A basic problem in the correlations between statistics and thermodynamics

By using a functional of probability distributions, several different statistical physics including extensive and nonextensive statistics are unified in a general formalism. It’s shown that the equivalence between the Maxent approach and the relation dU = TdS can be seen from the virtual work principle of mechanics. From dU = TdS, any entropic form which can exists for thermodynamical equilibrium system can be maximized in order to derive the probability distribution functions.     

Trace formula and Spectral Riemann Surfaces for a class of tri-diagonal matrices

For tri-diagonal matrices arising in the simplified Jaynes–Cummings model, we give an asymptotics of the eigenvalues, prove a trace formula and show that the Spectral Riemann Surface is irreducible.     

Fractional Nottale’s Scale Relativity and emergence of complexified gravity

Fractional calculus of variations has recently gained significance in studying weak dissipative and nonconservative dynamical systems ranging from classical mechanics to quantum field theories. In this paper, fractional Nottale’s Scale Relativity (NSR) for an arbitrary fractal dimension is introduced within the framework of fractional action-like variational approach recently introduced by the author. The formalism is based on fractional differential operators that generalize the differential operators of conventional NSR but that reduces to the standard formalism in the integer limit. Our main aim is to build the fractional setting for the NSR dynamical equations. Many interesting consequences arise, in particular the emergence of complexified gravity and complex time.     

New exact solutions to the -dimensional Konopelchenko–Dubrovsky equation

In this paper, the extended tanh method, the sech–csch ansatz, the Hirota’s bilinear formalism combined with the simplified Hereman form and the Darboux transformation method are applied to determine the traveling wave solutions and other kinds of exact solutions for the (2+1)-dimensional Konopelchenko–Dubrovsky equation and abundant new soliton solutions, kink solutions, periodic wave solutions and complexiton solutions are formally derived. The work confirms the significant features of the employed methods and shows the variety of the obtained solutions.     

The local Cauchy problem for multicomponent reactive flows in full vibrational non-equilibrium

We consider reactive mixtures of dilute polyatomic gases in full vibrational non-equilibrium. The governing equations are derived from the kinetic theory and possesses an entropy. We recast this system of conservation laws into a symmetric conservative form by using entropic variables. Following a formalism developed by the authors in a previous paper, the system is then rewritten into a normal form, that is, in the form of a quasilinear symmetric hyperbolic–parabolic system. Using a result of Vol'pert and Hudjaev, we prove local existence and uniqueness of a bounded smooth solution to the Cauchy problem. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Students’ perceived sociomathematical norms: The missing paradigm

This study proposes a framework for research which takes into account three aspects of sociomathematical norms: teachers’ endorsed norms, teachers’ and students’ enacted norms, and students’ perceived norms. We investigate these aspects of sociomathematical norms in two elementary school classrooms in relation to mathematically based and practically based explanations. Results indicate that even when the observed enacted norms are in agreement with the teachers’ endorsed norms, the students may not perceive these same norms. These results highlight the need to consider the students’ perspective when investigating sociomathematical norms.     

Students’ Activity in Computer-Supported Collaborative Problem Solving in Mathematics

The purpose of this study was to analyse secondary school students’ (N = 16) computer-supported collaborative mathematical problem solving. The problem addressed in the study was: What kinds of metacognitive processes appear during computer-supported collaborative learning in mathematics? Another aim of the study was to consider the applicability of networked learning in mathematics. The network-based learning environment Knowledge Forum (KF) was used to support students’ collaborative problem solving. The data consist of 188 posted computer notes, portfolio material such as notebooks, and observations. The computer notes were analysed through three stages of qualitative content analysis. The three stages were content analysis of computer notesin mathematical problem solving, content analysis of mathematical problem solving activity and content analysis of the students’ metacognitive activity. The results of the content analysis illustrate how networked discussions mediated mathematical knowledge and students’ questions, while the mathematical problem solving activity shows that the students co-regulate their thinking. The results of the content analysis of the students’ metacognitive activity revealed that the students use metacognitive knowledge and make metacognitive judgments and perform monitoring during networked discussions. In conclusion, the results of this study demonstrate that working with the networked technology contributes to the students’ use of their mathematical knowledge and stimulates them into making their thinking visible. The findings also show some metacognitive activity in the students’ computer-supported collaborative problem solving in mathematics.     

An Ekeland’s variational principle for set-valued mappings with applications

In this paper, we obtain a general Ekeland’s variational principle for set-valued mappings in complete metric space, which is different from those in [G.Y. Chen, X.X. Huang, Ekeland’s ε-variational principle for set-valued mapping, Mathematical Methods of Operations Research 48 (1998) 181–186; G.Y. Chen, X.X. Huang, S.H. Hou, General Ekeland’s Variational Principle for Set-Valued Mappings, Journal of Optimization Theory and Applications 106 (2000) 151–164; S.J. Li, W.Y. Zhang, On Ekeland’s variational Principle for set-valued mappings, Acta Mathematicae Application Sinica, English Series 23 (2007) 141–148]. By the result, we prove some existence results for a general vector equilibrium problem under nonconvex and compact or noncompact assumptions of its domain, respectively. Moreover, we give some equivalent results to the variational principle.     

A multisymplectic framework for classical field theory and the calculus of variations II: space + time decomposition

In a previous paper I laid the foundations of a covariant Hamiltonian framework for the calculus of variations in general. The purpose of the present work is to demonstrate, in the context of classical field theory, how this covariant Hamiltonian formalism may be space + time decomposed. It turns out that the resulting “instantaneous” Hamiltonian formalism is an infinite- dimensional version of Ostrogradski 's theory and leads to the standard symplectic formulation of the initial value problem. The salient features of the analysis are: (i) the instantaneous Hamiltonian formalism does not depend upon the choice of Lepagean equivalent; (ii) the space + time decomposition can be performed either before or after the covariant Legendre transformation has been carried out, with equivalent results; (iii) the instantaneous Hamiltonian can be recovered in natural way from the multisymplectic structure inherent in the theory; and (iv) the space + time split symplectic structure lives on the space of Cauchy data for the evolution equations, as opposed to the space of solutions thereof.     

An exact solution of start-up flow for the fractional generalized Burgers’ fluid in a porous half-space

Modified Darcy’s law for fractional generalized Burgers’ fluid in a porous medium is introduced. The flow near a wall suddenly set in motion for a fractional generalized Burgers’ fluid in a porous half-space is investigated. The velocity of the flow is described by fractional partial differential equations. By using the Fourier sine transform and the fractional Laplace transform, an exact solution of the velocity distribution is obtained. Some previous and classical results can be recovered from our results, such as the velocity solutions of the Stokes’ first problem for viscous Newtonian, second grade, Maxwell, Oldroyd-B or Burgers’ fluids.     

Computation of an infinite integral using series acceleration

A numerical computation in crystallography involves an infinite integral depending on one parameter. In a recent article in this journal this computational problem is addressed using Romberg’s method and tools for error control. One observe numerical difficulties with the reported approach both near the parameter’s endpoints and near the parameter interval’s midpoint. In this short note we will present an alternative approach making use of a known infinite series formulation of the problem at hand and a simple and efficient series acceleration technique. If some care is taken to avoid cancellations the numerical results are excellent for all values of the parameter. AMS subject classification 65B05, 65B10, 65D30     

A generalized mixed vector variational-like inequality problem

In this paper, we introduce relaxed η-α-P-monotone mapping, and by utilizing KKM technique and Nadler’s Lemma we establish some existence results for the generalized mixed vector variational-like inequality problem. Further, we give the concepts of η-complete semicontinuity and η-strong semicontinuity and prove the solvability for generalized mixed vector variational-like inequality problem without monotonicity assumption by applying the Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.     

Optimal Mal’tsev conditions for congruence modular varieties

For varieties, congruence modularity is equivalent to the tolerance intersection property, TIP in short. Based on TIP, it was proved in [5] that for an arbitrary lattice identity implying modularity (or at least congruence modularity) there exists a Mal’tsev condition such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal’tsev condition. However, the Mal’tsev condition constructed in [5] is not the simplest known one in general. Now we improve this result by constructing the best Mal’tsev condition and various related conditions. As an application, we give a particularly easy new proof of the result of Freese and Jónsson [11] stating that modular congruence varieties are Arguesian, and we strengthen this result by replacing “Arguesian” by “higher Arguesian” in the sense of Haiman [18]. We show that lattice terms for congruences of an arbitrary congruence modular variety can be computed in two steps: the first step mimics the use of congruence distributivity, while the second step corresponds to congruence permutability. Particular cases of this result were known; the present approach using TIP is even simpler than the proofs of the previous partial results.Dedicated to the memory of Ivan RivalReceived February 12, 2003; accepted in final form August 5, 2004.This revised version was published online in August 2005 with a corrected cover date.     

On an approach to the study of the Jaynes–Cummings sum in quantum optics

A new approach to the study of the Jaynes–Cummings sum, which determines the atomic inversion in quantum model of a single two-level atom interacting with a single mode of the quantized radiation field, based on the number theory theorems on approximation of trigonometric sums is presented.      

Strong law of large numbers for countable nonhomogeneous Markov chains

The aim of this paper is to establish a strong law of large numbers for the bivariate functions of countable nonhomogeneous Markov chains under the condition of uniform convergence in the Cesàro sense which differs from my previous results. As corollaries, we generalize one of the Liu and Liu’s results for the univariate functions case and obtain another Shannon–McMillan–Breiman theorem for this Markov chains.     

An improvement on Marotto’s theorem and its applications to chaotification of switching systems

The controversy surrounding the correctness of Marotto’s theorem continues over the last two decades, with many researchers claiming to have found an error in the proof. In this paper, we show that Marotto’s theorem is indeed correct for analyzing the existence of chaos in the sense of Li-Yorke even after relaxing certain assumptions in the proof. In addition, we extend the theory to derive the conditions for the existence of chaos in the sense of Devaney. We show that these results can be applied to study the chaotification of linear switching systems.     

Non-parametric efficiency, progress and regress measures for panel data: Methodological aspects

This purely methodological paper deals with the rôle of time in non-parametric efficiency analysis. Using both FDH and DEA technologies, it first shows how each observation in a panel can be characterized in efficiency terms vis-à-vis three different kinds of frontiers: (i) ‘contemporaneous’, (ii) ‘sequential’, and (iii) ‘intertemporal’. These are then compared with window analysis. Next, frontier shifts ‘outward’ and ‘inward’, interpreted as progress or regress are considered for the two kinds of technologies, and computational methods are described in detail for evaluating such shifts in either case. These are also contrasted with what is measured by the ‘Malmquist’ productivity index. Finally, an alternative way of identifying progress and regress, independent of the frontier notion and referring instead to some ‘benchmark’ notion, is extended here to panel data.     

On the solution of Dirichlet’s problem of the complex Monge–Ampère equation for the Cartan–Hartogs domain of the first type

The complex Monge–Ampère equation is a nonlinear equation with high degree; therefore getting its solution is very difficult. In the present paper how to get the solution of Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type is discussed, using an analytic method. Firstly, the complex Monge–Ampère equation is reduced to a nonlinear ordinary differential equation, then the solution of Dirichlet’s problem of the complex Monge–Ampère equation is reduced to the solution of a two-point boundary value problem for a nonlinear second-order ordinary differential equation. Secondly, the solution of Dirichlet’s problem is given as a semi-explicit formula, and in a special case the exact solution is obtained. These results may be helpful for a numerical method approach to Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type.