel summability method.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

Semiconductor final test scheduling with Sarsa(λ, k) algorithm

Semiconductor test scheduling problem is a variation of reentrant unrelated parallel machine problems considering multiple resource constraints, intricate {product, tester, kit, enabler assembly} eligibility constraints, sequence-dependant setup times, etc. A multi-step reinforcement learning (RL) algorithm called Sarsa(λ, k) is proposed and applied to deal with the scheduling problem with throughput related objective. Allowing enabler reconfiguration, the production capacity of the test facility is expanded and scheduling optimization is performed at the bottom level. Two forms of Sarsa(λ, k), i.e. forward view Sarsa(λ, k) and backward view Sarsa(λ, k), are constructed and proved equivalent in off-line updating. The upper bound of the error of the action-value function in tabular Sarsa(λ, k) is provided when solving deterministic problems. In order to apply Sarsa(λ, k), the scheduling problem is transformed into an RL problem by representing states, constructing actions, the reward function and the function approximator. Sarsa(λ, k) achieves smaller mean scheduling objective value than the Industrial Method (IM) by 68.59% and 76.89%, respectively for real industrial problems and randomly generated test problems. Computational experiments show that Sarsa(λ, k) outperforms IM and any individual action constructed with the heuristics derived from the existing heuristics or scheduling rules.     

Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem

In this paper we study generalised prime systems for which the integer counting function NP(x) is asymptotically well behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) is holomorphic for . We prove that for , for any ε>0, and also for ε=0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in NkP(x)−Ress=1(ζPk(s)xs/s), which is O(xθ) for some θ<1. Letting αk denote the infimum of such θ, we show that .     

Computational study of decaying annular vortex flow using the Rε/k − ε turbulence model

In this article, we present three dimensional CFD study of turbulent vortex flow in an annular passage using OpenFOAM 1.6. The vortex flow is generated by introducing the flow through a tangential entry to the passage. For the analysis presented in this article, turbulence was modeled using the R_ε/k − ε model, in addition, a comparison between such model with the standard k − ε model was conducted and discussed. The main characteristics of the flow such as vortex structure and recirculation zone were investigated. It was found that flow is subjected to Rankine vortex structure with three forced vortex regimes and a free vortex region near to the outer wall. The phenomenon of vortex decay was investigated by depicting the swirl number trend along the axial direction of the flow domain. It was found that the vortex decay is subjected to an exponential decay behavior. New coefficients for the exponential decay correlation were derived based on local values of velocity components in different radial planes.     

Domb's numbers and Ramanujan-Sato type series for 1/π

In this article, we construct a general series for . We indicate that Ramanujan's -series are all special cases of this general series and we end the paper with a new class of -series. Our work is motivated by series recently discovered by Takeshi Sato.     

The σ-isotypic decomposition and the σ-index of reversible-equivariant systems

This work is concerned with dynamical systems in presence of symmetries and reversing symmetries. We describe a construction process of subspaces that are invariant by linear Γ-reversible-equivariant mappings, where Γ is the compact Lie group of all the symmetries and reversing symmetries of such systems. These subspaces are the σ-isotypic components, first introduced by Lamb and Roberts in (1999) [10] and that correspond to the isotypic components for purely equivariant systems. In addition, by representation theory methods derived from the topological structure of the group Γ, two algebraic formulae are established for the computation of the σ-index of a closed subgroup of Γ. The results obtained here are to be applied to general reversible-equivariant systems, but are of particular interest for the more subtle of the two possible cases, namely the non-self-dual case. Some examples are presented.     

New approximation-solvability of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.     

The Fibonacci sequence modulo π, chaos and some rational recursive equations

In this paper, we provide the closed form solution to the inter-related equations

     

α Regularization of the POD-Galerkin dynamical systems of the Kuramoto-Sivashinsky equation

In order to improve the dynamics and stability of the POD-Galerkin models of strongly-stiff systems, an α-like regularization is suggested and assessed in the present article. In this method, the POD eigenmodes of the non-linear terms are replaced by their Helmholtz filtered counterparts, while the other terms are remained unchanged. As an example, the method is applied to the POD-Galerkin models of the one-dimensional Kuramoto-Sivashinsky (KS) equation in a full chaotic regime; and the fidelity of the original and regularized models to the direct numerical simulations (DNS) are investigated. Moreover, the effects of regularization on the dynamics of various terms, and whole of the systems, are analyzed via eigenvalue analysis of each term separately, and the total dynamical system as a whole. The numerical experiments show definite effectiveness of the method and excellent improvements in the predicted dynamics and stability, by minimum number of free parameters.     

On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI

Painlevé's transcendental differential equation PVI may be expressed as the consistency condition for a pair of linear differential equations with 2×2 matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators Γ? of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let P(t,∞):L²(0,∞)→L²(t,∞) be the orthogonal projection; then the Fredholm determinant τ(t)=det(I−P(t,∞)Γ?) defines the τ function, which is here expressed in terms of the solution of a matrix Gelfand-Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L²(0,∞); so det(I−Γ?P(t,∞)) can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé's equation with ?=1.