Education Commons^™

Hodge Theory On Transversely Symplectic Foliations, Yi Lin

Department of Mathematical Sciences Faculty Publications

In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic dδ-lemma for any such foliations with the (transverse) s-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds, and symplectic quasi-folds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries.

As an application, we show that on compact K-contact manifolds, the s-Lefschetz property implies a general result on the vanishing of cup products, and that the cup length of a 2n+1 dimensional compact K-contact manifold with the …

Cahost Facilitating The Johnson-Neyman Technique For Two-Way Interactions In Multiple Regression, Stephen W. Carden, Nicholas Holtzman, Michael Strube

Department of Mathematical Sciences Faculty Publications

When using multiple regression, researchers frequently wish to explore how the relationship between two variables is moderated by another variable; this is termed an interaction. Historically, two approaches have been used to probe interactions: the pick-a-point approach and the Johnson-Neyman (JN) technique. The pick-a-point approach has limitations that can be avoided using the JN technique. Currently, the software available for implementing the JN technique and creating corresponding figures lacks several desirable features–most notably, ease of use and figure quality. To fill this gap in the literature, we offer a free Microsoft Excel 2013 workbook, CAHOST (a concatenation of the first …

Vanishing Of Ext And Tor Over Fiber Products, Saeed Nasseh, Sean Sather-Wagstaff

Department of Mathematical Sciences Faculty Publications

Consider a non-trivial fiber product R=S×kT of local rings S, T with common residue field k. Given two finitely generate R-modules M and N, we show that if TorRi(M,N)=0=TorRi+1(M,N) for some i≥5, then pdR(M)≤1 or pdR(N)≤1. From this, we deduce several consequence, for instance, that R satisfies the Auslander-Reiten Conjecture.

Global Analysis Of A Stochastic Two-Scale Network Human Epidemic Dynamic Model With Varying Immunity Period, Divine Wanduku, G. S. Ladde

Department of Mathematical Sciences Faculty Publications

A stochastic SIR epidemic dynamic model with distributed-time-delay, for a two-scale dynamic population is derived. The distributed time delay is the varying naturally acquired immunity period of the removal class of individuals who have recovered from the infection, and have acquired natural immunity to the disease. We investigate the stochastic asymptotic stability of the disease free equilibrium of the epidemic dynamic model, and verify the impact on the eradication of the disease.

CO-Characterization Of Symplectic And Contact Embeddings And Lagrangian Rigidity, Stefan Müller

Department of Mathematical Sciences Faculty Publications

We present a novel C0-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by J.-C. Sikorav and Y. Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of C0-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from J-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a C0-characterization of contact embeddings …

The Gamma-Generalized Inverse Weibull Distribution With Applications To Pricing And Lifetime Data, Broderick O. Oluyede, Boikanyo Makubate, Divine Wanduku, Ibrahim Elbatal, Valeriia Sherina

Department of Mathematical Sciences Faculty Publications

A new distribution called the gamma-generalized inverse Weibull distribution which includes inverse exponential, inverse Rayleigh, inverse Weibull, Frechet, generalized inverse Weibull, gamma-exponentiated inverse exponential, exponentiated inverse exponential, Zografos and Balakrishnan-generalized inverse Weibull, Zografos and Balakrishnan-inverse Weibull, Zografos and Balakrishnan-generalized inverse exponential, Zografos and Balakrishnan-inverse exponential, Zografos and Balakrishnan-generalized inverse Rayleigh, Zografos and Balakrishnan-inverse Rayleigh, and Zografos and Balakrishnan-Fr'echet distributions as special cases is proposed and studied in detail. Some structural properties of this new distribution including density expansion, moments, Renyi entropy, distribution of the order statistics, moments of the order statistics and L-moments are presented. Maximum likelihood estimation technique is …

Ghost Series And A Motivated Proof Of The Andrews–Bressoud Identities, Shashank Kanade, James Lepowsky, Matthew C. Russell, Andrew Sills

Department of Mathematical Sciences Faculty Publications

We present what we call a “motivated proof” of the Andrews–Bressoud partition identities for even moduli. A “motivated proof” of the Rogers–Ramanujan identities was given by G.E. Andrews and R.J. Baxter, and this proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu. Recently, a “motivated proof” of the somewhat analogous Göllnitz–Gordon–Andrews identities has been found. In the present work, we introduce “shelves” of formal series incorporating what we call “ghost series,” which allow us to pass from one shelf to the next via natural recursions, leading to our motivated proof. We anticipate that …

Gorenstein Projective Precovers, Sergio Estrada, Alina Iacob, Katelyn A. Coggins

Department of Mathematical Sciences Faculty Publications

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.

On Gorenstein Fiber Products And Applications, Saeed Nasseh, Ryo Takahashi, Keller Vandebogert

Department of Mathematical Sciences Faculty Publications

We show that a non-trivial fiber product S×kT of commutative noetherian local rings S,T with a common residue field k is Gorenstein if and only if it is a hypersurface of dimension 1. In this case, both S and T are regular rings of dimension 1. We also give some applications of this result.

Totally Acyclic Complexes, Sergio Estrada, Xianhui Fu, Alina Iacob

Department of Mathematical Sciences Faculty Publications

It is known that over an Iwanaga–Gorenstein ring the Gorenstein injective (Gorenstein projective, Gorenstein flat) modules are simply the cycles of acyclic complexes of injective (projective, flat) modules. We consider the question: are these characterizations only working over Iwanaga–Gorenstein rings? We prove that if R is a commutative noetherian ring of finite Krull dimension then the following are equivalent: 1. R is an Iwanaga–Gorenstein ring. 2. Every acyclic complex of injective modules is totally acyclic. 3. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 4. Every acyclic complex of projective modules is totally acyclic. 5. …

A Zariski-Local Notion Of F-Total Acyclicity For Complexes Of Sheaves, Lars Winther Christensen, Sergio Estrada, Alina Iacob

Department of Mathematical Sciences Faculty Publications

We study a notion of total acyclicity for complexes of flat sheaves over a scheme. It is Zariski-local—i.e. it can be verified on any open affine covering of the scheme—and for sheaves over a quasi-compact semi-separated scheme it agrees with the categorical notion. In particular, it agrees, in their setting, with the notion studied by Murfet and Salarian for sheaves over a noetherian semi-separated scheme. As part of the study we recover, and in several cases extend the validity of, recent results on existence of covers and precovers in categories of sheaves. One consequence is the existence of an adjoint …

Gorenstein Flat And Projective (Pre)Covers, Sergio Estrada, Alina Iacob, Sinem Odabasi

Department of Mathematical Sciences Faculty Publications

We consider a right coherent ring R. We prove that the class of Gorenstein flat complexes is covering in the category of complexes of left R-modules Ch(R). When R is also left n-perfect, we prove that the class of Gorenstein projective complexes is special precovering in Ch(R).

Gorenstein Injective Envelopes And Covers Over Two Sided Noetherian Rings, Alina Iacob

Department of Mathematical Sciences Faculty Publications

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.

Multiple Solutions With Constant Sign Of A Dirichlet Problem For A Class Of Elliptic Systems With Variable Exponent Growth, Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao

Department of Mathematical Sciences Faculty Publications

We present here, in the system setting, a new set of growth conditions under which we manage to use a novel method to verify the Cerami compactness condition. By localization argument, decomposition technique and variational methods, we are able to show the existence of multiple solutions with constant sign for the problem without the well-known Ambrosetti--Rabinowitz type growth condition. More precisely, we manage to show that the problem admits four, six and infinitely many solutions respectively.