Mathematics Commons^™

Controlling Textures In Tsp Art, Robert Bosch, Robert Klock

Faculty & Staff Scholarship

By making simple modifications to how we calculate distances between points/cities, we control textures in TSP Art.

Supporting Student Success And Persistence In Stem With Active Learning Approaches In Emerging Scholars Classrooms, David Miller, Jessica Deshler, Tim Mceldowney, John Stewart, Edgar Fuller, Matt Pascal, Lynnette Michaluk

Faculty & Staff Scholarship

Over the last several decades, Emerging Scholars Programs (ESPs) have incorporated active learning strategies and challenging problems into collegiate mathematics, resulting in students, underrepresented minority (URM) students in particular, earning at least half of a letter grade higher than other students in Calculus. In 2009, West Virginia University (WVU) adapted ESP models for use in Calculus I in an effort to support the success and retention of URM STEM students by embedding group and inquiry-based learning into a designated section of Calculus I. Seats in the class were reserved for URM and first- generation students. We anticipated that supporting students …

Domino Steganography, Robert Bosch, Aaron Kreiner

Faculty & Staff Scholarship

No abstract provided.

Minimal Degrees Of Genocchi-Peano Functions: Calculus Motivated Number Theoretical Estimates, Krzysztof Ciesielski

Faculty & Staff Scholarship

A rational function of the form x α1 1 x α2 2 ⋯x αn n x β1 1 +x β2 2 +⋯+x βn n is a Genocchi-Peano example, GPE, provided it is discontinuous, but its restriction to any hyperplane is continuous. We show that the minimal degree D(n) of a GPE of n-variables equals 2 ⌊ e 2 e2−1 n⌋ + 2i for some i ∈ {0, 1, 2}. We also investigate the minimal degree Db(n) of a bounded GPE of n-variables and note that D(n) ≤ Db(n) ≤ n(n + 1). Finding better bounds for numbers Db(n) remains an …

Ssaw: A New Sequence Similarity Analysis Method Based On The Stationary Discrete Wavelet Transform, Jie Lin, Donald Adjeroh, Bing-Hua Jiang, Yue Jiang

Faculty & Staff Scholarship

Background: Alignment-free sequence similarity analysis methods often lead to significant savings in computational time over alignment-based counterparts.

Results: Anewalignment-freesequencesimilarityanalysismethod,calledSSAWisproposed.SSAWstandsfor Sequence Similarity Analysis using the Stationary Discrete Wavelet Transform (SDWT). It extracts k-mers from a sequence, then maps each k-mer to a complex number field. Then, the series of complex numbers formed are transformed into feature vectors using the stationary discrete wavelet transform. After these steps, the original sequence is turned into a feature vector with numeric values, which can then be used for clustering and/or classification.

Conclusions: Usingtwodifferenttypesofapplications,namely,clusteringandclassification,wecomparedSSAW against the the-state-of-the-art alignment free sequence analysis methods. SSAW demonstrates competitive or …

Graphene Nanoplatelets-Sericin Surface-Modified Gum Alloy For Improved Biological Response, Valentina Mitran, Valentina Dinca, Raluca Ion, Vasile D. Cojocaru, Patricia Neacsu, Cerasela Zoica Dinu, Laurentiu Rusen, Simona Brajnicov, Anca Bonciu, Maria Dinescu, Doina Raducanu, Ioan Dan

Faculty & Staff Scholarship

In this study a “Gum Metal” titanium-based alloy, Ti-31.7Nb-6.21Zr-1.4Fe-0.16O, was synthesized by melting and characterized in order to evaluate its potential for biomedical applications. The results showed that the newly developed alloy presents a very high strength, high plasticity and a low Young's modulus relative to titanium alloys currently used in medicine. For further bone implant applications, the newly synthesized alloy was surface modified with graphene nanoplatelets (GNP), sericin (SS) and graphene nanoplatelets/sericine (GNP–SS) composite films via Matrix Assisted Pulsed Laser Evaporation (MAPLE) technique. The characterization of each specimen was monitored by scanning electron microscopy (SEM), atomic force microscopy (AFM), …

Mechanism Of Byproducts Formation In The Isobutane/Butene Alkylation On Hy Zeolites, Zhiwei Chen, Feng Gao, Kun Ren, Quangui Wu, Yan Luo, Hongjun Zhou, Meng Zhang, Quan Xu

Faculty & Staff Scholarship

Submicron-size HY zeolites with a particles size of 200–700 nm were synthesized employing a crystal precipitation method in this study. The catalytic activity for the isobutane/butene alkylation was evaluated. The results indicated that butene conversion was above 90% and the selectivity of expected products (C8) was nearly at 90% within 72 h. The micropores-blocking and coverage of acid sites resulting from high hydrocarbons increased the difficulty for the diffusion of products to the bulk and inhibited the adsorption of reactant on activity sites, which caused deactivation of catalyst. The ultimate C12 content in alkylate oil, stemmed from trimerization of butene, …

Simultaneous Small Coverings By Smooth Functions Under The Covering Property Axiom, Krzysztof Ciesielski

Faculty & Staff Scholarship

The covering property axiom CPA is consistent with ZFC: it is satisfied in the iterated perfect set model. We show that CPA implies that for every ν ∈ ω ∪ {∞} there exists a family Fν ⊂ C ν (R) of cardinality ω1 < c such that for every g ∈ D ν (R) the set g \ S Fν has cardinality ≤ ω1. Moreover, we show that this result remains true for partial functions g (i.e., g ∈ D ν (X) for some X ⊂ R) if, and only if, ν ∈ {0, 1}. The proof of this result is based on the following theorem of independent interest (which, for ν 6= 0, seems to have been previously unnoticed): for every X ⊂ R with no isolated points, every ν-times differentiable function g : X → R admits a ν-times differentiable extension ¯g : B → R, where B ⊃ X is a Borel subset of R. The presented arguments rely heavily on a Whitney’s Extension Theorem for the functions defined on perfect subsets of R, which short but fully detailed proof is included. Some open questions are also posed.

The Effects Of Different Versions Of A Gateway Stem Course On Student Attitudes And Beliefs, Xiangming Wu, Jessica Deshler, Edgar Fuller

Faculty & Staff Scholarship

Background

Substantial research has been conducted focusing on student outcomes in mathematics courses in order to better understand the ways in which these outcomes depend on the underlying instructional methodologies found in the courses. From 2009 to 2014, the Mathematical Association of America (MAA) studied Calculus I instruction in United States (US) colleges and universities in the Characteristics of Successful Programs of College Calculus (CSPCC). One aspect of this study attempted to understand the impact of these courses on student experience.

Results

In this paper, we describe results from an examination of the effect of course structure on students’ attitudes …

Cluster-Based Network Proximities For Arbitrary Nodal Subsets, Kenneth S. Berenhaut, Peter S. Barr, Alyssa M. Kogel, Ryan L. Melvin

Faculty & Staff Scholarship

The concept of a cluster or community in a network context has been of considerable interest in a variety of settings in recent years. In this paper, employing random walks and geodesic distance, we introduce a unified measure of cluster-based proximity between nodes, relative to a given subset of interest. The inherent simplicity and informativeness of the approach could make it of value to researchers in a variety of scientific fields. Applicability is demonstrated via application to clustering for a number of existent data sets (including multipartite networks). We view community detection (i.e. when the full set of network nodes …

Existence Of Global Solutions For Nonlinear Magnetohydrodynamics With Finite Larmor Radius Corrections, Fariha Elsrrawi, Harumi Hattori

Faculty & Staff Scholarship

We discuss the existence of global solutions to the magnetohydrodynamics (MHD) equations, where the effects of finite Larmor radius corrections are taken into account. Unlike the usual MHD, the pressure is a tensor and it depends on not only the density but also the magnetic field. We show the existence of global solutions by the energy methods. Our techniques of proof are based on the existence of local solution by semigroups theory and a priori estimate.

On The Composition Of Derivatives, Krzysztof Ciesielski

Faculty & Staff Scholarship

We show that there exists a derivative f : [0, 1] → [0, 1] such that the graph of f ◦ f is dense in [0, 1]2 , so not a Gδ-set. In particular, f ◦ f is everywhere discontinuous, so not of Baire class 1, and hence it is not a derivative.

Lipschitz Restrictions Of Continuous Functions And A Simple Construction Of Ulam-Zahorski C1 Interpolation, Krzysztof Ciesielski

Faculty & Staff Scholarship

We present a simple argument that for every continuous function f : R → R its restriction to some perfect set is Lipschitz. We will use this result to provide an elementary proof of the C 1 free interpolation theorem, that for every continuous function f : R → R there exists a continuously differentiable function g : R → R which agrees with f on an uncountable set. The key novelty of our presentation is that no part of it, including the cited results, requires from the reader any prior familiarity with the Lebesgue measure theory.

A Continuous Tale On Continuous And Separately Continuous Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

Assume that you have developed a good set of tools allowing you to decide which real functions of one real variable, f : R → R, are continuous. (Q): How can such a tool-box be utilized to decide on the continuity of the functions g : R n → R of n real variables? This is one of the questions which must be faced by any student taking multivariable calculus. Of course, such a student is following the footsteps of many generations of mathematicians, which were, and still are, struggling with the same general question. The aim of this article …

On Closed Subsets Of ℝ And Of ℝ2 Admitting Peano Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

In this note we describe closed subsets of the real line P ⊂ R for which there exists a continuous function from P onto P 2 , called Peano function. Our characterization of those sets is based on the number of connected components of P. We also include a few remarks on compact subsets of R 2 admitting Peano functions, expressed in terms of connectedness and local connectedness.

Smooth Peano Functions For Perfect Subsets Of The Real Line, Krzysztof Ciesielski

Faculty & Staff Scholarship

In this paper we investigate for which closed subsets P of the real line R there exists a continuous map from P onto P 2 and, if such a function exists, how smooth can it be. We show that there exists an infinitely many times differentiable function f : R → R 2 which maps an unbounded perfect set P onto P 2 . At the same time, no continuously differentiable function f : R → R 2 can map a compact perfect set onto its square. Finally, we show that a disconnected compact perfect set P admits a continuous …

Sets Of Discontinuities Of Linearly Continuous Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

The class of linearly continuous functions f:Rⁿ-->R, that is, having continuous restrictions f|L to every straight line L, have been studied since the dawn of the twentieth century. In this paper we refine a description of the form that the sets D(f) of points of discontinuities of such functions can have. It has been proved by Slobodnik that D(f) must be a countable union of isometric copies of the graphs of Lipschitz functions h:K-->R, where K is a compact nowhere dense subset of R^n-1. Since the class Dⁿ of all …

Continuous And Smooth Images Of Sets, Krzysztof Ciesielski

Faculty & Staff Scholarship

This note shows that if a subset S of R is such that some continuous function f from R to R has the property "f[S] contains a perfect set," then some infinitely many times differentiable function g (from R to R) has the same property. Moreover, if f[S] is nowhere dense, then the g can have the stronger property "g[S] is perfect." The last result is used to show that it is consistent with ZFC (the usual axioms of set theory) that for each subset S of R of cardinality continuum there exists an infinitely many times differentiable function …

Functions Continuous On Twice Differentiable Curves, Discontinuous On Large Sets, Krzysztof Ciesielski

Faculty & Staff Scholarship

We provide a simple construction of a function F:R²-->R discontinuous on a perfect set P, while having continuous restrictions F|C for all twice differentiable curves C. In particular, F is separately continuous and linearly continuous. While it has been known that the projection \pi[P] of any such set P onto a straight line must be meager, our construction allows \pi[P] to have arbitrarily large measure. In particular, P can have arbitrarily large 1-Hausdorff measure, which is the best possible result in this direction, since any such P has Hausdorff dimension at most 1.

On Fundamental Groups Of Quotient Spaces, Jack S. Calcut, Robert E. Gompf, John D. Mccarthy

Faculty & Staff Scholarship

In classical covering space theory, a covering map induces an injection of fundamental groups. This paper reveals a dual property for certain quotient maps having connected fibers, with applications to orbit spaces of vector fields and leaf spaces in general.

Review: The Mathematics Of Sex: How Biology And Society Conspire To Limit Talented Women And Girls, Susan Jane Colley

Faculty & Staff Scholarship

No abstract provided.

Review: Pythagoras’ Revenge: A Mathematical Mystery; The Housekeeper And The Professor, Susan Jane Colley

Faculty & Staff Scholarship

No abstract provided.

On Additive Almost Continuous Functions Under Cpaprismgame, Krzysztof Ciesielski

Faculty & Staff Scholarship

We prove that the Covering Property Axiom CPA_prism^game, which holds in the iterated perfect set model, implies that there exists an additive discontinuous almost continuous function f from R to R whose graph is of measure zero. We also show that, under CPA_prism^game, there exists a Hamel basis H for which the set E⁺(H), of all linear combinations of elements from H with positive rational coefficients, is of measure zero. The existence of both of these examples follows from Martin's axiom, while it is unknown whether either of them can be constructed …

Separately Nowhere Constant Functions; N-Cube And \Alpha-Prism Densities, Krzysztof Ciesielski

Faculty & Staff Scholarship

A function f from a countable product X of of Polish spaces X_i into a Polish space is separately nowhere constant provided it is nowhere constant on every section of X. We show that every continuous separately nowhere constant function is one-to-one on a product of perfect subsets of X_i's. This result is used to distinguish between n-cube density notions for different n\leq\omega, where \omega-cube density is a basic notion behind the Covering Property Axiom CPA formulated by Ciesielski and Pawlikowski. We will also distinguish, for different values of \alpha<\omega₁, between the notions of \alpha-prism …

Algebras With Inner Mb-Representation, Krzysztof Ciesielski

Faculty & Staff Scholarship

We investigate algebras of sets, and pairs (A,I) consisting of an algebra A and an ideal I, which is a subset of A, that possess an inner MB-representation. We compare inner MB-representability of (A,I) with several properties of (A,I) considered by Baldwin. We show that A is inner MB-representable if and only if A =S(A \ H (A)), where S(^.) is a Marczewski operation defined below and H consists of sets that are hereditarily in A. We study uniqueness issue of the ideal in that representation.

Continuous Images Of Big Sets And Additivity Of S0 Under Cpaprism, Krzysztof Ciesielski

Faculty & Staff Scholarship

We prove that the Covering Property Axiom CPA_prism, which holds in the iterated perfect set model, implies the following facts.

• There exists a family G of uniformly continuous functions from R to [0,1] such that G has cardinality \omega₁ < \continuum and for every subset S of R of cardinality \continuum there exists a g in G with g[S]=[0,1].
• The additivity of the Marczewski's ideal s₀ is equal to \omega₁ < \continuum.

Crowded And Selective Ultrafilters Under The Covering Property Axiom, Krzysztof Ciesielski

Faculty & Staff Scholarship

In the paper we formulate an axiom CPA_{prism}^{game}, which is the most prominent version of the Covering Property Axiom CPA, and discuss several of its implications. In particular, we show that it implies that the following cardinal characteristics of continuum are equal to \omega₁, while \continuum=\omega₂: the independence number i, the reaping number r, the almost disjoint number a, and the ultrafilter base number u. We will also show that CPA_{prism}^{game} implies the existence of crowded and selective ultrafilters as well as nonselective P-points. In addition we prove that under CPA_{prism}^{game} every …

On Marczewski-Burstin Representations Of Certain Algebras Of Sets, Krzysztof Ciesielski

Faculty & Staff Scholarship

We show that the Generalized Continuum Hypothesis GCH (its appropriate part) implies that many natural algebras on R, including the algebra B of Borel sets and the interval algebra S, are outer Marczewski-Burstin representable by families of non-Borel sets. Also we construct, assuming again an appropriate part of GCH, that there are algebras on R which are not MB-representable. We prove that some algebras (including B and S) are not inner MB-representable. We give examples of algebras which are inner and outer MB-representable, or are inner but not outer MB-representable.

Measure Zero Sets Whose Algebraic Sum Is Non-Measurable, Krzysztof Ciesielski

Faculty & Staff Scholarship

In this note we will show that for every natural number n > 0 there exists an S ⊂ [0, 1] such that its n-th algebraic sum nS = S + ··· + S is a nowhere dense measure zero set,but its n+ 1-st algebraic sum nS +S is neither measurable nor it has the Baire property. In addition,the set S will be also a Hamel base,that is,a linear base of R over Q.

Measure Zero Sets With Non-Measurable Sum, Krzysztof Ciesielski

Faculty & Staff Scholarship

For any subset C of R there is a subset A of C such that A+A has inner measure zero and outer measure the same as C+C. Also, there is a subset A of the Cantor middle third set such that A+A is Bernstein in [0,2]. On the other hand there is a perfect set C such that C+C is an interval I and there is no subset A of C with A+A Bernstein in I.