Mathematics Commons^™

On The Boundary Blow-Up Solutions Of P(X)-Laplacian Equations With Gradient Terms, Yuan Liang, Qihu Zhang, Chunshan Zhao

Mathematical Sciences: Faculty Publications

In this paper we investigate boundary blow-up solutions of the problem

⎧⎩⎨⎪⎪−△p(x)u+f(x,u)=ρ(x,u)+K(|x|)|∇u|δ(|x|) in Ω, u(x)→+∞ as d(x, ∂Ω)→0,

where −△p(x)u=−div(|∇u|p(x)−2∇u) is called p(x)-Laplacian. The existence of boundary blow-up solutions is proved and the singularity of boundary blow-up solution is also given for several cases including the case of ρ(x,u) being a large perturbation (namely, ρ(x,u(x))f(x,u(x))→1 as x→∂Ω). In particular, we do not have the comparison principle.

Odd Fibbinary Numbers And The Golden Ratio, Linus Lindroos, Andrew Sills, Hua Wang

Mathematical Sciences: Faculty Publications

The fibbinary numbers are positive integers whose binary representation contains no consecutive ones. We prove the following result: If the jth odd fibbinary is the nth odd fibbinary number, then j={nɸ²]-1.

Weighted Inverse Weibull Distribution: Statistical Properties And Applications, Valeriia Sherina, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

In this paper, the weighted inverse Weibull (WIW) class of distributions is proposed and studied. This class of distributions contains several models such as: length-biased, hazard and reverse hazard proportional inverse Weibull, proportional inverse Weibull, inverse Weibull, inverse exponential, inverse Rayleigh, and Fr´echet distributions as special cases. Properties of these distributions including the behavior of the hazard function, moments, coefficients of variation, skewness, and kurtosis, R´enyi entropy and Fisher information are presented. Estimates of the model parameters via method of maximum likelihood (ML) are presented. Extensive simulation study is conducted and numerical examples are given.

Multiscale Geometric Modeling Of Macromolecules I: Cartesian Representation, Kelin Xia, Xin Feng, Zhan Chen, Yiying Tong, Guo-Wei Wei

Mathematical Sciences: Faculty Publications

This paper focuses on the geometric modeling and computational algorithm development of biomolecular structures from two data sources: Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB) in the Eulerian (or Cartesian) representation. Molecular surface (MS) contains non-smooth geometric singularities, such as cusps, tips and self-intersecting facets, which often lead to computational instabilities in molecular simulations, and violate the physical principle of surface free energy minimization. Variational multiscale surface definitions are proposed based on geometric flows and solvation analysis of biomolecular systems. Our approach leads to geometric and potential driven Laplace–Beltrami flows for biomolecular surface evolution and formation. The …

Theoretical Properties Of The Weighted Feller-Pareto Distributions, Oluseyi Odubote, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

In this paper, for the first time, a new six-parameter class of distributions called weighted Feller-Pareto (WFP) and related family of distributions is proposed. This new class of distributions contains several other Pareto-type distributions such as length-biased (LB) Pareto, weighted Pareto (WP I, II, III, and IV), and Pareto (P I, II, III, and IV) distributions as special cases. The pdf, cdf, hazard and reverse hazard functions, monotonicity properties, moments, entropy measures including Renyi, Shannon and s-entropies are derived.

Note On Rainbow Connection In Oriented Graphs With Diameter 2, Rebecca Holliday, Colton Magnant, Pouria Salehi Nowbandegani

Mathematical Sciences: Faculty Publications

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter $2$. For a tournament $T$ of diameter $2$, we show $2 \leq \overrightarrow{rc}(T) \leq 3$. Furthermore, we provide a general upper bound on the rainbow $k$-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of $k^{th}$ diameter $2$ has rainbow $k$-connection number at most approximately $k^{2}$.

Rainbow Generalizations Of Ramsey Theory - A Dynamic Survey, Shinya Fujita, Colton Magnant, Kenta Ozeki

Mathematical Sciences: Faculty Publications

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.

Building A Better Bijection Between Classes Of Compositions, James D. Diffenderfer

Mathematical Sciences: Faculty Publications

A bijective proof is given for the following theorem: The number of compositions of n into parts congruent to a (mod b) equals the number of compositions of n + b - a into parts congruent to b (mod a) that are greater than or equal to b. The bijection is then shown to preserve palindromicity.

A Decomposition Of Gallai Multigraphs, Alexander Halperin, Colton Magnant, Kyle Pula

Mathematical Sciences: Faculty Publications

An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees.

A New Class Of Generalized Dagum Distribution With Applications To Income And Lifetime Data, Broderick O. Oluyede, Shujiao Huang, Mavis Pararai

Mathematical Sciences: Faculty Publications

The generalized beta distribution of the second kind (GB2), McDonald [11], McDonald and Xu [12] is an important distribution with applications in finance and actuarial sciences, as well as economics, where Dagum distribution which is a sub-model of GB2 distribution plays an important role in size distribution of personal income. In this note, a new class of generalized Dagum distribution called gamma-Dagum distribution is presented. The gamma-Dagum (GD) distribution which includes the gamma-Burr III (GB III), gamma-Fisk or gamma-log logistic (GF of GLLog), Zografos and Balakrishnan-Dagum (ZB-D), ZB-Burr III (ZB-B III), ZB-Fisk of ZB-Log logistic (ZB-F or ZB-LLog), Burr III …

Discrete Fourier Restriction Associated With Schrödinger Equations, Yi Hu, Xiaochun Li

Mathematical Sciences: Faculty Publications

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain’s level set result on Strichartz estimates associated with Schrödinger equations on torus. Some sharp estimates on L 2(d+2) d norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on L 2 (Z).

Determinants Of Incidence And Hessian Matrices Arising From The Vector Space Lattice, Saeed Nasseh, Alexandra Seceleanu, Junzo Watanabe

Mathematical Sciences: Faculty Publications

We give explicit formulas for the determinants of the incidence and Hessian matrices arising from the interaction between the rank 1 and rank n−1 level sets of the subspace lattice of an n-dimensional finite vector space. Our exploration is motivated by the fact that both of these matrices arise naturally in the study of the combinatorial and algebraic Lefschetz properties for the vector space lattice and the graded Artinian Gorenstein algebra associated to it, respectively.

Exponentiated Kumaraswamy-Dagum Distribution With Applications To Income And Lifetime Data, Shujiao Huang, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

A new family of distributions called exponentiated Kumaraswamy-Dagum (EKD) distribution is proposed and studied. This family includes several well known sub-models, such as Dagum (D), Burr III (BIII), Fisk or Log-logistic (F or LLog), and new sub-models, namely, Kumaraswamy-Dagum (KD), Kumaraswamy-Burr III (KBIII), Kumaraswamy-Fisk or Kumaraswamy-Log-logistic (KF or KLLog), exponentiated Kumaraswamy-Burr III (EKBIII), and exponentiated Kumaraswamy-Fisk or exponentiated Kumaraswamy-Log-logistic (EKF or EKLLog) distributions. Statistical properties including series representation of the probability density function, hazard and reverse hazard functions, moments, mean and median deviations, reliability, Bonferroni and Lorenz curves, as well as entropy measures for this class of distributions and the …

Local Rings Of Embedding Codepth At Most 3 Have Only Trivial Semidualizing Complexes, Saeed Nasseh, Sean Sather-Wagstaff

Mathematical Sciences: Faculty Publications

We prove that a local ring R of embedding codepth at most 3 has at most two semidualizing complexes up to shift-isomorphism, namely, R itself and a dualizing R-complex if one exists.

The Apostles And Brothers Of Jesus, Andrew Sills

Mathematical Sciences: Faculty Publications

Excerpt: The Talpiot tomb, a tomb excavated outside Jerusalem in 1980 and brought to worldwide public attention in 2007, contained ten ossuaries, six of which were inscribed with names. The English equivalents of the names are Jesus son of Joseph, two Marys, a rare diminutive form of Joseph, a diminutive of Matthew, and a Judah son of Jesus. Because of the similarities between this collection of names and certain names appearing in the Christian New Testament, some are curious as to whether the Talpiot tomb may have once interred the remains of Jesus of Nazareth and some of his relatives. …

Mathematical Model Of Dynamic Protein Interactions Regulating P53 Protein Stability For Tumor Suppression, Hua Wang, Guang Peng

Mathematical Sciences: Faculty Publications

In the field of cancer biology, numerous genes or proteins form extremely complex regulatory network, which determines cancer cell fate and cancer cell survival. p53 is a major tumor suppressor that is lost in more than 50% of human cancers. It has been well known that a variety of proteins regulate its protein stability, which is essential for its tumor suppressive function. It remains elusive how we could understand and target p53 stabilization process through network analysis. In this paper we discuss the use of random walk and stationary distribution to measure the compound effect of a network of genes …

Multiple Solutions Of A P(X)-Laplacian Equation Involving Critical Nonlinearities, Yuan Liang, Xianbin Wu, Qihu Zhang, Chunshan Zhao

Mathematical Sciences: Faculty Publications

In this paper, we consider the existence of multiple solutions for the following p(x)-Laplacian equations with critical Sobolev growth conditions

−div(|∇u| p(x)−2 ∇u) + |u| ^p(x)−2 u = f(x, u) in Ω,

u = 0 on ∂Ω.

We show the existence of infinitely many pairs of solutions by applying the Fountain Theorem and the Dual Fountain Theorem respectively. We also present a variant of the concentration-compactness principle, which is of independent interest.

Discrete Fourier Restriction Associated With Kdv Equations, Yi Hu, Xiaochun Li

Mathematical Sciences: Faculty Publications

In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear term $F(u)\p_x u$ provided F∈C5 and the initial data ϕ∈Hs with s>1/2.

Comparing The Effects Of Interactive And Noninteractive Complementary Nutrients On Growth In A Chemostat, James P. Braselton, Martha L. Abell, Lorraine Braselton

Mathematical Sciences: Faculty Publications

We compare the effects of interactive and noninteractive complementary nutrients on the growth of an organism in the chemostat. We also compare these two situations to the case when the nutrients are substitutable. In previous studies, complementary nutrients have been assumed to be noninteractive. However, more recent research indicates that some complementary nutrient relationships are interactive. We show that interactive complementary and substitutable nutrients can lead to higher population densities than do noninteractive complementary nutrients. We numerically illustrate that if the washout rate is high, an organism can persist at higher densities when the complementary nutrients are interactive than when …

Rock-Paper-Scissors In The Chemostat, James P. Braselton, Martha L. Abell, Lorraine Braselton

Mathematical Sciences: Faculty Publications

Rock-Paper-Scissors is a game played by two players to determine a single winner. Biological relationships of Rock-Paper-Scissors are documented. In this paper, we form a continuous model of Rock-Papers-Scissors in the chemostat that coincides with the biology of such relationships. The basic models that we develop coincide with the observed phenomena. Because the model involves a system of seven nonlinear differential equations, global results are difficult to obtain. We present several numerical studies that are the result of a substantial number of numerical trials to illustrate the various possibilities that might occur in the context of the problem discussed here.

Sum Formula Of Multiple Hurwitz-Zeta Values, Jianqiang Zhao

Mathematical Sciences: Faculty Publications

Let s1,⋯,sd be d positive integers and define the multiple t-values of depth d byt(s1,⋯,sd)=∑n1>⋯>nd≥11(2n1−1)s1⋯(2nd−1)sd,which is equal to the multiple Hurwitz-zeta value 2−wζ(s1,⋯,sd;−12,⋯,−12) where w=s1+⋯+sd is called the weight. For d≤n, let T(2n,d) be the sum of all multiple t-values with even arguments whose weight is 2n and whose depth is d. In 2011, Shen and Cai gave formulas for T(2n,d) for d≤5 in terms of t(2n), t(2)t(2n−2) and t(4)t(2n−4). In this short note we generalize their results to arbitrary depth by using the theory of symmetric functions established by Hoffman (2012).

Extremal Values Of Ratios: Distance Problems Vs. Subtree Problems In Trees, László A. Székely, Hua Wang

Mathematical Sciences: Faculty Publications

The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374-385; Adv. Appl. Math. 34 (2005), 138-155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80 (1997), 37-56] determined extremal values of σT(w)/σT(u), σT(w)/σT(v), σ(T)/σT(v), and σ(T)/σT(w), where T is a tree on n vertices, v is in the centroid of …

The Mc-Dagum Distribution And Its Statistical Properties With Applications, Broderick O. Oluyede, Sasith Rajasooriya

Mathematical Sciences: Faculty Publications

In this paper, a new class of distributions called Mc-Dagum distribution is proposed. This class of distributions contains several distributions such as beta-Dagum, beta-Burr III, beta-Fisk, Dagum, Burr III and Fisk distributions as special cases. The hazard function, reverse hazard function, moments, mean residual life function, Renyi entropy and Fisher information are obtained. Lorenz, Bonferroni and Zenga curves are derived. Maximum likelihood estimates of the model parameters and numerical examples are given to illustrate the usefulness of the proposed class of distributions.

Numerical Studies Of The Generalized L₁ Greedy Algorithm For Sparse Signals, Fangjun Arroyo, Edward Arroyo, Xiezhang Li, Jiehua Zhu

Mathematical Sciences: Faculty Publications

The generalized l₁ greedy algorithm was recently introduced and used to reconstruct medical images in computerized tomography in the compressed sensing framework via total variation minimization. Experimental results showed that this algorithm is superior to the reweighted l₁-minimization and l₁ greedy algorithms in reconstructing these medical images. In this paper the effectiveness of the generalized l₁ greedy algorithm in finding random sparse signals from underdetermined linear systems is investigated. A series of numerical experiments demonstrate that the generalized l₁ greedy algorithm is superior to the reweighted l₁-minimization and l₁ greedy algorithms …

Rademacher's Infinite Partial Fractions Conjecture Is (Almost Certainly) False, Andrew Sills, Doron Zeilberger

Mathematical Sciences: Faculty Publications

In his book Topics in Analytic Number Theory, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating function for partitions of integers into at most N parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made towards proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this paper, we present a recurrence (alias difference equation) which provides a fast algorithm for calculating the Rademacher …

Distance-Based Graph Invariants Of Trees And The Harary Index, Stephan G. Wagner, Hua Wang, Xiao-Dong Zhang

Mathematical Sciences: Faculty Publications

Introduced in 1947, the Wiener index W(T) = ∑{u,v}⊆V(T) d(u, v) is one of the most thoroughly studied chemical indices. The extremal structures (in particular, trees with various constraints) that maximize or minimize the Wiener index have been extensively investigated. The Harary index H(T) = ∑{u,v}⊆V(T) , introduced in 1993, can be considered as the 'reciprocal analogue' of the Wiener index. From recent studies, it is known that the extremal structures of the Harary index and the Wiener index coincide in many instances, i.e., the graphs that maximize the Wiener index minimize the Harary index and vice versa. In this …

Greedy Trees, Subtrees And Antichains, Eric Ould Dadah Andriantiana, Stephan G. Wagner, Hua Wang

Mathematical Sciences: Faculty Publications

Greedy trees are constructed from a given degree sequence by a simple greedy algorithm that assigns the highest degree to the root, the second-, third-, ... highest degrees to the root's neighbors, and so on.

They have been shown to maximize or minimize a number of different graph invariants among trees with a given degree sequence. In particular, the total number of subtrees of a tree is maximized by the greedy tree. In this work, we show that in fact a much stronger statement holds true: greedy trees maximize the number of subtrees of any given order. This parallels recent …

Exponential Kleisli Monoids As Eilenberg-Moore Algebras, Dirk Hofmann, Frédéric Mynard, Gavin J. Seal

Mathematical Sciences: Faculty Publications

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

On Q-Analogs Of Wostenholme Type Congruences For Multiple Harmonic Sums, Jianqiang Zhao

Mathematical Sciences: Faculty Publications

Multiple harmonic sums are iterated generalizations of harmonic sums. Recently Dilcher has considered congruences involving q-analogs of these sums in depth one. In this paper we shall study the homogeneous case for arbitrary depth by using generating functions and shuffle relations of the q-analog of multiple harmonic sums. At the end, we also consider some non-homogeneous cases.

The Convergence Of Two Algorithms For Compressed Sensing Based Tomography, Xiezhang Li, Jiehua Zhu

Mathematical Sciences: Faculty Publications

The constrained total variation minimization has been developed successfully for image reconstruction in computed tomography. In this paper, the block component averaging and diagonally-relaxed orthogonal projection methods are proposed to incorporate with the total variation minimization in the compressed sensing framework. The convergence of the algorithms under a certain condition is derived. Examples are given to illustrate their convergence behavior and noise performance.