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Fourth Kenneth C. Schraut Memorial Lecture (Poster), University Of Dayton. Department Of Mathematics

Kenneth C. Schraut Memorial Lectures

No abstract provided.

2003 Undergraduate Mathematics Day Poster, University Of Dayton. Department Of Mathematics

Undergraduate Mathematics Day: Programs, Lectures, Promotional Materials

Poster of Undergraduate Mathematics Day events held Saturday, November 1, 2003.

2003 Program And Abstracts, University Of Dayton. Department Of Mathematics

Undergraduate Mathematics Day: Programs, Lectures, Promotional Materials

Program and abstracts of papers presented at the 2003 Undergraduate Mathematics Day at the University of Dayton

The Structure Of Residuated Lattices, Kevin K. Blount, Constantine Tsinakis

Mathematics Faculty Publications

A residuated lattice is an ordered algebraic structure [formula] such that is a lattice, is a monoid, and \ and / are binary operations for which the equivalences [formula] hold for all a,b,c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as "dividing" on the right by b and "dividing" on the left by a. The class of all residuated lattices is denoted by ℛℒ The study of such objects originated in the context of the theory of ring ideals in the 1930s. The …

Directed Acyclic Graphs, Stephen B. Maurer , '67

Mathematics & Statistics Faculty Works

No abstract provided.

Affine Manifolds, Log Structures, And Mirror Symmetry, Mark Gross, Bernd Siebert

Turkish Journal of Mathematics

We outline work in progress suggesting an algebro-geometric version of the Strominger-Yau-Zaslow conjecture. We define the notion of a toric degeneration, a special case of a maximally unipotent degeneration of Calabi-Yau manifolds. We then show how in this case the dual intersection complex has a natural structure of an affine manifold with singularities. If the degeneration is polarized, we also obtain an intersection complex, also an affine manifold with singularities, related by a discrete Legendre transform to the dual intersection complex. Finally, we introduce log structures as a way of reversing this construction: given an affine manifold with singularities with …

Symplectic Surgeries From Singularities, Ivan Smith, Richard Thomas

Turkish Journal of Mathematics

We describe a variety of symplectic surgeries (not a priori compatible with Kaehler structures) which are obtained by combining local Kaehler degenerations and resolutions of singularities. The effect of the surgeries is to replace configurations of Lagrangian spheres with symplectic submanifolds. We discuss several examples in detail, relating them to existence questions for symplectic manifolds with c_1 > 0, c_1 = 0, c_1 < 0 in four and six dimensions.

On Locally M-Pseudoconvex A^*-Algebras, A. El Kinani

Turkish Journal of Mathematics

We consider classical A^*-algebras in the context of locally pseudoconvex algebras. Results concerning the auxiliary topology and A^* -algebras of the first kind are given.

On Prime Submodules Of Finitely Generated Free Modules, Di̇lek Pusat Yilmaz

Turkish Journal of Mathematics

The aim of this work is to investigate prime submodules of finitely generated free modules over commutative domains in some special cases.

On L_{P}-Approximation By A Linear Combination Of A New Sequence Of Linear Positive Operators, P. N. Agrawal, Ali Jassim Mohammad

Turkish Journal of Mathematics

In the present paper, we study some direct results in L_{p} -approximation by a linear combination of a new sequence of linear positive operators. The error in the approximation is estimated in terms of the higher order integral modulus of smoothness using some properties of the Steklov means.

Asymptotic Solutions Of Diffusion Models For Risk Reserves, Sally S. L. Shao

Mathematics and Statistics Faculty Publications

We study a family of diffusion models for risk reserves which account for the investment income earned and for the inflation experienced on claim amounts. After we defined the process of the conditional probability of ruin over finite time and imposed the appropriate boundary conditions, classical results from the theory of diffusion processes turn the stochastic differential equation to a special class of initial and boundary value problems defined by a linear diffusion equation. Armedwith asymptotic analysis and perturbation theory, we obtain the asymptotic solutions of the diffusion models (possibly degenerate) governing the conditional probability of ruin over a finite …

Birkhoff-Kellogg Theorems On Invariant Directions For Multimaps, Ravi P. Agarwal, Donal O'Regan

Mathematics and System Engineering Faculty Publications

We establish Birkhoff-Kellogg type theorems on invariant directions for a general class of maps. Our results, in particular, apply to Kakutani, acyclic, O'Neill, approximable, admissible, and script U signCK maps.

A Non-Markovian Queueing System With A Variable Number Of Channels, Hong-Tam T. Rosson, Jewgeni H. Dshalalow

Mathematics and System Engineering Faculty Publications

In this paper we study a queueing model of type GI/M/m̃a/∞ with m parallel channels, sonic of which may suspend their service at specified random moments of time. Whether or not this phenomenon occurs depends on the queue length. The queueing process, which we target, turns out to be semi-regenerative, and we fully explore this utilizing semi-regenerative techniques. This is contrary to the more traditional supplementary variable approach and the less popular approach of combination semi-regenerative and supplementary variable technique. We pass to the limiting distribution of the continuous time parameter process through the embedded Markov chain for which we …

Existence Theory For Single And Multiple Solutions To Semipositone Discrete Dirichlet Boundary Value Problems With Singular Dependent Nonlinearities, Daqing Jiang, Lili Zhang, Donal O'Regan, Ravi P. Agarwal

Mathematics and System Engineering Faculty Publications

In this paper we establish the existence of single and multiple solutions to the semipositone discrete Dirichlet boundary value problem {Δ²y(i - 1) + μf(i, y(i)) = 0, i ∈ {1, 2, ..., T} y(0) = y(T + 1) = 0, where μ > 0 is a constant and our nonlinear term f(i, u) may be singular at u = 0.

Partitioning Regular Polygons Into Circular Pieces I: Convex Partitions, Mirela Damian, Joseph O'Rourke

Computer Science: Faculty Publications

We explore an instance of the question of partitioning a polygon into pieces, each of which is as “circular” as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a …

Regularity Of Minimizers Of The Calculus Of Variations In Carnot Groups Via Hypoellipticity Of Systems Of Hörmander Type, Luca Capogna, Nicola Garofalo

Mathematics Sciences: Faculty Publications

We prove the hypoellipticity for systems of Hörmander type with constant coefficients in Carnot groups of step 2. This result is used to implement blow-up methods and prove partial regularity for local minimizers of non-convex functionals, and for solutions of non-linear systems which appear in the study of non-isotropic metric structures with scalings. We also establish estimates of the Hausdorff dimension of the singular set.

Multiple Solutions For An Inhomogeneous Semilinear Elliptic Equation In Rn, Yinbin Deng, Yi Li, Xuejin Zhao

Mathematics and Statistics Faculty Publications

No abstract provided.

Stone Coalgebras, Clemens Kupke, Alexander Kurz, Yde Venema

Engineering Faculty Articles and Research

In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor. We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way. This yields a duality between the category of modal algebras and that of coalgebras over the Vietoris functor. Building on this idea, we introduce the notion of a Vietoris polynomial functor over the category of Stone spaces. …

A Contrast-Based Neural Control System For Ant Navigation, Joanna R. Wares, Predrag-Peter Ilich, Daniel P. Dougherty

Department of Math & Statistics Faculty Publications

A mathematical model for a neural control system of ant locomotion was developed. Contrast-based detectors using excitation and inhibition were tuned to specific contrast orientations. A control system using multiple orientation contrast detectors was then developed and optimized for a model ant, which could move via a biased random walk. The system allowed sufficient control to guide the ant through various mazes.

Uniqueness Of Primary Decompositions, Patrick F. Smith

Turkish Journal of Mathematics

Uniqueness properties of primary decompositions in modules over non-commutative rings are presented.

Pushouts Of Profinite Crossed Modules And Cat^1-Profinite Groups, Murat Alp, Özgün Gürmen

Turkish Journal of Mathematics

In this paper, we presented a brief review of crossed modules [7], cat^1-groups [6], pullback crossed modules [4], pullback cat^1-group [1], profinite crossed modules [5], cat^1-profinite groups [5], pullback profinite crossed modules [5], pullback cat^1-profinite groups [3]. We defined the pushout cat^1-profinite groups and gave the left adjoint constructions.

Polyhedral Approximations Of Riemannian Manifolds, Anton Petrunin

Turkish Journal of Mathematics

I'm trying to understand which Riemannian manifolds can be Lipschitz approximated by polyhedral spaces of the same dimension with curvature bounded below. The necessary conditions I found consist of some special inequality for curvature at each point (the geometric curvature bound). This inequality is also sufficient condition for local approximation. I conjecture that it is also a sufficient condition for global approximation, and I can prove it if the curvature bound is positive. In general I can prove it only with the additional assumption that tangent bundle of the manifold is stably trivial.

Prediction Intervals For The Binomial Distribution With Dependent Trials, Florian Sebastian Rueck

Masters Theses

"A generalization of a prediction interval procedure for the binomial distribution to the case of the binomial distribution with dependent trials is considered. Several different methods have been developed for obtaining prediction intervals for the binomial distribution. An unpublished study by Vlieger and Samaranayake has shown that two of these methods achieve coverage probabilities close to nominal levels. The proposed method is an extension of one of these methods and is based on the maximum likelihood predictive density proposed by Lejeune and Faulkenberry. A simulation study was carried out to investigate the coverage probabilities of the proposed prediction bounds.

This …

Computational Geometry Column 44, Joseph O'Rourke

Computer Science: Faculty Publications

The open problem of whether or not every pair of equal-area polygons has a hinged dissection is discussed.

On The Development Of The Intersection Of A Plane With A Polytope, Joseph O'Rourke

Computer Science: Faculty Publications

Define a “slice” curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a generalization of Cauchy's arm lemma to permit nonconvex “openings” of a planar convex chain.

Fiber Sums Of Genus 2 Lefschetz Fibrations, Denis Auroux

Turkish Journal of Mathematics

Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.

Duality And Fibrations On G_2 Manifolds, Sergei Gukov, Shing-Tung Yau, Eric Zaslow

Turkish Journal of Mathematics

We argue that G_2 manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M-five-brane fibers as target space. Mirror symmetry and various string and M-theory dualities involving G_2 manifolds may be incorporated into this framework. To give some examples, we construct two non-compact manifolds with G_2 structures: one with a K3 fibration, and one with a torus fibration and a metric of G_2 holonomy. Kaluza-Klein reduction of the latter solution gives abelian BPS monopoles in 3 + 1 dimensions.

On Cofinite Subgroups Of Mapping Class Groups, Mustafa Korkmaz

Turkish Journal of Mathematics

For every positive integer n, we exhibit a cofinite subgroup \Gamma_n of the mapping class group of a surface of genus at most two such that \Gamma_n admits an epimorphism onto a free group of rank n. We conclude that H^1 (\Gamma_n; {\mathbb Z}) has rank at least n and the dimension of the second bounded cohomology of each of these mapping class groups is the cardinality of the continuum. In the case of genus two, the groups \Gamma_n can be chosen not to contain the Torelli group. Similarly for hyperelliptic mapping class groups. We also exhibit an automorphism of …

Comparing Open Book And Heegaard Decompositions Of 3-Manifolds, J. Hyam Rubinstein

Turkish Journal of Mathematics

We study the maximal value of the Euler characteristic of the pages of all open book decompositions of closed orientable 3-manifolds. In particular, we describe some examples where the minimal genus Heegaard splittings of such 3-manifolds give rise to open book decompositions and other examples where the simplest open book decomposition has larger maximal Euler characteristic of pages than the smallest genus Heegaard splittings. Also, special properties of the Heegaard splitting associated to an open book decomposition are given. Techniques of minimal surface theory and hyperbolic geometry are shown to be useful for such problems.

Abelian Fibred Holomorphic Symplectic Manifolds, Justin Sawon

Turkish Journal of Mathematics

We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in this way, and are led to several natural conjectures. We then study the geometry of these fibrations. The expectation is that this point of view will prove useful in understanding holomorphic symplectic manifolds, and possibly lead to a classification.