Related papers: Neural ring homomorphism preserves mandatory sets …
Constraint Satisfaction Problem (CSP) is a fundamental algorithmic problem that appears in many areas of Computer Science. It can be equivalently stated as computing a homomorphism $\mbox{$\bR \rightarrow \bGamma$}$ between two relational…
The semiring-based constraint satisfaction problems (semiring CSPs), proposed by Bistarelli, Montanari and Rossi \cite{BMR97}, is a very general framework of soft constraints. In this paper we propose an abstraction scheme for soft…
Previous work on convexity of neural codes has produced codes that are open-convex but not closed-convex -- or vice-versa. However, why a code is one but not the other, and how to detect such discrepancies are open questions. We tackle…
A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of ${\rm Aut}(G)$. Regular…
We study the open, closed, and non-degenerate embedding dimensions of neural codes, which are the smallest respective dimensions in which one can find a realization of a code consisting of convex sets that are open, closed, or…
Given a bounded complex of finitely generated modules $M$ over a commutative noetherian local ring $R$, one assigns to it a variety, $\mathcal V_R(M)$, called the cohomological support variety of $M$ over $R$. The variety $\mathcal V_R(M)$…
Convex codes were recently introduced as models for neural codes in the brain. Any convex code $\C$ has an associated minimal embedding dimension $d(\C)$, which is the minimal Euclidean space dimension such that the code can be realized by…
Energy-minimizing constraint maps are a natural extension of the obstacle problem within a vectorial framework. Due to inherent topological constraints, these maps manifest a diverse structure that includes singularities similar to harmonic…
This paper investigates the algebraic structure of free convolutional codes over the finite local ring Z_{p^r}. We introduce a new structural invariant, the Residual Structural Polynomial, denoted by Delta_p(C) in F_p[D]. We construct this…
In this manuscript, we work over the non-chain ring $\mathcal{R} = \mathbb{F}_2[u]/\langle u^3 - u\rangle $. Let $m\in \mathbb{N}$ and let $L, M, N \subseteq [m]:=\{1, 2, \dots, m\}$. For $X\subseteq [m]$, define $\Delta_X:=\{v \in…
Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer…
A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative…
The authors develop a notion of homological prime spectrum for an arbitrary monoidal triangulated category, ${\mathbf C}$. Unlike the symmetric case due to Balmer, the homological primes of ${\mathbf C}$ are not defined as the maximal Serre…
Determining how the brain stores information is one of the most pressing problems in neuroscience. In many instances, the collection of stimuli for a given neuron can be modeled by a convex set in $\mathbb{R}^d$. Combinatorial objects known…
We give intrinsic characterizations of neural rings and homomorphisms between them. Also we introduce the notion of a basic monomial code map and characterize monomial code maps as compositions of basic monomial code maps. Finally, we…
The brain encodes spacial structure through a combinatorial code of neural activity. Experiments suggest such codes correspond to convex areas of the subject's environment. We present an intrinsic condition that implies a neural code may…
It is a well-known result that, in projective space over a field, every set-theoretical complete intersection of positive dimension in connected in codimension one (Hartshorne [H1,3.4.6] or [H2, Theorem 1.3]). Another important…
In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ${\mathbb P}_k$-free and ${\mathbb P}_k$-subgraph-free graphs. We consider the directed version of this…
A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring $R$. A sufficient condition is given, as well,…
This dissertation explores applications of discrete geometry in mathematical neuroscience. We begin with convex neural codes, which model the activity of hippocampal place cells and other neurons with convex receptive fields. In Chapter 4,…