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This paper discusses the RF design fundamentals of an RF communication system, including the transmission medium, wave propagation, free space path loss, the transmit and receive portion, link design and key components with the intent to provide practical knowledge on the process for designing an RF system.
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Catalin Cimpanu, writing for BleepingComputer: Mozilla will soon block the loading of data URIs in the Firefox navigation bar as part of a crackdown on phishing sites that abuse this protocol. The data: URI scheme (RFC 2397) was deployed in 1998 when developers were looking for ways to embed files in other files. What they came up with was the data: URI scheme that allows a developer to load a file represented as an ASCIIencoded octet stream inside another document. Since then, the URI scheme has become very popular with website developers as it allows them to embed textbased (CSS or JS) files or image (PNG, JPEG) files inside HTML documents instead of loading each resource via a separate HTTP request. This practice became hugely popular because search engines started ranking websites based on their page loading speed and the more HTTP requests a website made, the slower it loaded, and the more it affected a site's SERP position.
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An anonymous reader shares an Associated Press report: Leading researchers castigated a federal plan that would use artificial intelligence methods to scrutinize immigrants and visa applicants, saying it is unworkable as written and likely to be "inaccurate and biased" if deployed. The experts, a group of more than 50 computer and data scientists, mathematicians and other specialists in automated decisionmaking, urged the Department of Homeland Security to abandon the project, dubbed the "Extreme Vetting Initiative." That plan has its roots in President Donald Trump's repeated pledge during the 2016 campaign to subject immigrants seeking admission to the United States to more intense ideological scrutiny  or, as he put it, "extreme vetting." Over the summer, DHS published a "statement of objectives" for a system that would use computer algorithms to scan social media and other material in order to automatically flag undesirable entrants  and to continuously scan the activities of
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From a new wideranging interview of Elon Musk: An unfortunate fact of human nature is that when people make up their mind about something, they tend not to change it  even when confronted with facts to the contrary. "It's very unscientific," Musk says. "There's this thing called physics, which is this scientific method that's really quite effective for figuring out the truth." The scientific method is a phrase Musk uses often when asked how he came up with an idea, solved a problem or chose to start a business. Here's how he defines it for his purposes, in mostly his own words: 1. Ask a question. 2. Gather as much evidence as possible about it. 3. Develop axioms based on the evidence, and try to assign a probability of truth to each one. 4. Draw a conclusion based on cogency in order to determine: Are these axioms correct, are they relevant, do they necessarily lead to this conclusion, and with what probability? 5. Attempt to disprove the conclusion. Seek refutation from others to f
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An anonymous reader quotes a report from Bloomberg: The U.S. Federal Communications Commission under its Republican chairman plans to vote in December to kill the net neutrality rules passed during the Obama era, said two people briefed on the plans. Chairman Ajit Pai in April proposed gutting the rules that he blamed for depressing investment in broadband, and said he intended to "finish the job" this year. The chairman has decided to put his proposal to a vote at the FCC next month, said the people. The agency's monthly meeting is to be held Dec. 14. The people asked not to be identified because the plan hasn't been made public. It's not clear what language Pai will offer to replace the rules that passed with only Democratic votes at the FCC in 2015. He has proposed that the FCC end the designation of broadband companies such as AT&T Inc. and Comcast Corp. as common carriers. That would remove the legal authority that underpins the net neutrality rules. One of the people said Pai
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An anonymous reader shares a report: Amazon is giving Whole Foods shoppers an early gift for the holidays. The grocer announced Wednesday it's slashing prices again, this time on several "holiday staples," including sweet potatoes, canned pumpkin and turkey. If you're an Amazon Prime member, you'll pay even less for turkey: Whole Foods slashed turkey prices to $1.99 per pound (compared to $2.49 for nonPrime members), or $2.99 per pound for an organic turkey ($3.49 for nonPrime members).
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Reader cdreimer writes: According to a report in The Wall Street Journal (possibly paywalled), twothirds of Americans have reported being bullied in the workplace in the last year (up from half in 1989) and boorish behavior by bosses and coworkers are causing companies in lost productivity. The report reads: One of the first things visitors notice when they enter the Irvine, Calif., offices of Bryan Cave LLP is the granite plaque etched with the law firm's 10point code of civility. The gray slab, displayed in the law firm's reception area, proclaims that employees always say please and thank you, welcome feedback and acknowledge the contributions of others. Such rules may seem more at home in a kindergarten than a law firm, but Stuart Price, a longtime partner, says they serve as a daily reminder to keep things civil at work. Incivility  and its more extreme cousin, bullying  is becoming a bigger problem in workplaces. Nearly twothirds of Americans reported that they were bullie
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mirandakatz writes: This summer, Backchannel reported that Anthony Levandowski, the controversial engineer at the heart of the Uber/Waymo lawsuit, had filed paperwork for a new religion called the Way of the Future. Today, investigative reporter Mark Harris has all the details on what that AIbased religion actually likes  and Levandowski granted him his first interview about the new religion and his only public interview since Waymo filed its suit in February. As Levandowski tells him, we can see a hint of how a superhuman intelligence might treat humanity in our current relationships with animals  and that's why it's so important that we treat AI as a god, not a demon to be warded off. "Do you want to be a pet or livestock?" he asks. "We give pets medical attention, food, grooming, and entertainment. But an animal that's biting you, attacking you, barking and being annoying? I don't want to go there."
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An anonymous reader shares a Motherboard report: Every year, Forbes' 30 Under 30 list recognizes people blessed with both youth and exceptional talent in their field  including celebrities, startup founders, doctors, and artists. These are smart, savvy professionals  and when some of them include information security pros, they're bound to go poking around for vulnerabilities. That's what Yan Zhu, a privacy engineer who made the 2015 list, was doing when she found a gaping privacy hole in the way Forbes handles recipients' personal information. Once you make the list, Yan told me in a Twitter direct message, Forbes asks you to register for its annual Under 30 Summit conference. "They send you a link for conference registration, but it's not tied to your email address," she said. "So you can literally enter anyone's email address who is also a 30 Under 30 member and it shows you their personal info." That information carries over into all future years, she said.
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This article, based on a talk, treats some elementary, but not completely simple examples from probability. They concern multiple birthday coincidences, throwing dice, the combinatorics of the German card game "Doppelkopf", and the properties of products of uniformly distributed random numbers. The material, a lot of which was not taken from or found in other sources, should be of interest to all those who lecture or plan to lecture on probability, especially on an elementary or introductory level, and are looking for challenging examples or problems.
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We propose a general theory whose main component are functorial assignments $$\mathbf{\Sigma} \mapsto \Omega_{\mathbf{\Sigma}} \in \mathbf{E}(\mathbf{\Sigma}),$$ for a large class of functors $\mathbf{E}$ from a certain category of bordered surfaces (${\mathbf \Sigma}$'s) to a suitable a target category of topological vector spaces. The construction is done by summing appropriate compositions of the initial data over all homotopy classes of successive excisions of embedded pair of pants. We provide sufficient conditions to guarantee these infinite sums converge and as a result, we can generate mapping class group invariant vectors $\Omega_{\mathbf \Sigma}$ which we call amplitudes. The initial data encode the amplitude for pair of pants and tori with one boundary, as well as the "recursion kernels" used for glueing. We give this construction the name of "geometric recursion", abbreviated GR. As an illustration, we show how to apply our formalism to various spaces of continuous function
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Gammapositivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality. The idea behind it stems from work of Foata, Sch\"utzenberger and Strehl on the Eulerian polynomials; it was revived independently by Br\"and\'en and Gal in the course of their study of poset Eulerian polynomials and face enumeration of flag simplicial spheres, respectively, and has found numerous applications since then. This paper surveys some of the main results and open problems on gammapositivity, appearing in various combinatorial or geometric contexts, as well as some of the diverse methods that have been used to prove it.
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This paper concerns the viscous and nonresistive MHD systems which govern the motion of electrically conducting fluids interacting with magnetic fields. We consider an initialboundary value problem for both compressible and (nonhomogeneous and homogeneous) incompressible fluids in an infinite flat layer. We prove the global wellposedness of the systems around a uniform magnetic field which is vertical to the layer. Moreover, the solution converges to the steady state at an almost exponential rate as time goes to infinity. Our proof relies on a twotier energy method for the reformulated systems in Lagrangian coordinates.
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Augmented Lagrangian method (ALM) has been popularly used for solving constrained optimization problems. Its convergence and local convergence speed have been extensively studied. However, its global convergence rate is still open for problems with nonlinear inequality constraints. In this paper, we work on general constrained convex programs. For these problems, we establish the global convergence rate of ALM and its inexact variants. We first assume exact solution to each subproblem in the ALM framework and establish an $O(1/k)$ ergodic convergence result, where $k$ is the number of iterations. Then we analyze an inexact ALM that approximately solves the subproblems. Assuming summable errors, we prove that the inexact ALM also enjoys $O(1/k)$ convergence if smaller stepsizes are used in the multiplier updates. Furthermore, we apply the inexact ALM to a constrained composite convex problem with each subproblem solved by Nesterov's optimal firstorder method. We show that $O(\varepsilo
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In this paper, we show that the maximal divisible subgroup of groups $K_1$ and $K_2$ of an elliptic curve $E$ over a function field is uniquely divisible. Further those $K$groups modulo this uniquely divisible subgroup are explicitly computed. We also calculate the motivic cohomology groups of the minimal regular model of $E$, which is an elliptic surface over a finite field.
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We use the Haar function system in order to study the $L_2$ discrepancy of a class of digital $(0,n,2)$nets. Our approach yields exact formulas for this quantity, which measures the irregularities of distribution of a set of points in the unit interval. We will obtain such formulas not only for the classical digital nets, but also for shifted and symmetrized versions thereof. The basic idea of our proofs is to calculate all Haar coefficents of the discrepancy function exactly and insert them into Parseval's identity. We will also discuss reasons why certain (symmetrized) digital nets fail to achieve the optimal order of $L_2$ discrepancy and use the LittlewoodPaley inequality in order to obtain results on the $L_p$ discrepancy for all $p\in (1,\infty)$.
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We investigate the precoding, remote radio head (RRH) selection and signal splitting in the simultaneous wireless information and power transferring (SWIPT) cloud radio access networks \mbox{(CRANs)}. The objective is to minimize the power consumption of the SWIPT CRAN. Different from the existing literature, we consider the nonlinear fronthaul power consumption and the multiple antenna RRHs. By switching off the unnecessary RRHs, the group sparsity of the precoding coefficients is introduced, which indicates that the precoding process and the RRH selection are coupled. In order to overcome these issues, a group sparse precoding and signal splitting algorithm is proposed based on the majorizationminimization framework, and the convergence behavior is established. Numerical results are used to verify our proposed studies.
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In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multivalued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show that the same upper bound holds for the dimension of the singular set of an area minimizing $n$dimensional rectifiable current of arbitrary codimension. In either case, the dimension bound is sharp. We develop estimates to study the asymptotic behaviour of a multivalued Dirichlet energy minimizer on approach to its singular set. Our estimates imply that a Dirichlet energy minimizer at ${\mathcal H}^{n2}$ a.e. point of its singular set has a unique set of homogeneous multivalued cylindrical tangent functions (blowups) to which the minimizer, modulo a set of singlevalued harmonic functions, decays exponentially fast upon rescaling. A corollary is that the singular set
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For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none as forbidden. We examine structure of graphs whose degree sequences contain either forced or forbidden edges. Among the things we show, we determine the structure of the forced or forbidden edge sets, the relationship between the sizes of forced and forbidden sets for a sequence, and the resulting structural consequences to their realizations. This includes showing that the diameter of every realization of a degree sequence containing forced or forbidden edges is no greater than 3, and that these graphs are maximally edgeconnected.
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The change in Holographic entanglement entropy (HEE) for small fluctuations about pure anti De Sitter (AdS) is obtained by a perturbative expansion of the area functional in terms of the change in the bulk metric and the embedded extremal surface. However, it is known that change in the embedding appears in second order or higher. It was shown that these changes in the embedding can be calculated in the $2+1$ dimensional case by solving a generalized geodesic deviation equation. We generalize this result to arbitrary dimensions by deriving an inhomogeneous form of the Jacobi equation for minimal surfaces. The solutions of this equation map a minimal surface in a given space time to a minimal surface in a space time which is a perturbation over the initial space time. Using this we perturbatively calculate the changes in HEE upto second order for boosted black brane like perturbations over $AdS 4$ .
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We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for secondorder elliptic problems. Our method is obtained by inserting the $L^2$orthogonal projection onto the approximate space for a numerical trace into all facet integrals in the usual HDG formulation. The orders of convergence for all variables are optimal if we use polynomials of degree $k+l$, $k+1$ and $k$, where $k$ and $l$ are any nonnegative integers, to approximate the vector, scalar and trace variables, which implies that our method can achieve superconvergence for the scalar variable without postprocessing. Numerical results are presented to verify the theoretical results.
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In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the timeharmonic Maxwell's equations in waveguide configurations.The implementation of high order curlconforming finite elements is quite delicate, especially in the threedimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (this http URL). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easytouse but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell's equations in the timeharmonic regime, which is an underdeveloped issue in the literature, particularly for hi
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The embedded discontinuous Galerkin (EDG) method by Cockburn et al. [SIAM J. Numer. Anal., 2009, 47(4), 26862707] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from discontinuous functions to continuous ones, and adopts piecewise polynomials of equal degrees on simplex meshes for all variables. In this paper, we analyze a new EDG method for second order elliptic problems on polygonal/polyhedral meshes. By using piecewise polynomials of degrees $k+1$, $k+1$, $k$ ($k\geq 0$) to approximate the potential, numerical trace and flux, respectively, the new method is shown to yield optimal convergence rates for both the potential and flux approximations. Numerical experiments are provided to confirm the theoretical results.
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Let $D$ be a bounded domain in $\mathbb C^n$. We study approximation of (not necessarily bounded from above) $m$subharmonic function $D$ by continuous $m$subharmonic ones defined on neighborhoods of $\overline{D}$. We also consider the existence of a $m$subharmonic function on $D$ whose boundary values coincides with a given real valued continuous function on $\partial D$ except for a sufficiently small subset of $\partial D.$
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Kenmotsu's formula describes surfaces in Euclidean 3space by their mean curvature functions and Gauss maps. In Lorentzian 3space, AkutagawaNishikawa's formula and Magid's formula are Kenmotsutype formulas for spacelike surfaces and for timelike surfaces, respectively. We apply them to a few problems concerning rotational or helicoidal surfaces with constant mean curvature. Before that, we show that the three formulas above can be written in a unified single equation.
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This paper analyzes a timestepping discontinuous Galerkin method for modified anomalous subdiffusion problems with two time fractional derivatives of orders $ \alpha $ and $ \beta $ ($ 0 < \alpha < \beta < 1 $). The stability of this method is established, the temporal accuracy of $ O(\tau^{m+1\beta/2}) $ is derived, where $m$ denotes the degree of polynomials for the temporal discretization. It is shown that, even the solution has singularity near $ t = {0+} $, this temporal accuracy can still be achieved by using the graded temporal grids. Numerical experiments are performed to verify the theoretical results.
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Servicing the school transportation demand safely with a minimum number of buses is one of the highest financial goals for school transportation directors. To achieve that objective, a good and efficient way to solve the routing and scheduling problem is required. Due to the growth of the computing power, the spotlight has been shed on solving the combined problem of the school bus routing and scheduling. A recent attempt tried to model the routing problem by maximizing the trip compatibilities with the hope of requiring fewer buses in the scheduling problem. However, an overcounting problem associated with trip compatibility could diminish the performance of this approach. An extended model is proposed in this paper to resolve this issue along with an iterative solution algorithm. This extended model is an integrated model for multischool bus routing and scheduling problem. The result shows better solutions for 8 test problems can be found with a fewer number of buses (up to 25%) an
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We consider SturmLiouville operators on geometrical graphs without cycles (trees) with singular potentials from the class $W_2^{1}$. We suppose that the potentials are known on a part of the graph, and study the socalled partial inverse problem, which consists in recovering the potentials on the remaining part of the graph from some parts of several spectra. The main results of the paper are the uniqueness theorem and a constructive procedure for the solution of the partial inverse problem. Our method is based on the completeness and the Rieszbasis property of special systems of vector functions, and the reduction of the partial inverse problem to the complete one on a part of the graph.
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A new lower bound for the maximal length of a multivector is obtained. It is much closer to the best known upper bound than previously reported lower bound estimates. The maximal length appears to be unexpectedly large for $n$vectors, with n>2, since the few exactly known values seem to grow linearly with vector space dimension, whereas the new lower bound has a polynomial order equal to n1 like the best known upper bound. This result has implications for quantum chemistry.
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This work presents an extension of discretely entropy stable discontinuous Galerkin (DG) methods to the resistive magnetohydrodynamics (MHD) equations. Although similar to the compressible NavierStokes equations at first sight, there are some important differences concerning the resistive MHD equations that need special focus. The continuous entropy analysis of the ideal MHD equations, which are the advective parts of the resistive MHD equations, shows that the divergencefree constraint on the magnetic field components must be incorporated as a nonconservative term in a form either proposed by Powell or Janhunen. Consequently, this nonconservative term needs to be discretized, such that the approximation is consistent with the entropy. As an extension of the ideal MHD system, we address in this work the continuous analysis of the resistive MHD equations and show that the entropy inequality holds. Thus, our first contribution is the proof that the resistive terms are symmetric and p
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The supply of electrical energy is being increasingly sourced from renewable generation resources. The variability and uncertainty of renewable generation, compared to a dispatchable plant, is a significant dissimilarity of concern to the traditionally reliable and robust distribution systems. In order to reach the optimal operation of community Microgrids including various Distributed Energy Resource, the stochastic nature of renewable generation should be considered in the decisionmaking process. To this end, this paper proposes a stochastic scenario based model for optimal dynamic energy management of Microgrids with the goal of cost and emission minimization as well as reliability maximization. In the proposed model, the uncertainties of load consumption and also, the available output power of wind and photovoltaic units are modeled by a scenariobased stochastic programming. Through this method, the inherent stochastic nature of the proposed problem is released and the proble
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Distributed optimization algorithms are essential for training machine learning models on very largescale datasets. However, they often suffer from communication bottlenecks. Confronting this issue, a communicationefficient primaldual coordinate ascent framework (CoCoA) and its improved variant CoCoA+ have been proposed, achieving a convergence rate of $\mathcal{O}(1/t)$ for solving empirical risk minimization problems with Lipschitz continuous losses. In this paper, an accelerated variant of CoCoA+ is proposed and shown to possess a convergence rate of $\mathcal{O}(1/t^2)$ in terms of reducing suboptimality. The analysis of this rate is also notable in that the convergence rate bounds involve constants that, except in extreme cases, are significantly reduced compared to those previously provided for CoCoA+. The results of numerical experiments are provided to show that acceleration can lead to significant performance gains.
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Let $H$ be a transfer Krull monoid over a finite ablian group $G$ (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit $a \in H$ can be written as a product of irreducible elements, say $a = u_1 \ldots u_k$, and the number of factors $k$ is called the length of the factorization. The set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. It is classical that the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths depends only on the group $G$, and a standing conjecture states that conversely the system $\mathcal L (H)$ is characteristic for the group $G$. Let $H'$ be a further transfer Krull monoid over a finite ablian group $G'$ and suppose that $\mathcal L (H)= \mathcal L (H')$. We prove that, if $G\cong C_n^r$ with $r\le n3$ or ($r\ge n1\ge 2$ and $n$ is a prime power), then $G$ and $G'$ are isomorphic.
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We prove that any rigid representation of $\pi_1\Sigma_g$ in $\mathrm{Homeo}_+(S^1)$ with Euler number at least $g$ is necessarily semiconjugate to a discrete, faithful representation into $\mathrm{PSL}(2,\mathbb{R})$. Combined with earlier work of Matsumoto, this precisely characterizes Fuchsian actions by a topological rigidity property. Though independent, this work can be read as an introduction to the companion paper {\em rigidity and geometricity for surface group actions on the circle}, by the same authors.
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Dimension profiles were introduced by Falconer and Howroyd to provide formulae for the boxcounting and packing dimensions of the orthogonal projections of a set E or a measure on Euclidean space onto almost all mdimensional subspaces. The original definitions of dimension profiles are somewhat awkward and not easy to work with. Here we rework this theory with an alternative definition of dimension profiles in terms of capacities of E with respect to certain kernels, and this leads to the boxcounting dimensions of projections and other images of sets relatively easily. We also discuss other uses of the profiles, such as the information they give on exceptional sets of projections and dimensions of images under certain stochastic processes. We end by relating this approach to packing dimension.
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The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set $P$ in a Cartesian square $\mathbb{F}_p^n\times\mathbb{F}_p^n$. More precisely, we perform horizontal and vertical sums and differences on $P$, that is, operations on the second coordinate when the first one is fixed, or vice versa. The structure we find is the zero set of a family of bilinear forms on a Cartesian product of vector subspaces. The codimensions of the subspaces and the number of bilinear forms involved are bounded by a function $c(\delta)$ of the density $\delta=\lvert P\rvert/p^{2n}$ only. The proof uses various tools of additive combinatorics, such as the (linear) Bogolyubov theorem, the density increment method, as well as the BalogSzem\'er\'ediGowers and FreimanRuzsa theorems.
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Bender and Canfield proved in 1991 that the generating series of maps in higher genus is a rational function of the generating series of planar maps. In this paper, we give the first bijective proof of this result. Our approach starts with the introduction of a canonical orientation that enables us to construct a bijection between $4$valent bicolorable maps and a family of unicellular blossoming maps.
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In a recent paper, Duval, Goeckner, Klivans and Martin disproved the longstanding conjecture by Stanley, that every CohenMacaulay simplicial complex is partitionable. We construct counterexamples to this conjecture that are even \emph{balanced}, i.e., their underlying graph has a minimal coloring. This answers a question by Duval et al. in the negative.
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We here present a general and tractable model for lineofsight (LOS) scenarios, which is based on the product of two independent and nonidentically distributed $\kappa$$\mu$ shadowed random variables. Simple closedform expressions for the probability density function, cumulative distribution function and momentgenerating function are derived, which are as tractable as the corresponding expressions derived from a product of Nakagami$m$ random variables. This model simplifies the challenging characterization of LOS product channels, as well as combinations of LOS with nonLOS ones. It also shows a higher flexibility when fitting measurements with respect to exact approaches based on the Rician distribution.
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We consider a distribution grid used to charge electric vehicles subject to voltage stability and various other constraints. We model this as a class of resourcesharing networks known as bandwidthsharing networks in the communication network literature. Such networks have proved themselves to be an effective flowlevel model of data traffic in wired and wireless networks. We focus on resource sharing networks that are driven by a class of greedy control rules that can be implemented in a decentralized fashion. For a large number of such control rules, we can characterize the performance of the system, subject to voltage stability constraints, by a fluid approximation. This leads to a set of dynamic equations that take into account the stochastic behavior of cars. We show that the invariant point of these equations is unique and can be computed by solving a specific ACOPF problem, which admits an exact convex relaxation. For the class of weighted proportional fairness control, we show
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In this paper we introduce the polynomials $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$ given by $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{xr}{nk} \ (n\ge 0)$, $D_0^{(r)}(x)=1,\ D_1^{(r)}(x)=x$ and $D_{n+1}^{(r)}(x)=xD_n^{(r)}(x)n(n+2r)D_{n1}^{(r)}(x)\ (n\ge 1).$ We show that $\{D_n^{(r)}(x)\}$ are orthogonal polynomials for $r>\frac 12$, and establish many identities for $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$, especially obtain a formula for $d_n^{(r)}(x)^2$ and the linearization formulas for $d_m^{(r)}(x)d_n^{(r)}(x)$ and $D_m^{(r)}(x)D_n^{(r)}(x)$. As an application we extend recent work of Sun and Guo.
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We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real $r\in (0,1)$ and an integer $d\geq 2$, let $N(r,d)$ denote the minimum number of points inside the $d$dimensional unit cube $[0,1]^d$ such that they intersect every axisaligned box inside $[0,1]^d$ of volume greater than $r$. We prove an upper bound on $N(r,d)$, matching a lower bound of Aistleitner et al. up to a multiplicative constant depending only on $r$. This fully determines the rate of growth of $N(r,d)$ if $r\in(0,1)$ is fixed.
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The SturmLiouville operator with singular potentials on the lasso graph is considered. We suppose that the potential is known a priori on the boundary edge, and recover the potential on the loop from a part of the spectrum and some additional data. We prove the uniqueness theorem and provide a constructive algorithm for the solution of this partial inverse problem.
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In this paper we present aposteriori KAM results for existence of $d$dimensional isotropic invariant tori for $n$DOF Hamiltonian systems with additional $nd$ independent first integrals in involution. We carry out a covariant formulation that does not require the use of actionangle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) isoenergetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.
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Usually, the additivity property is inherent to the systems obeying linear equations of motion. In this paper I will show that in the systems obeying the nonlinear Schrodinger equation with the general form of nonlinearity, the additivity property is valid at least for the basic integrals of motion calculated for the nonlinear perturbations against a stationary background solution.
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We study robust PCA for the fully observed setting, which is about separating a low rank matrix $\boldsymbol{L}$ and a sparse matrix $\boldsymbol{S}$ from their sum $\boldsymbol{D}=\boldsymbol{L}+\boldsymbol{S}$. In this paper, a new algorithm, termed accelerated alternating projections, is introduced for robust PCA which accelerates existing alternating projections proposed in [Netrapalli, Praneeth, et al., 2014]. Let $\boldsymbol{L}_k$ and $\boldsymbol{S}_k$ be the current estimates of the low rank matrix and the sparse matrix, respectively. The algorithm achieves significant acceleration by first projecting $\boldsymbol{D}\boldsymbol{S}_k$ onto a low dimensional subspace before obtaining the new estimate of $\boldsymbol{L}$ via truncated SVD. Exact recovery guarantee has been established which shows linear convergence of the proposed algorithm. Empirical performance evaluations establish the advantage of our algorithm over other stateoftheart algorithms for robust PCA.
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We characterize even measures $\mu=wdx+\mu_s$ on the real line with finite entropy integral $\int_{R} \frac{\log w(t)}{1+t^2}dt>\infty$ in terms of $2\times 2$ Hamiltonian generated by $\mu$ in the sense of inverse spectral theory. As a corollary, we obtain criterion for spectral measure of Krein string to have converging logarithmic integral.
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This paper provides a simple method to extract the zeros of some weight two Eisenstein series of level $N$ where $N=2,3,5$ and $7$. We show that $\tilde{E}_N$ is integral over the polynomial ring $M(Sl_2(\mathbb{Z}))=\bigoplus_{k\ge 0}M_k(Sl_2(\mathbb{Z}))$ for these values of $N$; and thus the zeros of $\tilde{E}_N$ are `controlled' by those of $E_4$ and $E_6$ in the fundamental domain of $\Gamma_0(N)$.
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We investigate a descent on simple graphs, starting with the complete graph on $n$ vertices and ending up with the cycle graph by removing one edge after another. We obtain quantitative results showing that graphs with small diameter must have some eigenvalues of large algebraic degree.
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We consider a group of computation units trying to cooperatively solve a distributed optimization problem with shared linear equality and inequality constraints. Assuming that the computation units are communicating over a network whose topology is described by a timeinvariant directed graph, by combining saddlepoint dynamics with Lie bracket approximation techniques we derive a methodology that allows to design distributed continuoustime optimization algorithms that solve this problem under minimal assumptions on the graph topology as well as on the structure of the constraints. We discuss several extensions as well as special cases in which the proposed procedure becomes particularly simple.
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Heavytailed errors impair the accuracy of the least squares estimate, which can be spoiled by a single grossly outlying observation. As argued in the seminal work of Peter Huber in 1973 [{\it Ann. Statist.} {\bf 1} (1973) 799821], robust alternatives to the method of least squares are sorely needed. To achieve robustness against heavytailed sampling distributions, we revisit the Huber estimator from a new perspective by letting the tuning parameter involved diverge with the sample size. In this paper, we develop nonasymptotic concentration results for such an adaptive Huber estimator, namely, the Huber estimator with the tuning parameter adapted to sample size, dimension, and the variance of the noise. Specifically, we obtain a subGaussiantype deviation inequality and a nonasymptotic Bahadur representation when noise variables only have finite second moments. The nonasymptotic results further yield two conventional normal approximation results that are of independent interest, th
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A graph $\Gamma$ labelled by a set $S$ defines a group $G(\Gamma)$ whose generators are the set of labels $S$ and whose relations are all words which can be read on closed paths of this graph. We introduce the notion of aspherical graph and prove that such a graph defines an aspherical group presentation. This result generalizes a theorem of Dominik Gruber on graphs satisfying graphical $C(6)$condition.
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A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that persistence of a compact geodesic locally contractible space $X$ carries a lot of geometric information. However, by definition the corresponding Rips complexes have uncountably many vertices. In this paper we show that nonetheless, the whole persistence of $X$ may be obtained by an appropriate finite sample (subset of $X$), and that persistence of any subset of $X$ is well interleaved with the persistence of $X$. It follows that the persistence of $X$ is the minimum of persistences obtained by all finite samples. Furthermore, we prove a much improved Stability theorem for such approximations. As a special case we provide for each $r>0$ a density $s>0$, so that for eac
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This paper derives the Feynman rules for the diagrammatic perturbation expansion of the effective action around an arbitrary solvable problem. The perturbation expansion around a Gaussian theory is well known and composed of oneline irreducible diagrams only. For the expansions around an arbitrary, nonGaussian problem, we show that a more general class of irreducible diagrams remains in addition to a second set of diagrams that has no analogue in the Gaussian case. The effective action is central to field theory, in particular to the study of phase transitions, symmetry breaking, effective equations of motion, and renormalization. We exemplify the method on the Ising model, where the effective action amounts to the Gibbs free energy, recovering the ThoulessAndersonPalmer meanfield theory in a fully diagrammatic derivation. Higher order corrections follow with only minimal effort compared to existing techniques. Our results show further that the Plefka expansion and the hightemper
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We show existence of a unique solution and a comparison theorem for a onedimensional backward stochastic differential equation with jumps that emerge from a L\'evy process. The considered generators obey a timedependent extended monotonicity condition in the yvariable and have linear timedependent growth. Within this setting, the results generalize those of Royer (2006), Yin and Mao (2008) and, in the $L^2$case with linear growth, those of Kruse and Popier (2016). Moreover, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we consider BSDEs where the Poisson random measure admits only jumps of size larger than $1/n$. We show convergence of their solutions to those of the original BSDE, as $n \to \infty.$ The proofs only rely on It\^o's formula and the BihariLaSalle inequality and do not use Girsanov transforms.
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The Lowest Landau Level (LLL) equation emerges as an accurate approximation for a class of dynamical regimes of BoseEinstein Condensates (BEC) in twodimensional isotropic harmonic traps in the limit of weak interactions. Building on recent developments in the field of spatially confined extended Hamiltonian systems, we find a fully nonlinear solution of this equation representing periodically modulated precession of a single vortex. Motions of this type have been previously seen in numerical simulations and experiments at moderately weak coupling. Our work provides the first controlled analytic prediction for trajectories of a single vortex, suggests new targets for experiments, and opens up the prospect of finding analytic multivortex solutions.
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Recovery of population size history from sequence data and testing of hypotheses about features of population size history are important problems in population genetics. Inference commonly relies on a coalescentbased model linking the population size history to genealogies. We consider the problem of recovering the true population size history from two possible alternatives on the basis of coalescent time data. We give exact expressions for the probability of selecting the correct alternative in a variety of biologically interesting cases as a function of the separation between the alternative size histories, the number of genealogies and loci sampled, and the sampling times. The results are applied to human population history. As coalescent times are inferred from sequence data rather than directly observed, the inferential limits we give can be viewed as optimistic.
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The first part of this article gives error bounds for approximations of Markov kernels under FosterLyapunov conditions. The basic idea is that when both the approximating kernel and the original kernel satisfy a FosterLyapunov condition, the longtime dynamics of the two chains  as well as the invariant measures, when they exist  will be close in a weighted total variation norm, provided that the approximation is sufficiently accurate. The required accuracy depends in part on the Lyapunov function, with more stable chains being more tolerant of approximation error. We are motivated by the recent growth in proposals for scaling Markov chain Monte Carlo algorithms to large datasets by defining an approximating kernel that is faster to sample from. Many of these proposals use only a small subset of the data points to construct the transition kernel, and we consider an application to this class of approximating kernel. We also consider applications to distribution approximations in G
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Accommodating MachinetoMachine applications and their requirements is one of the challenges on the way from LTE towards 5G networks. The envisioned high density of devices, alongside with their sporadic and synchronized transmission patterns, might create signaling storms and overload in the current LTE network. Here, we address the notorious random access (RA) challenge, namely, scalability of the radio link connection establishment protocol in LTE networks. We revisit the binary countdown technique for contention resolution (BCCR), and apply it to the LTE RA procedure. We analytically investigate the performance gains and tradeoffs of applying BCCR in LTE. We further simulatively compare BCCR RA with the stateoftheart RA techniques, and demonstrate its advantages in terms of delay and throughput.
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This work is about a slowfast data assimilation system when only slow components are observable. First, we obtain its low dimensional reduction via an invariant slow manifold. Second, we prove that the low dimensional filter on the slow manifold approximates the original filter in a suitable metric. Finally, we illustrate this approximate filter numerically in an example.
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Let $B_{H}(t), t\geq [0,T], T\in(0,\infty)$ be the standard Multifractional Brownian Motion(mBm), in this contribution we are concerned with the exact asymptotics of \begin{eqnarray*} \mathbb{P}\left\{\sup_{t\in[0,T]}B_{H}(t)>u\right\} \end{eqnarray*} as $u\rightarrow\infty$. Mainly depended on the structures of $H(t)$, the results under several important cases are investigated.
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