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- Absorbing sets (1)
- Bernstein–Sato polynomial (1)
- Center manifold (1)
- Conjugacy growth (1)
- Contraction mapping theorem (1)
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- D-module (1)
- Decoding (1)
- Decomposition kinetics (1)
- Equilibrium points (1)
- Fractional and nonlocal gradients (1)
- Graph product (1)
- Growth-efficiency trade-off (1)
- Homogenization (1)
- Invariant foliation (1)
- Invariant manifold (1)
- Iterative decoding (1)
- Localization (1)
- Low-density parity-check code (1)
- Mathematical oncology (1)
- Matrix analysis (1)
- Methods in prime characteristic (1)
- Microbial adaptation (1)
- Microbial model (1)
- Nash blowup (1)
- Nonlocal function spaces (1)
- Nonlocal variational problems (1)
- Numerical calculation (1)
- Optimization (1)
- Prime characteristic (1)
- Quasiconvexity (1)
Articles 1 - 16 of 16
Full-Text Articles in Physical Sciences and Mathematics
Differentiating By Prime Numbers, Jack Jeffries
Differentiating By Prime Numbers, Jack Jeffries
Department of Mathematics: Faculty Publications
It is likely a fair assumption that you, the reader, are not only familiar with but even quite adept at differentiating by x. What about differentiating by 13? That certainly didn’t come up in my calculus class! From a calculus perspective, this is ridiculous: are we supposed to take a limit as 13 changes? One notion of differentiating by 13, or any other prime number, is the notion of p-derivation discovered independently by Joyal [Joy85] and Buium [Bui96]. p-derivations have been put to use in a range of applications in algebra, number theory, and arithmetic geometry. Despite the wide range …
A Variational Theory For Integral Functionals Involving Finite-Horizon Fractional Gradients, Javier Cueto, Carolin Carolin, Hidde Schönberger
A Variational Theory For Integral Functionals Involving Finite-Horizon Fractional Gradients, Javier Cueto, Carolin Carolin, Hidde Schönberger
Department of Mathematics: Faculty Publications
The center of interest in this work are variational problems with integral functionals depending on nonlocal gradients with finite horizon that correspond to truncated versions of the Riesz fractional gradient. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural …
Idempotent Completions Of Equivariant Matrix Factorization Categories, Michael K. Brown, Mark E. Walker
Idempotent Completions Of Equivariant Matrix Factorization Categories, Michael K. Brown, Mark E. Walker
Department of Mathematics: Faculty Publications
We prove that equivariant matrix factorization categories associated to henselian local hypersurface rings are idempotent complete, generalizing a result of Dyckerhoff in the non- equivariant case.
Analysis Of Syndrome-Based Iterative Decoder Failure Of Qldpc Codes, Kirsten D. Morris, Tefjol Pllaha, Christine A. Kelley
Analysis Of Syndrome-Based Iterative Decoder Failure Of Qldpc Codes, Kirsten D. Morris, Tefjol Pllaha, Christine A. Kelley
Department of Mathematics: Faculty Publications
Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code’s graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. In this paper we examine the failure inducing sets of QLDPC codes under syndrome-based iterative decoding, and their connection to absorbing sets in classical LDPC codes.
Computation Of The Basic Reproduction Numbers For Reaction-Diffusion Epidemic Models, Chayu Yang, Jin Wang
Computation Of The Basic Reproduction Numbers For Reaction-Diffusion Epidemic Models, Chayu Yang, Jin Wang
Department of Mathematics: Faculty Publications
We consider a class of k-dimensional reaction-diusion epidemic models (k = 1; 2; • • • ) that are developed from autonomous ODE systems. We present a computational approach for the calculation and analysis of their basic reproduction numbers. Particularly, we apply matrix theory to study the relationship between the basic reproduction numbers of the PDE models and those of their underlying ODE models. We show that the basic reproduction numbers are the same for these PDE models and their associated ODE models in several important scenarios. We additionally provide two numerical examples to verify our analytical results.
Pull-Push Method: A New Approach To Edge-Isoperimetric Problems, Sergei L. Bezrukov, Nikola Kuzmanovski, Jounglag Lim
Pull-Push Method: A New Approach To Edge-Isoperimetric Problems, Sergei L. Bezrukov, Nikola Kuzmanovski, Jounglag Lim
Department of Mathematics: Faculty Publications
We prove a generalization of the Ahlswede-Cai local-global principle. A new technique to handle edge-isoperimetric problems is introduced which we call the pull-push method. Our main result includes all previously published results in this area as special cases with the only exception of the edge-isoperimetric problem for grids. With this we partially answer a question of Harper on local-global principles. We also describe a strategy for further generalization of our results so that the case of grids would be covered, which would completely settle Harper’s question.
When Are The Natural Embeddings Of Classical Invariant Rings Pure?, Melvin Hochster, Jack Jeffries, Vaibhav Pandey, Anurag K. Singh
When Are The Natural Embeddings Of Classical Invariant Rings Pure?, Melvin Hochster, Jack Jeffries, Vaibhav Pandey, Anurag K. Singh
Department of Mathematics: Faculty Publications
Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as inWeyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; …
Listening For Common Ground In High School And Early Collegiate Mathematics, Gail Burrill, Henry Cohn, Yvonne Lai, Dev P. Sinha, Ji Y. Son, Katherine F. Stevenson
Listening For Common Ground In High School And Early Collegiate Mathematics, Gail Burrill, Henry Cohn, Yvonne Lai, Dev P. Sinha, Ji Y. Son, Katherine F. Stevenson
Department of Mathematics: Faculty Publications
Solutions to pressing and complex social challenges require that we reach for common ground. Only through cooperation among people with a broad range of backgrounds and expertise can progress be made on issues as challenging as improving student success in mathematics. In this spirit, the AMS Committee on Education held a forum in May 2022 entitled The Evolving Curriculum in High School and Early Undergraduate Mathematical Sciences Education.1 This article is a report on that forum by the authors listed above, who were among the organizers and presenters.
Theory Of Invariant Manifold And Foliation And Uniqueness Of Center Manifold Dynamics, Bo Deng
Theory Of Invariant Manifold And Foliation And Uniqueness Of Center Manifold Dynamics, Bo Deng
Department of Mathematics: Faculty Publications
Here we prove that the dynamics on any two center-manifolds of a fixed point of any Ck,1 dynamical system of finite dimension with k ≥ 1 are Ck-conjugate to each other. For pedagogical purpose, we also extend Perron’s method for differential equations to diffeomorphisms to construct the theory of invariant manifolds and invariant foliations at fixed points of dynamical systems of finite dimensions.
Nash Blowups Of Toric Varieties In Prime Characteristic, Daniel Duarte, Jack Jeffries, Luis Núñez-Betancourt
Nash Blowups Of Toric Varieties In Prime Characteristic, Daniel Duarte, Jack Jeffries, Luis Núñez-Betancourt
Department of Mathematics: Faculty Publications
We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially.
Extremal Absorbing Sets In Low-Density Parity-Check Codes, Emily Mcmillon, Allison Beemer, Christine A. Kelley
Extremal Absorbing Sets In Low-Density Parity-Check Codes, Emily Mcmillon, Allison Beemer, Christine A. Kelley
Department of Mathematics: Faculty Publications
Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of b for a given a for which an (a, b)-absorbing set may exist. We identify certain cases of extremal …
Formal Conjugacy Growth In Graph Products I, Laura Ciobanu,, Susan Hermiller, Valentin Mercier
Formal Conjugacy Growth In Graph Products I, Laura Ciobanu,, Susan Hermiller, Valentin Mercier
Department of Mathematics: Faculty Publications
In this paper we give a recursive formula for the conjugacy growth series of a graph product in terms of the conjugacy growth and standard growth series of subgraph products. We also show that the conjugacy and standard growth rates in a graph product are equal provided that this property holds for each vertex group. All results are obtained for the standard generating set consisting of the union of generating sets of the vertex groups.
Bernstein-Sato Theory For Singular Rings In Positive Characteristic, Jack Jack, Luis Núñez-Betancourt, Eamon Quinlan-Gallego
Bernstein-Sato Theory For Singular Rings In Positive Characteristic, Jack Jack, Luis Núñez-Betancourt, Eamon Quinlan-Gallego
Department of Mathematics: Faculty Publications
The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic.
In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical …
Tri-Plane Diagrams For Simple Surfaces In S4, Manuel Aragón, Zack Dooley, Alexander Goldman, Yucong Lei, Isaiah Martinez, Nicholas Meyer, Devon Peters, Scott Warrander, Ana Wright, Alex Zupan
Tri-Plane Diagrams For Simple Surfaces In S4, Manuel Aragón, Zack Dooley, Alexander Goldman, Yucong Lei, Isaiah Martinez, Nicholas Meyer, Devon Peters, Scott Warrander, Ana Wright, Alex Zupan
Department of Mathematics: Faculty Publications
Meier and Zupan proved that an orientable surface K in S4 admits a tri-plane diagram with zero crossings if and only if K is unknotted, so that the crossing number of K is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in S4, proving that c(Pn,m) = max{1, |n−m|}, where Pn,m denotes the connected sum of n unknotted projective planes with normal Euler number +2 and m unknotted projective planes with normal Euler number −2. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane …
Decomposition Rate As An Emergent Property Of Optimal Microbial Foraging, Stefano Manzoni, Arjun Chakrawal, Glenn Ledder
Decomposition Rate As An Emergent Property Of Optimal Microbial Foraging, Stefano Manzoni, Arjun Chakrawal, Glenn Ledder
Department of Mathematics: Faculty Publications
Decomposition kinetics are fundamental for quantifying carbon and nutrient cycling in terrestrial and aquatic ecosystems. Several theories have been proposed to construct process-based kinetics laws, but most of these theories do not consider that microbial decomposers can adapt to environmental conditions, thereby modulating decomposition. Starting from the assumption that a homogeneous microbial community maximizes its growth rate over the period of decomposition, we formalize decomposition as an optimal control problem where the decomposition rate is a control variable. When maintenance respiration is negligible, we find that the optimal decomposition kinetics scale as the square root of the substrate concentration, resulting …
Modeling And Analyzing Homogeneous Tumor Growth Under Virotherapy, Chayu Yang, Jin Wang
Modeling And Analyzing Homogeneous Tumor Growth Under Virotherapy, Chayu Yang, Jin Wang
Department of Mathematics: Faculty Publications
We present a mathematical model based on ordinary differential equations to investigate the spatially homogeneous state of tumor growth under virotherapy. The model emphasizes the interaction among the tumor cells, the oncolytic viruses, and the host immune system that generates both innate and adaptive immune responses. We conduct a rigorous equilibrium analysis and derive threshold conditions that determine the growth or decay of the tumor under various scenarios. Numerical simulation results verify our analytical predictions and provide additional insight into the tumor growth dynamics.