Entire Blow-Up Solutions of Semilinear Elliptic Equations and Systems

We examine two problems concerning semilinear elliptic equations. We consider single equations of the form Δu = p(x)uα + q(x)uβ for 0 \u3cα ≤ β ≤1 and systems Δu = p(| x |) f (v), Δv = q(| x |)g(u) , both in Euclidean n -space, n ≥ 3 . These types of problems arise in steady state diffusion, the electric potential of some bodies, subsonic motion of gases, and control theory. For the single equation case, we present sufficient conditions on p and q to guarantee existence of nonnegative bounded solutions on the entire space. We also give alternative conditions that ensure existence of nonnegative radial solutions blowing up at infinity. Similarly, for systems, we provide conditions on p,q, f , and g that guarantee existence of nonnegative solutions on the entire space. The main requirement for f and g will be closely related to a growth requirement known as the Keller-Osserman condition. Further, we demonstrate the existence of solutions blowing up at infinity and describe a set of initial conditions that would generate such solutions. Lastly, we examine several specific examples numerically to graphically demonstrate the results of our analysis

Fusion frame constructions and frame partitions

Fusion frames consist of a sequence of subspaces from a Hilbert space and corresponding positive weights so that the sum of weighted orthogonal projections onto these subspaces is an invertible operator on the space. Despite extensive literature on fusion frames, and several construction methods for unit-weight fusion frames with prescribed subspace dimensions and fusion frame operator spectra, there do not exist such constructions for prescribed non-unit weights. There are also very few constructions which allow one to control geometric properties among the subspaces. First we will adapt a flexible construction technique known as spectral tetris to provide the first constructions of the most general classes of fusion frames: fusion frames with arbitrary non-unit weights. Moreover, we provide for the first time necessary and sufficient conditions for when a fusion frame can be constructed via spectral tetris methods. Then we present a new alternative construction leveraging Naimark complements to build a large class of fusion frames whose principal angles between any two subspaces are constant. Changing focus to frame partitions, the celebrated Rado-Horn theorem is an often-applied tool which provides a tight bound on the minimal number of linearly indpendent sets in a partition of vectors. This theorem has been rediscovered many times over, but no existing results describe how to find such partitions. We present an alternative proof of the Rado-Horn theorem and then adapt our proof's ideas to capture much more spanning and independence information compared with existing Rado-Horn results. We further describe how to build partitions with these properties

Tremain Equiangular Tight Frames

Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph

Hadamard Equiangular Tight Frames

An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. They are often represented as the columns of a short, fat matrix. In certain applications we want this matrix to be flat, that is, have the property that all of its entries have modulus one. In particular, real flat ETFs are equivalent to self-complementary binary codes that achieve the Grey-Rankin bound. Some flat ETFs are (complex) Hadamard ETFs, meaning they arise by extracting rows from a (complex) Hadamard matrix. These include harmonic ETFs, which are obtained by extracting the rows of a character table that correspond to a difference set in the underlying finite abelian group. In this paper, we give some new results about flat ETFs. One of these results gives an explicit Naimark complement for all Steiner ETFs, which in turn implies that all Kirkman ETFs are possibly-complex Hadamard ETFs. This in particular produces a new infinite family of real flat ETFs. Another result establishes an equivalence between real flat ETFs and certain types of quasi-symmetric designs, resulting in a new infinite family of such designs

Polyphase Equiangular Tight Frames and Abelian Generalized Quadrangles

An equiangular tight frame (ETF) is a type of optimal packing of lines in a finite-dimensional Hilbert space. ETFs arise in various applications, such as waveform design for wireless communication, compressed sensing, quantum information theory and algebraic coding theory. In a recent paper, signature matrices of ETFs were constructed from abelian distance regular covers of complete graphs. We extend this work, constructing ETF synthesis operators from abelian generalized quadrangles, and vice versa. This produces a new infinite family of complex ETFs as well as a new proof of the existence of certain generalized quadrangles. This work involves designing matrices whose entries are polynomials over a finite abelian group. As such, it is related to the concept of a polyphase matrix of a finite filter bank

The Impact of COVID-19 on Perceived Barriers and Facilitators to the Healthfulness of Communities With Low-Income

Background: The COVID-19 pandemic brought new challenges affecting the wellbeing of individuals in communities with low income. Understanding where people live and how those environments can facilitate or hinder living a healthy lifestyle is essential for developing interventions that target behavior change and health promotion. Objective: This study compares Extension Nutrition Educators’ (NEs) perceptions of the barriers and facilitators impacting the healthfulness of the environment of communities with low income in eleven states before and during the COVID-19 pandemic