arXiv subjects

Joel Friedman

Publications and source records attributed to Joel Friedman.

At least 19 recordsLinked to original sources

arXiv2026

Duality and a Canonical Sheaf in Periodic Riemann Functions

Let $f\colon{\mathbb Z}^2\to{\mathbb Z}$ be a Riemann function whose weight $W$ is a perfect matching. Then there is a family of sheaves of $k$-vector spaces $\{{{M}}_{W,{\bf d}}\}_{{\bf d}\in{\mathbb Z}^2}$ on a five-point topological that models $f$ in that $f({\bf d})=b^0({{M}}_{W,{\bf d}})$ and that $$ b^1({{M}}_{W,{\bf d}})= f^\wedge_{\bf K}({\bf d}-{\bf K}) $$ for any ${\bf K}\in{\mathbb Z}^2$. Hence a Riemann-Roch formula for $f$ is equivalent to an Euler characteristic computation of ${{M}}_{W,{\bf d}}$. If $f$ and $W$ are $r$-periodic, then the sheaves ${{M}}_{W,{\bf d}}$ become ${{O}}_r$-modules of finite type for a natural sheaf of rings ${{O}}={{O}}_r$. We show that in this case there is a ``canonical ${{O}}$-module'' $\omega=\omega_W$ and a pairing for $i=0,1$, $$ H^i(M_{W,{\bf 0}}\otimes F) \times {\rm Ext}^{1-i}(F,M_{W^\wedge_{\bf L},{\bf K}})\to H^1(\omega)\cong k $$ that is perfect when ${\bf L}={\bf K}+{\bf 1}$ and ${{F}}$ is a certain type of line bundle or a certain type of skyscraper sheaf. In particular when ${{F}}$ is a line bundle, we realize the above formula for $b^1({{M}}_{W,{\bf d}})$ as a duality theorem akin to Serre duality. We show that canonical ${{O}}$-module $\omega_W$ is a rather exceptional element in a family of tensor products of two modules ${{M}}\otimes_{{O}}{{M}}'$, where ${{M}}$ and ${{M}}'$ vary over ${{O}}_r$-modules of the form ${{M}}_{W',{\bf d}}$. This article doesn't assume any background in sheaf theory; rather we describe all our sheaves as a ``diagrams of vector spaces,'' where each diagram is essentially a sheaf of vector spaces on a fixed topological space of five points.

math.CO↗

arXiv2026

Urschel Nodal Domains via Perturbation Theory

We prove several types of Courant nodal domain theorems for generalized Laplacians on graphs, based on an invariant introduced by Urschel, which we call the "Urschel number", denoted ${\rm UN}({\bf f})$, of an eigenvector ${\bf f}$. We refine Urschel's invariant, and use perturbation techniques to obtain some new results. First, we show the existence of mutually orthogonal eigenvectors, such that if the $k$-th eigenvalue has multiplicity $m$, then for $0\le j\le m-1$, ${\rm UN}({\bf f}_{k+j})\le k+\min(j,(m-1)-j)$. Second, for a simple $k$-th eigenvalue, we classify the zeroes of ${\bf f}_k$ as either "shallow or "deep"; we obtain a number of results that say, roughly speaking, the more shallow vertices ${\bf f}_k$ has, the more control we have over our new invariants based on Urschel's. Our new invariants of an eigenvector, ${\bf f}_k$, are a sequence of integers whose minimum value is ${\rm UN}({\bf f}_k)$ and whose maximum, denoted ${\rm UN}_{\max{}}({\bf f}_k)$, is the maximum number of nodal domains of any possible positive/negative signing or "charge" of the zeroes of ${\bf f}_k$. An example of our second type of result is that if ${\bf f}_k$ has no deep vertices, then ${\rm UN}_{\max{}}({\bf f}_k)\le k$. We provide a number of examples to illustrate our main results, and how they differ from the situation in analysis. We also describe a minor improvement of the Gladwell-Zhu theorem for an orthonormal eigenbasis in the presence of eigenvalues of sufficient multiplicity.

math.CO↗

arXiv2023

Coordination and Discoordination in Linear Algebra, Linear Information Theory, and Coded Caching

In the first part of this paper we develop some theorems in linear algebra applicable to information theory when all random variables involved are linear functions of the individual bits of a source of independent bits. We say that a collection of subspaces of a vector space are "coordinated" if the vector space has a basis such that each subspace is spanned by its intersection with the basis. We measure the failure of a collection of subspaces to be coordinated by an invariant that we call the "discoordination" of the family. We develop some foundational results regarding discoordination. In particular, these results give a number of new formulas involving three subspaces of a vector space. We then apply a number of our results, along with a method of Tian to obtain some new lower bounds in a special case of the basic coded caching problem. In terms of the usual notation for these problems, we show that for $N=3$ documents and $K=3$ caches, we have $6M+5R\ge 11$ for a scheme that achieves the memory-rate pair $(M,R)$, assuming the scheme is linear. We also give a new caching scheme for $N=K=3$ that achieves the pair $(M,R) = (1/2,5/3)$.

cs.IT↗

arXiv2022

Surface state evolution induced by magnetic order in axion insulator candidate EuIn2As2

Gapping of Dirac surface states through time reversal symmetry breaking may realize the axion insulator state in condensed matter. Despite tremendous efforts, only a few material systems fall in this category of intrinsic magnetic topological insulators (TI). Recent theoretical calculations proposed the antiferromagnetic EuIn$_2$As$_2$ to be a topologically non-trivial magnetic insulator with gapped surface states. Here we use scanning tunneling microscopy and spectroscopy (STM/STS) complemented with density-functional theory (DFT) calculations and modelling to probe the surface electronic states in EuIn$_2$As$_2$. We find a spin-orbit induced bulk gap of ~120 meV located only a few meV above the Fermi energy, within which topological surface states reside. Temperature dependent measurements provide evidence of the partial gapping (~40 meV) of the surface states at low temperatures below the AFM order, which decreases with increasing temperature but remains finite above $T_N$

cond-mat.str-el↗

arXiv2022

Sheaves and Duality in the Two-Vertex Graph Riemann-Roch Theorem

For each graph on two vertices, and each divisor on the graph in the sense of Baker-Norine, we describe a sheaf of vector spaces on a finite category whose zeroth Betti number is the Baker-Norine "Graph Riemann-Roch" rank of the divisor plus one. We prove duality theorems that generalize the Baker-Norine "Graph Riemann-Roch" Theorem.

math.CO↗

arXiv2022

Euler Characteristics and Duality in Riemann Functions and the Graph Riemann-Roch Rank

By a {\em Riemann function} we mean a function $f\colon{\mathbb Z}^n\to{\mathbb Z}$ such that $f({\bf d})=f(d_1,\ldots,d_n)$ is equals $0$ for ${\rm deg}({\bf d})=d_1+\cdots+d_n$ sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$ -- the {\em offset of $f$} -- for ${\rm deg}({\bf d})$ sufficiently large. By adding $1$ to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions. For such an $f$, for any ${\bf K}\in{\mathbb Z}^n$ there is a unique Riemann function $f^\wedge_{\bf K}$ such that for all ${\bf d}\in{\mathbb Z}^n$ we have $$ f({\bf d}) - f^\wedge_{\bf K}({\bf K}-{\bf d}) = {\rm deg}({\bf d})+C $$ which we call a {\em generalized Riemann-Roch formula}. We show that any such equation can be viewed as an Euler charactersitic equation of sheaves of a particular simple type that we call {\em diagrams}. This article does not assume any prior knowledge of sheaf theory. To certain Riemann functions $f\colon{\mathbb Z}^2\to{\mathbb Z}$ there is a simple family of diagrams $\{\mathcal{M}_{W,{\bf d}}\}_{{\bf d}\in{\mathbb Z}^2}$ such that $f({\bf d})=b^0({\mathcal{M}}_{W,{\bf d}})$ and $f^\wedge_{\bf K}({\bf K}-{\bf d})=b^1({\mathcal{M}}_{W,{\bf d}})$. Furthermore we give a canonical isomorphism $$ H^1({\mathcal{M}}_{W,{\bf d}})^* \to H^0({\mathcal{M}}_{W',{\bf K}-{\bf d}}) $$ where $W'$ is the weight of $f^\wedge_{\bf K}$. General Riemann functions $f\colon{\mathbb Z}^2\to{\mathbb Z}$ are similarly modeled with formal differences of diagrams. Riemann functions ${\mathbb Z}^n\to{\mathbb Z}$ are modeled using their restrictions to two of their variables. These constructions involve some ad hoc choices, although the equivalence class of virtual diagram obtained is independent of the ad hoc choices.

math.CO↗

arXiv2022

Generalized Riemann Functions, Their Weights, and the Complete Graph

By a {\em Riemann function} we mean a function $f\colon{\mathbb Z}^n\to{\mathbb Z}$ such that $f({\bf d})$ is equals $0$ for $d_1+\cdots+d_n$ sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$, for $d_1+\cdots+d_n$ sufficiently large. By adding $1$ to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions. To each Riemann function we associate a related function $W\colon{\mathbb Z}^n\to{\mathbb Z}$ via Möbius inversion that we call the {\em weight} of the Riemann function. We give evidence that the weight seems to organize the structure of a Riemann function in a simpler way: first, a Riemann function $f$ satisfies a Riemann-Roch formula iff its weight satisfies a simpler symmetry condition. Second, we will calculate the weight of the Baker-Norine rank for certain graphs and show that the weight function is quite simple to describe; we do this for graphs on two vertices and for the complete graph. For the complete graph, we build on the work of Cori and Le Borgne who gave a linear time method to compute the Baker-Norine rank of the complete graph. The associated weight function has a simple formula and is extremely sparse (i.e., mostly zero). Our computation of the weight function leads to another linear time algorithm to compute the Baker-Norine rank, via a formula likely related to one of Cori and Le Borgne, but seemingly simpler, namely $$ r_{{\rm BN},K_n}({\bf d}) = -1+\biggl| \biggl\{ i=0,\ldots,{\rm deg}({\bf d}) \ \Bigm| \ \sum_{j=1}^{n-2} \bigl( (d_j-d_{n-1}+i) \bmod n \bigr) \le {\rm deg}({\bf d})-i \biggr\} \biggr|. $$ Our study of weight functions leads to a natural generalization of Riemann functions, with many of the same properties exhibited by Riemann functions.

math.CO↗

arXiv2022

Coexisting Kondo hybridization and itinerant f-electron ferromagnetism in UGe2

Kondo hybridization in partially filled f-electron systems conveys significant amount of electronic states sharply near the Fermi energy leading to various instabilities from superconductivity to exotic electronic orders. UGe2 is a 5f heavy fermion system, where the Kondo hybridization is interrupted by the formation of two ferromagnetic phases below a 2nd order transition Tc ~ 52 K and a crossover transition Tx ~ 32 K. These two ferromagnetic phases are concomitantly related to a spin-triplet superconductivity that only emerges and persists inside the magnetically ordered phase at high pressure. The origin of the two ferromagnetic phases and how they form within a Kondo-lattice remain ambiguous. Using scanning tunneling microscopy and spectroscopy, we probe the spatial electronic states in the UGe2 as a function of temperature. We find a Kondo resonance and sharp 5f-electron states near the chemical potential that form at high temperatures above Tc in accordance with our density functional theory (DFT) + Gutzwiller calculations. As temperature is lowered below Tc, the resonance narrows and eventually splits below Tx dumping itinerant f-electron spectral weight right at the Fermi energy. Our findings suggest a Stoner mechanism forming the highly polarized ferromagnetic phase below Tx that itself sets the stage for the emergence of unconventional superconductivity at high pressure.

cond-mat.str-el↗

arXiv2021

A Note on the Trace Method for Random Regular Graphs

The main goal of this note is to illustrate the advantage of analyzing the non-backtracking spectrum of a regular graph rather than the ordinary spectrum. We show that by switching to non-backtracking spectrum, the method of proof used in [Puder 2015, arXiv::1212.5216] yields a bound of $2\sqrt{d-1}+\frac{2}{\sqrt{d-1}}$ instead of the original $2\sqrt{d-1}+1$ on the second largest eigenvalue of a random $d$-regular graph.

math.CO↗

arXiv2021

Inhomogeneous Kondo-lattice in geometrically frustrated Pr$_{2}$Ir$_{2}$O$_{7}$

Magnetic fluctuations induced by geometric frustration of local Ir-spins disturb the formation of long range magnetic order in the family of pyrochlore iridates, R$_{2}$Ir$_{2}$O$_{7}$ (R = lanthanide)$^{1}$. As a consequence, Pr$_{2}$Ir$_{2}$O$_{7}$ lies at a tuning-free antiferromagnetic-to-paramagnetic quantum critical point and exhibits a diverse array of complex phenomena including Kondo effect, biquadratic band structure, metallic spin-liquid (MSL), and anomalous Hall effect$^{2-5}$. Using spectroscopic imaging with the scanning tunneling microscope, complemented with machine learning K-means clustering analysis, density functional theory, and theoretical modeling, we probe the local electronic states in single crystal of Pr$_{2}$Ir$_{2}$O$_{7}$ and discover an electronic phase separation. Nanoscale regions with a well-defined Kondo resonance are interweaved with a non-magnetic metallic phase with Kondo-destruction. Remarkably, the spatial nanoscale patterns display a correlation-driven fractal geometry with power-law behavior extended over two and a half decades, consistent with being in proximity to a critical point. Our discovery reveals a new nanoscale tuning route, viz. using a spatial variation of the electronic potential as a means of adjusting the balance between Kondo entanglement and geometric frustration.

cond-mat.str-el↗

arXiv2019

On the Relativized Alon Second Eigenvalue Conjecture I: Main Theorems, Examples, and Outline of Proof

This is the first in a series of six articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. Many of the techniques we develop hold whether or not the base graph is regular. Our first main theorem in this series of articles is that if the base graph is $d$-regular, then for any $ε>0$, as the degree, $n$, of the covering map tends to infinity, some new adjacency eigenvalue of the map is larger in absolute value that $2(d-1)^{1/2}+ε$ with probability at most order $1/n$. Our second main theorem is that if, in addition, the base graph is Ramanujan, then this probability is bounded above and below by $1/n$ to the power of a positive integer that we call the {\em tangle power} of the model, i.e., of the probability spaces of random covering maps of degree $n$. The tangle power is fairly easy to bound from below, and at times to compute exactly; it measures the probability that certain {\em tangles} appear in the random covering graph, where a {\em tangle} is a local event that forces the covering graph to have a new eigenvalue strictly larger than $2(d-1)^{1/2}$. Our main theorems are relativizations of Alon's conjecture on the second eigenvalue of random regular graphs of large degree. In this first article of the series, we introduce all the terminology needed in this series, motivate this terminology, precisely state all the results in the remaining articles, and make some remarks about their proofs. As such, this article provides an overview of the entire series of articles; furthermore, the rest of the articles in this series may be read independently of one another.

cs.DM↗

arXiv2019

On the Relativized Alon Eigenvalue Conjecture II: Asymptotic Expansion Theorems for Walks

This is the second in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. The first main result in this article concerns the function $f(k,n)$ defined as the number of SNBC (strictly non-backtracking closed) walks of length $k$ of a given homotopy type in a random covering graph of degree $n$ of a fixed graph. We prove the existence of asymptotic expansions in powers of $1/n$ for $f(k,n)$, where the coefficients---functions of $k$---are proven to have some desirable properties; namely, these coefficients are approximately a sum of polynomials times exponential functions. The second main result is a generalization of the first, where the number of SNBC walks of length $k$ is multiplied by an indicator function that the covering graph contains a certain type of {\em tangle}; the second result requires more terminology, although its proof uses the same basic tools used to prove the first result. % The motivation for the second main result will be clear in % the third article in this series of articles. The results in this article are mostly straightforward generalizations of methods used in previous works. However, this article (1) "factors" these methods into a number of short, conceptually simple, and independent parts, (2) writes each independent part in more general terms, and (3) significantly simplifies of one of the previous computations. As such we expect that this article will make it easier to apply trace methods to related models of random graphs.

cs.DM↗

arXiv2019

On the Relativized Alon Second Eigenvalue Conjecture III: Asymptotic Expansions for Tangle-Free Hashimoto Traces

This is the third in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this paper we consider random graphs that are random covering graphs of large degree $n$ of a fixed base graph. We prove the existence of asympototic expansion in $1/n$ for the expected value of the number of strictly non-backtracking closed walks of length $k$ times the indicator function that the graph is free of certain {\em tangles}; moreover, we prove that the coefficients of these expansions are "nice functions" of $k$, namely approximately equal to a sum of polynomials in $k$ times exponential functions of $k$. Our results use the methods of Friedman used to resolve Alon's original conjecture, combined with the results of Article~II in this series of articles. One simplification in this article over the previous methods of Friedman is that the "regularlized traces" used in this article, which we call {\em certified traces}, are far easier to define and work with than the previously utilized {\em selective traces}.

cs.DM↗

arXiv2019

A Relativized Alon Second Eigenvalue Conjecture for Regular Base Graphs IV: An Improved Sidestepping Theorem

This is the fourth in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this paper we prove a {\em Sidestepping Theorem} that is more general and easier to use than earlier theorems of this kind. Such theorems concerns a family probability spaces $\{\mathcal{M}_n\}$ of $n\times n$ matrices, where $n$ varies over some infinite set, $N$, of natural numbers. Many trace methods use simple "Markov bounds" to bound the expected spectral radius of elements of ${\mathcal{M}}_n$: this consists of choosing one value, $k=k(n)$, for each $n\in N$, and proving expected spectral radius bounds based on the expected value of the trace of the $k=k(n)$-power of elements of ${\mathcal{M}}_n$. {\em Sidestepping} refers to bypassing such simple Markov bounds, obtaining improved results using a number of values of $k$ for each fixed $n\in N$. In more detail, if the $M\in {\mathcal{M}}_n$ expected value of ${\rm Trace}(M^k)$ has an asymptotic expansion in powers of $1/n$, whose coefficients are "well behaved" functions of $k$, then one can get improved bounds on the spectral radius of elements of ${\mathcal{M}}_n$ that hold with high probability. Such asymptotic expansions are shown to exist in the third article in this series for the families of matrices that interest us; in the fifth and sixth article in this series we will apply the Sidestepping Theorem in this article to prove the main results in this series of articles. This article is independent of all other articles in this series; it can be viewed as a theorem purely in probability theory, concerning random matrices or, equivalently, the $n$ random variables that are the eigenvalues of the elements of ${\mathcal{M}}_n$.

math.PR↗

arXiv2019

On the Relativized Alon Second Eigenvalue Conjecture VI: Sharp Bounds for Ramanujan Base Graphs

This is the sixth in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this article we show that if the fixed graph is regular Ramanujan, then the {\em algebraic power} of the model of random covering graphs is $+\infty$. This implies a number of interesting results, such as (1) one obtains the upper and lower bounds---matching to within a multiplicative constant---for the probability that a random covering map has some new adjacency eigenvalue outside the Alon bound, and (2) with probability smaller than any negative power of the degree of the covering map, some new eigenvalue fails to be within the Alon bound without the covering map containing one of finitely many "tangles" as a subgraph (and this tangle containment event has low probability).

cs.DM↗

arXiv2019

On the Relativized Alon Second Eigenvalue Conjecture V: Proof of the Relativized Alon Conjecture for Regular Base Graphs

This is the fifth in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this article we use the results of Articles~III and IV in this series to prove that if the base graph is regular, then as the degree, $n$, of the covering map tends to infinity, some new adjacency eigenvalue has absolute value outside the Alon bound with probability bounded by $O(1/n)$. In addition, we give upper and lower bounds on this probability that are tight to within a multiplicative constant times the degree of the covering map. These bounds depend on two positive integers, the \emph{algebraic power} (which can also be $+\infty$) and the \emph{tangle power} of the model of random covering map. We conjecture that the algebraic power of the models we study is always $+\infty$, and in Article~VI we prove this when the base graph is regular and \emph{Ramanujan}. When the algebraic power of the model is $+\infty$, then the results in this article imply stronger results, such as (1) the upper and lower bounds mentioned above are matching to within a multiplicative constant, and (2) with probability smaller than any negative power of the degree, the some new eigenvalue fails to be within the Alon bound only if the covering map contains one of finitely many "tangles" as a subgraph (and this event has low probability).

cs.DM↗

arXiv2019

Inner Rank and Lower Bounds for Matrix Multiplication

We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$ matrix multiplication over an arbitrary field is at least $2n^2-n+1$. While inner rank does not provide improvements to currently known lower bounds, we argue that this notion merits further study.

cs.CC↗

arXiv2018

Visualizing Uniaxial-strain Manipulation of Antiferromagnetic Domains in Fe1+YTe Using a Spin-polarized Scanning Tunneling Microscope

The quest to understand correlated electronic systems has pushed the frontiers of experimental measurements toward the development of new experimental techniques and methodologies. Here we use a novel home-built uniaxial-strain device integrated into our variable temperature scanning tunneling microscope that enables us to controllably manipulate in-plane uniaxial strain in samples and probe their electronic response at the atomic scale. Using scanning tunneling microscopy with spin-polarization techniques, we visualize antiferromagnetic domains and their atomic structure in Fe1+yTe samples, the parent compound of iron-based superconductors, and demonstrate how these domains respond to applied uniaxial strain. We observe the bidirectional antiferromagnetic domains in the unstrained sample, with an average domain size of 50 to 150 nm, to transition into a single unidirectional domain under applied uniaxial strain. The findings presented here open a new direction to utilize a valuable tuning parameter in scanning tunneling microscopy, as well as other spectroscopic techniques, both for tuning the electronic properties as for inducing symmetry breaking in quantum material systems.

cond-mat.str-el↗