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Some data is linearly additive, other data is not. In this paper, I discuss types of data based on the boundedness of the data and their linearity. 1) Unbounded data can be linear. 2) Oneside bounded data is usually log transformed to be linear. 3) Twoside bounded data is not linear. 4) Untidy data do not fit in these categories. An example of twosided bounded data is probabilities which should be transformed into a linear probability space by taking the logarithm of the odds ratio (log10 odds) which is termed Weight (W). Calculations of means and standard deviation is more accurate when calculated as W values than when calculated as probabilities. A methods to analyze untidy data is discussed.
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In this work we compute the Dixmier invariants and bitangents of the plane quartics with 3,6 or 9cyclic automorphisms, we find that a quartic curve with 6cyclic automorphism will have 3 horizontal bitangents which form an asysgetic triple. We also discuss the linear matrix representation problem of such curves, and find a degree 6 equation of 1 variable which solves the symbolic solution of the linear matrix representation problem for the curve with 6cyclic automorphism.
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In order to study gravitational waves in any realistic astrophysical scenario, one must consider geometry perturbations up to second order. Here, we present a general technique for studying linear and quadratic perturbations on a spacetime with torsion. Besides the standard metric mode, a "torsionon" perturbation mode appears. This torsional mode will be able to propagate only in a certain kind of theories.
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A FogRadio Access Network (FRAN) is studied in which cacheenabled Edge Nodes (ENs) with dedicated fronthaul connections to the cloud aim at delivering contents to mobile users. Using an informationtheoretic approach, this work tackles the problem of quantifying the potential latency reduction that can be obtained by enabling DevicetoDevice (D2D) communication over outofband broadcast links. Following prior work, the Normalized Delivery Time (NDT)  a metric that captures the high signaltonoise ratio worstcase latency  is adopted as the performance criterion of interest. Joint edge caching, downlink transmission, and D2D communication policies based on compressandforward are proposed that are shown to be informationtheoretically optimal to within a constant multiplicative factor of two for all values of the problem parameters, and to achieve the minimum NDT for a number of special cases. The analysis provides insights on the role of D2D cooperation in improving the de
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We consider the motion of a point mass in a onedimensional viscous compressible barotropic fluid. The fluidpoint mass system is governed by the barotropic compressible NavierStokes equations and Newton's equation of motion. Our main result concerns the long time behavior of the fluid and the point mass; it gives pointwise convergence estimates of the density and the velocity of the fluid to their equilibrium values. As a corollary, it shows that the fluid velocity $U(x,t)$ and the point mass velocity $V(t)=U(h(t)\pm 0,t)$, where $h(t)$ is the location of the point mass, decay differently as $U(\cdot,t)_{L^{\infty}(\mathbb{R}\backslash \{ h(t) \})}\approx t^{1/2}$ and $V(t)\lesssim t^{3/2}$. This discrepancy between the decay rates of $U(\cdot,t)_{L^{\infty}(\mathbb{R}\backslash \{ h(t) \})}$ and $V(t)=U(h(t)\pm 0,t)$ is due to the hyperbolicparabolic nature of the problem: The fluid velocity decays slower on the characteristics $x=\pm ct$, where $c$ is the speed o
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We prove linear inviscid damping near a general class of monotone shear flows in a finite channel, in Gevrey spaces. It is an essential step towards proving nonlinear inviscid damping for general shear flows that are not close to the Couette flow, which is a major open problem in 2d Euler equations.
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In this paper, we realize polynomial $\H$modules $\Omega(\lambda,\alpha,\beta)$ from irreducible twisted HeisenbergVirasoro modules $\A_{\alpha,\beta}$. It follows from $\H$modules $\Omega(\lambda,\alpha,\beta)$ and $\mathrm{Ind}(M)$ that we obtain a class of natural nonweight tensor product modules $\big(\bigotimes_{i=1}^m\Omega(\lambda_i,\alpha_i,\beta_i)\big)\otimes \mathrm{Ind}(M)$. Then we give the necessary and sufficient conditions under which these modules are irreducible and isomorphic, and also give that the irreducible modules in this class are new.
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In this work, we revisit a discrete implementation of the primaldescent dualascent gradient method applied to saddlepoint optimization problems. Through a short proof, we establish linear (exponential) convergence of the algorithm for stronglyconvex cost functions with Lipschitz continuous gradients. Unlike previous studies, we provide explicit upper bounds on the stepsize parameters for stable behavior and on the resulting convergence rate and highlight the importance of having two different primal and dual stepsizes. We also explain the effect of the augmented Lagrangian penalty term on the algorithm stability and performance for the distributed minimization of aggregate cost functions over multiagent networks.
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We introduce a method to improve the tractability of the wellknown Sample Average Approximation (SAA) without compromising important theoretical properties, such as convergence in probability and the consistency of an independent and identically distributed (iid) sample. We consider each scenario as a polyhedron of the mix of firststage and secondstage decision variables. According to John's theorem, the LownerJohn ellipsoid of each polyhedron will be unique which means that different scenarios will have correspondingly different LownerJohn ellipsoids. By optimizing the objective function regarding both feasible regions of the polyhedron and its unique LownerJohn ellipsoid, respectively, we obtain a pair of optimal values, which would be a coordinate on a twodimensional plane. The scenarios, whose coordinates are close enough on the plane, will be treated as one scenario; thus our method reduces the sample size of an iid sample considerably. Instead of using a large iid sample d
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This paper considers dilations and translations of lines in the Desargues affine plane. A dilation of a line transforms each line into a parallel line whose length is a multiple of the length of the original line. In addition to the usual Playfair axiom for parallel lines in an affine plane, further conditions are given for distinct lines to be parallel in the Desargues affine plane. This paper introduces the dilation of parallel lines in a finite Desargues affine plane that is a bijection of the lines. Two main results are given in this paper, namely, each dilation in a finite Desarguesian plane is an isomorphism between skew fields constructed over isomorphic lines and each dilation in a finite Desarguesian plane occurs in a Pappian space.
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It is often necessary to disclose training data to the public domain, while protecting privacy of certain sensitive labels. We use information theoretic measures to develop such privacy preserving data disclosure mechanisms. Our mechanism involves perturbing the data vectors in a manner that strikes a balance in the privacyutility tradeoff. We use maximal information leakage between the output data vector and the confidential label as our privacy metric. We first study the theoretical BernoulliGaussian model and study the privacyutility tradeoff when only the mean of the Gaussian distributions can be perturbed. We show that the optimal solution is the same as the case when the utility is measured using probability of error at the adversary. We then consider an application of this framework to a data driven setting and provide an empirical approximation to the Sibson mutual information. By performing experiments on the MNIST and FERG datasets, we show that our proposed framework a
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In this paper, we construct some families of infinitely many hyperelliptic curves of genus $2$ with exactly two rational points. In the proof, we first show that the MordellWeil ranks of these hyperelliptic curves are $0$ and then determine the sets of rational points by using the LutzNagell type theorem for hyperelliptic curves which was proven by Grant.
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Linux Journal celebrates 25 years since it began publishing. "Most magazines have the life expectancy of a house plant. Such was the betting line for Linux Journal when it started in April 1994. Our budget was a shoestring. The closest our owner, SSC (Specialized System Consultants) came to the magazine business was with the reference cards it published for UNIX, C, VI, Java, Bash and so on."
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Linux.com interviews Richard Hughes about the Linux Vendor Firmware Service (LVFS), which has recently joined the Linux Foundation as a new project. Hughes is the founder and maintainer of the project. "The shortterm goal was to get 95% of updatable consumer hardware supported. With the recent addition of HP that's now a realistic target, although you have to qualify the 95% with 'new consumer nonenterprise hardware sold this year' as quite a few vendors will only support hardware no older than a few years at most, and most still charge for firmware updates for enterprise hardware. My longterm goal is for the LVFS to be seen like a boring, critical part of infrastructure in Linux, much like you’d consider an NTP server for accurate time, or a PGP keyserver for trust. With the recent Spectre and Meltdown issues hitting the industry, firmware updates are no longer seen as something that just adds support for new hardware or fixes the occasional hardware issue. Now the EFI BIO
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As Apple continues to fight legislation that would make it easier for consumers to repair their iPhones, MacBooks, and other electronics, the company appears to be able to implement many of the requirements of the legislation, according to an internal presentation obtained by Motherboard. According to the presentation, titled “Apple Genuine Parts Repair” and dated April 2018, the company has begun to give some repair companies access to Apple diagnostic software, a wide variety of genuine Apple repair parts, repair training, and notably places no restrictions on the types of repairs that independent companies are allowed to do. The presentation notes that repair companies can “keep doing what you’re doing, with … Apple genuine parts, reliable parts supply, and Apple process and training.” This is, broadly speaking, what right to repair activists have been asking state legislators to require companies to offer for years. At this point, Apple’s fight against right to repair is basi
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A user by the name of grem75 has uploaded two screenshots of KDE 0.1 to imgur, and they offer a very intriguing look at just how far we’ve come. I’ve only found this RPM, no source unfortunately. This is installed on Red Hat 4.1 with Qt 1.33. Impressive amount of progress for being so early in development. The project had been announced in October 1996, this package was built in February 1997. There really were no complete desktop environments available for Linux at the time, most distros shipped with FVWM and some assortment of applications from various toolkits. Gnome didn’t start until August of 1997. XFCE existed, but was just a panel for FVWM. I’ve recently made the jump from Windows 10 to KDE Neon on my laptop, and after so many rocky years through KDE 4.x, I have to say the KDE desktop environment currently exists in an incredibly polished and attractive state, striking a perfect balance between attractiveness, usability, and customisability. KDE is curre
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There’s another mobile operating system on the rise, but this one is special for a few reasons. First, it’s not necessarily trying to unseat iOS and Android — it’s designed to run on feature phones. It also has received significant investment from Google, and in most cases, Assistant and other Google applications are preinstalled. The operating system in question is ‘KaiOS,’ and it’s already shipping on a handful of phones, including the 4G version of the Nokia 8810 and the Jio JioPhone. I’ve been using KaiOS for a while on the Maxcom MK241, and while it’s definitely better than the average feature phone, it still has rough edges. A KaiOS device is definitely on my list of devices, since it’s a popular operating system I haven’t yet had the chance to try. I like the idea of having a more focused, less capable device, with better battery life and less distractions.
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We study the limiting behavior of the eigenvalues of KreinFellerOperators with respect to weakly convergent probability measures. Therefore, we give a representation of the eigenvalues as zeros of measure theoretic sine functions. Further, we make a proposition about the limiting behavior of the previously determined eigenfunctions. With the main results we finally determine the speed of convergence of eigenvalues and functions for sequences which converge to invariant measures on the Cantor set.
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Facebook seems interested in speeding up blockchain smart contracts—but w
0206 MIT Technology 13304This is a review of the geometry of quantum states using elementary methods and pictures. Quantum states are represented by a convex body, often in high dimensions. In the case of nqubits, the dimension is exponentially large in n. The space of states can be visualized, to some extent, by its simple cross sections: Regular simplexes, balls and hyperoctahedra. When the dimension gets large there is a precise sense in which the space of states resembles, almost in every direction, a ball. The ball turns out to be a ball of rather low purity states. We also address some of the corresponding, but harder, geometric properties of separable and entangled states and entanglement witnesses.
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An anonymous reader quotes a report from NBC New York: Colin Kroll, the cofounder of HQ Trivia and Vine, died of an accidental overdose, the city's medical examiner announced Tuesday. According to the autopsy results, Kroll died of "acute intoxication due to the combined effects of fentanyl, fluoroisobutyryl fentanyl, heroin, and cocaine." Kroll, 34, was found dead in his SoHo, Manhattan, apartment on Dec. 16, 2018. Police responded to a 911 call for a welfare check at the Spring Street apartment where they found Kroll unconscious and unresponsive in a bedroom of the apartment, a New York Police Department spokesman previously told NBC News. Kroll was named the chief executive of HQ Trivia, a phonebased trivia platform, in September. Prior to that, Kroll cofounded Vine, the popular shortform video service acquired in 2012 by Twitter. Vine was discontinued four years later.
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This is a review of the geometry of quantum states using elementary methods and pictures. Quantum states are represented by a convex body, often in high dimensions. In the case of nqubits, the dimension is exponentially large in n. The space of states can be visualized, to some extent, by its simple cross sections: Regular simplexes, balls and hyperoctahedra. When the dimension gets large there is a precise sense in which the space of states resembles, almost in every direction, a ball. The ball turns out to be a ball of rather low purity states. We also address some of the corresponding, but harder, geometric properties of separable and entangled states and entanglement witnesses.
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Some new results provide opportunities to ensure the exponential convergence to a unique quasistationary distribution in the total variation norm, for quite general strong Markov processes. Specifically, nonreversible processes with discontinuous trajectories are concerned, which seems to be a substantial breakthrough. Considering jumps driven by Poisson Point Processes in four different applications, we intend to illustrate the potential of these results and motivate an original yet apparently very technical criterion. Such criterion is expected to be much easier to verify than an implied property essential for our proof, namely a comparison of the asymptotic extinction rate between different initial conditions. Keywords : continuoustime and continuousspace Markov process , jumps , quasistationary distribution , survival capacity , Qprocess , Harris recurrence
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In this paper, we present a general framework to construct recurrent fractal interpolation surfaces (RFISs) on rectangular grids. Then we introduce bilinear RFISs, which are easy to be generated while there are no restrictions on interpolation points and vertical scaling factors. We also obtain the box dimension of bilinear RFISs under certain constraints, where the main assumption is that vertical scaling factors have uniform sums under a compatible partition.
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We study (smooth, complex) Fano 4folds X having a rational contraction of fiber type, that is, a rational map X>Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a nonzero, nonbig movable divisor on X. Our main result is that if Y is not P^1 or P^2, then the Picard number rho(X) of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4fold X has a dominant rational map X>Z, regular and proper on an open subset of X, with dim(Z)=3, then either X is a product of surfaces, or rho(X) is at most 12. These results are part of a program to study Fano 4folds with large Picard number via birational geometry.
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Self dual symmetric Rspaces have special curves, called circles, introduced by Burstall, Donaldson, Pedit and Pinkall in 2011, whose definition does not involve the choice of any Riemannian metric. We characterize the elements of the big transformation group G of a self dual symmetric Rspace M as those diffeomorphisms of M sending circles in circles. Besides, although these curves belong to the realm of the invariants by G, we manage to describe them in Riemannian geometric terms: Given a circle c in M, there is a maximal compact subgroup K of G such that c, except for a projective reparametrization, is a diametrical geodesic in M (or equivalently, a diagonal geodesic in a maximal totally geodesic flat torus of M), provided that M carries the canonical symmetric Kinvariant metric. We include examples for the complex quadric and the split standard or isotropic Grassmannians.
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We construct a family of $(2,n)$almost Grassmannian structures of regularity $C^1$, each admitting a oneparameter group of strongly essential automorphisms, and each not flat on any neighborhood of the higherorder fixed point. This shows that Theorem 1.3 of [9] does not hold assuming only $C^1$ regularity of the structure (see also [2, Prop 3.5]).
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This paper is the continuation of the study on discrete harmonic analysis related to Jacobi expansions initiated in [1]. Considering the operator $\mathcal{J}^{(\alpha,\beta)}=J^{(\alpha,\beta)}I$, where $J^{(\alpha,\beta)}$ is the threeterm recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we focus on the study of weighted inequalities for the Riesz transform associated with it.
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Inspired by a similar analysis for the vacuum conformal Einstein field equations by Paetz [Ann. H. Poincar\'e 16, 2059 (2015)], in this article we show how to construct a system of quasilinear wave equations for the geometric fields associated to the conformal Einstein field equations coupled to matter models whose energymomentum tensor has vanishing trace. In this case, the equation of conservation for the energymomentum tensor is conformally invariant. Our analysis includes the construction of a subsidiary evolution system which allows to prove the propagation of the constraints. We discuss how the underlying structure behind these systems of equations is the integrability conditions satisfied by the conformal field equations. The main result of our analysis is that both the evolution and subsidiary equations for the geometric part of the conformal Einsteintracefree matter field equations close without the need of any further assumption on the matter models other than the vanishin
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The classic Fatou lemma states that the lower limit of a sequence of integrals of functions is greater or equal than the integral of the lower limit. It is known that Fatou's lemma for a sequence of weakly converging measures states a weaker inequality because the integral of the lower limit is replaced with integral of the lower limit in two parameters, where the second parameter is the argument of the functions. This paper provides sufficient conditions when Fatou's lemma holds in its classic form for a sequence of weakly converging measures. The functions can take both positive and negative values. The paper also provides similar results for sequences of setwise converging measures. It also provides Lebesgue's and monotone convergence theorem for sequences of weakly and setwise converging measures. The obtained results are used to prove broad sufficient conditions for the validity of optimality equations for averagecosts Markov decision processes.
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It is known that there exist CalabiYau structures on the complexifications of symmetric spaces of compact type. In this paper, we describe the CalabiYau structures of the complexified symmetric spaces in terms of the Schwarz's theorem in detail. We consider the case where the CalabiYau structure arises from the Riemannian metric corresponding to the Stenzel metric. In the complexified symmetric spaces equipped with such a CalabiYau structure, we give constructions of special Lagrangian submanifolds of any phase which are invariant under the actions of symmetric subgroups of the isometry group of the original symmetric space of compact type.
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Statistics has made tremendous advances since the times of Fisher, Neyman, Jeffreys, and others, but the fundamental questions about probability and inference that puzzled our founding fathers still exist and might even be more relevant today. To overcome these challenges, I propose to look beyond the two dominating schools of thought and ask what do scientists need out of statistics, do the existing frameworks meet these needs, and, if not, how to fill the void? To the first question, I contend that scientists seek to convert their data, posited statistical model, etc., into calibrated degrees of belief about quantities of interest. To the second question, I argue that any framework that returns additive beliefs, i.e., probabilities, necessarily suffers from false confidencecertain false hypotheses tend to be assigned high probabilityand, therefore, risks making systematically misleading conclusions. This reveals the fundamental importance of nonadditive beliefs in the context
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A connected graph G is called matching covered if every edge of G is contained in a perfect matching. Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition. It was introduced by Norine and intended as a tool for the structural study of matching covered graphs, especially in the context of Pfaffian orientations. Norine conjectured that graphs of high perfect matching width would contain a large grid as a matching minor, similar to the result on treewidth by Robertson and Seymour. In this paper we obtain the first results on perfect matching width since its introduction. For the restricted case of bipartite graphs, we show that perfect matching width is equivalent to directed treewidth and thus the Directed Grid Theorem by Kawarabayashi and Kreutzer for directed \treewidth implies Norine's conjecture.
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Cloud providers have recently introduced lowpriority machines to reduce the cost of computations. Exploiting such opportunity for machine learning tasks is challenging inasmuch as lowpriority machines can elastically leave (through preemption) and join the computation at any time. In this paper, we design a new technique called coded elastic computing enabling distributed machine learning computations over elastic resources. The proposed technique allows machines to transparently leave the computation without sacrificing the algorithmlevel performance, and, at the same time, flexibly reduce the workload at existing machines when new machines join the computation. Thanks to the redundancy provided by encoding, our approach is able to achieve similar computational cost as the original (uncoded) method when all machines are present; the cost gracefully increases when machines are preempted and reduces when machines join. We test the performance of the proposed technique on two miniben
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Herein, we focus on explicit constructions of $\ell\times\ell$ binary kernels with small scaling exponent for $\ell \le 64$. In particular, we exhibit a sequence of binary linear codes that approaches capacity on the BEC with quasilinear complexity and scaling exponent $\mu < 3$. To the best of our knowledge, such a sequence of codes was not previously known to exist. The principal challenges in establishing our results are twofold: how to construct such kernels and how to evaluate their scaling exponent. In a single polarization step, an $\ell\times\ell$ kernel $K_\ell$ transforms an underlying BEC into $\ell$ bitchannels $W_1,W_2,\ldots,W_\ell$. The erasure probabilities of $W_1,W_2,\ldots,W_\ell$, known as the polarization behavior of $K_\ell$, determine the resulting scaling exponent $\mu(K_\ell)$. We first introduce a class of selfdual binary kernels and prove that their polarization behavior satisfies a strong symmetry property. This reduces the problem of constructing $K_\
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A twotype version of the frog model on $\mathbb{Z}^d$ is formulated, where active type $i$ particles move according to lazy random walks with probability $p_i$ of jumping in each time step ($i=1,2$). Each site is independently assigned a random number of particles. At time 0, the particles at the origin are activated and assigned type 1 and the particles at one other site are activated and assigned type 2, while all other particles are sleeping. When an active type $i$ particle moves to a new site, any sleeping particles there are activated and assigned type $i$, with an arbitrary tiebreaker deciding the type if the site is hit by particles of both types in the same time step. We show that the event $G_i$ that type $i$ activates infinitely many particles has positive probability for all $p_1,p_2\in(0,1]$ ($i=1,2$). Furthermore, if $p_1=p_2$, then the types can coexist in the sense that $\mathbb{P}(G_1\cap G_2)>0$. We also formulate several open problems. For instance, we conjectur
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We describe the possible linear actions of a cyclic group $G = \mathbb{Z} /n$ on a complex torus, using the cyclotomic exact sequence for the group algebra $\mathbb{Z} [G]$. The main application is devoted to a structure theorem for BagneraDe Franchis Manifolds, but we also give an application to hypergeometric integrals.
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Amplitudecoherent (AC) detection is an efficient detection technique that can simplify the receiver design while providing reliable symbol error rate (SER). Therefore, this work considers AC detector design and SER analysis using Mary amplitude shift keying (MASK) modulation over Rician fading channels. More specifically, we derive the optimum, nearoptimum and a suboptimum AC detectors and compare their SER to the coherent, noncoherent and the heuristic AC detectors. Moreover, the analytical SER of the heuristic detector is derived using two different approaches for single and multiple receiving antennas. One of the derived expressions is expressed in terms of a single integral that can be evaluated numerically, while the second approach gives a closedform analytical expression for the SER, which is also used to derive a simple formula for the asymptotic SER at high signaltonoise ratios (SNRs). The obtained analytical and simulation results show that the SER of the AC and coheren
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We study the convergence of volumenormalized Betti numbers in BenjaminiSchramm convergent sequences of nonpositively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X \neq \mathbb H^3$, and $(M_n)$ is any BenjaminiSchramm convergent sequence of finite volume $X$manifolds, then the normalized Betti numbers $b_k(M_n)/vol(M_n)$ converge for all $k$. As a corollary, if $X$ has higher rank and $(M_n)$ is any sequence of distinct, finite volume $X$manifolds, the normalized Betti numbers of $M_n$ converge to the $L^2$ Betti numbers of $X$. This extends our earlier work with Nikolov, Raimbault and Samet, where we proved the same convergence result for uniformly thick sequences of compact $X$manifolds.
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This article analyses geometric constructions by origami when up to $n$ simultaneous folds may be done at each step. It shows that any arbitrary angle can be $m$sected if the largest prime factor of $m$ is $p\le n+2$. Also, the regular $m$gon can be constructed if the largest prime factor of $\phi(m)$ is $q\le n+2$, where $\phi$ is Euler's totient function.
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We study the longterm qualitative behavior of randomly perturbed dynamical systems. More specifically, we look at limit cycles of certain stochastic differential equations (SDE) with Markovian switching, in which the process switches at random times among different systems of SDEs, when the switching is fast varying and the diffusion (white noise) term is slowly changing. The system is modeled by $$ dX^{\varepsilon,\delta}(t)=f(X^{\varepsilon,\delta}(t), \alpha^\varepsilon(t))dt+\sqrt{\delta}\sigma(X^{\varepsilon,\delta}(t), \alpha^\varepsilon(t))dW(t) , \ X^\varepsilon(0)=x, $$ where $\alpha^\varepsilon(t)$ is a finite state space, Markov chain with generator $Q/\varepsilon=\big(q_{ij}/\varepsilon\big)_{m_0\times m_0}$ with $Q$ being irreducible. The relative changing rates of the switching and the diffusion are highlighted by the two small parameters $\varepsilon$ and $\delta$. We associate to the system the averaged ordinary differential equation (ODE) \[ d\bar X(t)=\bar f(\bar X(t
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To increase the reliability of simulations by particle methods for incompressible viscous flow problems, convergence studies and improvements of accuracy are considered for a fully explicit particle method for incompressible NavierStokes equations. The explicit particle method is based on a penalty problem, which converges theoretically to the incompressible NavierStokes equations, and is discretized in space by generalized approximate operators defined as a wider class of approximate operators than those of the smoothed particle hydrodynamics (SPH) and moving particle semiimplicit (MPS) methods. By considering an analytical derivation of the explicit particle method and truncation error estimates of the generalized approximate operators, sufficient conditions of convergence are conjectured.Under these conditions, the convergence of the explicit particle method is confirmed by numerically comparing errors between exact and approximate solutions. Moreover, by focusing on the trunca
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This paper advocates a pair of strategies in nonorthogonal multiple access (NOMA) in unmanned aerial vehicles (UAVs) communications, where multiple UAVs play as new aerial communications platforms for serving terrestrial NOMA users. A new multiple UAVs framework with invoking stochastic geometry technique is proposed, in which a pair of practical strategies are considered: 1) UAVCentric strategy for offloading actions and 2) UserCentric strategy for providing emergency communications. In order to provide practical insights for the proposed NOMA assisted UAV framework, an imperfect successive interference cancelation (ipSIC) scenario is taken into account. For both UAVCentric strategy and UserCentric strategy, we derive new exact expressions for the coverage probability. We also derive new analytical results for orthogonal multiple access (OMA) for providing a benchmark scheme. The derived analytical results in both UserCentric strategy and UAVCentric strategy explicitly indicate
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Risk estimation is at the core of many learning systems. The importance of this problem has motivated researchers to propose different schemes, such as cross validation, generalized cross validation, and Bootstrap. The theoretical properties of such estimates have been extensively studied in the lowdimensional settings, where the number of predictors $p$ is much smaller than the number of observations $n$. However, a unifying methodology accompanied with a rigorous theory is lacking in highdimensional settings. This paper studies the problem of risk estimation under the highdimensional asymptotic setting $n,p \rightarrow \infty$ and $n/p \rightarrow \delta$ ($\delta$ is a fixed number), and proves the consistency of three risk estimates that have been successful in numerical studies, i.e., leaveoneout cross validation (LOOCV), approximate leaveoneout (ALO), and approximate message passing (AMP)based techniques. A corner stone of our analysis is a bound that we obtain on the dis
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We review and extend several recent results on the existence of the ground state for the nonlinear Schr\"odinger (NLS) equation on a metric graph. By ground state we mean a minimizer of the NLS energy functional constrained to the manifold of fixed $L^2$norm. In the energy functional we allow for the presence of a potential term, of deltainteractions in the vertices of the graph, and of a powertype focusing nonlinear term. We discuss both subcritical and critical nonlinearity. Under general assumptions on the graph and the potential, we prove that a ground state exists for sufficiently small mass, whenever the constrained infimum of the quadratic part of the energy functional is strictly negative.
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In this paper we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics \cite{Silling2000} or nonlocal diffusion models \cite{Rossi}. We derive nonlocal versions of the EulerLagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure we show regularity of solutions for certain EulerLagrange equations. No growth assumptions are needed for the existence and regularity of minimizers results, in contrast with the classical theory.
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We derive a formula of ChernGaussBonnet type for the Euler characteristic of a four dimensional manifoldwithboundary in terms of the geometry of the LoewnerNirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the boundary is umbilic, then the sum of the renormalized volume and the boundary integral is a conformal invariant. Analogous results are proved for asymptotically hyperbolic metrics in dimension four for which the second elementary symmetric function of the eigenvalues of the Schouten tensor is constant. Extensions and generalizations of these results are discussed. Finally, a general result is proved identifying the infinitesimal anomaly of the renormalized volume of an asymptotically hyperbolic metric in terms of its renormalized volume coefficients, and used to outline alternate proofs of the conformal invariance of the renormalized volume plus boundary integral.
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Let $M$ be a Tmotive. We introduce the notion of duality for $M$. Main results of the paper (we consider uniformizable $M$ over $F_q[T]$ of rank $r$, dimension $n$, whose nilpotent operator $N$ is 0): 1. Algebraic duality implies analytic duality (Theorem 5). Explicitly, this means that the lattice of the dual of $M$ is the dual of the lattice of $M$, i.e. the transposed of a Siegel matrix of $M$ is a Siegel matrix of the dual of $M$. 2. Let $n=r1$. There is a 1  1 correspondence between pure Tmotives (all they are uniformizable), and lattices of rank $r$ in $C^n$ having dual (Corollary 8.4).
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The Wasserstein distances are a set of metrics on probability distributions supported on $\mathbb{R}^d$ with applications throughout statistics and machine learning. Often, such distances are used in the context of variational problems, in which the statistician employs in place of an unknown measure a proxy constructed on the basis of independent samples. This raises the basic question of how well measures can be approximated in Wasserstein distance. While it is known that an empirical measure comprising i.i.d. samples is rateoptimal for general measures, no improved results were known for measures possessing smooth densities. We prove the first minimax rates for estimation of smooth densities for general Wasserstein distances, thereby showing how the curse of dimensionality can be alleviated for sufficiently regular measures. We also show how to construct discretely supported measures, suitable for computational purposes, which enjoy improved rates. Our approach is based on novel bo
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Recent progress in Zauner's conjecture has leveraged deep conjectures in algebraic number theory to promote numerical line packings to exact and verifiable solutions to the line packing problem. We introduce a numericaltoexact technique in the real setting that does not require such conjectures. Our approach is completely reproducible, matching Sloane's database of putatively optimal numerical line packings with Mathematica's builtin implementation of cylindrical algebraic decomposition. As a proof of concept, we promote a putatively optimal numerical packing of eight points in the real projective plane to an exact packing, whose optimality we establish in a forthcoming paper.
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Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a YangBaxter equation for a colored fivevertex model. As a biproduct, we construct Demazure atoms on Kashiwara's $\mathcal{B}_\infty$ crystal and give new algorithms for computing LascouxSch\"utzenberger keys.
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Isogeometric Analysis allows highorder discretizations of boundary value problems using a number of degrees of freedom which is as small as for a loworder method. To be able to choose high spline degrees in practice, suitable numerical solvers are required. In nontrivial cases, the computational domain is typically decomposed into several patches, where for each patch a separate isogeometric discretization is chosen. If the discretization agrees on the interfaces between the patches, the coupling can be done in a conforming way. Otherwise, nonconforming discretizations (utilizing discontinuous Galerkin approaches) are required. The author and his coworkers have previously introduced multigrid solvers for Isogeometric Analysis. In the present paper, these results are extended to the nonconforming case. Moreover, it is shown that the solves get even more powerful if the proposed smoother is combined with a (standard) GaussSeidel smoother.
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In this article, we introduce Brownian motion on stable looptrees using resistance techniques. We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally loglogarithmic, and globally logarithmic around leading terms of $r^{\alpha}$ and $t^{\frac{\alpha}{\alpha + 1}}$ respectively. These volume fluctuations are the same order as for the Brownian continuum random tree, but the upper volume fluctuations (and corresponding lower heat kernel fluctuations) are different to those of stable trees.
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We compute the DonaldsonThomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in arXiv:1603.05271, we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the KatzKlemmVafa formula for primitive curve classes which is independent of the computation of KawaiYoshioka.
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SQUID (Superconducting QUantum Interference Device) metamaterials, subject to a timeindependent (dc) flux gradient and driven by a sinusoidal (ac) flux field, support chimera states that can be generated with zero initial conditions. The dc flux gradient and the amplitude of the ac flux can control the number of desynchronized clusters of such a generated chimera state (i.e., its `heads') as well as their location and size. The combination of three measures, i.e., the synchronization parameter averaged over the period of the driving flux, the incoherence index, and the chimera index, is used to predict the generation of a chimera state and its multiplicity on the parameter plane of the dc flux gradient and the ac flux amplitude. Moreover, the fullwidth halfmaximum of the distribution of the values of the synchronization parameter averaged over the period of the ac driving flux, allows to distinguish chimera states from nonchimera, partially synchronized states.
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