Transformation of Quasiconvex Functions to Eliminate Local Minima

Quasiconvex functions present some difficulties in global optimization, because their graph contains “flat parts”; thus, a local minimum is not necessarily the global minimum. In this paper, we show that any lower semicontinuous quasiconvex function may be written as a composition of two functions, one of which is nondecreasing, and the other is quasiconvex with the property that every local minimum is global minimum. Thus, finding the global minimum of any lower semicontinuous quasiconvex function is equivalent to finding the minimum of a quasiconvex function, which has no local minima other than its global minimum. The construction of the decomposition is based on the notion of “adjusted sublevel set.” In particular, we study the structure of the class of sublevel sets, and the continuity properties of the sublevel set operator and its corresponding normal operator.     

Singular sources mining using evidential conflict analysis

Singular sources mining is essential in many applications like sensor fusion or dataset analysis. A singular source of information provides pieces of evidence that are significantly different from the majority of the other sources. In the Dempster-Shafer theory, the pieces of evidence collected by a source are summarized by basic belief assignments (bbas). In this article, we propose to mine singular sources by analyzing the conflict between their corresponding bbas. By viewing the conflict as a function of parameters called discounting rates, new developments are obtained and a criterion that weights the contribution of each bba to the conflict is introduced. The efficiency and the robustness of this criterion is demonstrated on several sets of bbas with various specificities.     

A convergent decomposition algorithm for support vector machines

In this work we consider nonlinear minimization problems with a single linear equality constraint and box constraints. In particular we are interested in solving problems where the number of variables is so huge that traditional optimization methods cannot be directly applied. Many interesting real world problems lead to the solution of large scale constrained problems with this structure. For example, the special subclass of problems with convex quadratic objective function plays a fundamental role in the training of Support Vector Machine, which is a technique for machine learning problems. For this particular subclass of convex quadratic problem, some convergent decomposition methods, based on the solution of a sequence of smaller subproblems, have been proposed. In this paper we define a new globally convergent decomposition algorithm that differs from the previous methods in the rule for the choice of the subproblem variables and in the presence of a proximal point modification in the objective function of the subproblems. In particular, the new rule for sequentially selecting the subproblems appears to be suited to tackle large scale problems, while the introduction of the proximal point term allows us to ensure the global convergence of the algorithm for the general case of nonconvex objective function. Furthermore, we report some preliminary numerical results on support vector classification problems with up to 100 thousands variables.     

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.     

The dependence of the height of a Lorenz curve of a Zipf function on the size of the system

The Lorenz curve of a Zipf function describes, graphically, the relation between the fraction of the items and the fraction of the sources producing these items. Hence it generalizes the so-called 80/20-rule to general fractions.In this paper we examine the relation of such Lorenz curves with the size of the system (expressed by the total number of sources). We prove that the height of such a Lorenz curve is an increasing function of the total number of sources.In other words, we show that the share of items as a function of the corresponding share of sources increases with increasing size of the system. This conclusion is opposite (but not in contradiction) to a conclusion of Aksnes and Sivertsen (studied in an earlier paper of Egghe) but where “share of sources” is replaced by “number of sources”.     

An axiomatic derivation of the coding-theoretic possibilistic entropy

We re-take the possibilistic (strictly non-probabilistic) model for information sources and information coding put forward in (Fuzzy Sets and Systems 132–1 (2002) 11–32); the coding-theoretic possibilistic entropy is defined there as the asymptotic rate of compression codes, which are optimal with respect to a possibilistic (not probabilistic) criterion. By proving a uniqueness theorem, in this paper we provide also an axiomatic derivation for such a possibilistic entropy, and so are able to support its use as an adequate measure of non-specificity, or rather of “possibilistic ignorance”, as we shall prefer to say. We compare our possibilistic entropy with two well-known measures of non-specificity: Hartley measure as found in set theory and U-uncertainty as found in possibility theory. The comparison allows us to show that the latter possesses also a coding-theoretic meaning.     

On the Description of the Content-Equality by a Relation

In the geometry of polyhedra we understand by an elementary content-functional a real valued, non-negative, finite additive measure on the set of polyhedra which is invariant under isometries. There are close relations between the content-measurement and the relation of equidecomposability. Two polyhedra are called equidecomposable if they are decomposed into pairwise congruent pieces. For an example we consider the set of all polygons in the euclidean plane. It is well known that planar polygons have the same area if and only if they are equidecomposable. In the three-dimensional euclidean space one also can describe the content-equality of polyhedra by a relation. Two polyhedra have the same volume if they are equidecomposable with respect to equiaffine mappings (see [3]). In [4] the concept of an invariant content of polyhedra in a topological Klein space is introduced. Each regular closed quasicompact set ot the space is called polyhedron. Under this supposition two polyhedra have equal contents if they are equivalent by decomposition. The relation “equivalent by decomposition” is closely related to the relation “equidecomposable”.     

Measuring the fuzziness of human thoughts: An application of fuzzy sets to sociological research

Conventionally, sociologists measure the membership of an individual to a group by a “0 or 1” characteristic function. But when the definition of that group is fuzzy and an individual is neither a full member nor a nonmember, this dichotomous characteristic function may distort the reality. Instead of the “0 or 1” characteristic function by classical set theory, fuzzy set theory introduces a membership function which is a gradation from 0 to 1 to measure the degree to which an object (an individual) belongs to a concept (a group). Based on the rationale of fuzzy set theory, we suggest some new methods of data collection and analysis. Among several noteworthy findings, two points are emphasized: 1) the fuzzy set is an appropriate way of measuring the fuzziness of human thought; and 2) it allows one to relax the conventional assumption that all individuals have identical distributions and deviations around their means.     

A classification of teacher interventions in mathematics teaching

One of the best-known quotes in pedagogical literature is Maria Montessoris “Help me to do it myself.” This citation can be applied to many open questions. For example, how to help students working autonomously on cognitively demanding tasks is not only an unanswered question in didactical literature, but there has also been relatively little research done in this area. This article reflects upon qualitatively oriented studies from the German research project DISUM and selected literature about “teacher interventions”. Based on this, we propose, from a mathematically didactic point of view, a multi-dimensional framework, which allows us to identify central aspects of teacher interventions.     

About a generalized model of lymphoma

In this work we introduce and analyze a generalized model of precursor T-lymphoblastic lymphoma as a competition between two clonotypes of naïve T-cells, one “normal” and one tumorous. It is modeled as a continuous-time bivariate Markov process. Using an expansion of the master equation a deterministic approximation and the Fokker–Planck equation are derived. For a deterministic model we show existence and uniqueness of global solutions and positive invariance of the first quadrant of the phase space. Stability analysis of the model is performed, finding conditions guaranteeing existence of a unique, positive, steady state, which is proved to be globally stable. It is shown that expectations of fluctuations for both clonotypes tend to zero for large time. We also present numerical simulations in which two types of behavior of solutions are observed: either both clonotypes survive in the repertoire or the “normal” clonotype becomes extinct. Comparing this result with the rules of maintenance of naïve T-cell repertoire, which say that clonotypes with more specific set of receptors have longer life-span, it seems that “normal” clonotype follows them, whereas the tumorous one violates them and tends to the maximum possible expansion. The model supports the hypothesis of mutated precursor cells as an origin of cancer.     

Almost classical solutions to the total variation flow

The paper examines the one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of “almost classical” solutions we are able to determine evolution of facets – flat regions of solutions. A key element of our approach is the natural regularity determined by the nonlinear elliptic operator, for which x ² is an example of an irregular function. Such a point of view allows us to construct solutions. We apply this idea to numerical simulations for typical initial data. Due to the nature of Dirichlet data, any monotone function is an equilibrium. We prove that each solution reaches such a steady state in finite time.     

Unifying local–global type properties in vector optimization

It is well-known that all local minimum points of a semistrictly quasiconvex real-valued function are global minimum points. Also, any local maximum point of an explicitly quasiconvex real-valued function is a global minimum point, provided that it belongs to the intrinsic core of the function’s domain. The aim of this paper is to show that these “local min–global min” and “local max–global min” type properties can be extended and unified by a single general local–global extremality principle for certain generalized convex vector-valued functions with respect to two proper subsets of the outcome space. For particular choices of these two sets, we recover and refine several local–global properties known in the literature, concerning unified vector optimization (where optimality is defined with respect to an arbitrary set, not necessarily a convex cone) and, in particular, classical vector/multicriteria optimization.     

Generalized nonlinear ordinary differential equations linear in measures

Summary In this paper one treats the initial value problem for nonlinear measure differential equations. Under various hypotheses one gets existence of global solutions and sometimes uniqueness too. There is also an example giving nonexistence or bifurcation depending on the initial datum and the measure in the equation. The results are generalizations to the nonlinear case of earlier linear results by the author. One main feature is the procedure which shows how the solutions act at points where the involved measures have point masses of arbitrary magnitude.     

Mass and spring dimer Fermi–Pasta–Ulam–Tsingou nanopterons with exponentially small,nonvanishing ripples

We study traveling waves in mass and spring dimer Fermi–Pasta–Ulam–Tsingou (FPUT) lattices in the long wave limit. Such lattices are known to possess nanopteron traveling waves in relative displacement coordinates. These nanopteron profiles consist of the superposition of an exponentially localized “core,” which is close to a Korteweg–de Vries solitary wave, and a periodic “ripple,” whose amplitude is small beyond all algebraic orders of the long wave parameter, although a zero amplitude is not precluded. Here we deploy techniques of spatial dynamics, inspired by results of Iooss and Kirchgässner, Iooss and James, and Venney and Zimmer, to construct mass and spring dimer nanopterons whose ripples are both exponentially small and also nonvanishing. We first obtain “growing front” traveling waves in the original position coordinates and then pass to relative displacement. To study position, we recast its traveling wave problem as a first-order equation on an infinite-dimensional Banach space; then we develop hypotheses that, when met, allow us to reduce such a first-order problem to one solved by Lombardi. A key part of our analysis is then the passage back from the reduced problem to the original one. Our hypotheses free us from working strictly with lattices but are easily checked for FPUT mass and spring dimers. We also give a detailed exposition and reinterpretation of Lombardi's methods, to illustrate how our hypotheses work in concert with his techniques, and we provide a dialog with prior methods of constructing FPUT nanopterons, to expose similarities and differences with the present approach.     

A queueing theoretic methodology for the analysis of separable conflict resolution algorithms with variable length elementary events

We describe a queueing theoretic approach to the delay analysis for the class of synchronous random-access protocols consisting of a Capetanakis-type Tree Algorithm for conflict resolution and a window algorithm for channel access. Our method features a stochastic decomposition, in which a major component of the delay is viewed as a discrete time queueing problem, where each window (selected by the channel access algorithm) becomes a customer requiring service in the form of conflict resolution. This technique is sufficiently powerful to give us the distribution of the packet delay in steady state. In this paper, we extend our method to allow the durations of elementary algorithmic steps to take on a general distribution (rather than being constants), which allows us to provide a unified treatment of channels with shared errors, some types of explicit reservation systems, and Local Area Networks with carrier sensing and/or collision detection, possibly in combination with variable size packets.     

基于GMCR-NPAWLAK混合模型的冲突分析研究

冲突分析图模型中,决策者的态度只有肯定和否定两种,实际问题中往往不止两种;新PAWLAK冲突模型(NPAWLAK模型)将冲突系统中决策者的三种态度扩展到决策争端的三种程度,符合实际情况,因而研究冲突系统中决策者的偏好排序和全局可行方案对决策者的策略选择具有重要意义。本文在NPAWLAK模型的基础上,引入冲突分析图模型理论(GMCR),提出GMCR-NPAWLAK冲突分析混合模型。该混合模型首先拓展和改进的策略优先排序法,实现了冲突系统中各决策者的客观偏好排序;同时,模型给出了全局可行方案的算法,该算法依据决策者的偏好排序分析结果找出系统的全局可行方案。最后,本文以某企业劳资关系的NPAWLAK冲突为例,对冲突系统进行建模和偏好分析,得到了冲突各方的偏好序列和全局可行方案,同时验证了混合模型的有效性。     

On co-isosimple modules and co-isoradical of modules

Abstract

In this paper we study “co-isosimple” modules, that is, those modules which are isomorphic to all of its non-zero quotients modules. This allows us to define and study “isomaximal” submodules, “isomax” modules and the “co-isoradical” of a module. We study some of its basic properties and we give a characterization of left V-ring using these concepts.

Communicated by Alberto Facchini     

Extension of the hierarchy for k-OBDDs of small width

In this paper we explore the well-known k-OBDD model of branching programs. We develop a method of representation of the k-OBDD computation process as an “automata-communication protocol” computation process. Our method allows us to extend the hierarchy proved by Bolling-Sauerhoff-Sieling-Wegener in 1996 for k-OBDDs. Moreover, using the PJM function (a modification of well-known PJ and ISA functions), we prove a new hierarchy.