arXiv Math @arxiv_math@qoto.org

This paper summarizes the author's insights and experiences of teaching mathematics to incarcerated students within the U.S. prison system.

We provide an alternative unified approach for proving the Pythagorean theorem (in dimension $2$ and higher), the law of sines and the law of cosines, based on the concept of shape derivative. The idea behind the proofs is very simple: we translate a triangle along a specific direction and compute the resulting change in area. Equating the change in area to zero yields the statements of the three aforementioned theorems.

This note points out a gap in the proof of one of the technical results in the paper "Asymptotic Invariants of Base Loci", that appeared in Ann. Inst. Fourier (Grenoble) 56 (2006), 1701-1734. We provide a correct proof of this result.

We correct an error in [I. Kangasniemi, and J. Onninen, On the heterogeneous distortion inequality. Math. Ann. 384 (2022), no. 3-4, 1275-1308.]

We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations which sometimes occur at the cores of these objects. These are topologically protected by an 't Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects, and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the "symmetry breaking long exact sequence" (SBLES). The mathematical backbone of the SBLES is the Smith homomorphism, a family of maps between twisted bordism groups. Though many examples have been studied, we give the first completely general account of the Smith homomorphism. We lift it to a map of Thom spectra and identify the cofiber, producing a long exact sequence of twisted bordism groups; the SBLES is the Anderson dual of that long exact sequence. Our work develops further the theory of higher Berry phase and its bulk-boundary correspondence and serves as a new computational tool for classifying symmetry protected topological phases.

Diffusion Models (DMs) have recently set state-of-the-art on many generation benchmarks. However, there are myriad ways to describe them mathematically, which makes it difficult to develop a simple understanding of how they work. In this survey, we provide a concise overview of DMs from the perspective of dynamical systems and Ordinary Differential Equations (ODEs) which exposes a mathematical connection to the highly related yet often overlooked class of energy-based models, called Associative Memories (AMs). Energy-based AMs are a theoretical framework that behave much like denoising DMs, but they enable us to directly compute a Lyapunov energy function on which we can perform gradient descent to denoise data. We then summarize the 40 year history of energy-based AMs, beginning with the original Hopfield Network, and discuss new research directions for AMs and DMs that are revealed by characterizing the extent of their similarities and differences

By incorporating two gauge connections, transgression forms provide a generalization of Chern-Simons actions that are genuinely gauge-invariant. In this work, we show that, when defined on a manifold with a boundary, the Hamiltonian formulation of a transgression field theory can be consistently carried out without the need to implement regularizing boundary terms at the level of first-class constraints. By considering boundary variations of the relevant functionals in the Poisson brackets, the surface integral in the very definition of a transgression action can be translated into boundary contributions in the generators of gauge transformations and diffeomorphisms. This prescription systematically leads to the corresponding surface charges of the theory, reducing to the general expression for conserved charges in (higher-dimensional) Chern-Simons theories when one of the gauge connections in the transgression form is set to zero.

This paper refers to an imaging problem in the presence of nonlinear materials. Specifically, the problem we address falls within the framework of Electrical Resistance Tomography and involves two different materials, one or both of which are nonlinear. Tomography with nonlinear materials in the early stages of developments, although breakthroughs are expected in the not-too-distant future. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted p_0-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the $p_0$-Laplacian in inverse problems with nonlinear materials. Moreover, when $p_0=2$, this result allows all the imaging methods and algorithms developed for linear materials to be brought into the arena of problems with nonlinear materials. The main result of this work is that for "small" Dirichlet data, (i) one material can be replaced by a perfect electric conductor and (ii) the other material can be replaced by a material giving rise to a weighted $p_0$-Laplace problem.

In this paper, we establish lower bounds on Weil height of algebraic integers in terms of the low lying zeros of the Dedekind zeta-function. As a result, we prove Lehmer's conjecture for certain infinite non-Galois extensions conditional on GRH. We also introduce and study a condition on prime ideals with small norms for arbitrary infinite extensions, in the spirit of a prime splitting condition for infinite Galois extensions introduced by E. Bombieri and U. Zannier.

We generalize Artin-Ihara L-functions for graphs to hypergraphs by exploring several analogous notions, such as (unramified) Galois coverings and Frobenius elements. To a hypergraph $H$, one can naturally associate a bipartite graph $B_H$ encoding incidence relations of $H$. We study Artin-Ihara $L$-functions of hypergraphs $H$ by using Artin-Ihara $L$-functions of associated bipartite graphs $B_H$. As a result, we prove various properties for Artin-Ihara L-functions for hypergraphs. For instance, we prove that the Ihara zeta function of a hypergraph $H$ can be written as a product of Artin-Ihara $L$-functions.

We prove the Ramanujan and Sato-Tate conjectures for Bianchi modular forms of weight at least 2. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ of parallel weight, where $F$ is any CM field. We deduce these theorems from a new potential automorphy theorem for the symmetric powers of 2-dimensional compatible systems of Galois representations of parallel weight.

In this perspective, we introduce recent research into the structure and function of complex investor networks supporting sustainability efforts. Using the case of solar, wind and hydro energy technologies, this perspective explores the complexity in low-carbon finance markets, defined as markets that direct capital flows towards low-carbon technologies, using network approaches to study their structure and dynamics. Investors are modeled as nodes which form a network or higher-order network connected by edges representing projects in which joint funding or security-related insurance was provided or other investment-related interaction occurred. We review the literature on investor networks generally, particularly in the case of complex networks, and address areas where these ideas were applied in this emerging field. The complex investor dynamics which emerge from the extant funding scenarios are not well understood. These dynamics have the potential to result in interesting non-linear behaviour, growth, and decline, which can be studied, explained and controlled using the tools of network science.

Both Feynman integrals and holographic Witten diagrams can be represented as multivariable hypergeometric functions of a class studied by Gel'fand, Kapranov & Zelevinsky known as GKZ or $\mathcal{A}$-hypergeometric functions. Among other advantages, this formalism enables the systematic construction of highly non-trivial weight-shifting operators known as 'creation' operators. We derive these operators from a physics perspective, highlighting their close relation to the spectral singularities of the integral as encoded by the facets of the Newton polytope. Many examples for Feynman and Witten diagrams are given, including novel weight-shifting operators for holographic contact diagrams. These in turn allow momentum-space exchange diagrams of different operator dimensions to be related while keeping the spacetime dimension fixed. In contrast to previous constructions, only non-derivative vertices are involved.

The quantum rate-distortion function plays a fundamental role in quantum information theory, however there is currently no practical algorithm which can efficiently compute this function to high accuracy for moderate channel dimensions. In this paper, we show how symmetry reduction can significantly simplify common instances of the entanglement-assisted quantum rate-distortion problems, allowing for more efficient computation regardless of the numerical algorithm being used. For some of these problem instances, symmetry reduction allows us to derive closed-form expressions for the quantum rate-distortion function. Additionally, we propose an inexact variant of the mirror descent algorithm to compute the quantum rate-distortion function with provable sublinear convergence rates. We show how this mirror descent algorithm is related to Blahut-Arimoto and expectation-maximization methods previously used to solve similar problems in information theory. Using these techniques, we present the first numerical experiments to compute a multi-qubit quantum rate-distortion function, and show that our proposed algorithm solves faster and to higher accuracy when compared to existing methods.

The goal of this manuscript to establish the best possible estimate on coefficient functionals like Hermitian-Toeplitz determinant of secoend order involving logarithmic coefficients, initial logarithmic inverse coefficients and initial order Schwarzian derivatives of the Ozaki close-to-convex functions.

The development of solutions to the kinetics of homomolecular self-assembly into amyloid fibrils using fixed-point methods, and their subsequent application to the analysis of in vitro kinetic experiments, has led to numerous advances in our understanding of the fundamental chemical mechanisms behind amyloidogenic disorders such as Alzheimer's and Parkinson's diseases. However, as our understanding becomes more detailed and new data become available, kinetic models need to increase in complexity. The resulting rate equations are no longer amenable to extant solution methods, hindering ongoing efforts to elucidate the mechanistic determinants of aggregation in living systems. Here, we demonstrate that most linear self-assembly reactions are described by the same unusual class of singularly perturbed rate equations, that cannot be solved by normal singular perturbation techniques such as renormalization group. We instead develop a new method based on Lie symmetry that can reliably solve this class of equations, and use it in conjunction with experimental data to determine the kinetics of co-aggregation of the Alzheimer's disease-associated Abeta42, Abeta40 and Abeta38 peptides. Our method also rationalizes several successful earlier solutions for homomolecular self-assembly kinetics whose mathematical justification was previously unclear. Alongside its generality and mathematical clarity, its much greater accuracy and simplicity compared to extant methods will enable its rapid and widespread adoption by researchers modelling filamentous self-assembly kinetics.

In this note we study $T^2$-invariant pluriclosed metrics on the Kodaira-Thurston surface. We obtain a characterization of $T^2$-invariant Vaisman metrics, and notice that the Kodaira-Thurston surface admits Vaisman metrics with non-constant scalar curvature. Then we study the behaviour of the Vaisman condition in relation to the pluriclosed flow. As a consequence, we show that if the initial metric on the Kodaira-Thurston surface is a $T^2$-invariant Vaisman metric, then the pluriclosed flow preserves the Vaisman condition, extending to the non-constant scalar curvature case the previous results in [6].