Affine Mappings in Iterated Function Systems

Today the reproduction of pictures in journals and books is no longer possible without “information compression”. This important process needs “Iterated Function Systems”-in short IFS. Doing so, affine mappings and their properties play a fundamental role. In this paper we distinguish between distance contracting and area contracting mappings and we investigate the connections between. The main goal is to find a criterion which allows to decide if the iteration process of an IFS converges or not.     

Orthocomplemented difference lattices in association with generalized rings

Orthocomplemented difference lattices (ODLs) are orthocomplemented lattices endowed with an additional operation of “abstract symmetric difference”. In studying ODLs as universal algebras or instances of quantum logics, several results have been obtained (see the references at the end of this paper where the explicite link with orthomodularity is discussed, too). Since the ODLs are “nearly Boolean”, a natural question arises whether there are “nearly Boolean rings” associated with ODLs. In this paper we find such an association — we introduce some difference ring-like algebras (the DRAs) that allow for a natural one-to-one correspondence with the ODLs. The DRAs are defined by only a few rather plausible axioms. The axioms guarantee, among others, that a DRA is a group and that the association with ODLs agrees, for the subrings of DRAs, with the famous Stone (Boolean ring) correspondence.     

Determining new possible weight values in PROMETHEE: a procedure based on data envelopment analysis

Operational research (OR) offers efficient tools to support managers in strategic decision-making processes. Data envelopment analysis (DEA) and multiple criteria decision aid (MCDA) are two important research areas in OR. These two domains are both based on the evaluation of “objects” according to multiple “points of views”. Within the MCDA framework, choosing appropriate weights for the different criteria often arises as a problem itself for decision makers. As a consequence, researchers have developed original methodologies to help them during this elicitation phase. In this work, we aim to investigate how DEA can be used to propose weights in the context of the PROMETHEE II method. More precisely, we suggest an extension of the so-called “decision maker brain” used in the GAIA plane (also known as PROMETHEE VI) based on DEA. The underlying idea is based on the computation of weights in PROMETHEE (GAIA brain) which are compatible with the DEA analysis. We end this paper with a numerical example.     

竞争型“以旧换再”闭环供应链策略选择及生产决策研究

针对两阶段闭环供应链系统,研究了古诺竞争型闭环供应链中的“以旧换再”策略选择问题。研究发现:(1)企业如何及何时实施“以旧换再”策略取决于自身及竞争企业的再制造水平。再制造水平不仅影响了企业的“以旧换再”数量,同时还会影响产品市场份额及利润。(2)“以旧换再”策略可以提高企业竞争力,增加企业产品市场份额和提高收益;(3)提高 “以旧换再”补贴及再制造产品接受度, 降低“二手市场”价格,均可以降低企业实施“以旧换再”策略和提高企业的“以旧换再”数量。     

Coarse grid correction by aggregation / disaggregation with application in image reconstruction

In algebraic reconstruction of images in computerized tomography we are dealing with rectangular, large, sparse and ill-conditioned linear systems of equations. In this paper we describe a two-grid algorithm for solving such kind of linear systems, which uses Kaczmarz's projection method as relaxation. The correction step is performed with a special “local” aggregation / disaggregation type procedure. In this respect, we have to solve a small sized minimization problem associated to each coarse grid pixel. The information so obtained is then “distributed” to the neighbour fine grid pixels. Some image reconstruction experiments are also presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetry principles via interactions and symmetry-violations

In this work we would like to continue our ideas concerning the principle mentioned at the end of [Appl. Math. Comput. 113 (2000) 289]. From now on we will call this principle the principle of symmetry of states. We recall that:Principle of symmetry of states. Any system whose states have the same symmetry can take part in “interaction-chain”, until the symmetry of the states will be maximal and the system will be stable. In this case, the system has a minimal number of different states. Each “intermediate system” can be viewed like a homomorphic image of the anterior.We recall the fact that we have used the automata theory for modelling the systems. The “homomorphic image” words mentioned in the statement of our principle refer to automata-homomorphism. We mention also that we have applied this principle concerning radioactivity, making a conjecture, that we have considered each radioactive nucleus like an automaton and for each radioactive chain (viewed as an interaction-chain) applied our principle.Our purpose in this work is to try to find other examples of interactions which are in concordance with our principle (are all the interactions in concordance with this principle ?).     

A Sino-German semi-virtual seminar in mathematics education

In summer 2006 the University of Education in Weingarten, Germany, and East China Normal University, Shanghai, performed a semi-virtual seminar with mathematics students on “Mathematics and Architecture”. The goal was the joint development of teaching materials for German or Chinese school, based on different buildings such as “Nanpu Bridge”, or the “Eiffel Tower”. The purpose of the seminar was to provide a learning environment for students supported by using information and communication technology (ICT) to understand how the hidden mathematics in buildings should be related to school mathematics; to experience the multicultural potential of the international language “Mathematics”; to develop “media competence” while communicating with others and using technologies in mathematics education; and to recognize the differences in teaching mathematics between the two cultures. In this paper we will present our ideas, experiences and results from the seminar.     

Is the definition of mathematics as used in the PISA assessment Framework applicable to the HarmoS Project?

The project known as the “Harmonisation of the Obligatory School”, or in its shortened form as “HarmoS”, has a high priority for Switzerland's educational policy in the coming years. Its purpose is to determine levels of competency, valid throughout Switzerland, for specific areas of study and including the subject of mathematics. The general theoretical basis of the overall HarmoS Project is constituted by the expertise written under the direction of Eckhard Klieme and entitled “Zur Entwicklung nationaler Bildungsstandards” (Klieme 2003) [i.e. “On the Development of National Education Standards”]. The proposal announcing the HarmoS partial project devoted to Mathematics includes references to the results and subsequent analysis of PISA 2003. It thus seems appropriate for us to begin our work on HarmoS with a critical consideration of the definition of mathematics and mathematical literacy as they are used in the PISA Study. In a first part, we want to describe the core ideas of HarmoS. In a second part, we will address the meaning of general educational goals for the development of competency models and education standards to the extent that it is necessary to properly locate our problem. In a third part we will analyse the concept of mathematics which is at the basis of the PISA Study (OECD 2004) and more precisely defined in the publication “Assessment Framework” (OECD 2003) In the fourth and last part, we will try to provide a differentiated answer to the question posed in the title of this paper.     

A presentation for reduced extended affine weyl groups

In 1985 K. Saito [Sal] introduced the concept of an extended affine Weyl group (EAWG), the Weyl group of an extended affine root system (EARS). In [A2, Section 5J, we gave a presentation called “a presentation by conjugation” for the class of EAWGs of index zero, a subclass of EAWGs. In this paper we will give a presentation wh.ich we call a “generalized present.ation by conjugation” for the class of reduced EAWGs. If the extended affine Weyl group is of index zero this presentation reduces to “a presentation by conjugation”. Our main result states that when the nullity of the EARS is 2, these two presentations coincide that is, EAWGs of nullity 2 have “a presentation by conjugation”. In [ST] another presentation for EAWGs of nullity 2 is given.     

Quantum Transport in Degenerate Systems

Transport in nonequilibrium degenerate quantum systems is investigated. The transfer rate depends on the parameters of the system. In this paper we investigate the dependence of the flow (transfer rate) on the angle between “bright” vectors (which define the interaction of the system with the environment). We show that in some approximation for the system under investigation the flow is proportional to the cosine squared of the angle between the “bright” vectors. Earlier the author has shown that in this degenerate quantum system excitation of nondecaying quantum “dark” states is possible; moreover, the effectiveness of this process is proportional to the sine squared of the angle between the “bright” vectors (this phenomenon was discussed as a possible model of excitation of quantum coherence in quantum photosynthesis). Thus quantum transport and excitation of dark states are competing processes; “dark” states can be considered as a result of leakage of quantum states in a quantum thermodynamic machine which performs the quantum transport.     

A necessary and sufficient condition of convexity for SOC reformulation of trust-region subproblem with two intersecting cuts

We consider the extended trust-region subproblem with two linear inequalities. In the “nonintersecting” case of this problem, Burer et al. have proved that its semi-definite programming relaxation with second-order-cone reformulation (SDPR-SOCR) is a tight relaxation. In the more complicated “intersecting” case, which is discussed in this paper, so far there is no result except for a counterexample for the SDPR-SOCR. We present a necessary and sufficient condition for the SDPR-SOCR to be a tight relaxation in both the “nonintersecting” and “intersecting” cases. As an application of this condition, it is verified easily that the “nonintersecting” SDPR-SOCR is a tight relaxation indeed. Furthermore, as another application of the condition, we prove that there exist at least three regions among the four regions in the trust-region ball divided by the two intersecting linear cuts, on which the SDPR-SOCR must be a tight relaxation. Finally, the results of numerical experiments show that the SDPR-SOCR can work efficiently in decreasing or even eliminating the duality gap of the nonconvex extended trust-region subproblem with two intersecting linear inequalities indeed.     

Graph congruences and what they connote

Abstract

The algebraic notion of a “congruence” seems to be foreign to contemporary graph theory. We propound that it need not be so by developing a theory of congruences of graphs: a congruence on a graph G = (V, E) being a pair (～, ) of which ～ is an equivalence relation on V and is a set of unordered pairs of vertices of G with a special relationship to ～ and E. Kernels and quotient structures are used in this theory to develop homomorphism and isomorphism theorems which remind one of similar results in an algebraic context. We show that this theory can be applied to deliver structural decompositions of graphs into “factor” graphs having very special properties, such as the result that each graph, except one, is a subdirect product of graphs with universal vertices. In a final section, we discuss corresponding concepts and briefly describe a corresponding theory for graphs which have a loop at every vertex and which we call loopy graphs. They are in a sense more “algebraic” than simple graphs, with their meet-semilattices of all congruences becoming complete algebraic lattices.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Interplay between local and international journals: The case of Sicily, 1880–1920

In 1884, Giovan Battista Guccia founded first the Circolo Matematico di Palermo and then some years later its journal, the Rendiconti del Circolo Matematico di Palermo. Although historians of mathematics have published a number of works on the Circolo and the Rendiconti, there are very few systematic studies on mathematics in Sicilian periodicals. In our paper, we shall investigate the relationships between the “international” Rendiconti and the “local” proceedings published by the Sicilian academies located in Catania, Messina, and Palermo. What is the image of mathematics that emerges from these journals? What is the presence of Sicilian mathematicians among the authors in the different cases? May we recognize a Sicilian dynamics in mathematics and, if so, in what sense?     

Generation of nonlinear modulated waves in a modified Noguchi electrical transmission network

In the present work, we study the generation of nonlinear modulated waves in a modified version of Noguchi electrical tr ansmission network. In the continuum limit, we have considered the semi discrete approximation and showed that wave modulations in the network are governed by a generalized Chen-Lee-Liu (G-CLL) equation whose “self steepening” parameter is free from line parameters. We have investigated the effects of the “self steepening” parameter of the equation on the dynamics of modulated waves propagating in the network and shown that it can be adequately used either to enhance or to soften the instability of the nonlinear Schrödinger (NLS) equation. Our investigations showed that the introduction of the “self steepening” term in the NLS equation of the network allows bright and dark solitonlike waves to coexist in the same modulational stable and unstable frequency regions of the NLS. Our analytical studies of the G-CLL equation of the network showed that the amplitude of the solitonlike waves propagating in the network decreases as the “self steepening” parameter of the G-CLL equation increases.     

Nonlinear Resonances Leading to Strong Pulse Attenuation in Granular Dimer Chains

We study nonlinear resonances in granular periodic one-dimensional chains. Specifically, we consider a diatomic (“dimer”) chain composed of alternating “heavy” and “light” spherical beads with no precompression. In a previous work (Jayaprakash et al. in Phys. Rev. E 83(3):036606, 2011) we discussed the existence of families of solitary waves in these systems that propagate without distortion of their waveforms. We attributed this dynamical feature to “antiresonance” in the dimer that led to the complete elimination of radiating waves in the trail of the propagating solitary wave. Antiresonances were associated with certain symmetries of the velocity waveforms of the dimer beads. In this work we report on the opposite phenomenon: the break of waveform symmetries, leading to drastic attenuation of traveling pulses due to radiation of traveling waves to the far field. We use the connotation of “resonance” to describe this dynamical phenomenon resulting in maximum amplification of the amplitudes of radiated waves that emanate from the propagating pulse. Each antiresonance can be related to a corresponding resonance in the appropriate parameter plane. We study the nonlinear resonance mechanism numerically and analytically and show that it can lead to drastic attenuation of pulses propagating in the dimer. Furthermore, we estimate the discrete values of the normalized mass ratio between the light and heavy beads of the dimer for which resonances are realized. Finally, we show that by adding precompression the resonance mechanism gradually degrades, as does the capacity of the dimer to passively attenuate propagating pulses.     

Bidimensional allocation of seats via zero-one matrices with given line sums

In some proportional electoral systems with more than one constituency the number of seats allotted to each constituency is pre-specified, as well as, the number of seats that each party has to receive at a national level. “Bidimensional allocation” of seats to parties within constituencies consists of converting the vote matrix V into an integer matrix of seats “as proportional as possible” to V, satisfying constituency and party totals and an additional “zero-vote zero-seat” condition. In the current Italian electoral law this Bidimensional Allocation Problem (or Biproportional Apportionment Problem—BAP) is ruled by an erroneous procedure that may produce an infeasible allocation, actually one that is not able to satisfy all the above conditions simultaneously. In this paper we focus on the feasibility aspect of BAP and, basing on the theory of (0,1)-matrices with given line sums, we formulate it for the first time as a “Matrix Feasibility Problem”. Starting from some previous results provided by Gale and Ryser in the 60’s, we consider the additional constraint that some cells of the output matrix must be equal to zero and extend the results by Gale and Ryser to this case. For specific configurations of zeros in the vote matrix we show that a modified version of the Ryser procedure works well, and we also state necessary and sufficient conditions for the existence of a feasible solution. Since our analysis concerns only special cases, its application to the electoral problem is still limited. In spite of this, in the paper we provide new results in the area of combinatorial matrix theory for (0,1)-matrices with fixed zeros which have also a practical application in some problems related to graphs.     

The role of normally hyperbolic invariant manifolds (NHIMS) in the context of the phase space setting for chemical reaction dynamics

In this paper we give an introduction to the notion of a normally hyperbolic invariant manifold (NHIM) and its role in chemical reaction dynamics.We do this by considering simple examples for one-, two-, and three-degree-of-freedom systems where explicit calculations can be carried out for all of the relevant geometrical structures and their properties can be explicitly understood. We specifically emphasize the notion of a NHIM as a “phase space concept”. In particular, we make the observation that the (phase space) NHIM plays the role of “carrying” the (configuration space) properties of a saddle point of the potential energy surface into phase space.We also consider an explicit example of a 2-degree-of-freedom system where a “global” dividing surface can be constructed using two index one saddles and one index two saddle. Such a dividing surface has arisen in several recent applications and, therefore, such a construction may be of wider interest.     

Statistician Heal Thyself: Have We Lost the Plot?

In 1984, Cleveland suggested that statisticians have an important role in changing the use of graphics in science for the better. Thirty years later, we compared graphs published in top-rated applied science and statistics journals, evaluated for overall quality and against five principles of graphical excellence. Nearly 40% of the 97 graphs we sampled were rated as poor, with no striking differences between the applied science and statistics graphs. Better use of graphs requires better definition of variables, units of measurement, scales, groups, and other graphical elements, and more routine use of grid lines on a “standard” set of graphical forms. Progress over the next 30 years needs to be supported by changes in software defaults.