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A double extension ($D$-extension) of a Lie algebra ${\mathfrak a}$ with a non-degenerate invariant symmetric bilinear form $B_{\mathfrak a}$, briefly a NIS Lie algebra, is an enlargement of ${\mathfrak a}$ by means of a central extension and a derivation; the affine Kac-Moody algebras are the most known examples. Let ${\mathfrak a}$ be a restricted Lie algebra equipped with a NIS $B_{\mathfrak a}$. Suppose $\mathfrak a$ has a restricted derivation $D$ such that $B_{\mathfrak a}$ is $D$-invariant. We show that the double extension of $\mathfrak a$ caries a $p$-mapping constructed by means of $B_{\mathfrak a}$ and $D$. We show that, the other way round, any restricted NIS Lie algebra can be obtained as a $D$-extension of another restricted NIS Lie algebra of codimension 2 provided that the center is not trivial together with an extra condition pertaining to the central element. We give examples of $D$-extensions of restricted Lie algebras in small characteristic related with Manin tripl
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We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator, H_\#, with a potential given by a sum of translates of identical atomic potential wells of depth $\lambda^2$, centered on a subset of a honeycomb structure with a zigzag edge. In the strong binding regime ($\lambda$ large) there exist {\it edge states}, solutions of $H_\#\Psi=E\Psi$, which are localized transverse to the edge and are propagating (plane-wave like) parallel to the edge. Upon $\lambda-$ dependent rescaling, these edge-states are well-approximated by, and converge (as $\lambda\uparrow\infty$) to the flat-band of edge states of a limiting discrete " tight-binding " model .
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The task of non-adaptive group testing is to identify up to $d$ defective items from $N$ items, where a test is positive if it contains at least one defective item, and negative otherwise. If there are $t$ tests, they can be represented as a $t \times N$ measurement matrix. We have answered the question of whether there exists a scheme such that a larger measurement matrix, built from a given $t\times N$ measurement matrix, can be used to identify up to $d$ defective items in time $O(t \log_2{N})$. In the meantime, a $t \times N$ nonrandom measurement matrix with $t = O \left(\frac{d^2 \log_2^2{N}}{(\log_2(d\log_2{N}) - \log_2{\log_2(d\log_2{N})})^2} \right)$ can be obtained to identify up to $d$ defective items in time $\mathrm{poly}(t)$. This is much better than the best well-known bound, $t = O \left( d^2 \log_2^2{N} \right)$. For the special case $d = 2$, there exists an efficient nonrandom construction in which at most two defective items can be identified in time $4\log_2^2{N}$ u
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In this paper, we propose an efficient beam training technique for millimeter-wave (mmWave) communications. When some mobile users are under high mobility, the beam training should be performed frequently to ensure the accurate acquisition of the channel state information. In order to reduce the resource overhead caused by frequent beam training, we introduce a dedicated beam training strategy which sends the training beams separately to a specific high mobility user (called a target user) without changing the periodicity of the conventional beam training. The dedicated beam training requires small amount of resources since the training beams can be optimized for the target user. In order to satisfy the performance requirement with low training overhead, we propose the optimal training beam selection strategy which finds the best beamforming vectors yielding the lowest channel estimation error based on the target user's probabilistic channel information. Such dedicated beam training is
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Pickrell has fully characterized the unitarily invariant probability measures on infinite Hermitian matrices, and an alternative proof of this classification has been found by Olshanski and Vershik. Borodin and Olshanski deduced from this proof that under any of these invariant measures, the extreme eigenvalues of the minors, divided by the dimension, converge almost surely. In this paper, we prove that one also has a weak convergence for the eigenvectors, in a sense which is made precise. After mapping Hermitian to unitary matrices via the Cayley transform, our result extends a convergence proven in our paper with Maples and Nikeghbali, for which a coupling of the Circular Unitary Ensemble of all dimensions is considered.
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A property, or statistical functional, is said to be elicitable if it minimizes expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limits of empirical risk minimization. While several recent papers have asked which properties are elicitable, we instead advocate for a more nuanced question: how many dimensions are required to indirectly elicit a given property? This number is called the elicitation complexity of the property. We lay the foundation for a general theory of elicitation complexity, including several basic results about how elicitation complexity behaves, and the complexity of standard properties of interest. Building on this foundation, we establish several upper and lower bounds for the broad class of Bayes risks. We apply these results by proving tight complexity bounds, with respect to identifiable properties, for variance, financial risk measures, entropy, norms, and new properties of interest. We then
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One form of characterizing the expressiveness of a piecewise linear neural network is by the number of linear regions, or pieces, of the function modeled. We have observed substantial progress in this topic through lower and upper bounds on the maximum number of linear regions and a counting procedure. However, these bounds only account for the dimensions of the network and the exact counting may take a prohibitive amount of time, therefore making it infeasible to benchmark the expressiveness of networks. In this work, we approximate the number of linear regions of specific rectifier networks with an algorithm for probabilistic lower bounds of mixed-integer linear sets. In addition, we present a tighter upper bound that leverages network coefficients. We test both on trained networks. The algorithm for probabilistic lower bounds is several orders of magnitude faster than exact counting and the values reach similar orders of magnitude, hence making our approach a viable method to compar
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The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the birational geometry of the moduli space $\overline{\mathrm{M}}_{0,n}$ of $n$-tuples of points in $\mathbb{P}^1$. In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely $d=2$, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, $d=3$, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we expla
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Dimension-varying linear systems are investigated. First, a dimension-free state space is proposed. A cross dimensional distance is constructed to glue vectors of different dimensions together to form a cross-dimensional topological space. This distance leads to projections over different dimensional Euclidean spaces and the corresponding linear systems on them, which provide a connection among linear systems with different dimensions. Based on these projections, an equivalence of vectors and an equivalence of matrices over different dimensions are proposed. It follows that the dynamics on quotient space is obtained, which provides a proper model for cross-dimensional systems. Finally, using the lift of dynamic systems on quotient space to Euclidean spaces of different dimensions, a cross-dimensional model is proposed to deal with the dynamics of dimension-varying process of linear systems. On the cross-dimensional model a control is designed to realize the transfer between models on E
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We prove an L2 recovery bound for a family of sparse estimators defined as minimizers of some empirical loss functions -- which include hinge loss and logistic loss. More precisely, we achieve an upper-bound for coefficients estimation scaling as (k*/n)\log(p/k*): n,p is the size of the design matrix and k* the dimension of the theoretical loss minimizer. This is done under standard assumptions, for which we derive stronger versions of a cone condition and a restricted strong convexity. Our bound holds with high probability and in expectation and applies to an L1-regularized estimator and to a recently introduced Slope estimator, which we generalize for classification problems. Slope presents the advantage of adapting to unknown sparsity. Thus, we propose a tractable proximal algorithm to compute it and assess its empirical performance. Our results match the best existing bounds for classification and regression problems.
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We consider in this paper the Grenander estimator of unbounded, in general, nonincreasing densities on the interval [0; 1] without any smoothness assumptions. For fixed number n of i.i.d. random vari- ables X1;X2; : : : ;Xn with values in [0; 1] and the nonincreasing den- sity function f(x), 0 < x < 1, we prove an inequality bounding the weighted integrated square error of the Grenander estimator with probability one by the classical Kolmogorov-Smirnov statistic. Fur- ther, we consider some interesting implications of the latter inequality
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Some exotic compact objects possess evanescent ergosurfaces: timelike submanifolds on which a Killing vector field, which is timelike everywhere else, becomes null. We show that any manifold possessing an evanescent ergosurface but no event horizon exhibits a linear instability of a peculiar kind: either there are solutions to the linear wave equation which concentrate a finite amount of energy into an arbitrarily small spatial region, or the energy of waves measured by a stationary family of observers can be amplified by an arbitrarily large amount. In certain circumstances we can rule out the first type of instability. We also provide a generalisation to asymptotically Kaluza-Klein manifolds. This instability bears some similarity with the "ergoregion instability" of Friedman, and we use many of the results from the recent proof of this instability by Moschidis.
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Statistical analysis on networks has received growing attention due to demand from various emerging applications. In dynamic networks, one of the key interests is to model the event history of time-stamped interactions amongst nodes. We propose to model dynamic directed communication networks via multivariate counting processes. A pseudo partial likelihood approach is exploited to capture the network dependence structure. Asymptotic results of the resulting estimation are established. Numerical results are performed to demonstrate effectiveness of our proposal.
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Inspired by Conway's surreal numbers, we study real closed fields whose value group is isomorphic to the additive reduct of the field. We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields.
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The necessary and sufficient conditions under which a given family $\mathcal{F}$ of subsets of finite set $X$ coincides with the family $\mathbf B_X$ of all balls generated by some ultrametric $d$ on $X$ are found. It is shown that the representing tree of the ultrametric space $(\mathbf B_{X}, d_H)$ with the Hausdorff distance $d_H$ can be obtained from the representing tree $T_X$ of ultrametric space $(X, d)$ by adding a leaf to every internal vertex of $T_X$.
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In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier-Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.
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A formal inverse of a given automatic sequence (the sequence of coefficients of the composition inverse of its associated formal power series) is also automatic. The comparison of properties of the original sequence and its formal inverse is an interesting problem. Such an analysis has been done before for the Thue{Morse sequence. In this paper, we describe arithmetic properties of formal inverses of the generalized Thue-Morse sequences and formal inverses of two modifications of the Rudin{Shapiro sequence. In each case, we give the recurrence relations and the automaton, then we analyze the lengths of strings of consecutive identical letters as well as the frequencies of letters. We also compare the obtained results with the original sequences.
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Given a tuple $E=(E_1,\dots,E_g)$ of $d\times d$ matrices, the collection of those tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $\| \sum E_j\otimes X_j\|\le 1$ is called a spectraball $\mathcal B_E$. Likewise, given a tuple $B=(B_1,\dots,B_g)$ of $e\times e$ matrices the collection of tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $I + \sum B_j\otimes X_j +\sum B_j^* \otimes X_j^*\succeq 0$ is a free spectrahedron $\mathcal D_B$. Assuming $E$ and $B$ are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $p:\mathcal B_E\to \mathcal D_B$ normalized by $p(0)=0$ and $p'(0)=I$ if and only if $\mathcal B_E=\mathcal B_B$ and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant algebraic representation.
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Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable-regular LDPC codes with a given variable degree and girth. The derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets. Considering that non-zero codewords are a special case of trapping sets, we also derive tight upper bounds on the minimum weight of such codewords, i.e., the minim
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We propose a classification of symmetric conservative clones with a finite carrier. For the study, we use the functional Galois connection $(Inv_Q, Pol_Q)$, which is a natural modification of the connection $(Inv, Pol)$ based on the preservation relation between functions $f$ on a set $A$ (of all finite arities) and sets of functions $h\in A^Q$ for an arbitrary set $Q$.
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Covert communication is necessary when revealing the mere existence of a message leaks sensitive information to an attacker. Consider a network link where an authorized transmitter Jack sends packets to an authorized receiver Steve, and the packets visit Alice, Willie, and Bob, respectively, before they reach Steve. Covert transmitter Alice wishes to alter the packet stream in some way to send information to covert receiver Bob without watchful and capable adversary Willie being able to detect the presence of the message. In our previous works, we addressed two techniques for such covert transmission from Alice to Bob: packet insertion and packet timing. In this paper, we consider covert communication via bit insertion in packets with available space (e.g., with size less than the maximum transmission unit). We consider three scenarios: 1) packet sizes are independent and identically distributed (i.i.d.) with a probability mass function (pmf) whose support is a set of one bit spaced va
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Let $\mathcal{F}_{25}$ be the family of irreducible highest weight modules for the Virasoro algebra of central charge $25$ which are not isomorphic to Verma modules. Let $L(25,0)$ be the Virasoro vertex operator algebra of central charge 25. We prove that the fusion rules for the $L(25,0)$-modules in $\mathcal{F}_{25}$ are in correspondence with the tensor rules for the irreducible finite dimensional representations of $sl(2, \mathbb{C})$, extending the known correspondence between modules for the Virasoro algebras of dual central charges 1 and 25.
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We continue the construction of a Lagrangian description of irreducible half-integer higher-spin representations of the Poincare group with an arbitrary Young tableaux having $k$ rows, on a basis of the BRST--BFV approach suggested for bosonic fields in our first article (Nucl. Phys. B862 (2012) 270, [arXiv:1110.5044[hep-th]). Starting from a description of fermionic mixed-symmetry higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space associated with a special Poincare module, we realize a conversion of the initial operator constraint system (constructed with respect to the relations extracting irreducible Poincare-group representations) into a system of first-class constraints. To do this, we find, in first time, by means of generalized Verma module the auxiliary representations of the constraint subsuperalgebra, to be isomorphic due to Howe duality to $osp(k|2k)$ superalgebra, and containing the subsystem of second-class constraints in terms of new o
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In this article, by introducing a new method in estimating the counting function of the auxiliary function, we prove a new generalization of uniqueness theorems for meromorphic mappings into $\P^n(\C )$ which share few hyperplanes regardless of multiplicities. Our result improves the previous result in this topic. Moreover our proof is simpler than the previous proofs.
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In this article we formulate a group of birational transformations which is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations to a mutation-periodic quiver on a torus. This group provides four types of generalizations of Jimbo-Sakai's $q$-Painlev\'e VI equation as translations of the affine Weyl group. Then the known three systems are obtained again, the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy and a similarity reduction of the $q$-Drinfeld-Sokolov hierarchy.
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We define and study a new class of bialgebras, which generalize certain Turner double algebras related to generic blocks of symmetric groups. Bases and generators of these algebras are given. We investigate when the algebras are symmetric, which is relevant to block theory of finite groups. We then establish a double centralizer property related to blocks of Schur algebras.
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The initial value problem for two-dimensional Zakharov-Kuznetsov equation is shown to be globally well-posed in $H^s({\mathbb{R}^2})$ for all $\frac{5}{7}<s<1$ via using $I$-method in the context of atomic spaces. By means of the increment of modified energy, the exsitence of global attractor for weakly damped, forced Zakharov-Kuznetsov equation is also established in $H^s({\mathbb{R}^2})$ for $\frac{10}{11}<s<1$.
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We study a three-dimensional fluid model describing rapidly rotating convection that takes place in tall columnar structures. The purpose of this model is to investigate the cyclonic and anticyclonic coherent structures. Global existence, uniqueness, continuous dependence on initial data, and large-time behavior of strong solutions are shown provided the model is regularized by a weak dissipation term.
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We consider the well-posedness and numerical approximation of a Hamilton--Jacobi equation on an evolving hypersurface in $\mathbb R^3$. Definitions of viscosity sub- and supersolutions are extended in a natural way to evolving hypersurfaces and provide uniqueness by comparison. An explicit in time monotone numerical approximation is derived on evolving interpolating triangulated surfaces. The scheme relies on a finite volume discretisation which does not require acute triangles. The scheme is shown to be stable and consistent leading to an existence proof via the proof of convergence. Finally an error bound is proved of the same order as in the flat stationary case.
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Leafless elementary trapping sets (LETSs) are known to be the problematic structures in the error floor region of low-density parity-check (LDPC) codes over the additive white Gaussian (AWGN) channel under iterative decoding algorithms. While problems involving the general category of trapping sets, and the subcategory of elementary trapping sets (ETSs), have been shown to be NP-hard, similar results for LETSs, which are a subset of ETSs are not available. In this paper, we prove that, for a general LDPC code, finding a LETS of a given size a with minimum number of unsatisfied check nodes b is NP-hard to approximate with any guaranteed precision. We also prove that finding the minimum size a of a LETS with a given b is NP-hard to approximate. Similar results are proved for elementary absorbing sets, a popular subcategory of LETSs.
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The component-by-component (CBC) algorithm is a method for constructing good generating vectors for lattice rules for the efficient computation of high-dimensional integrals in the "weighted" function space setting introduced by Sloan and Wo\'zniakowski. The "weights" that define such spaces are needed as inputs into the CBC algorithm, and so a natural question is, for a given problem how does one choose the weights? This paper introduces two new CBC algorithms which, given bounds on the mixed first derivatives of the integrand, produce a randomly shifted lattice rule with a guaranteed bound on the root-mean-square error. This alleviates the need for the user to specify the weights. We deal with "product weights" and "product and order dependent (POD) weights". Numerical tables compare the two algorithms under various assumed bounds on the mixed first derivatives, and provide rigorous upper bounds on the root-mean-square integration error.
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Higher-order connectivity patterns such as small induced sub-graphs called graphlets (network motifs) are vital to understand the important components (modules/functional units) governing the configuration and behavior of complex networks. Existing work in higher-order clustering has focused on simple homogeneous graphs with a single node/edge type. However, heterogeneous graphs consisting of nodes and edges of different types are seemingly ubiquitous in the real-world. In this work, we introduce the notion of typed-graphlet that explicitly captures the rich (typed) connectivity patterns in heterogeneous networks. Using typed-graphlets as a basis, we develop a general principled framework for higher-order clustering in heterogeneous networks. The framework provides mathematical guarantees on the optimality of the higher-order clustering obtained. The experiments demonstrate the effectiveness of the framework quantitatively for three important applications including (i) clustering, (ii)
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We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle may not be negligible in comparison to the finescale. This motivates our study of a periodic, directed, and weighted graph embedded in ${\mathbb R}^d$ and the scaling limit of the associated continuous-time random walk $Z(t)$ on the graph's nodes, which jumps along the graph's edges with jump rates given by the edge weights. We show that the scaled process $\varepsilon^2 Z(t/\varepsilon^2)$ converges to a linear drift $\bar{U}t$ and the case of interest to us is that of null drift $\bar{U}=0$. In this case, we show that $\varepsilon Z(t/\varepsilon^2)$ converges weakly to a Brownian motion. The diffusivity of the limiting Brownian motion can be computed by solving a set of linear algebra problems. As we allow for jump rates to be irreversible, our f
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This paper studies two families of piecewise constant functions which are determined by the $(n-2)$-skeleta of collections of honeycomb tessellations of $\mathbb{R}^{n-1}$ with standard permutohedra. The union of the codimension $1$ cones obtained by extending the facets which are incident to a vertex of such a tessellation is called a blade. We prove ring-theoretically that such a honeycomb, with 1-skeleton built from a cyclic sequence of segments in the root directions $e_i-e_{i+1}$, decomposes locally as a Minkowski sum of isometrically embedded components of hexagonal honeycombs: tripods and one-dimensional subspaces. For each triangulation of a cyclically oriented polygon there exists such a factorization. This consequently gives resolution to an issue proposed and developed by A. Ocneanu, to develop a structure theory for an object he introduced during his study of higher Lie theory: permutohedral blades. We introduce a certain canonical basis for a vector space spanned by piecew
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Sampling a signal below the Shannon-Nyquist rate causes aliasing, meaning different frequencies to become indistinguishable. It is also well-known that recovering spectral information from a signal using a parametric method can be ill-posed or ill-conditioned and therefore should be done with caution. We present an exponential analysis method to retrieve high-resolution information from coarse-scale measurements, using uniform downsampling. We exploit rather than avoid aliasing. While we loose the unicity of the solution by the downsampling, it allows to recondition the problem statement and increase the resolution. Our technique can be combined with different existing implementations of multi-exponential analysis (matrix pencil, MUSIC, ESPRIT, APM, generalized overdetermined eigenvalue solver, simultaneous QR factorization,...) and so is very versatile. It seems to be especially useful in the presence of clusters of frequencies that are difficult to distinguish from one another.
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This paper deals with the establishment of Input-to-State Stability (ISS) properties for infinite dimensional systems with respect to both boundary and distributed disturbances. First, an ISS estimate is established with respect to finite dimensional boundary disturbances for a class of Riesz-spectral boundary control systems satisfying certain eigenvalue constraints. Second, a concept of weak solutions is introduced in order to relax the disturbances regularity assumptions required to ensure the existence of strong solutions. The proposed concept of weak solutions, that applies to a large class of boundary control systems which is not limited to the Riesz-spectral ones, provides a natural extension of the concept of both strong and mild solutions. Assuming that an ISS estimate holds true for strong solutions, we show the existence, the uniqueness, and the ISS property of the weak solutions.
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We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the second author to show that these equivalence relations are invariantly universal, in the sense of [3], and thus in particular complete analytic. We also introduce a new kind of Borel reducibility relation for standard Borel G-spaces, which requires the preservation of stabilizers, and explain its connection with the notion of full embeddings commonly considered in category theory.
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In this article, we study a model problem featuring a L\'evy process in a domain with semi-transparent boundary by considering the following perturbed fractional Laplacian operator \[(-\D)^t + (-\D)_{\Omega}^{{s}/{2}} \ b (-\D)_{\Omega}^{{s}/{2}} + q, \quad 0<s<t<1\] on a bounded Lipschitz domain $\Omega \subset \R^n$. While the non-locality of the fraction Laplacian $(-\Delta)^t$ depends on entire $\mathbb{R}^n$, in its non-local perturbation the non-locality depends on the domain $\Omega$ through the regional fractional Laplacian term $(-\Delta)^{s/2}_{\Omega}$ and $b$ exhibits the semi-transparency of the process. We analyze the well-posedness of the model and certain qualitative property like unique continuation property, Runge approximation scheme considering its regional non-local perturbation. Then we move into studying the inverse problem and find that by knowing the corresponding Dirichlet to Neumann map (D-N map) of $\mathscr{L}_{b,c}$ on the exterior domain $\R^n \s
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A group $G$ has cube-free order if no prime to the third power divides $|G|$. We describe an algorithm that given two cube-free groups $G$ and $H$ of known order, decides whether $G\cong H$, and, if so, constructs an isomorphism $G\to H$. If the groups are input as permutation groups, then our algorithm runs in time polynomial in the input size, improving on the previous super-polynomial bound. An implementation of our algorithm is provided for the computer algebra system {\sf GAP}.
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In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in optimal control problems. Using the notion of $L$-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic invariants of Jacobi curves.
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Approximate duality of frame pairs have been investigated by Christensen and Laugesen in (Sampl. Theory Signal Image Process., 9, 2011, 77-90), with the motivation to obtain an important applications in Gabor systems, wavelets and general frame theory. In this paper we obtain some of the known results in approximate duality of frames to K-g-frames. We also obtain new K-g-frames and approximate K-g-duals from a K-g-frame and an approximate K-g-dual. Finally, we give an equivalent condition under which the subsequence of a K-g-frame still to be a K-g-frame.
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We prove that the cone over a Dirichlet arrangement is supersolvable if and only if its Orlik-Solomon algebra is Koszul. This was previously shown for four other classes of arrangements. We exhibit an infinite family of cones over Dirichlet arrangements that are combinatorially distinct from these other four classes.
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Let $p:\mathbb{C} \rightarrow \mathbb{C}$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p'(z) = 0$, are contained in the convex hull of its roots. Recent quantitative versions of B{\o}gvad, Khavinson & Shapiro and Totik show that if almost all roots are contained in a bounded convex domain $K \subset \mathbb{C}$, then almost all roots of the derivative $p'$ are in a $\varepsilon-$neighborhood $K_{\varepsilon}$ (in a precise sense). We prove a quantitative version: if a polynomial $p$ has $n$ roots in $K$ and $\lesssim c_{K, \varepsilon} (n/\log{n})$ roots outside of $K$, then $p'$ has at least $n-1$ roots in $K_{\varepsilon}$. This establishes, up to a logarithm, a conjecture of the first author: we also discuss an open problem whose solution would imply the full conjecture.
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An equivalent definition of the Fibonacci numbers is that they are the unique sequence such that every integer can be written uniquely as a sum of non-adjacent terms. We can view this as we have bins of length 1, we can take at most one element from a bin, and if we choose an element from a bin we cannot take one from a neighboring bin. We generalize to allowing bins of varying length and restrictions as to how many elements may be used in a decomposition. We derive conditions on when the resulting sequences have uniqueness of decomposition, and (similar to the Fibonacci case) when the number of summands converges to a Gaussian; the main tool in the proofs here is the Lyaponuv Central Limit Theorem.
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In this article we present a simple and unified probabilistic approach to prove nonexistence of positive super-solutions for system of equations involving potential terms and fractional Laplacian in the exterior domain. The class of problems we consider in this article is quite general compared to the literature.
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In this short note we focus on two recently introduced parameters in the literature, namely `local dimension' (a parameter on partially ordered sets) and `local boxicity' (a parameter on graphs). First we establish a relation between them and then we give an upper bound for both the parameters in terms of the maximum degree of a graph (for local dimension we consider the comparability graph of a poset). Finally, we show that the local boxicity of a graph is upper bounded by its `product dimension'.
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We extend a classical theorem by H. Lewy to planar $\sigma$-harmonic mappings, that is mappings $U$ whose components $u^1$ and $u^2$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u^i)=0$ , for $i=1,2$. A similar result is established for pairs of solutions of certain second order non--divergence equations.
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Changes to network structure can substantially affect when and how widely new ideas, products, and conventions are adopted. In models of biological contagion, interventions that randomly rewire edges (making them "longer") accelerate spread. However, there are other models relevant to social contagion, such as those motivated by myopic best-response in games with strategic complements, in which individual's behavior is described by a threshold number of adopting neighbors above which adoption occurs (i.e., complex contagions). Recent work has argued that highly clustered, rather than random, networks facilitate spread of these complex contagions. Here we show that minor modifications of prior analyses, which make them more realistic, reverse this result. The modification is that we allow very rarely below threshold adoption, i.e., very rarely adoption occurs, where there is only one adopting neighbor. To model the trade-off between long and short edges we consider networks that are the
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Let $s_1, s_2, \ldots$ be the sequence of positive integers, arranged in increasing order, that are representable by any binary quadratic form of fixed discriminant $D$. We show that \[ \limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log s_n} \ge \frac{\varphi(|D|)}{2|D|(1+\log \varphi(|D|))}\gg \frac{1}{\log \log |D|}, \] improving a lower bound of $\frac{1}{|D|}$ of Richards (1982). In the special case of sums of two squares, we improve Richards's bound of $1/4$ to $\frac{195}{449}=0.434\ldots$. We also generalize Richards's result in another direction and establish a lower bound on long gaps between sums of two squares in certain sparse sequences.
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We prove that the existence of a term $s$ satisfying $s(r,a,r,e) = s(a,r,e,a)$ in a general algebraic structure is equivalent to an existence of a term $t$ satisfying $t(x,x,y,y,z,z)=t(y,z,z,x,x,y)$. As a consequence of a general version of this theorem and previous results we get that each strongly connected digraph of algebraic length one, which is compatible with an operation $t$ satisfying an identity of the from $t(\ldots)=t(\ldots)$, has a loop.
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We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level l projective unitary positive-energy representations of any given loop group $LG$. The construction is based on certain supersymmetric conformal field theory models associated with LG in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.
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We study homotopically non-trivial spheres of Legendrians in the standard contact R3 and S3. We prove that there is a homotopy injection of the contactomorphism group of S3 into some connected components of the space of Legendrians induced by the natural action. We also provide examples of loops of Legendrians that are non-trivial in the space of formal Legendrians, and thus non-trivial as loops of Legendrians, but which are trivial as loops of smooth embeddings for all the smooth knot types.
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Meet semidistributive varieties are in a sense the last of the most important classes in universal algebra for which it is unknown whether it can be characterized by a strong Maltsev condition. We present a new, relatively simple Maltsev condition characterizing the meet-semidistributive varieties, and provide a candidate for a strong Maltsev condition.
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We construct pairs of marked groups with isomorphic Cayley graphs but different Borel chromatic numbers for the free parts of their shift graphs. This answers a question of Kechris and Marks. We also show that these graphs have different Baire measurable and measure chromatic numbers, answering analogous versions of the question.
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We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $\omega$. We show that if $\mathcal J$ is an arbitrary $F_\sigma$ ideal, or is any finite or countably iterated Fubini product of $F_\sigma$ ideals, then there are no analytic infinite $\mathcal J$-MAD families, and assuming Projective Determinacy there are no infinite projective $\mathcal J$-MAD families; and under the full Axiom of Determinacy + $V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal J$-mad families. These results apply in particular when $\mathcal J$ is the ideal of finite sets $\mathrm{Fin}$, which corresponds to the classical notion of MAD families. The proofs combine ideas from invariant descriptive set theory and forcing.
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We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of $(\mathbb{N}-\tfrac12)\times(\mathbb{N}-\tfrac12)$ inside triangles with vertices $(0, 0), (0, \lambda \sqrt{\beta})$ and $(\lambda/{\sqrt{\beta}}, 0)$ with respect to $\beta>0$, for fixed $\lambda\geq 0$. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice $(\mathbb{N}+\sigma)\times(\mathbb{N}+\tau)$, for $\sigma, \tau >-1$. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, $\sigma=\tau=-\tfrac12$, is a critic
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The reciprocal degree resistance distance index of a connected graph $G$ is defined as $RDR(G)=\sum\limits_{\{u,v\}\subseteq V(G)}\frac {d_G(u)+d_G(v)}{r_G(u,v)}$, where $r_G(u,v)$ is the resistance distance between vertices $u$ and $v$ in $G$. Let $\mathscr {U}_n$ denote the set of unicyclic graphs with $n$ vertices. We study the graph with maximum reciprocal degree resistance distance index among all graphs in $\mathscr {U}_n$, and characterize the corresponding extremal graph.
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We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.
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Last month when Boeing and SpaceX announced the first astronauts who will fly on their commercial crew spacecraft, several newspapers across the U.S. began publishing an op-ed that criticized the process by which Boeing competitor SpaceX fuels its Falcon 9 rocket. "The first op-ed appeared in a Memphis newspaper a week before the commercial crew announcement," reports Ars Technica. "In recent weeks, copies of the op-ed have also appeared in the Houston Chronicle, various Alabama newspapers, Albuquerque Journal, Florida Today, and The Washington Times." Ars Technica reports: All of these op-eds were bylined by "retired spacecraft operator" Richard Hagar, who worked for NASA during the Apollo program and now lives in Tennessee. (Based upon his limited social media postings, Hagar appears to be more interested in conservative politics than in space these days). Each op-ed cites Hagar's work on NASA's recovery from the Apollo 1 fire and the hard lessons NASA learned that day about human sp
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The Commodore Datasette recording format is heavily optimized for data safety and can compensate for many typical issues of cassette tape, like incorrect speed, inconsistent speed (wow/flutter), and small as well as longer dropouts. This makes the format more complex and way less efficient than, for example, "Turbo Tape" or all other custom formats used by commercial games. Let's explore the format by writing a minimal tape loader for the C64, optimized for size, which can decode correct tapes, but does not support error correction. I'm no expert, but sometimes I wonder if modern computer classes and schools in general are on the right track by focusing solely on modern systems like Chromebooks and iPads. Wouldn't it be better to teach kids programming in BASIC, with limited resources, on, say, C64 emulators?
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Apple chief executive Tim Cook urged consumers not to believe the dominant tech industry narrative that the data collected about them will lead to better services. From a report: In an interview with "Vice News Tonight" that aired Tuesday, Cook highlighted his company's commitment to user privacy, positioning Apple's business as one that stands apart from tech giants that compile massive amounts of personal data and sell the ability to target users through advertising [The link may be paywalled; alternative source]. "The narrative that some companies will try to get you to believe is: I've got to take all of our data to make my service better," he said. "Well, don't believe them. Whoever's telling you that, it's a bunch of bunk." [...] Cook said in the interview that he is "exceedingly optimistic" that the topic of data privacy has reached an elevated level of public debate. "When the free market doesn't produce a result that's great for society you have to ask yourself what do we need
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An anonymous reader quotes a report from Motherboard: Apple has introduced software locks that will effectively prevent independent and third-party repair on 2018 MacBook Pro computers, according to internal Apple documents obtained by Motherboard. The new system will render the computer "inoperative" unless a proprietary Apple "system configuration" software is run after parts of the system are replaced. According to the document, which was distributed to Apple's Authorized Service Providers late last month, this policy will apply to all Apple computers with the "T2" security chip, which is present in 2018 MacBook Pros as well as the iMac Pro. The software lock will kick in for any repair which involves replacing a MacBook Pro's display assembly, logic board, top case (the keyboard, touchpad, and internal housing), and Touch ID board. On iMac Pros, it will kick in if the Logic Board or flash storage are replaced. The computer will only begin functioning again after Apple or a member o
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According to a study published in the journal Molecules, researchers found that six common artificial sweeteners approved by the FDA and 10 sport supplements that contained them were found to be toxic to the digestive gut microbes of mice. CNBC reports: Researchers from Ben-Gurion University of the Negev in Israel and Nanyang Technological University in Singapore tested the toxicity of aspartame, sucralose, saccharine, neotame, advantame, and acesulfame potassium-k. They observed that when exposed to only 1 milligram per milliliter of the artificial sweeteners, the bacteria found in the digestive system became toxic. According to the study, the gut microbial system "plays a key role in human metabolism," and artificial sweeteners can "affect host health, such as inducing glucose intolerance." Additionally, some of the effects of the new FDA-approved sweeteners, such as neotame, are still unknown. However, the study found that mice treated with the artificial sweetener neotame had diffe
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An anonymous reader quotes a report from TechCrunch: Quantum computing represents tremendous promise to completely alter technology as we've known it, allowing operations that weren't previously possible with traditional computing. The downside of these powerful machines is that they could be strong enough to break conventional cryptography schemes. Today, BlackBerry announced a new quantum-resistant code signing service to help battle that possibility. The solution, which will be available next month, is actually the product of a partnership between BlackBerry and Isara Corporation, a company whose mission is to build quantum-safe security solutions. BlackBerry is using Isara's cryptographic libraries to help sign and protect code as security evolves. "By adding the quantum-resistant code signing server to our cybersecurity tools, we will be able to address a major security concern for industries that rely on assets that will be in use for a long time. If your product, whether it's a
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