Mathematics Commons^™

A Simple Model Of Cortical Dynamics Explains Variability And State Dependence Of Sensory Responses In Urethane-Anesthetized Auditory Cortex, Carina Curto, Shuzo Sakata, Stephan Marguet, Vladimir Itskov, Kenneth D. Harris

Department of Mathematics: Faculty Publications

The responses of neocortical cells to sensory stimuli are variable and state dependent. It has been hypothesized that intrinsic cortical dynamics play an important role in trial-to-trial variability; the precise nature of this dependence, however, is poorly understood. We show here that in auditory cortex of urethane-anesthetized rats, population responses to click stimuli can be quantitatively predicted on a trial-by-trial basis by a simple dynamical system model estimated from spontaneous activity immediately preceding stimulus presentation. Changes in cortical state correspond consistently to changes in model dynamics, reflecting a nonlinear, self-exciting system in synchronized states and an approximately linear system in …

Parameterizing The Growth-Decline Boundary For Uncertain Population Projection Models, Joan Lubben, Derek Boeckner, Richard Rebarber, Stuart Townley, Brigitte Tenhumberg

Department of Mathematics: Faculty Publications

We consider discrete time linear population models of the form n(t + 1) = An(t) where A is a population projection matrix or integral projection operator, and represents a structured population at time t. It is well known that the asymptotic growth or decay rate of n(t) is determined by the leading eigenvalue of A. In practice, population models have substantial parameter uncertainty, and it might be difficult to quantify the effect of this uncertainty on the leading eigenvalue. For a large class of matrices and integral operators A, we …

Analysis Of Connections Between Pseudocodewords, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

The role of pseudocodewords in causing noncodeword outputs in linear programming (LP) decoding, graph cover decoding, and iterative message-passing decoding is investigated. The three main types of pseudocodewords in the literature — linear programming pseudocodewords, graph cover pseudocodewords, and computation tree pseudocodewords — are reviewed and connections between them are explored. Some discrepancies in the literature on minimal and irreducible pseudocodewords are highlighted and clarified, and a value for the minimal degree cover necessary to realize an LP pseudocodeword is found. Additionally, some conditions for the existence of connected realizations of graph cover pseudocodewords are given. This allows for further …

Connections Between Computation Trees And Graph Covers, Deanna Dreher, Judy L. Walker

Department of Mathematics: Faculty Publications

Connections between graph cover pseudocodewords and computation tree pseudocodewords are investigated with the aim of bridging the gap between the theoretically attractive analysis of graph covers and the more intractable analysis of iterative message-passing algorithms that are intuitively linked to graph covers. Both theoretical results and numerous examples are presented.

Cell Groups Reveal Structure Of Stimulus Space, Carina Curto, Vladimir Itskov

Department of Mathematics: Faculty Publications

An important task of the brain is to represent the outside world. It is unclear how the brain may do this, however, as it can only rely on neural responses and has no independent access to external stimuli in order to ‘‘decode’’ what those responses mean. We investigate what can be learned about a space of stimuli using only the action potentials (spikes) of cells with stereotyped—but unknown—receptive fields. Using hippocampal place cells as a model system, we show that one can (1) extract global features of the environment and (2) construct an accurate representation of space, up to an …

Supplementary Text To Accompany “Cell Groups Reveal Structure Of Stimulus Space”, Carina Curto, Vladimir Itskov

Department of Mathematics: Faculty Publications

Here we present a brief exposition of some material from algebraic topology that we use in our methods. We include it for completeness, as it may not be familiar for many readers. In particular, we define simplicial complexes, simplicial homology groups, and state the theorem cited in the Results section. See [Bott and Tu, 1982, Ewald, 1996, Hatcher, 2002] for more details.

Threefold Flops Via Matrix Factorization, Carina Curto, David R. Morrison

Department of Mathematics: Faculty Publications

The structure of birational maps between algebraic varieties becomes increasingly complicated as the dimension of the varieties increases. There is no birational geometry to speak of in dimension one: if two complete algebraic curves are birationally isomorphic then they are biregularly isomorphic. In dimension two we encounter the phenomenon of the blowup of a point, and every birational isomorphism can be factored into a sequence of blowups and blowdowns. In dimension three, however, we first encounter birational maps which are biregular outside of a subvariety of codimension two (called the center of the birational map). When the center has a …

A Universal Theory Of Decoding And Pseudocodewords, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

The discovery of turbo codes [5] and the subsequent rediscovery of low-density parity-check (LDPC) codes [9, 18] represent a major milestone in the field of coding theory. These two classes of codes can achieve realistic bit error rates, between 10−5 and 10−12, with signalto- noise ratios that are only slightly above the minimum possible for a given channel and code rate established by Shannon’s original capacity theorems. In this sense, these codes are said to be near-capacity-achieving codes and are sometimes considered to have solved (in the engineering sense, at least) the coding problem for the additive white Gaussian noise …

Nonbinary Quantum Error-Correcting Codes From Algebraic Curves, Jon-Lark Kim, Judy L. Walker

Department of Mathematics: Faculty Publications

We give a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia- Stichtenoth tower of function fields which are constructible in polynomial time.

Binary quantum error-correcting codes have been constructed in several ways. One interesting construction uses algebraic-geometry codes [2], [6], [7], [12], with the main idea being to apply the binary CSS construction [4], [5], [16] to the asymptotically good algebraic-geometry codes arising from the Garcia-Stichtenoth [11] tower of function fields over F_q2 (where q is a …

Local Cohomology And Support For Triangulated Categories, Dave Benson, Srikanth Iyengar, Henning Krause

Department of Mathematics: Faculty Publications

We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects whose triangulated support and cohomological support differ. In the …

Matrix Model Superpotentials And Ade Singularities, Carina Curto

Department of Mathematics: Faculty Publications

We use F. Ferrari’s methods relating matrix models to Calabi–Yau spaces in order to explain much of Intriligator and Wecht’s ADE classification of N = 1 superconformal theories which arise as RG fixed points of N = 1 SQCD theories with adjoints. We find that ADE superpotentials in the Intriligator–Wecht classification exactly match matrix model superpotentials obtained from Calabi–Yau with corresponding ADE singularities. Moreover, in the additional Ô, Â, Dˆ and Ê cases we find new singular geometries. These “hat” geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for …

Towards Universal Cover Decoding, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

Low complexity decoding of low-density paritycheck (LDPC) codes may be obtained from the application of iterative message-passing decoding algorithms to the bipartite Tanner graph of the code. Arguably, the two most important decoding algorithms for LDPC codes are the sum-product decoder and the min-sum (MS) decoder. On a bipartite graph without cycles (a tree), the sum-product decoder minimizes the probability of bit error, while the min-sum decoder minimizes the probability of word error [9]. While the behavior of sum-product and min-sum is easily understood when operating on trees, their behavior becomes much more difficult to characterize when the Tanner graph …

Ldpc Codes From Voltage Graphs, Christine A. Kelley, Judy L. Walker

Department of Mathematics: Faculty Publications

Several well-known structure-based constructions of LDPC codes, for example codes based on permutation and circulant matrices and in particular, quasi-cyclic LDPC codes, can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph theory can be used to give simple proofs of many known properties of these codes. In addition, the notion of abelianinevitable cycle is introduced and the subgraphs giving rise to these cycles are classified. We also indicate how, by using more sophisticated voltage assignments, new classes of good LDPC codes may be obtained.

Average Min-Sum Decoding Of Ldpc Codes, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

Simulations have shown that the outputs of minsum (MS) decoding generally behave in one of two ways: the output either eventually stabilizes at a codeword or eventually cycles through a finite set of vectors that may include both codewords and non-codewords. This inconsistency in MS across iterations has significantly contributed to the difficulty in studying the performance of this decoder. To overcome this problem, a new decoder, average min-sum (AMS), is proposed; this decoder outputs the average of the min-sum output vectors over a finite set of iterations. Simulations comparing MS, AMS, linear programming (LP) decoding, and maximum likelihood (ML) …

Valuations For Spike Train Prediction, Vladimir Itskov, Carina Curto, Kenneth D. Harris

Department of Mathematics: Faculty Publications

The ultimate product of an electrophysiology experiment is often a decision on which biological hypothesis or model best explains the observed data. We outline a paradigm designed for comparison of different models, which we refer to as spike train prediction. A key ingredient of this paradigm is a prediction quality valuation that estimates how close a predicted conditional intensity function is to an actual observed spike train. Although a valuation based on log likelihood (L) is most natural, it has various complications in this context. We propose that a quadratic valuation (Q) can be used as an alternative to L. …

A Universal Theory Of Pseudocodewords, Nathan Axvig, Emily Price, Eric T. Psota, Deanna Turk, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

Three types of pseudocodewords for LDPC codes are found in the literature: graph cover pseudocodewords, linear programming pseudocodewords, and computation tree pseudocodewords. In this paper we first review these three notions and known connections between them. We then propose a new decoding rule — universal cover decoding — for LDPC codes. This new decoding rule also has a notion of pseudocodeword attached, and this fourth notion provides a framework in which we can better understand the other three.

Mathematical Analysis Of Pde Systems Which Govern °Uid Structure Interactive Phenomena, George Avalos, Roberto Triggiani

Department of Mathematics: Faculty Publications

In this paper, we review and comment upon recently derived results for time dependent partial differential equation (PDE) models, which have been used to describe the various fluid-structure interactions which occur in nature. For these fluid-structure PDEs, this survey is particularly focused on the authors' results of (i) semigroup wellposedness, (ii) stability, and (iii) backward uniqueness.

The Time Invariance Principle, Ecological (Non)Chaos, And A Fundamental Pitfall Of Discrete Modeling, Bo Deng

Department of Mathematics: Faculty Publications

This paper is to show that most discrete models used for population dynamics in ecology are inherently pathological that their predications cannot be independently verified by experiments because they violate a fundamental principle of physics. The result is used to tackle an on-going controversy regarding ecological chaos. Another implication of the result is that all continuous dynamical systems must be modeled by differential equations. As a result it suggests that researches based on discrete modeling must be closely scrutinized and the teaching of calculus and differential equations must be emphasized for students of biology.

Promoting Undergraduate Research In Mathematics At The University Of Nebraska – Lincoln, Judy L. Walker, Glenn Ledder, Richard Rebarber, Gordon S. Woodward

Department of Mathematics: Faculty Publications

The Department of Mathematics at the University of Nebraska – Lincoln (UNL) has several programs which promote undergraduate research in a variety of ways. Two of these are summer programs which draw from a national applicant pool: The Nebraska REU in Applied Mathematics (Section 1) is a traditional NSF-funded REU site, and Nebraska IMMERSE (Section 2) offers a summer “bridge” program (with a research bent) for students about to start graduate school in mathematics. IMMERSE is a relatively new program, started in 2004 as part of the department’s Mentoring through Critical Transition Points (MCTP) grant from NSF. The MCTP grant …

Characterizations Of Pseudo-Codewords Of Ldpc Codes, Ralf Koetter, Wen-Cheng W. Li, Pascal O. Vontobel, Judy L. Walker

Department of Mathematics: Faculty Publications

An important property of high-performance, low complexity codes is the existence of highly efficient algorithms for their decoding. Many of the most efficient, recent graph-based algorithms, e.g. message passing algorithms and decoding based on linear programming, crucially depend on the efficient representation of a code in a graphical model. In order to understand the performance of these algorithms, we argue for the characterization of codes in terms of a so called fundamental cone in Euclidean space which is a function of a given parity check matrix of a code, rather than of the code itself. We give a number of …

S-Extremal Additive F4 Codes, Evangeline P. Bautista, Philippe Gaborit, Jon-Lark Kim, Judy L. Walker

Department of Mathematics: Faculty Publications

Binary self-dual codes and additive self-dual codes over F₄ have in common interesting properties, for example, Type I, Type II, shadows, etc. Recently Bachoc and Gaborit introduced the notion of s-extremality for binary self-dual codes, generalizing Elkies' study on the highest possible minimum weight of the shadows of binary self-dual codes. In this paper, we introduce a concept of s-extremality for additive self-dual codes over F₄, give a bound on the length of these codes with even distance d, classify them up to minimum distance d = 4, give possible lengths and (shadow) weight …

Norming Algebras And Automatic Complete Boundedness Of Isomorphisms Of Operator Algebras, David R. Pitts

Department of Mathematics: Faculty Publications

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if A1 and A2 are operator algebras, then any bounded epimorphism of A1 onto A2 is completely bounded provided that A2 contains a norming C*-subalgebra. We use this result to give some insights into Kadison’s Similarity Problem: we show that every faithful bounded homomorphism of a C*-algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a C-algebra is similar to a -representation precisely when the image operator algebra -norms itself. We give …

Robustness With Respect To Sampling For Stabilization Of Riesz Spectral Systems, Richard Rebarber, Stuart Townley

Department of Mathematics: Faculty Publications

We suppose that a continuous-time feedback is input–output stabilizing for an infinite-dimensional system. We address the question of whether the sampled-data controller obtained by applying idealized sample-and-hold to this continuous-time feedback is also input–output stabilizing if the sampling time is small enough. This question has been previously addressed for fairly general systems under various conditions. In this note, we restrict our attention to Riesz spectral systems, for which we generalize the existing results. Specifically, we give two relatively simple conditions which, combined, are sufficient for the sampled-data controller to be stabilizing. The first condition is a spectrum decomposition for the …

Asymptotic Stability Of A Fluid-Structure Semigroup, George Avalos

Department of Mathematics: Faculty Publications

The strong stability problem for a fluid-structure interactive partial differential equation (PDE) is considered. The PDE comprises a coupling of the linearized Stokes equations to the classical system of elasticity, with the coupling occurring on the boundary interface between the fluid and solid media. It is now known that this PDE may be modeled by a $C_{0}$-semigroup of contractions on an appropriate Hilbert space. However, because of the nature of the unbounded coupling between fluid and structure, the resolvent of the semigroup generator will \emph{not} be a compact operator. In consequence, the classical solution to the stability problem, by means …

Coordinate Systems And Bounded Isomorphisms, David R. Pitts, David R. Pitts

Department of Mathematics: Faculty Publications

For a Banach D-bimoduleMover an abelian unital C*-algebraD, we define E1(M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual M#. Equip E1(M) with the weak*-topology. We develop general properties of E1(M). It is properly viewed as a coordinate system for M when M C, where C is a unital C*-algebra containing D as a regular MASA with the extension property; moreover, E1(C) coincides with Kumjian’s twist in the context of C*-diagonals. We identify the C*-envelope of a subalgebra A of a C*-diagonal when D A C. For triangular subalgebras, each containing …

Reversals And Transpositions Over Finite Alphabets, A. J. Radcliffe, A. D. Scott, E. L. Wilmer

Department of Mathematics: Faculty Publications

Extending results of Christie and Irving, we examine the action of reversals and transpositions on finite strings over an alphabet of size k. We show that determining reversal, transposition, or signed reversal distance between two strings over a finite alphabet is NP-hard, while for “dense” instances we give a polynomial-time approximation scheme. We also give a number of extremal results, as well as investigating the distance between random strings and the problem of sorting a string over a finite alphabet.

Matrix Model Superpotentials And Calabi–Yau Spaces: An A-D-E Classification, Carina Curto

Department of Mathematics: Faculty Publications

We use F. Ferrari’s methods relating matrix models to Calabi-Yau spaces in order to explain Intriligator and Wecht’s ADE classification of N = 1 superconformal theories which arise as RG fixed points of N = 1 SQCD theories with adjoints. The connection between matrix models and N = 1 gauge theories can be seen as evidence for the Dijkgraaf–Vafa conjecture. We find that ADE superpotentials in the Intriligator–Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yau’s with corresponding ADE singularities. Moreover, in the additional Ô, Â, Dˆ and Ê cases we find new singular geometries. These ‘hat’ geometries are …

Pseudo-Codewords Of Cycle Codes Via Zeta Functions, Ralf Koetter, Wen-Cheng W. Li, Pascal O. Vontobel, Judy L. Walker

Department of Mathematics: Faculty Publications

Cycle codes are a special case of low- density parity-check (LDPC) codes and as such can be decoded using an iterative message-passing decod- ing algorithm on the associated Tanner graph. The existence of pseudo-codewords is known to cause the decoding algorithm to fail in certain instances. In this paper, we draw a connection between pseudo- codewords of cycle codes and the so-called edge zeta function of the associated normal graph and show how the Newton polyhedron of the zeta function equals the fundamental cone of the code, which plays a crucial role in characterizing the performance of iterative de- coding …

Weak Solutions To The Cauchy Problem Of A Semilinear Wave Equation With Damping And Source Terms, Petronela Radu

Department of Mathematics: Faculty Publications

In this paper we prove local existence of weak solutions for a semilinear wave equation with power-like source and dissipative terms on the entire space ℝⁿ. The main theorem gives an alternative proof of the local in time existence result due to J. Serrin, G. Todorova and E. Vitillaro, and also some extension to their work. In particular, our method shows that sources that are not locally Lipschitz in L² can be controlled without any damping at all. If the semilinearity involving the displacement has a “good” sign, we obtain global existence of solutions.

Absolutely Continuous Representations And A Kaplansky Density Theorem For Free Semigroup Algebras, Kenneth R. Davidson, Jiankui Li, David R. Pitts

Department of Mathematics: Faculty Publications

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra An. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a ∗-extendible represen- tation . A ∗-extendible representation of An is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous func- tionals are intimately related to maps which intertwine a given ∗-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is …