Technical Reports

Technical Reports 2014

Rainbow copies of C4 in edge-colored hypercubes

J. Balogh* M. Delcourt B. Lidický C. Palmer§

Abstract: For positive integers k and d such that 4≤kd and k≠5, we determine the maximum number of rainbow colored copies of C4 in a k-edge-coloring of the d-dimensional hypercube Qd. Interestingly, the k-edge-colorings of Qd yielding the maximum number of rainbow copies of C4 also have the property that every copy of C4 which is not rainbow is monochromatic.

Keywords: hypercube and rainbow-coloring

AMS Subject Classification: 05C15,  05C35

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*Department of Mathematics, University of Illinois, Urbana, IL 61801, USA and Bolyai Institute, University of Szeged, Szeged, Hungary Research is partially supported NSF CAREER Grant DMS-0745185, Arnold O. Beckman Research Award (UIUC Campus Research Board 13039) and Marie Curie FP7-PEOPLE-2012-IIF 327763.

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA Research supported by NSF Graduate Research Fellowship DGE 1144245 and DMS 0838434 EMSW21MCTP: Research Experience for Graduate Students.

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA Research is partially supported by NSF grant DMS-1266016.

§Department of Mathematical Sciences, University of Montana, Missoula, Montana 59801, USA Work partly done while visiting University of Illinois. Research supported by Hungarian National Science Fund (OTKA), grant NK 78439.

Topological orderings of weighted directed acyclic graphs

Dániel Gerbner* Balázs Keszegh Cory Palmer Dömötör Pálvölgyi§

Abstract: We call a topological ordering of a weighted directed acyclic graph non-negative if the sum of weights on the vertices in any prefix of the ordering is non-negative. We investigate two processes for constructing non-negative topological orderings of weighted directed acyclic graphs. The first process is called a mark sequence and the second is a generalization called a mark-unmark sequence. We answer a question of Erickson by showing that every non-negative topological ordering that can be realized by a mark-unmark sequence can also be realized by a mark sequence. We also investigate the question of whether a given weighted directed acyclic graph has a non-negative topological ordering. We show that even in the simple case when every vertex is a source or a sink the question is NP-complete.

Keywords: topological ordering, directed acyclic graph

AMS Subject Classification: 05C85, 05C20

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 *Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest H-1364, Hungary. Research supported by Hungarian National Science Fund (OTKA), grant PD 109537.

Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest H-1364, Hungary. Research supported by Hungarian National Science Fund (OTKA), grant PD 108406 and grant NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003) and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences

Department of Mathematical Sciences, University of Montana, Missoula, Montana 59801, USA. Research supported by Hungarian National Science Fund (OTKA), grant NK 78439.

§Eötvös Loránd University, Faculty of Science, Department of Computer Science, Pázmány Péter 1/C, Budapest H-1117, Hungary. Research supported by Hungarian National Science Fund (OTKA), grant PD 104386 and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

Cyclic Length in the Tame Brauer Group of the Function Field of a p-adic Curve

Eric Brussel, Kelly Mckinnie, And Eduardo Tengan

Abstract: Let F be the function field of a smooth curve over the p-adic number field Qp. We show that for each prime-to-p number n the n-torsion subgroup H2(F, μn) = nBr(F) is generated by Z/n-cyclic classes; in fact the Z/n-length is equal to two. It follows that the Brauer dimension of F is two (first proved in [20]), and any F-division algebra of period n and index n2 is decomposable.

AMS Subject Classification: 16K50

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Brauer groups on K3 surfaces and arithmetic applications

Kelly McKinnie, Justin Sawon, Sho Tanimoto, and Anthony Várilly-Alvarado

Abstract: For a prime p, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to sublattices of index p of the transcendental lattice TS of S; we classify these lattices up to isomorphism using Nikulin’s discriminant form technique. We then study geometric realizations of p-torsion Brauer elements as Brauer-Severi varieties in a few cases via projective duality. We use one of these constructions for an arithmetic application, giving new kinds of counter-examples to weak approximation on K3 surfaces of degree two.

AMS Subject Classification: 16K50

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A Modification of Ripley’s K Function to Measure Aggregation About a Mass

Michael O. Smith1, Jackson Ball2,3, Benjamin B. Holloway2,3, Ferenc Erdelyi4, Gabor Szabo4, Emily Stone1,2, Jonathan Graham1 and J. Josh Lawrence2,3

1Department of Mathematical and Statistical Sciences,
University of Montana, Missoula, MT 59812
2Center for Structural and Functional Neuroscience,
University of Montana, Missoula, MT 59812
3Department of Biomedical and Pharmaceutical Sciences,
University of Montana, Missoula, MT 59812
4Laboratory of Molecular Biology and Genetics,
Institute of Experimental Medicine,
Hungarian Academy of Sciences, H-1450 Budapest, Hungary

We present a methodology that detects spatial patterns of 3-dimensional point processes that include a mass within the study region. Spatial patterns such as clustering, randomness, or repulsion are considered in respect to the mass surface. Our method closely resembles Ripley’s K Function but is modified to discern the pattern about the mass surface. We walk though the definition and derivation of Ripley’s K Function and then follow this process to define the Modified K function. We develop this novel function according to the definition: the Modified K function times the intensity is the expected number of events within a distance h of a mass.
Special consideration of edge effects is taken in order to make the function invariant to the location of the mass within the study region. Significance of spatial patterns is determined using Monte Carlo confidence envelopes similar to Ripley’s K Function. Simulations are performed to inform researchers how the Modified K function performs under different types of aggregation. Finally, we apply the Modified K function to neuroscience as a novel analysis tool by examining the spatial pattern of neurotransmitter release sites as events about a neuron. Supplemental materials for this article are available online.

Key Words: 3-Dimensions; Clustering; Point Process; Spatial Pattens; Spatial Statistics

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Modeling Simple Experimental Platform for In Vitro Study of Drug Elution from Drug Eluting Stents (DES)

Abstract: We present a simple model of experimental setup for in vitro study of drug release from drug eluting stents and drug propagation in artificial tissue samples representing blood vessels. The model is further reduced using the assumption on vastly different characteristic diffusion times in the stent coating and in the artificial tissue. The model is used to derive a relationship between the times at which the measurements have to be taken for two experimental platforms, with corresponding artificial tissue samples made of different materials with different drug diffusion coefficients, to properly compare the drug release characteristics of drug eluting stents.

Key words: model reduction, asymptotic expansion, drug eluting stents, experimental measurements

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A kinetic model for the frequency dependence of cholinergic modulation at hippocampal GABAergic synapses.

Abstract: In this paper we develop a simple model of presynaptic neuromodulation of GABA signalling to decipher paired whole-cell recordings of frequency dependent cholinergic neuromodulation at CA1 PV BC-pyramidal cell synapses. Variance-mean analysis is employed to normalize the data, which is then used to estimate parameters in the mathematical model. Various parameterizations and hidden parameter dependencies are investigated using Monte Carlo Markov Chain parameter estimation techniques. This analysis reveals that frequency dependence of cholinergic modulation requires both calcium-dependent recovery from depression and mAChR-induced inhibition of presynaptic calcium channels. A reduction in calcium entry into the presynaptic terminal in the kinetic model accounted for the frequency-dependent e®ects of mAChR activation.

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Generalized Correlation Integral Vectors: A New Distance Concept for Chaotic Dynamical Systems

Abstract: Several concepts of dimension have been developed to characterize properties of chaotic trajectories. To estimate parameters of chaotic dynamical systems a measure to quantify the likelihood function of chaotic variability (the 'distance' between di?erent trajectories) is needed. We review problems encountered by previously used method and propose a method related to the correlation dimension concept. The major advantage of the new construct is its insensitivity with respect to varying initial values, to the choice of a solver, numeric tolerances, etc. A way to create the statistical likelihood for model parameters is presented, together with a sound framework for Markov chain Monte Carlo sampling. The methodology is illustrated using computational examples for the Lorenz 63 and Lorenz 95 systems.

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Indigenous Universalities and Peculiarities of Innovation

Abstract: Rooted in diverse cultures and in distinct regions of the world, Indigenous communities and people have for centuries negotiated unique relationships with their surroundings, including new forms of conquest that continue to emerge today. The intent of this book is to showcase instances of Indigenous innovation in sustainability, ecological stewardship, and oral knowledge that demonstrate cosmological conceptions of time and place and critical practices that can be shared in order to shape new conceptualizations of relationships within the world that are transformative and just.

The book is based on empirical and/or other research that has been collaboratively conducted with Indigenous communities, narratives and counter-narratives, qualitative fieldwork reflections, and theoretical perspectives. Authors should center their work around major definitions and ideas of innovation within Indigenous contexts that are respectful of and exemplify local knowledge. At the same time, authors are asked to engage in conscientious discussion regarding dominant discourse that posits Western modes of progress that impact those Indigenous communities and any tensions that result between communities and province, state, and/or country.

Keywords: Indigeneous Education; Traditional Knowledge; Innovation 

AMS Subject Classification: n/a

Book (in preparation) in Advances in Innovation Education, vol.2, Sense Publishers Pdf (26 KB)

Exploring Creativity: From the Mathematics Classroom to the Mathematician's Mind

Abstract: Learners of mathematics do not typically experience mathematics as a creative subject, yet research mathematicians often describe their field as a highly creative endeavour (Burton, 2004). The term creativity has sometimes come to imply eminent acts/products/achievements, yet research suggests that creative thinking is an everyday occurrence (Craft, 2002). In this working group we sought to capture the essence of mathematical creativity as seen through the eyes of mathematicians and described by current research, and express it in ways that might also be applicable to learners of mathematics including, but not restricted to, students described as highly able. Our initial questions for consideration included: What is mathematical creativity? Does it differ from other kinds of creativity? How can we observe it in learners? Is creativity necessary for mathematics research? How can creativity be enhanced in classroom mathematics learning? Are some students more mathematically creative than others? Time was allowed during the working group meetings for those participants who wished to be generative; in particular, the construction or sharing of classroom tasks that had potential for occasioning creative behaviour was a focus for some participants. Such tasks could be illustrative to teachers who wish to provide potentially rich learning environments to students, and samples are included in this report.

Keywords: Mathematical Proof; Algorithms; Visual proofs

AMS Subject Classification:97

Preprint of Proceedings of the 2013 Annual Canadian Mathematics Education Study Group, Working Group on Creativity Pdf (318 KB)

Cultural Nuances of Mathematics: The Case of Proof

Abstract: Las pruebas han sido el núcleo de las matemáticas que comienzan con los Elementos de Euclides, que ejemplifica el carácter axiomático-deductivo de la escritura matemática formal. Sin embargo, varias reorganizaciones de los elementos eran necesarios por la talla de David Hilbert, con el fin de eliminar pequeños defectos en la superestructura deductivo. Por ejemplo, el criterio de Side-Angle-Side de la congruencia de dos triángulos que se presentan en el libro que revela las fallas de tratar de forzar un argumento deductive artificial, donde no existe uno. En su lugar, adaptamos dos milenios más tarde como un axioma [una verdad evidente por sí misma] para realizar la reorganización lógica sonido. Comentaristas como los Proclo erudito bizantino tienen anotado los elementos con observaciones sobre las fallas en varios argumentos de reducción al absurdo forzados en numerosas proposiciones en el libro. Si uno examina el libro IV de cerca, hay numerosas pruebas que son esencialmente pruebas de existencia establecidos por regla y compass construcciones. Esto indica que axiomatizar aunque esencial para la creación de una estructura lógica formal, viene con su parte de los defectos que eventualmente necesitan reorganización. Más importante aún, los defectos revelan una dimensión humanista a la noción de prueba. Ante esta exposición de motivos, la cuestión que aborda en este trabajo es: ¿Hay otras tradiciones de prueba que revelan matices culturales de la dimensión humanística de hacer matemáticas? (Sriraman, 2008).

Keywords: Mathematical Proof; Algorithms; Visual proofs

AMS Subject Classification: 97

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The Roeper School: A Model for Holistic Development of High Ability

Abstract: How can we design schools that energetically promote intellectual development while also attending to the social, emotional, and ethical growth of students? In today’s frenzied climate of accountability driven school reform it is difficult to establish anything more than achievement of superficial knowledge and skill. Fortunately, there is a vibrant example of holistic, student-centered education that engenders dynamic, multidimensional student growth. The Roeper School enables students to develop strong intrinsic motivation as they discover aspirations and develop talents consistent with those aspirations. Simultaneously, from a very young age students take considerable responsibility for their own actions and for the processes that go on in their school.Following the Roeper philosophy each student generates a long-term sense of purposeful direction, a strong sense of intrapersonal awareness, impressive creative and critical thinking skills, and a finely tuned sense of ethical responsibility. Upon graduation Roeper students are well prepared to find or create highly productive niches in the world of work and rewarding personal lives while serving as mature, ethical citizens of a complex, 21st-century, globalized society.

This book includes descriptions of the multidimensional education the Roeper School provides. The perspectives in the volume are diverse, coming from leading researchers and theorists in the field of gifted education as well as teachers, administrators, alumni, and current students from the school itself. Overall, the book provides a beacon of hope for 21st-century education.

Keywords: Schools; Talent Development; Alternative Schools

AMS Subject Classification: 97

Publisher preview of TOC Pdf (330 KB)

Commentary on Probabilistic Thinking: Presenting Plural Perspectives

Abstract: Those of you familiar with research investigating probabilistic thinking in the field of mathematics education, might, at this point in the book, be expecting a “wish list” for future research, which has become customary (e.g., Kapadia and Borovcnik 1991; Jones et al. 2007; Shaughnessy 1992); however, we will not be adding to the list of wish lists. Instead, we have decided to, in this commentary, highlight some of the overarching themes that have emerged from the significant amount of research housed in this volume. Themes emerging from each of the four main perspectives—Mathematics and Philosophy, Psychology, Stochastics and Mathematics Education—are now commented on in turn.

Keywords: Probabilistic thinking

AMS Subject Classification: 97

Publisher offprint from E.Chernoff & B. Sriraman (Eds). Probabilistic Thinking: Presenting Plural Perspectives. Full off print not available due to publisher embargo. Pdf (28 KB)

A Brief Overview and Critique of Perspective II on Probabilistic and Statistical Reasoning

Abstract: In this overview and critique of perspective II, we briefly focus on several ways that Models & Modeling Perspectives (MMP) can be used to provide a unifying theoretical framework for developing both: (a) a coherent summary of the kind of research reported in this book, and (b) a useful list of testable claims that appear to be priorities to investigate in future research emanating from the kind of studies reported here. In particular, MMP was developed to identify: (a) emerging new types of mathematical thinking that appear to be needed beyond mathematics classrooms—including in situations involving new sciences (e.g., social sciences, engineering) not traditionally emphasized in K-12 curriculum materials, (b) new ways to operationally define important achievements currently being ignored in both school testing programs and documents specifying standards for teaching and learning, and (c) new teaching and learning opportunities made possible by new model-development tools for students.

Keywords: Probabilistic reasoning; Statistical reasoning; Modeling perspectives

AMS Subject Classification: 97

Publisher offprint from E.Chernoff & B. Sriraman (Eds). Probabilistic Thinking: Presenting Plural Perspectives. Full off print not available due to publisher embargo. Pdf (28 KB)

Commentary on Perspective I: The Humanistic Dimensions of Probability

Abstract: In this commentary to perspective 1 of probabilistic thinking, we highlight the salient aspects of the history and philosophy of probability. In particular we examine historical problems from probability whose solutions improved over time, as well as the need to take into account the subjectivist approach to probability besides the classical approaches.

Keywords: History and philosophy of probability; Paradoxes; Historical problems in probability

AMS Subject Classification: 97

Publisher offprint from E.Chernoff & B. Sriraman (Eds). Probabilistic Thinking: Presenting Plural Perspectives. Full off print not available due to publisher embargo. Pdf (28 KB)