An iterative method for a Cauchy problem for the heat equation

An iterative method for reconstruction of the solution to a parabolic initial boundary value problem of second order from Cauchy data is presented. The data are given on a part of the boundary. At each iteration step, a series of well-posed mixed boundary value problems are solved for the parabolic operator and its adjoint. The convergence proof of this method in a weighted L2-space is included.

On Distributed Oblivious Transfer

The paper has been presented at the International Conference Pioneers of Bulgarian Mathematics, Dedicated to Nikola Obreshkoff and Lubomir Tschakaloff , Sofia, July, 2006. The material in this paper was presented in part at INDOCRYPT 2002

On Proactive Verifiable Secret Sharing Schemes

The paper has been presented at the International Conference Pioneers of Bulgarian Mathematics, Dedicated to Nikola Obreshkoff and Lubomir Tschakaloff , Sofia, July, 2006. The material in this paper was presented in part at the 11th Workshop on Selected Areas in Cryptography (SAC) 2004

When does a Bounded Domain Cover a Projective Manifold? (Survey)

* The research has been partially supported by Bulgarian Funding Organizations, sponsoring the Algebra Section of the Mathematics Institute, Bulgarian Academy of Sciences, a Contract between the Humboldt Univestit¨at and the University of Sofia, and Grant MM 412 / 94 from the Bulgarian Board of Education and Technology

Mathematica Balkanica New Series Book Volume 24 2010

Muharem Avdispahic 1 Coordinator of the TEMPUS Project SEE Doctoral Studies in Mathematical Sciences (144703-TEMPUS-2008-BA-TEMPUS-JPCR) The main goals of the TEMPUS Project ”SEE Doctoral Studies in Math- ematical Sciences”, funded by European Commission under the TEMPUS IV first call, consist of the development of a model of structured doctoral studies in Mathematical Sciences involving the network of Western Balkans universi- ties, the curricula design based on the existing strenghts and tendencies in the areas of Pure Mathematics, Applied Mathematics and Theoretical Computer Science and the first phase of implementation of the agreed model during the SEE Doctoral Year in Mathematical Sciences 2011. A decisive step in this direction was ”SEE Young Researchers Workshop” held in Ohrid, FYR Macedonia, September 16-20, 2009, as a part of the Math- ematical Society of South-Eastern Europe (MASSEE) International Congress on Mathematics - MICOM 2009. MICOM 2009 continued the tradition of two previous highly successful MASSEE congresses that took place in Bulgaria in 2003 and in Cyprus in 2006. This volume of the journal Mathematika Balkanica contains the talks de- livered at Ohrid Workshop by South-Eastern European PhD students in various stage of their research towards a doctoral degree in mathematics or informat- ics. Facilitating publication efforts of young researchers from the universities of Sarajevo, Tuzla, Belgrade, Skopje, Stip, Graz, and Sofia fully coincides with MASSEE goals to promote, organize and support scientific, research and edu- cational activities in South-Eastern Europe. The consent of the Editorial Board of Mathematica Balkanica to publish ”SEE Young Researchers Workshop” contributions aptly meets intentions of European reform processes aimed at creating the European Higher Education Area and European Research Area. It is an encouragement to these young researchers in the first place and at the same time an encouragement to their institutions in overcoming fragmentation and enhancing their capacities through fostering reciprocal development of human resources.

Percent in a Nutshell …

Ironically, the “learning of percent” is one of the most problematic aspects of school mathematics. In our view, these difficulties are not associated with the arithmetic aspects of the “percent problems”, but mostly with two methodological issues: firstly, providing students with a simple and accurate understanding of the rationale behind the use of percent, and secondly - overcoming the psychological complexities of the fluent and comprehensive understanding by the students of the sometimes specific wordings of “percent problems”. Before we talk about percent, it is necessary to acquaint students with a much more fundamental and important (regrettably, not covered by the school syllabus) classical concepts of quantitative and qualitative comparison of values, to give students the opportunity to learn the relevant standard terminology and become accustomed to conventional turns of speech. Further, it makes sense to briefly touch on the issue (important in its own right) of different representations of numbers. Percent is just one of the technical, but common forms of data representation: p% = p × % = p × 0.01 = p × 1/100 = p/100 = p × 10-2 "Percent problems” are involved in just two cases: I. The ratio of a variation m to the standard M II. The relative deviation of a variation m from the standard M The hardest and most essential in each specific "percent problem” is not the routine arithmetic actions involved, but the ability to figure out, to clearly understand which of the variables involved in the problem instructions is the standard and which is the variation. And in the first place, this is what teachers need to patiently and persistently teach their students. As a matter of fact, most primary school pupils are not yet quite ready for the lexical specificity of “percent problems”. ....Math teachers should closely, hand in hand with their students, carry out a linguistic analysis of the wording of each problem ... Schoolchildren must firmly understand that a comparison of objects is only meaningful when we speak about properties which can be objectively expressed in terms of actual numerical characteristics. In our opinion, an adequate acquisition of the teaching unit on percent cannot be achieved in primary school due to objective psychological specificities related to this age and because of the level of general training of students. Yet, if we want to make this topic truly accessible and practically useful, it should be taught in high school. A final question to the reader (quickly, please): What is greater: % of e or e% of Pi

За извънкласните форми на обучение по математика

Stojanka Valcheva, Vladimir Todorov - In the present talk we discuss some problems that arise out of the limiting the time designed for teaching Mathematics in the secondary school in Bulgaria.

Mode-locking intracavity dynamics:modeling nonlinear power and energy fluctuations

Modern high-power, pulsed lasers are driven by strong intracavity fluctuations. Critical in driving the intracavity dynamics is the nontrivial phase profiles generated and their periodic modification from either nonlinear mode-coupling, spectral filtering or dispersion management. Understanding the theoretical origins of the intracavity fluctuations helps guide the design, optimization and construction of efficient, high-power and high-energy pulsed laser cavities. Three specific mode-locking component are presented for enhancing laser energy: waveguide arrays, spectral filtering and dispersion management. Each component drives a strong intracavity dynamics that is captured through various modeling and analytic techniques.

Generalized KPCA by adaptive rules in feature space

Principal component analysis (PCA) is well recognized in dimensionality reduction, and kernel PCA (KPCA) has also been proposed in statistical data analysis. However, KPCA fails to detect the nonlinear structure of data well when outliers exist. To reduce this problem, this paper presents a novel algorithm, named iterative robust KPCA (IRKPCA). IRKPCA works well in dealing with outliers, and can be carried out in an iterative manner, which makes it suitable to process incremental input data. As in the traditional robust PCA (RPCA), a binary field is employed for characterizing the outlier process, and the optimization problem is formulated as maximizing marginal distribution of a Gibbs distribution. In this paper, this optimization problem is solved by stochastic gradient descent techniques. In IRKPCA, the outlier process is in a high-dimensional feature space, and therefore kernel trick is used. IRKPCA can be regarded as a kernelized version of RPCA and a robust form of kernel Hebbian algorithm. Experimental results on synthetic data demonstrate the effectiveness of IRKPCA. © 2010 Taylor & Francis.

Causal attributions for success or failure by passing and failing students in College Algebra

Success in mathematics has been identified as a predictor of baccalaureate degree completion. Within the coursework of college mathematics, College Algebra has been identified as a high-risk course due to its low success rates. ^ Research in the field of attribution theory and academic achievement suggests a relationship between a student's attributional style and achievement. Theorists and researchers contend that attributions influence individual reactions to success and failure. They also report that individuals use attributions to explain and justify their performance. Studies in mathematics education identify attribution theory as the theoretical orientation most suited to explain academic performance in mathematics. This study focused on the relationship among a high risk course, low success rates, and attribution by examining the difference in the attributions passing and failing students gave for their performance in College Algebra. ^ The methods for the study included a pilot administration of the Causal Dimension Scale (CDSII) which was used to conduct reliability and principal component analyses. Then, students (n = 410) self-reported their performance on an in-class test and attributed their performance along the dimensions of locus of causality, stability, personal controllability, and external controllability. They also provided open-ended attribution statements to explain the cause of their performance. The quantitative data compared the passing and failing groups and their attributions for performance on a test using One-Way ANOVA and Pearson chi square procedures. The open-ended attribution statements were coded in relation to ability, effort, task difficulty, and luck and compared using a Pearson chi square procedure. ^ The results of the quantitative data comparing passing and failing groups and their attributions along the dimensions measured by the CDSII indicated statistical significance in locus of causality, stability, and personal controllability. The results comparing the open-ended attribution statements indicated statistical significance in the categories of effort and task difficulty. ^

O comportamento pró-social de crianças com sintomatologia do transtorno da conduta

Pro-social behaviors are seen regularly throughout our daily lives, as we often witness people giving alms, helping a neighbor move, donating blood, or taking care of a friend's children, among others. From an evolutionary perspective, such behaviors occur because they have a high adaptive value to our species, precisely due to our high degree of dependence on group living for survival. Probably, for this same reason, since children have shown a preference for prosocial behaviors over antisocial behaviors, this preference becomes more visible as we grow. However, children with symptoms of conduct disorder show a pattern of aggressive, impulsive and more selfish behaviors than children without such symptoms. Furthermore, these children also experience environments in which antisocial behaviors are more frequent and intense compared to the general population. Priming experiments are one way of measuring the influence of simple environmental cues on our behavior. For example, driving faster when listening to music, religious people help more on religious elements, like the bible, and children are more cooperative after playing games of an educational nature. Thus, the objectives of the current study were to: evaluate whether there is any difference in generosity, through sharing behavior, among children with and without symptoms of conduct disorder; analyze the influence of prosocial priming on sharing behavior on children with and without symptoms of conduct disorder; and finally, analyze from an evolutionary perspective, the reasons given by children with and without symptoms of conduct disorder for sharing or not sharing with their best friend in a classroom environment. To address this question, the teachers of these children were asked to respond to an inventory that was designed to signal the presence or absence of symptoms of conduct disorder. Children identified as having or not having symptoms of conduct disorder could then undergo an experimental (with priming) or control (no priming) condition. Under the experimental condition, the children were asked to watch two short videos showing scenes of helping and sharing among peers, to perform a distraction activity, and finally to chose two of four different materials presented by the researcher and decide how much of these two materials they would like to share with their best friend in the classroom. Then the children were asked about their reasons for sharing or not sharing. Children subjected to the control condition performed the same activities as in the xi experimental condition, but did not watch the video first. The results showed a notable difference in the effect of priming in accordance with the child's stage of development; a difference in the amount of material donated to a best friend by children with and without symptoms of conduct disorder, and a change in this observed difference with the influence of pro-social priming; and finally, a convergence in the thinking of children regarding their reasons for sharing with evolutionary theory. The results of this study also indicate the importance of individual factors, developmental stage, environmental and evolutionary conditions in the pro-social behavior of children with and without symptoms of conduct disorder.

O eterno retorno como paródia e como colagem na filosofia da diferença de Gilles Deleuze

Gilles Deleuze hás commented on many philosophers, but his relationship with Nietzsche plays a singular role in his thought: appropriating the concept of the “eternal return” to think the central axis of his thesis, Difference and repetition (1968). Terms “difference” and “repetition” appeared associated to eternal return in his Nietzsche and philosophy (1962). Our dissertation thesis analyzes the presentations of that concept in bothworks. Chapter one presents the style construction and critical, methodological aspects of Nietzschean philosophy, fundamental elements to understand Deleuze’s interpretation. It subsequently analyzes the first presentation of that concept, expressed in the following terms: the aesthetic existence, either innocent or justified from the figure of game. We will see how the image of game implies another concept of chance, that leads Deleuze to think of an affirmative philosophical “type”, capable of creating new values. Chapter two evaluates the existential, “ethical-selective”, “physicalcosmological” character of the concept of eternal return, as much as the difficulties it imposes upon Nietzsche’s interpreter. We present afterwards Deleuzian comprehension of eternal return as a “parody” or a “simulacrum of doctrine”. Chapter three analyzes that interpretive position as a transvaluation of values from a rearrange of perspectives in order to overcome the negative comprehensions of existence. We want to question the way Deleuze builds another image of thought from the concept of eternal return – an image that, by a sort of “colagem” and selective elimination of the negativity, proposes a historiographic work and unfolds a lineage of thinkers of immanence and difference, a detour from the thought of identity, the same and the similar. We want thus to understand Deleuze’s critique of “dogmatic image of thought”.

O eterno retorno como paródia e como colagem na filosofia da diferença de Gilles Deleuze

Gilles Deleuze hás commented on many philosophers, but his relationship with Nietzsche plays a singular role in his thought: appropriating the concept of the “eternal return” to think the central axis of his thesis, Difference and repetition (1968). Terms “difference” and “repetition” appeared associated to eternal return in his Nietzsche and philosophy (1962). Our dissertation thesis analyzes the presentations of that concept in bothworks. Chapter one presents the style construction and critical, methodological aspects of Nietzschean philosophy, fundamental elements to understand Deleuze’s interpretation. It subsequently analyzes the first presentation of that concept, expressed in the following terms: the aesthetic existence, either innocent or justified from the figure of game. We will see how the image of game implies another concept of chance, that leads Deleuze to think of an affirmative philosophical “type”, capable of creating new values. Chapter two evaluates the existential, “ethical-selective”, “physicalcosmological” character of the concept of eternal return, as much as the difficulties it imposes upon Nietzsche’s interpreter. We present afterwards Deleuzian comprehension of eternal return as a “parody” or a “simulacrum of doctrine”. Chapter three analyzes that interpretive position as a transvaluation of values from a rearrange of perspectives in order to overcome the negative comprehensions of existence. We want to question the way Deleuze builds another image of thought from the concept of eternal return – an image that, by a sort of “colagem” and selective elimination of the negativity, proposes a historiographic work and unfolds a lineage of thinkers of immanence and difference, a detour from the thought of identity, the same and the similar. We want thus to understand Deleuze’s critique of “dogmatic image of thought”.

Sequências de Fibonacci e a Razão Áurea - aplicações no ensino básico

This thesis aims to present a study of the Fibonacci sequence, initiated from a simple problem of rabbits breeding and the Golden Ratio, which originated from a geometrical construction, for applications in basic education. The main idea of the thesis is to present historical records of the occurrence of these concepts in nature and science and their influence on social, cultural and scientific environments. Also, it will be presented the identification and the characterization of the basic properties of these concepts and howthe connection between them occurs,and mainly, their intriguing consequences. It is also shown some activities emphasizing geometric constructions, links to other mathematics areas, curiosities related to these concepts and the analysis of questions present in vestibular (SAT-Scholastic Aptitude Test) and Enem(national high school Exam) in order to show the importance of these themes in basic education, constituting an excellent opportunity to awaken the students to new points of view in the field of science and life, from the presented subject and to promote new ways of thinking mathematics as a transformative science of society.

A generalised formulation for computing the microbuckling load in periodic layered materials

Acknowledgments The financial support of the part of this research by The Royal Society, The Royal Academy of Engineering and The Carnegie Trust for the Universities of Scotland is gratefully acknowledged.