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Nadler introduced certain Lagrangian singularities indexed by trees, and determined their microlocal sheaves to be the category of modules over the corresponding tree quiver. Another family of spaces indexed by trees: the tree plumbings of spheres. The Fukaya-Seidel category of the Lefschetz fibration with this plumbing as fiber and all spheres as vanishing cycles is well known to also be modules over the tree quiver. Here we upgrade this matching of categories to a matching of geometry.
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Rapid growth of the Internet of Things (IoT) necessitates employing privacy-preserving techniques to protect users' sensitive information. Even when user traces are anonymized, statistical matching can be employed to infer sensitive information. In our previous work, we have established the privacy requirements for the case that the user traces are instantiations of discrete random variables and the adversary knows only the structure of the dependency graph, i.e., whether each pair of users is connected. In this paper, we consider the case where data traces are instantiations of Gaussian random variables and the adversary knows not only the structure of the graph but also the pairwise correlation coefficients. We establish the requirements on anonymization to thwart such statistical matching, which demonstrate the (significant) degree to which knowledge of the pairwise correlation coefficients further significantly aids the adversary in breaking user anonymity.
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We study the inhomogeneous Curie-Weiss model with external field, where the inhomogeneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their weights. In this model, the sum of the spins obeys a central limit theorem outside the critical line. We derive a Berry-Esseen rate of convergence for this limit theorem using Stein's method for exchangeable pairs. For this, we, amongst others, need to generalize this method to a multidimensional setting with unbounded random variables.
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Many Random Number Generators (RNG) are available nowadays; they are divided in two categories, hardware RNG, that provide "true" random numbers, and algorithmic RNG, that generate pseudo random numbers (PRNG). Both types usually generate random numbers $(X_n)$ as independent uniform samples in a range $0,\cdots,2^{b-1}$, with $b = 8, 16, 32$ or $b = 64$. In applications, it is instead sometimes desirable to draw random numbers as independent uniform samples $(Y_n)$ in a range $1, \cdots, M$, where moreover M may change between drawings. Transforming the sequence $(X_n)$ to $(Y_n)$ is sometimes known as scaling. We discuss different methods for scaling the RNG, both in term of mathematical efficiency and of computational speed.
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Optimizing nonlinear systems involving expensive (computer) experiments with regard to conflicting objectives is a common challenge. When the number of experiments is severely restricted and/or when the number of objectives increases, uncovering the whole set of optimal solutions (the Pareto front) is out of reach, even for surrogate-based approaches. As non-compromising Pareto optimal solutions have usually little point in applications, this work restricts the search to relevant solutions that are close to the Pareto front center. The article starts by characterizing this center. Next, a Bayesian multi-objective optimization method for directing the search towards it is proposed. A criterion for detecting convergence to the center is described. If the criterion is triggered, a widened central part of the Pareto front is targeted such that sufficiently accurate convergence to it is forecasted within the remaining budget. Numerical experiments show how the resulting algorithm, C-EHI, be
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Let $M\subset K$ be number fields. We consider the relation of relative power integral bases of $K$ over $M$ with absolute power integral bases of $K$ over $Q$. We show how generators of absolute power integral bases can be calculated from generators of relative ones. We apply our ideas in infinite families of octic fields with quadratic subfields.
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We consider the open quantum random walks on the crystal lattices and investigate the central limit theorems for the walks. On the integer lattices the open quantum random walks satisfy the central limit theorems as was shown by Attal, {\it et al}. In this paper we prove the central limit theorems for the open quantum random walks on the crystal lattices. We then provide with some examples for the Hexagonal lattices. We also develop the Fourier analysis on the crystal lattices. This leads to construct the so called dual processes for the open quantum random walks. It amounts to get Fourier transform of the probability densities, and it is very useful when we compute the characteristic functions of the walks. In this paper we construct the dual processes for the open quantum random walks on the crystal lattices providing with some examples.
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In a recent paper of the author with D. Dummit and H. Kisilevsky, we constructed a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and proved a simple criterion characterizing such matrices. We then analyzed the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$, respectively. In this paper, the goal is to construct and study the finite-field analogues of these residue matrices, the "$d$th-power residue matrices", using the general $d$th-power residue symbol over a finite field.
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Fixing a subgroup $\Gamma$ in a group $G$, the full commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ of $G$ with $[\Gamma: \Gamma \cap \Delta][\Delta : \Gamma \cap \Delta] \leq n$. For pairs $\Gamma \leq G$, where $G$ is a Chevalley group scheme defined over $\mathbb{Z}$ and $\Gamma$ is an arithmetic lattice in $G$, we give precise estimates for the full commensurability growth, relating it to subgroup growth and a computable invariant that depends only on $G$.
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Given a nontrivial homogeneous ideal $I\subseteq k[x_1,x_2,\ldots,x_d]$, a problem of great recent interest has been the comparison of the $r$th ordinary power of $I$ and the $m$th symbolic power $I^{(m)}$. This comparison has been undertaken directly via an exploration of which exponents $m$ and $r$ guarantee the subset containment $I^{(m)}\subseteq I^r$ and asymptotically via a computation of the resurgence $\rho(I)$, a number for which any $m/r > \rho(I)$ guarantees $I^{(m)}\subseteq I^r$. Recently, a third quantity, the symbolic defect, was introduced; as $I^t\subseteq I^{(t)}$, the symbolic defect is the minimal number of generators required to add to $I^t$ in order to get $I^{(t)}$. We consider these various means of comparison when $I$ is the edge ideal of certain graphs by describing an ideal $J$ for which $I^{(t)} = I^t + J$. When $I$ is the edge ideal of an odd cycle, our description of the structure of $I^{(t)}$ yields solutions to both the direct and asymptotic containme
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The Gross-Keating invariant of a half-integral matrix over a $p$-adic integer ring is a fundamental concept in the study of quadratic forms, and has important applications for Siegel modular forms and arithmetic geometry. We introduce the Mathematica package computeGK, a computer program for calculating the Gross-Keating invariant and the Siegel series of a half-integral matrix over $\mathbb{Z}_p$, as well as other related quantities. As a by-product, we obtain a table of the arithmetic intersection numbers related to the classical modular polynomials using the explicit formula of Gross and Keating.
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Let $\mathcal{A}$ be an abelian variety over a number field, with a good reduction at a prime ideal containing a prime number $p$. Denote by ${\rm A}$ an abelian variety over a finite field of characteristic $p$, obtained by the reduction of $\mathcal{A}$ at the prime ideal. In this paper we derive an algorithm which allows to decompose of the group scheme ${\rm A}[p]$ into indecomposable quasi-polarized ${\rm BT}_1$-group schemes. This can be done for the unramified $p$ on the basis of its decomposition into prime ideals in the endomorphism algebra of ${\rm A}$. We also compute all types of such correspondence for abelian varieties of dimension up to $5$. As a consequence we establish the relation between the decompositions of prime $p$ and the corresponding pairs of $p$-rank and $a$-number of an abelian variety ${\rm A}$.
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We study the rate of weak convergence of Markov chains to diffusion processes under suitable but quite general assumptions. We give an example in the financial framework, applying the convergence analysis to a multiple jumps tree approximation of the CIR process. Then, we combine the Markov chain approach with other numerical techniques in order to handle the different components in jump-diffusion coupled models. We study the speed of convergence of this hybrid approach and we provide an example in finance, applying our results to a tree-finite difference approximation in the Heston or Bates model.
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A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent to some other edges appeared previously. In this paper, we focus on complete bipartite graphs and trees. For complete bipartite graphs, we obtain an exact formula for their shelling numbers. And for trees, we propose a simple method to count shellings and bound shelling numbers using vertex degrees and diameter.
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We propose a new unified formulation of the current algebra theory in terms of supergeometry. We take a QP-structure, i.e. a differential graded symplectic structure, on a graded cotangent bundle as a fundamental framework. A Poisson bracket in a current algebra is constructed by the so called derived bracket of the graded Poisson structure induced from the above symplectic structure. By taking a canonical transformation on a QP-manifold, correct anomalous terms in physical theories are derived. A large class of current algebras with and without anomalous terms are constructed from this structure. Moreover, a new class of current algebras related higher structures are obtained systematically.
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We present Deep Graph Infomax (DGI), a general approach for learning node representations within graph-structured data in an unsupervised manner. DGI relies on maximizing mutual information between patch representations and corresponding high-level summaries of graphs---both derived using established graph convolutional network architectures. The learnt patch representations summarize subgraphs centered around nodes of interest, and can thus be reused for downstream node-wise learning tasks. In contrast to most prior approaches to graph representation learning, DGI does not rely on random walks, and is readily applicable to both transductive and inductive learning setups. We demonstrate competitive performance on a variety of node classification benchmarks, which at times even exceeds the performance of supervised learning.
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We consider Jacobi matrices $J$ whose parameters have the power asymptotics $\rho_n=n^{\beta_1} \left( x_0 + \frac{x_1}{n} + {\rm O}(n^{-1-\epsilon})\right)$ and $q_n=n^{\beta_2} \left( y_0 + \frac{y_1}{n} + {\rm O}(n^{-1-\epsilon})\right)$ for the off-diagonal and diagonal, respectively. We show that for $\beta_1 > \beta_2$, or $\beta_1=\beta_2$ and $2x_0 > |y_0|$, the matrix $J$ is in the limit circle case and the convergence exponent of its spectrum is $1/\beta_1$. Moreover, we obtain upper and lower bounds for the upper density of the spectrum. When the parameters of the matrix $J$ have a power asymptotic with one more term, we characterise the occurrence of the limit circle case completely (including the exceptional case $\lim_{n\to \infty} |q_n|\big/ \rho_n = 2$) and determine the convergence exponent in almost all cases.
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We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with rational singularities with components equivalent to derived categories of local finite dimensional algebras. First, we discuss how a semiorthogonal decomposition of a resolution of singularities of a surface $X$ may induce a semiorthogonal decomposition of $X$. In the case when $X$ has cyclic quotient singularities, we introduce the condition of adherence for the components of the resolution that allows to identify the components of the induced decomposition with derived categories of explicit local finite dimensional algebras. Further, we present an obstruction in the Brauer group of $X$ to the existence of such semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of $X$. We illustrate the theory by exhibiting a semiorthogona
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Coarse-graining is central to reducing dimensionality in molecular dynamics, and is typically characterized by a mapping which projects the full state of the system to a smaller class of variables. While extensive literature has been devoted to coarse-graining starting from reversible systems, not much is known in the non-reversible setting. In this article, starting with a non-reversible dynamics, we introduce and study an effective dynamics which approximates the (non-closed) projected dynamics. Under fairly weak conditions on the system, we prove error bounds on the trajectorial error between the projected and the effective dynamics. In addition to extending existing results to the non-reversible setting, our error estimates also indicate that the notion of mean force motivated by this effective dynamics is a good one.
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Randomly initialized first-order optimization algorithms are the method of choice for solving many high-dimensional nonconvex problems in machine learning, yet general theoretical guarantees cannot rule out convergence to critical points of poor objective value. For some highly structured nonconvex problems however, the success of gradient descent can be understood by studying the geometry of the objective. We study one such problem -- complete orthogonal dictionary learning, and provide converge guarantees for randomly initialized gradient descent to the neighborhood of a global optimum. The resulting rates scale as low order polynomials in the dimension even though the objective possesses an exponential number of saddle points. This efficient convergence can be viewed as a consequence of negative curvature normal to the stable manifolds associated with saddle points, and we provide evidence that this feature is shared by other nonconvex problems of importance as well.
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A criterion for elements of free Zinbiel algebras to be Lie or Jordan is established. This criterion is used in studying speciality problems of Tortkara algebras. We construct a base of free special Tortkara algebras. Further, we prove analogue of classical Cohn's and Shirshov's theorems for Tortkara algebras.
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The connection between inequalities in additive combinatorics and analogous versions in terms of the entropy of random variables has been extensively explored over the past few years. This paper extends a device introduced by Ruzsa in his seminal work introducing this correspondence. This extension provides a toolbox for establishing the equivalence between sumset inequalities and their entropic versions. It supplies simpler proofs of known results and opens a path for obtaining new ones. Some new examples in nonabelian groups illustrate the power of the device.
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An axial algebra is a commutative non-associative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion law as the Griess algebra. The $2$-generated axial algebras of Monster type, called Norton-Sakuma algebras, have been fully classified and are one of nine isomorphism types. In this paper, we enumerate and construct the $3$-generated axial algebras of Monster type which do not contain a $5\textrm{A}$, or $6\textrm{A}$ subalgebra.
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For any congruence subgroup of the modular group, we extend the region of convergence of the Euler product of the Selberg zeta function beyond the boundary Re s = 1, if it is attached with a nontrivial irreducible unitary representation. The region is determined by the size of the lowest eigenvalue of the Laplacian, and it extends to Re s $\geqslant$ 3/4 under Selberg's eigenvalue conjecture. More generally, for any unitary representation we establish the relation between the behavior of the partial Euler product in the critical strip and the estimate of the error term in the prime geodesic theorem. For the trivial representation, the proof essentially exploits the idea of the celebrated work of Ramanujan.
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We propose and test exact quantization conditions for the $N$-particle quantum elliptic Ruijsenaars-Schneider integrable system, as well as its Calogero-Moser limit, based on the conjectural correspondence to the five-dimensional $\mathcal{N} = 1$ $SU(N)$ gauge theory in the Nekrasov-Shatashvili limit. We discuss two natural sets of quantization conditions, related by the electro-magnetic duality, and the importance of non-perturbative corrections in the Planck constant. We also comment on the eigenfunction problem, by reinterpreting the Separation of Variables approach in gauge theory terms.
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The efficient and accurate QR decomposition for matrices with hierarchical low-rank structures, such as HODLR and hierarchical matrices, has been challenging. Existing structure-exploiting algorithms are prone to numerical instability as they proceed indirectly, via Cholesky decompositions or a block Gram-Schmidt procedure. For a highly ill-conditioned matrix, such approaches either break down in finite-precision arithmetic or result in significant loss of orthogonality. Although these issues can sometimes be addressed by regularization and iterative refinement, it would be more desirable to have an algorithm that avoids these detours and is numerically robust to ill-conditioning. In this work, we propose such an algorithm for HODLR matrices. It achieves accuracy by utilizing Householder reflectors. It achieves efficiency by utilizing fast operations in the HODLR format in combination with compact WY representations and the recursive QR decomposition by Elmroth and Gustavson. Numerical
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Composite convex optimization problems which include both a nonsmooth term and a low-rank promoting term have important applications in machine learning and signal processing, such as when one wishes to recover an unknown matrix that is simultaneously low-rank and sparse. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal methods for composite optimization and even simple subgradient methods. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, which are often applied to a smooth approximation of the nonsmooth objective, are slow since their runtime scales with both the large Lipshitz parameter of the smoothed gradient vector and with $1/\epsilon$. In this paper we develop efficient algorithms for \textit{stochastic} optimization of a strongly-convex objective which includes both a nonsmooth term and a low-rank promoting term. In
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We relate the recognition principle for infinite $\mathbf P^1$-loop spaces to the theory of motivic fundamental classes of D\'eglise, Jin, and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with $\mathbf A^n/(\mathbf A^n-0)$, and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite E-correspondences for E a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calm\`es and Fasel's category of finite Milnor-Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of E-module spectra factors through the category of finite E-correspondences.
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The goal of the present paper is to push Sz.-Nagy--Foias model theory for a completely nonunitary Hilbert-space contraction operator $T$, to the case of a commuting pair of contraction operators $(T_1, T_2)$ having product $T = T_1 T_2$ which is completely nonunitary. The idea is to use the Sz.-Nagy-Foias functional model for $T$ as the model space also for the commutative tuple ($T_1, T_2)$ with $T = T_1 T_2$ equal to the usual Sz.-Nagy--Foias model operator, and identify what added structure is required to classify such commutative contractive factorizations $T = T_1 T_2$ up to unitary equivalence. In addition to the characteristic function $\Theta_T$, we identify additional invariants $({\mathbb G}, {\mathbb W})$ which can be used to construct a functional model for the commuting pair $(T_1, T_2)$ and which have good uniqueness properties: if two commutative contractive pairs $(T_1, T_2)$ and $(T'_1, T'_2)$ are unitarily equivalent, then their characteristic triples $(\Theta, {\math
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Stabilizer codes are a powerful method for implementing fault-tolerant quantum memory and in the case of topological codes, they form useful models for topological phases of matter. In this paper, we discuss the theory of stabilizer codes as a discrete version of a linear gauge structure, a concept we introduce here. A linear gauge structure captures all the familiar ingredients of $U(1)$ gauge theory including a generalization of charge conservation, Maxwell equations and topological terms. However, this perspective does not depend on any particular Hamiltonian or Lagrangian, which as a consequence is agnostic to the source of the gauge redundancy. Based upon the connection to stabilizer codes, we view the source of the gauge redundancy as an ambiguity in the tensor product structure of the Hilbert space. That is, the gauge provides a set of equivalent but arbitrary ways to factorize the Hilbert space. We apply this formalism to the $d=2$ and $3$ toric code as well as the paradigm fra
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Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We investigate the existence (or lack thereof) of proper reductions of an ideal of $R$ and the number of generators needed for a reduction in the case $k$ is a finite field. When $R$ is one-dimensional, we give a formula for the smallest integer $n$ for which every ideal has an $n$-generated reduction. It follows that in a one-dimensional local Noetherian ring every ideal has a principal reduction if and only if the number of maximal ideals in the normalization of the reduced quotient of $R$ is at most $|k|$. In higher dimensions, we show that for any positive integer, there exists an ideal of $R$ that does not have an $n$-generated reduction and that if $n \geq \dim R$ this ideal can be chosen to be $\mathfrak{m}$-primary. In the case where $R$ is a two-dimensi
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We present a new technique to apply finite element methods to partial differential equations over curved domains. A change of variables along a coordinate transformation satisfying only low regularity assumptions can translate a Poisson problem over a curved physical domain to a Poisson problem over a polyhedral parametric domain. This greatly simplifies both the geometric setting and the practical implementation, at the cost of having globally rough non-trivial coefficients and data in the parametric Poisson problem. Our main result is that a recently developed broken Bramble-Hilbert lemma is key in harnessing regularity in the physical problem to prove higher-order finite element convergence rates for the parametric problem. Numerical experiments are given which confirm the predictions of our theory.
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We develop a Gr\"obner basis theory for a class of algebras that generalizes both PBW-algebras and rings of differential algebras on smooth varieties. Emphasis lies on methods to compute filtrations and graded structures defined by weight vectors. The approach is tailored for bifiltered D-modules satisfying properties of mixed Hodge modules. As a key ingredient in functors of such modules our theory applies to compute the order filtration on pieces of a V-filtration.
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In this work we consider the gravitating vortex equations. These equations couple a metric over a compact Riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section --- the Higgs field ---, and have a symplectic interpretation as moment-map equations. As a particular case of the gravitating vortex equations on $\mathbb{P}^1$, we find the Einstein--Bogomol'nyi equations, previously studied in the theory of cosmic strings in physics. We prove two main results in this paper. Our first main result gives a converse to an existence theorem of Y. Yang for the Einstein--Bogomol'nyi equations, establishing in this way a correspondence with Geometric Invariant Theory for these equations. In particular, we prove a conjecture by Y. Yang about the non-existence of cosmic strings on $\mathbb{P}^1$ superimposed at a single point. Our second main result is an existence and uniqueness result for the gravitating vortex equations in genus greater than one.
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This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.
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In this work we develop a Hessian-based sampling method for the construction of goal-oriented reduced order models with high-dimensional parameter inputs. Model reduction is known very challenging for high-dimensional parametric problems whose solutions also live in high-dimensional manifolds. However, the manifold of some quantity of interest (QoI) depending on the parametric solutions may be low-dimensional. We use the Hessian of the QoI with respect to the parameter to detect this low-dimensionality, and draw training samples by projecting the high-dimensional parameter to a low-dimensional subspace spanned by the eigenvectors of the Hessian corresponding to its dominating eigenvalues. Instead of forming the full Hessian, which is computationally intractable for a high-dimensional parameter, we employ a randomized algorithm to efficiently compute the dominating eigenpairs of the Hessian whose cost does not depend on the nominal dimension of the parameter but only on the intrinsic di
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In many tomographic imaging problems the data consist of integrals along lines or curves. Increasingly we encounter "rich tomography" problems where the quantity imaged is higher dimensional than a scalar per voxel, including vectors tensors and functions. The data can also be higher dimensional and in many cases consists of a one or two dimensional spectrum for each ray. In many such cases the data contain not just integrals along rays but the distribution of values along the ray. If this is discretized into bins we can think of this as a histogram. In this paper we introduce the concept of "histogram tomography". For scalar problems with histogram data this holds the possibility of reconstruction with fewer rays. In vector and tensor problems it holds the promise of reconstruction of images that are in the null space of related integral transforms. For scalar histogram tomography problems we show how bins in the histogram correspond to reconstructing level sets of function, while mom
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In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type $FP_3$.
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Transatomic Power, an MIT spinout that drew wide attention and millions in funding, is shutting down almost two years after the firm backtracked on bold claims for its design of a molten-salt reactor. From a report: The company, founded in 2011, plans to announce later today that it's winding down. Transatomic had claimed its technology could generate electricity 75 times more efficiently than conventional light-water reactors, and run on their spent nuclear fuel. But in a white paper published in late 2016, it backed off the latter claim entirely and revised the 75 times figure to "more than twice," a development first reported by MIT Technology Review. Those downgrades forced the company to redesign its system. That delayed plans to develop a demonstration reactor, pushing the company behind rival upstarts like TerraPower and Terrestrial Energy, says Leslie Dewan, the company's cofounder and chief executive. The longer timeline and reduced performance advantage made it harder to rais
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An anonymous reader shares a report: Set against the austere peaks of the Western Brooks Range, the lake, about 20 football fields in size, looked like it was boiling. Its waters hissed, bubbled and popped as a powerful greenhouse gas escaped from the lake bed. Some bubbles grew as big as grapefruits, visibly lifting the water's surface several inches and carrying up bits of mud from below. This was methane. As the permafrost thaws across the fast-warming Arctic, it releases carbon dioxide, the top planet-warming greenhouse gas, from the soil into the air. Sometimes, that thaw spurs the growth of lakes in the soft, sunken ground, and these deep-thawing bodies of water tend to unleash the harder-hitting methane gas. But not this much of it. This lake, which Katey Walter Anthony, an ecologist who has studied 300 lakes across the tundras of the Arctic, dubbed Esieh Lake, looked different. And the volume of gas wafting from it could deliver the climate system another blow if lakes like thi
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Despite its advanced age, Cobol is still the most prevalent programming language in the financial-services industry world-wide. Software programmed in Cobol powers millions of banking transactions every day and underpins critical computer mainframes. WSJ: And Cobol isn't going away anytime soon. Banks and other companies have come to the uncomfortable realization that ripping out old mainframes is pricey and complicated. Transitioning to new systems is likely to take years, and besides, a lot of the older tech works just fine. The problem is that Cobol isn't popular with new programmers. So, with a generation of Cobol specialists retiring, there is a continuing hunt to find a new generation of programmers to service this technology. In Texas, Mr. Hinshaw's (an anecdote in the story) company, the Cobol Cowboys, a group of mostly older programmers, is training U.S. military veterans in the programming language. Accenture is coaching hundreds of Cobol programmers every year in India and t
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The Fedora Project announced Tuesday the beta availability of Fedora 29 -- the latest version of the free and open-source Fedora OS. From a report: It features updated packages, improved support for Raspberry Pi, and more. "Highlighting Fedora 29 Beta is the addition of modularity across all Fedora editions. First delivered in Fedora 28 Server, modularity enables multiple versions of the same software (like Node.js) to be selected on a per-system basis, with parallel installation done through containers. This can provide some users the ability to use tried-and-true versions of software while enabling other users to work with just-released innovation without impacting the overall stability of the Fedora operating system," says Matthew Miller, Fedora Project Leader. Miller further says, "The importance of ARM to IoT has not been lost on Fedora, and Fedora 29 Beta aims to make the Fedora operating system a home for both ARM and IoT. These features start with enhanced ZRAM support for swap
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Earlier this year, Mozilla announced Firefox Monitor, a service that will inform you if your online accounts were hacked in a recent data breach. It's now available to general public. A report adds: For the new security-focused tool, Mozilla partnered with Troy Hunt, the renowned security expert behind Have I Been Pwned? (HIBP), which is a database of data breaches that allows anyone to discover whether one of their online accounts has been compromised. The first iteration of Firefox Monitor is, for all intents and purposes, a clone of HIBP. After you enter your email address and hit the scan button, you're told which online services have leaked your personal details (if any). You can also sign up to be notified of any future data breaches involving one or more of your email addresses.
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Earlier this year, Google updated its financial services-ad policies to ban any advertising about cryptocurrency-related content, including initial coin offerings (ICOs), wallets, and trading advice. Google appears to be reversing course with a new policy starting in October that will allow regulated cryptocurrency exchanges to buy ads in the U.S. and Japan. Advertising about ICOs, wallets and trading advice are reportedly still not allowed. CNBC reports: Google's updated policy applies to advertisers all over the world, though the ads can only run in the U.S. and Japan, and interested parties will need to apply for certification to serve ads in each country individually. Google's move follows Facebook, which started allowing pre-approved cryptocurrency advertisers in June. Google parent company Alphabet gets roughly 86 percent of its total revenue from advertising. The company booked more than $54 billion in ad revenue in the first half of 2018.
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The Internet Society, a global non-profit organization dedicated to the open development, evolution and use of the Internet, today announced that it is partnering with Facebook to develop Internet Exchange Points (IXP) throughout Africa. From a press release: An Internet Exchange Point is where multiple local and international networks, ISPs and content providers interconnect their networks together to efficiently exchange Internet traffic through an arrangement commonly referred to as Peering. Currently, 42% of countries in Africa lack IXPs, which means that most of their domestic Internet traffic is exchanged through points outside their respective country, usually through satellite or submarine fiber across multiple international hubs to reach their destination. This can result in poor end-user experiences and discourages hosting content locally, which are some of the key factors towards the development of the local Internet ecosystem. Peering at IXPs helps keep domestic Internet tr
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We review in detail the Batalin-Vilkovisky formalism and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an $L_\infty$-algebra and how quasi-isomorphisms between $L_\infty$-algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern-Simons theories and give some useful shortcuts in usually rather involved computations.
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Let X and X' be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles, in the sense of Henning Meyer's graduate thesis, between the tropicalization of the intersection product of X and X' and the stable intersection of trop(X) and trop(X'), when restricted to (the inverse image under the tropicalization map of) a connected component C of the intersection of trop(X) and trop(X'). This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection in a toric tropical variety, and check that this definition is compatible with the intersection product in loc.cit.. As a result we get a numerical equivalence (after a compactification and restricting to C) between the intersection product of X and X' and the stable intersection of trop(X) and trop(X') via the compactified stable intersection. In particular, when X and X' have complementary codim
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This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an orbifold with another triangulation. This method is used to show that eventually periodic continued fractions have partial quotients which have exponential growth when iteratively multiplied by $n$, for $n$ any fixed, natural number.
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For a given decreasing positive real function $\psi$, let $\mathcal{A}_n(\psi)$ be the set of real numbers for which there are infinitely many integer polynomials $P$ of degree up to $n$ such that $\left\lvert P(x) \right\rvert \leq \psi(\operatorname{H}(P))$. A theorem by Bernik states that $\mathcal{A}_n(\psi)$ has Hausdorff dimension $\frac{n+1}{w+1}$ in the special case $\psi(r) = r^{-w}$, while a theorem by Beresnevich, Dickinson and Velani implies that the Hausdorff measure $\operatorname{\mathcal{H}}^g(\mathcal{A}_n(\psi))=\infty$ when a certain series diverges. In this paper we prove the convergence counterpart of this result when $P$ has bounded discriminant, which leads to a complete solution when $n = 3$ and $\psi(r) = r^{-w}$.
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In this paper we propose a distributed implementation of the relaxed Alternating Direction Method of Multipliers algorithm (R-ADMM) for optimization of a separable convex cost function, whose terms are stored by a set of interacting agents, one for each agent. Specifically the local cost stored by each node is in general a function of both the state of the node and the states of its neighbors, a framework that we refer to as `partition-based' optimization. This framework presents a great flexibility and can be adapted to a large number of different applications. We show that the partition-based R-ADMM algorithm we introduce is linked to the relaxed Peaceman-Rachford Splitting (R-PRS) operator which, historically, has been introduced in the literature to find the zeros of sum of functions. Interestingly, making use of non expansive operator theory, the proposed algorithm is shown to be provably robust against random packet losses that might occur in the communication between neighboring
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Motivated by high-dimensional nonlinear optimization problems as well as ill-posed optimization problems arising in image processing, we consider a bilevel optimization model where we seek among the optimal solutions of the inner level problem, a solution that minimizes a secondary metric. Our goal is to address the high-dimensionality of the bilevel problem, and the nondifferentiability of the objective function. Minimal norm gradient, sequential averaging, and iterative regularization are some of the recent schemes developed for addressing the bilevel problem. But none of them address the high-dimensional structure and nondifferentiability. With this gap in the literature, we develop a randomized block coordinate iterative regularized gradient descent scheme (RB-IRG). We establish the convergence of the sequence generated by RB-IRG to the unique solution of the bilevel problem of interest. Furthermore, we derive a rate of convergence $\mathcal{O} \left(\frac{1}{{k}^{0.5-\delta}}\righ
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In this paper an algorithm designed for large databases is introduced for the enhancement of pass rates in mathematical university lower division courses with several sections. Using integer programming techniques, the algorithm finds the optimal pairing of students and lecturers in order to maximize the success chances of the students' body. The students-lecturer success probability is computed according to their corresponding profiles stored in the data bases.
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This paper presents a coordinate-free formulation of virtual holonomic constraints for underactuated Lagrangian control systems on Riemannian manifolds. It is shown that when a virtual constraint enjoys a regularity property, the constrained dynamics are described by an affine connection dynamical system. The affine connection of the constrained system has an elegant relationship to the Riemannian connection of the original Lagrangian control system. Necessary and sufficient conditions are given for the constrained dynamics to be Lagrangian. A key condition is that the affine connection of the constrained dynamics be metrizable. Basic results on metrizability of affine connections are first reviewed, then employed in three examples in order of increasing complexity. The last example is a double pendulum on a cart with two different actuator configurations. For this control system, a virtual constraint is employed which confines the second pendulum to within the upper half-plane.
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We consider the problem of defining a strategy consisting of a set of facilities taking into account also the location where they have to be assigned and the time in which they have to be activated. The facilities are evaluated with respect to a set of criteria. The plan has to be devised respecting some constraints related to different aspects of the problem such as precedence restrictions due to the nature of the facilities. Among the constraints, there are some related to the available budget. We consider also the uncertainty related to the performances of the facilities with respect to considered criteria, and plurality of stakeholders participating to the decision. The considered problem can be seen as the combination of some prototypical operations research problems: knapsack problem, location problem and project scheduling. Indeed, the basic brick of our model is a variable $x_{ilt}$ which takes value 1 if facility $i$ is activated in location $l$ at time $t$, and $0$ otherwise.
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A numerical semigroup is a subset of the non-negative integers that is closed under addition. For a randomly generated numerical semigroup, the expected number of minimum generators can be expressed in terms of a doubly-indexed sequence of integers, denoted $h_{n, i}$, that count generating sets with certain properties. We prove a recurrence that implies the sequence $h_{n,i}$ is eventually quasipolynomial when the second parameter is fixed.
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The Frobenius test exponent $\operatorname{Fte}(R)$ of a local ring $(R,\mathfrak{m})$ of prime characteristic $p > 0$ is the smallest $e_0 \in \mathbb{N}$ such that for every ideal $\mathfrak{q}$ generated by a (full) system of parameters, the Frobenius closure $\mathfrak{q}^F$ has $(\mathfrak{q}^F)^{[p^{e_0}]} = \mathfrak{q}^{[p^{e_0}]}$. We establish a suffcient condition for $\operatorname{Fte}(R)<\infty$ and use it to show that if $R$ is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e. $H^j_{\mathfrak{m}}(R) / 0^F_{H^j_{\mathfrak{m}}(R)}$ is finite length for $0 \le j < d$, then $\operatorname{Fte}(R)<\infty$.
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During the last decades, significant advances have been made in the area of power system stability and control. Nevertheless, when this analysis is carried out by means of decentralized conditions in a general network, it has been based on conservative assumptions such as the adoption of lossless networks. In the current paper, we present a novel approach for decentralized stability analysis and control of power grids through the transformation of both the network and the bus dynamics into the system reference frame. In particular, the aforementioned transformation allows us to formulate the network model as an input-output system that is shown to be passive even if the network's lossy nature is taken into account. We then introduce a broad class of bus dynamics that are viewed as multivariable input/output systems compatible with the network formulation, and appropriate passivity conditions are imposed on those that guarantee stability of the power network. We discuss the opportunitie
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This paper is devoted to the analysis of problems of optimal control of ensembles governed by the Liouville (or continuity) equation. The formulation and study of these problems have been put forward in recent years by R.W. Brockett, with the motivation that ensemble control may provide a more general and robust control framework. Following Brockett's formulation of ensemble control, a Liouville equation with unbounded drift function, and a class of cost functionals that include tracking of ensembles and different control costs is considered. For the theoretical investigation of the resulting optimal control problems, a well-posedness theory in weighted Sobolev spaces is presented for the Liouville and transport equations. Then, a class of non-smooth optimal control problems governed by the Liouville equation is formulated and existence of optimal controls is proved. Furthermore, optimal controls are characterised as solutions to optimality systems; such a characterisation is the key t
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A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of $p$-adic abelian integrals.
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