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L$_2$ regularization and weight decay regularization are equivalent for standard stochastic gradient descent (when rescaled by the learning rate), but as we demonstrate this is \emph{not} the case for adaptive gradient algorithms, such as Adam. While common deep learning frameworks of these algorithms implement L$_2$ regularization (often calling it "weight decay" in what may be misleading due to the inequivalence we expose), we propose a simple modification to recover the original formulation of weight decay regularization by decoupling the weight decay from the optimization steps taken w.r.t. the loss function. We provide empirical evidence that our proposed modification (i) decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam, and (ii) substantially improves Adam's generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperfo
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We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold $M$ measures the minimal size of possibly ideal triangulations of $M$ "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, the main difference being that ideal simplices are now allowed to appear in representatives of the fundamental class. We show that the ideal simplicial volume is bounded above by the ordinary simplicial volume, and that it vanishes if and only if the ordinary simplicial volume does. We show that, for manifolds with amenable boundary, the ideal simplicial volume coincides with the classical one, whereas for hyperbolic manifolds with geodesic boundary it can be strictly smaller. We compute the ideal simplicial volume of an infinite family of hyperbolic $3$manifolds with geodesic boundary, for which the exact value of the classical simplicial volume is not known, and we exhi
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Genuine highdimensional entanglement, i.e. the property of having a high Schmidt number, constitutes a resource in quantum communication, overcoming limitations of lowdimensional systems. States with a positive partial transpose (PPT), on the other hand, are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question, whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e. linear, scaling in local dimension should be possible, albeit without providing an insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to answer a recent question by Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally,
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Arbitrary order dissipative and conservative Hermite methods for the scalar wave equation achieving $\mathcal{O}(2m)$ orders of accuracy using $\mathcal{O}(m^d)$ degrees of freedom per node in $d$ dimensions are presented. Stability and error analyses as well as implementation strategies for accelerators are also given.
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We prove that if $G$ is a group of finite Morley rank which acts definably and generically sharply $n$transitively on a connected abelian group $V$ of Morley rank $n$ with no involutions, then there is an algebraically closed field $F$ of characteristic $\ne 2$ such that $V$ has a structure of a vector space of dimension $n$ over $F$ and $G$ acts on $V$ as the group $\operatorname{GL}_n(F)$ in its natural action on $F^n$.
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Let $\mathcal{X}$ be a class of left $R$modules, $\mathcal{Y}$ be a class of right $R$modules. In this paper, we introduce and study Gorenstein $(\mathcal{X}, \mathcal{Y})$flat modules as a common generalization of some known modules such as Gorenstein flat modules \cite{EJT93}, Gorenstein $n$flat modules \cite{SUU14}, Gorenstein $\mathcal{B}$flat modules \cite{EIP17}, Gorenstein ACflat modules \cite{BEI17}, $\Omega$Gorenstein flat modules \cite{EJ00} and so on. We show that the class of all Gorenstein $(\mathcal{X}, \mathcal{Y})$flat modules have a strong stability. In particular, when $(\mathcal{X}, \mathcal{Y})$ is a perfect (symmetric) duality pair, we give some functorial descriptions of Gorenstein $(\mathcal{X}, \mathcal{Y})$flat dimension, and construct a hereditary abelian model structure on $R$Mod whose cofibrant objects are exactly the Gorenstein $(\mathcal{X}, \mathcal{Y})$flat modules. These results unify the corresponding results of the aforementioned modules.
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Some geometric properties of a normalized hyperBessel functions are investigated. Especially we focus on the radii of starlikeness, convexity, and uniform convexity of hyperBessel functions and we show that the obtained radii satisfy some transcendental equations. In addition, we give some bounds for the first positive zero of normalized hyperBessel functions, Redheffertype inequalities, and bounds for this function. In this study we take advantage of EulerRayleigh inequalities and LaguerreP\'{o}lya class of real entire functions, intensively.
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It is pointed out that the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N2h}}{e^{n^{N}x}1}$ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page $332$ of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters $N$ and $h$ that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for $N$ odd and $m>0$, gives a relation between $\zeta(2m+1)$ and $\zeta(2Nm+1)$. A result complementary to the aforementioned generalization is obtained for any even $N$ and $m\in\mathbb{Z}$. It generalizes a transformation of Wigert and can be regarded as a formula for $\zeta\left(2m+1\frac{1}{N}\right)$. Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind etafunction $\eta(z)$, Zudilin
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Let $k\geq 2, n\geq 1$ be integers. Let $f: \mathbb{R}^{n} \to \mathbb{C}$. The $k$th GowersHostKra norm of $f$ is defined recursively by \begin{equation*} \ f\_{U^{k}}^{2^{k}} =\int_{\mathbb{R}^{n}} \ T^{h}f \cdot \bar{f} \_{U^{k1}}^{2^{k1}} \, dh \end{equation*} with $T^{h}f(x) = f(x+h)$ and $\f\_{U^1} =  \int_{\mathbb{R}^{n}} f(x)\, dx $. These norms were introduced by Gowers in his work on Szemer\'edi's theorem, and by HostKra in ergodic setting. It's shown by Eisner and Tao that for every $k\geq 2$ there exist $A(k,n)< \infty$ and $p_{k} = 2^{k}/(k+1)$ such that $\ f\_{U^{k}} \leq A(k,n)\f\_{p_{k}}$, with $p_{k} = 2^{k}/(k+1)$ for all $f \in L^{p_{k}}(\mathbb{R}^{n})$. The optimal constant $A(k,n)$ and the extremizers for this inequality are known. In this exposition, it is shown that if the ratio $\ f \_{U^{k}}/\f\_{p_{k}}$ is nearly maximal, then $f$ is close in $L^{p_{k}}$ norm to an extremizer.
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We define enumerative invariants associated to a hybrid Gauged Linear Sigma Model. We prove that in the relevant special cases, these invariants recover both the GromovWitten type invariants defined by ChangLi and FanJarvisRuan using cosection localization as well as the FJRW type invariants constructed by PolishchukVaintrob. The invariants are defined by constructing a "fundamental factorization" supported on the moduli space of LandauGinzburg maps to a convex hybrid model. This gives the kernel of a FourierMukai transform; the associated map on Hochschild homology defines our theory.
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Dynamical measurement schemes are an important tool for the investigation of quantum manybody systems, especially in the age of quantum simulation. Here, we address the question whether generic measurements can be implemented efficiently if we have access to a certain set of experimentally realizable measurements and can extend it through time evolution. For the latter, two scenarios are considered (a) evolution according to unitary circuits and (b) evolution due to Hamiltonians that we can control in a timedependent fashion. We find that the time needed to realize a certain measurement to a predefined accuracy scales in general exponentially with the system size  posing a fundamental limitation. The argument is based, on the construction of $\varepsilon$packings for manifolds of observables with identical spectra and a comparison of their cardinalities to those of $\varepsilon$coverings for quantum circuits and unitary timeevolution operators. The former is related to the study
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Given a graph $G$ and a positive integer $k$, the \emph{GallaiRamsey number} is defined to be the minimum number of vertices $n$ such that any $k$edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the GallaiRamsey numbers for books $B_{m} = K_{2} + \overline{K_{m}}$ and prove sharp results for $m \leq 5$.
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We introduce a parametrized version of the Wadge game for functions and show that each lower cone in the Weihrauch degrees is characterized by such a game. These parametrized Wadge games subsume the original Wadge game, the eraser and backtrack games as well as Semmes's tree games. In particular, we propose that the lower cones in the Weihrauch degrees are the answer to Andretta's question on which classes of functions admit game characterizations. We then discuss some applications of such parametrized Wadge games. Using machinery from Weihrauch reducibility theory, we introduce games characterizing every (transfinite) level of the Baire hierarchy via an iteration of a pruning derivative on countably branching trees.
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We study the geometric engineering of gauge theories with gauge group Spin(4) and SO(4) using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graph the affine $A_1$ Dynkin diagram. There are eight different ways to engineer such collisions using decorated Kodaira fibers. The MordellWeil group of the elliptic fibration is required to be trivial for Spin(4) and Z/2Z for SO(4). Each of these models have two possible crepant resolutions connected by a flop. We also compute a generating function for the Euler characteristic of such elliptic fibrations over a base of arbitrary dimensions. In the case of a threefold, we also compute the triple intersection numbers of the fibral divisors. In the case of CalabiYau threefolds, we also compute their Hodge numbers, and check the cancellations of anomalies in a sixdimensional supergravity theory.
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We provide an explicit expression for the Pleijel constant for the planar disk and some of its sectors, as well as for $N$dimensional rectangles. In particular, the Pleijel constant for the disk is equal to 0.4613019... Also, we characterize the Pleijel constant for some rings and annular sectors in terms of asymptotic behavior of zeros of certain crossproducts of Bessel functions.
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In this paper we study paraKenmotsu manifolds. We characterize this manifolds by tensor equations and study their properties. We are devoted to a study of $\eta$Einstein manifolds. We show that a conformally flat paraKenmotsu manifold is a space of constant negative curvature $1$ and we prove that if a paraKenmotsu manifold is a space of constant $\varphi$paraholomorphic sectional curvature $H$, then it is a space of constant curvature and $H=1$. Finally the object of the present paper is to study a 3dimensional paraKenmotsu manifold, satisfying certain curvature conditions. Among other, it is proved that any 3dimensional paraKenmotsu manifold with $\eta$parallel Ricci tensor is of constant scalar curvature and any 3dimensional paraKenmotsu manifold satisfying cyclic Ricci tensor is a manifold of constant negative curvature $1$.
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We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of nonuniform orbifolds are convergent to this space in a geometric ("BenjaminiSchramm") sense for hyperbolic threespace and a product of hyperbolic planes. We also deal with arbitrary sequences of maximal arithmetic threedimensional hyperbolic lattices defined over a quadratic or cubic field. A motivating application is the study of Betti numbers of Bianchi groups.
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The problem of optimal power allocation among light emitting diode (LED) transmitters in a visible light positioning (VLP) system is considered for the purpose of improving localization performance of visible light communication (VLC) receivers. Specifically, the aim is to minimize the Cram\'{e}rRao lower bound (CRLB) on the localization error of a VLC receiver by optimizing LED transmission powers in the presence of practical constraints such as individual and total power limitations and illuminance constraints. The formulated optimization problem is shown to be convex and thus can efficiently be solved via standard tools. We also investigate the case of imperfect knowledge of localization parameters and develop robust power allocation algorithms by taking into account both overall system uncertainty and individual parameter uncertainties related to the location and orientation of the VLC receiver. In addition, we address the total power minimization problem under predefined accuracy
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Inspired by Clarkson's inequalities for $L^p$ and continuing work from \cite{CR}, this paper computes the optimal constant $C$ in the weak parallelogram laws $$ \f + g \^r + C\f  g\^r \leq 2^{r1}\big( \f\^r + \g\^r \big), $$ $$ \f + g \^r + C\f g \^r \geq 2^{r1}\big( \f\^r + \g \^r \big)$$ for the $L^p$ spaces, $1 < p < \infty$.
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In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in $\mathbb{Z}^{n}$ with radius $r\ge2$ and dimension $n\ge3$. Besides its own interest in coding theory and discrete geometry, this conjecture is also strongly related to the degreediameter problems of abelian Cayley graphs. Although there are many papers on this topic, the GolombWelch conjecture is far from being solved. In this paper, we prove the nonexistence of linear perfect Lee codes by introducing some new algebraic methods. Using these new methods, we show the nonexistence of linear perfect Lee codes of radii $r=2,3$ in $\mathbb{Z}^n$ for infinitely many values of the dimension $n$. In particular, there does not exist linear perfect Lee codes of radius $2$ in $\mathbb{Z}^n$ for all $3\le n\le 100$ except 8 cases.
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The notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel; and has been mainly studied on proper geodesic metric spaces. We present here a generalization of the metric compactification that can be applied to infinitedimensional Banach spaces. Thereafter we give a complete description of the metric compactification of infinitedimensional $\ell_{p}$ spaces for all $1\leq p < \infty$. We also give a full characterization of the metric compactification of infinitedimensional Hilbert spaces.
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We consider the optimal stopping problem with nonlinear $f$expectation (induced by a BSDE) without making any regularity assumptions on the reward process $\xi$. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$Snell envelope of $\xi$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $\xi$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American options with irregular payoff in an imperfect market model.
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BoixDe StefaniVanzo defined the notion of level for a smooth projective hypersurface over a finite field in terms of the stabilisation of a chain of ideals previously considered by \`AlvarezMontanerBlickleLyubeznik, and showed that in the case of an elliptic curve the level is 1 if and only if it is ordinary and 2 otherwise. Here we extend their theorem to the case of CalabiYau hypersurfaces by relating their level to the $F$jumping exponents of BlickleMusta\c{t}\u{a}Smith and the HartshorneSpeiserLyubeznik numbers of Musta\c{t}\u{a}Zhang.
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Contingency and accident are two important notions in philosophy and philosophical logic. Their meanings are so close that they are mixed sometimes, in both everyday discourse and academic research. This indicates that it is necessary to study them in a unified framework. However, there has been no logical research on them together. In this paper, we propose a language of a bimodal logic with these two concepts, investigate its modeltheoretical properties such as expressivity and frame definability. We axiomatize this logic over various classes of frames, whose completeness proofs are shown with the help of a crucial schema. The interactions between contingency and accident can sharpen our understanding of both notions. Then we extend the logic to a dynamic case: public announcements. By finding the required reduction axioms, we obtain a complete axiomatization, which gives us a good application to Moore sentences.
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Crowdsourcing relies on people's contributions to meet product or systemlevel objectives. Crowdsourcingbased methods have been implemented in various cyberphysical systems and realtime markets. This paper explores a framework for Crowdsourced Energy Systems (CES), where smallscale energy generation or energy trading is crowdsourced from distributed energy resources, electric vehicles, and shapable loads. The merits/pillars of energy crowdsourcing are discussed. Then, an operational model for CESs in distribution networks with different types of crowdsourcees is proposed. The model yields a market equilibrium depicting traditional and distributed generator and load setpoints. Given these setpoints, crowdsourcing incentives are designed to steer crowdsourcees to the equilibrium. As the number of crowdsourcees and energy trading transactions scales up, a secure energy trading platform is required. To that end, the presented framework is integrated with a lightweight Blockchain implem
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In this paper, we consider a class of the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + x^{b} u^\alpha u = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] with $0<b<\min\{2,d\}$ and $\alpha_\star\leq \alpha <\alpha^\star$ where $\alpha_\star =\frac{42b}{d}$ and $\alpha^\star=\frac{42b}{d2}$ if $d\geq 3$ and $\alpha^\star = \infty$ if $d=1,2$. In the masscritical case $\alpha=\alpha_\star$, we prove that if $u_0$ has negative energy and satisfies either $xu_0 \in L^2$ with $d\geq 1$ or $u_0$ is radial with $d\geq 2$, then the corresponding solution blows up in finite time. Moreover, when $d=1$, we prove that if the initial data (not necessarily radial) has negative energy, then the corresponding solution blows up in finite time. In the mass and energy intercritical case $\alpha_\star< \alpha <\alpha^\star$, we prove the blowup below ground state for radial initial data with $d\geq 2$. This result extends the one of Farah in \ci
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We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical expressions for evaluating the operators away from the boundary. When two particle are located close to each other, we use a truncated series expansion to compute the hydrodynamic interaction. On the other hand, we use the standard spectrally accurate quadrature scheme to evaluate smooth integrals on the farfield, and accelerate the resulting discrete sums using the fast multipole method (FMM). We employ this discretization scheme to analyze several boundary integral formulations of interest including those arising in porous media flow, active matter and magnetohydrodynamics of rigid particles. We provide numerical results verifying the accuracy and scaling of their evaluation.
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Let $\Omega$ be a starshaped bounded domain in $(\mathbb{S}^{n}, ds^{2})$ with smooth boundary. In this article, we give a sharp lower bound for the first nonzero eigenvalue of the Steklov eigenvalue problem in $\Omega.$ This result is the generalization of a result given by Kuttler and Sigillito for a starshaped bounded domain in $\mathbb{R}^2.$ Further, we also obtain a two sided bound for the first nonzero eigenvalue of the Steklov problem on the ball in $\mathbb{R}^n$ with rotationally invariant metric and with bounded radial curvature.
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For wideband spectrum sensing, compressive sensing has been proposed as a solution to speed up the high dimensional signals sensing and reduce the computational complexity. Compressive sensing consists of acquiring the essential information from a sparse signal and recovering it at the receiver based on an efficient sampling matrix and a reconstruction technique. In order to deal with the uncertainty, improve the signal acquisition performance, and reduce the randomness during the sensing and reconstruction processes, compressive sensing requires a robust sampling matrix and an efficient reconstruction technique. In this paper, we propose an approach that combines the advantages of a Circulant matrix with Bayesian models. This approach is implemented, extensively tested, and its results have been compared to those of l1 norm minimization with a Circulant or random matrix based on several metrics. These metrics are Mean Square Error, reconstruction error, correlation, recovery time, sam
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We point out that when a quadratic type Li\'enard equation is suitably interpreted shows branching due to the double valuedness of the governing Hamiltonian. Under certain approximation of the guiding coupling constant we derive its quantum counterpart that is guided by a momentumdependent mass function.
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In this note, we prove that the centralizer lattice ${\mathfrak C}(G)$ of a group $G$ cannot be written as a union of two proper intervals. In particular, it follows that ${\mathfrak C}(G)$ has no breaking point. As an application, we show that the generalized quaternion $2$groups are not capable.
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In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $\alpha>1$ on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both a weighted concentration integral operator, and a SturmLiouville differential operator. Different from existing works on multidimensional PSWFs, the ball PSWFs are defined as a generalisation of orthogonal {\em ball polynomials} in primitive variables with a tuning parameter $c>0$, through a "perturbation" of the SturmLiouville equation of the ball polynomials. From this perspective, we can explore some interesting intrinsic connections between the ball PSWFs and the finite Fourier and Hankel transforms. We provide an efficient and accurate algorithm for computing the ball PSWFs and the associated eigenvalues, and present various numerical results to illustrate the efficiency of the method. Under this uniform framework, we can recover the existing PSWFs by suitable variable substitutio
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For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the socalled adapted solutions or adapted Msolutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forwardbackward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted Msolutions is established by means of the socalled representation partial differential equations and (forward) stochastic differential equations. Wellposedness of the representation partial differential equations are also proved in certain sense.
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This may not come as a huge surprise, but it's going to be pricey if you break Apple's fully sealed and densely packed new speaker. From a report: Repair pricing for the HomePod was posted to Apple's website this week, and the number is high enough that it's clear you should invest in a warranty if you're worried about breaking one: an outofwarranty repair from Apple will cost $279 in the US, which is 80 percent of the price of a brandnew HomePod. So you're not so much repairing it as getting a small discount on a new one.
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In a piece this month, The New Yorker argues that online food discovery and delivery platforms are bad for restaurants. From the report: In recent years, online platforms like Uber Eats, Seamless, and GrubHub (which merged with Seamless, in 2013) have turned delivery from a small segment of the restaurant industry, dominated by pizza, to a booming new source of sales for food establishments of all stripes. When the average consumer logs in to the Caviar app to order a Mulberry & Vine salad for the office or a grain bowl on the way home from work, she might reasonably assume that her order is benefitting the restaurant's bottom line. But Gauthier, like many other restaurant owners I've spoken to in recent months, paints a more complicated picture. "We know for a fact that as delivery increases, our profitability decreases," she said. For each order that Mulberry & Vine sends out, between twenty and forty per cent of the revenue goes to thirdparty platforms and couriers. (Gauthi
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UPSat is the first open source  both hardware and software  satellite to have ever been launched in orbit. Pierros Papadeas, the Director of Operations for Libre Space Foundation, which helped build the UPSat, talked about the project at FOSDEM, a noncommercial, volunteerorganized European event focused on free and opensource software development. You can watch the talk here; and read an interview of him with folks at FOSDEM ahead of the talk here. Two excerpts from the interview: Q: What challenges did you encounter while designing, building, testing and eventually launching UPSat in orbit? PP: The challenges where numerous, starting with the financial ones. Lack of appropriate funding led us to invest heavily in the project (through Libre Space Foundation funds) to ensure its successful completion. Countless volunteer participation was also key to the success. On the technical side, with minimal documentation and knowledge sharing around space projects we had to reinvent the
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Linus has released 4.16rc1 and closed the merge window for this development cycle. "I don't want to jinx anything, but things certainly look a lot better than with 4.15. We have no (known) nasty surprises pending, and there were no huge issues during the merge window. Fingers crossed that this stays fairly calm and sane."
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How can we approximate sparse graphs and sequences of sparse graphs (with average degree unbounded and $o(n)$)? We consider convergence in the first $k$ moments of the graph spectrum (equivalent to the numbers of closed $k$walks) appropriately normalized. We introduce a simple, easy to sample, random graph model that captures the limiting spectra of many sequences of interest, including the sequence of hypercube graphs. The Random Overlapping Communities (ROC) model is specified by a distribution on pairs $(s,q)$, $s \in \mathbb{Z}_+, q \in (0,1]$. A graph on $n$ vertices with average degree $d$ is generated by repeatedly picking pairs $(s,q)$ from the distribution, adding an Erd\H{o}sR\'{e}nyi random graph of edge density $q$ on a subset of vertices chosen by including each vertex with probability $s/n$, and repeating this process so that the expected degree is $d$. Our proof of convergence to a ROC random graph is based on the Stieltjes moment condition. We also show that the model
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Recently Ruckle \cite{RuckleArithmeticalSummability} introduced the theory of arithmetical summability suggested by the sum $ \sum_{km}f(k) $ as $ k $ ranges over the divisors of $m$ including $ 1 $ and $ m .$ Following Ruckle \cite{RuckleArithmeticalSummability} we construct the sequence space $ AS(G) $ and $ AC(G) $ of arithmetic summable and arithmetic convergent sequences in the sense of geometric calculus and derive interesting results in the geometric field.
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We complete the study of the asymptotic behavior, as $p\rightarrow +\infty$, of the positive solutions to \[ \left\{\begin{array}{lr}\Delta u= u^p & \mbox{in}\Omega\\ u=0 &\mbox{on}\partial \Omega \end{array}\right. \] when $\Omega$ is any smooth bounded domain in $\mathbb R^2$, started in [4]. In particular we show quantization of the energy to multiples of $8\pi e$ and prove convergence to $\sqrt{e}$ of the $L^{\infty}$norm, thus confirming the conjecture made in [4].
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With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[\sqrt{\Delta+m^2}\, u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $V$ is a bounded potential, not necessarily continuous, and $F$ the primitive of $f$. We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.
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It is wellknown that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities are sufficiently smooth and uniformly bounded away from zero. We show that a uniform lower bound, whose size we sharply characterize, is in general necessary for asymptotic equivalence to hold.
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Let $E$ be a Moran set on $\mathbb{R}^1$ associated with a closed interval $J$ and two sequences $(n_k)_{k=1}^\infty$ and $(\mathcal{C}_k=(c_{k,j})_{j=1}^{n_k})_{k\geq1}$. Let $\mu$ be the infinite product measure (Moran measure) on $E$ associated with a sequence $(\mathcal{P}_k)_{k\geq1}$ of positive probability vectors with $\mathcal{P}_k=(p_{k,j})_{j=1}^{n_k},k\geq 1$. We assume that \[ \inf_{k\geq1}\min_{1\leq j\leq n_k}c_{k,j}>0,\;\inf_{k\geq1}\min_{1\leq j\leq n_k}p_{k,j}>0. \] For every $n\geq 1$, let $\alpha_n$ be an $n$ optimal set in the quantization for $\mu$ of order $r\in(0,\infty)$ and $\{P_a(\alpha_n)\}_{a\in\alpha_n}$ an arbitrary Voronoi partition with respect to $\alpha_n$. For every $a\in\alpha_n$, we write $I_a(\alpha,\mu):=\int_{P_a(\alpha_n)}d(x,\alpha_n)^rd\mu(x)$ and \[ \underline{J}(\alpha_n,\mu):=\min_{a\in\alpha_n}I_a(\alpha,\mu),\; \overline{J}(\alpha_n,\mu):=\max_{a\in\alpha_n}I_a(\alpha,\mu). \] We show that $\underline{J}(\alpha_n,\mu),\overline{J}(
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Implicational bases are objects of interest in formal concept analysis and its applications. Unfortunately, even the smallest base, the DuquenneGuigues base, has an exponential size in the worst case. In this paper, we use results on the average number of minimal transversals in random hypergraphs to show that the base of proper premises is, on average, of quasipolynomial size.
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BBC reports: The Information Commissioner's Office (ICO) took down its website after a warning that hackers were taking control of visitors' computers to mine cryptocurrency. Security researcher Scott Helme said more than 4,000 websites, including many government ones, were affected. He said the affected code had now been disabled and visitors were no longer at risk. The ICO said: "We are aware of the issue and are working to resolve it." Mr Helme said he was alerted by a friend who had received a malware warning when he visited the ICO website. He traced the problem to a website plugin called Browsealoud, used to help blind and partially sighted people access the web. The cryptocurrency involved was Monero  a rival to Bitcoin that is designed to make transactions in it "untraceable" back to the senders and recipients involved. The plugin had been tampered with to add a program, Coinhive, which "mines" for Monero by running processorintensive calculations on visitors' computers. T
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This paper develops coding techniques to reduce the running time of distributed learning tasks. It characterizes the fundamental tradeoff to compute gradients (and more generally vector summations) in terms of three parameters: computation load, straggler tolerance and communication cost. It further gives an explicit coding scheme that achieves the optimal tradeoff based on recursive polynomial constructions, coding both across data subsets and vector components. As a result, the proposed scheme allows to minimize the running time for gradient computations. Implementations are made on Amazon EC2 clusters using Python with mpi4py package. Results show that the proposed scheme maintains the same generalization error while reducing the running time by $32\%$ compared to uncoded schemes and $23\%$ compared to prior coded schemes focusing only on stragglers (Tandon et al., ICML 2017).
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The support of any module over a commutative ring is defined as the collection of all prime ideals of the ring at which the localization of the module is nonzero. For finitely generated modules, the support is the collection of all prime ideals containing the annihilator of the module. In this article, we raise the natural question that over which commutative rings, the support of every module is the collection of all the prime ideals of its annihilator. We completely classify such rings, and in the process it also comes out that it is enough to require that only for countably generated modules, the support is the collection of all prime ideals containing the annihilator of the module.
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We obtain compact orientable embedded surfaces with constant mean curvature $0<H<\frac{1}{2}$ and arbitrary genus in $\mathbb{S}^2\times\mathbb{R}$. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature $\frac{1}{2}$ tangent along an equator. This is a particular case of a conjugate Plateau construction of doubly periodic surfaces with constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$, $\mathbb{H}^2\times\mathbb{R}$, and $\mathbb{R}^3$ with bounded height and enjoying the symmetries of certain tessellations of $\mathbb{S}^2$, $\mathbb{H}^2$, and $\mathbb{R}^2$ by regular polygons.
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Fractional difference sequence spaces have been studied in the literature recently. In this work, some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators on some difference sequence spaces of fractional orders are established. Some classes of compact operators on those spaces are characterized. The results of this work are more general and comprehensive then many other studies in literature.
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In this study, we give the Sturm comparison theorems for discrete fractional SturmLiouville (DFSL) equations within RiemannLiouville and Gr\"unwaldLetnikov sense. The emergence of SturmLiouville equations began as one dimensional Schr\"odinger equation in quantum mechanics and one of the most important results is Sturm comparison theorems [27]. These theorems give information about the properties of zeros of two equations having different potentials.
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The notions of complementability and maximality were introduced in 1974 by Ornstein and Weiss in the context of the automorphisms of a probability space, in 2008 by Brossard and Leuridan in the context of the Brownian filtrations, and in 2017 by Leuridan in the context of the polyadic filtrations indexed by the nonpositive integers. We present here some striking analogies and also some differences existing between these three contexts.
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We extend results of Chang and Ran regarding large dimensional families of immersed curves of positive genus in projective space in two directions. In one direction, we prove a sharp bound for the dimension of a complete family of smooth rational curves immersed into projective space, completing the picture in projective space. In another direction, we isolate the necessary positivity condition on the tangent bundle of projective space used to run the argument, which allows us to rule out large dimensional families of immersed curves of positive genus in generalized flag varieties.
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This paper analyzes the iterationcomplexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs. More specifically, the objective function is of the form $f + h$ where $f$ is a differentiable function whose gradient is Lipschitz continuous and $h$ is a closed convex function with bounded domain. The method, basically, consists of applying an accelerated inexact proximal point method for solving approximately a sequence of quadratic penalized subproblems associated to the linearly constrained problem. Each subproblem of the proximal point method is in turn approximately solved by an accelerated composite gradient method. It is shown that the proposed scheme generates a $\rho$approximate stationary point in at most ${\cal{O}}(1/\rho^{3})$. Finally, numerical results showing the efficiency of the proposed method are also given.
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A cognitive radio system has the ability to observe and learn from the environment, adapt to the environmental conditions, and use the radio spectrum more efficiently. It allows secondary users (SUs) to use the primary users (PUs) channels when they are not being utilized. Cognitive radio involves three main processes: spectrum sensing, deciding, and acting. In the spectrum sensing process, the channel occupancy is measured with spectrum sensing techniques in order to detect unused channels. In the deciding process, sensing results are analyzed and decisions are made based on these results. In the acting process, actions are made by adjusting the transmission parameters to enhance the cognitive radio performance. One of the main challenges of cognitive radio is the wideband spectrum sensing. Existing spectrum sensing techniques are based on a set of observations sampled by an ADC at the Nyquist rate. However, those techniques can sense only one channel at a time because of the hardware
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The goal of this paper is to develop numerical methods computing a few smallest elasticity transmission eigenvalues, which are of practical importance in inverse scattering theory. The problem is challenging since it is nonlinear, nonselfadjoint, and of fourth order. We construct a nonlinear function whose values are generalized eigenvalues of a series of selfadjoint fourth order problems. The roots of the function are the transmission eigenvalues. Using an $H^2$conforming finite element for the selfadjoint fourth order eigenvalue problems, we employ a secant method to compute the roots of the nonlinear function. The convergence of the proposed method is proved. In addition, a mixed finite element method is developed for the purpose of verification. Numerical examples are presented to verify the theory and demonstrate the effectiveness of the two methods.
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In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in smallstrain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e., a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discretetime incremental minimum problems. The main difficulty in the passage to the continuoustime limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulati
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This work explores a distributed computing setting where $K$ nodes are assigned fractions (subtasks) of a computational task in order to perform the computation in parallel. In this setting, a wellknown main bottleneck has been the internode communication cost required to parallelize the task, because unlike the computational cost which could keep decreasing as $K$ increases, the communication cost remains approximately constant, thus bounding the total speedup gains associated to having more computing nodes. This bottleneck was substantially ameliorated by the recent introduction of coded MapReduce techniques which allowed each node  at the computational cost of having to preprocess approximately $t$ times more subtasks  to reduce its communication cost by approximately $t$ times. In reality though, the associated speed up gains were severely limited by the requirement that larger $t$ and $K$ necessitated that the original task be divided into an extremely large number of subt
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Consider the first exit time of onedimensional Brownian motion from a random passageway. We discuss a Brownian motion with two timedependent random boundaries in quenched sense. Showing $t^{1}\ln\bfP^x(\forall_{s\in[0,t]}a+\beta W_s\leq B_s\leq b+\beta W_sW)$ converges to a finite positive constant $\gamma(ba,\beta)$ almost surely if $a<x<b$ and $\{W_s\}_{s\geq 0}$ is another onedimensional Brownian motion independent of $\{B_s\}_{s\geq 0}, B_0=x, W_0=0.$ $P(\cdotW)$ is the probability condition on a realization of $\{W_s\}_{s\geq 0}$. Some properties of binary function $\gamma(ba,\beta)$ are also found.
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Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some $\ell_{p}$ type fractional difference sequence spaces via Euler gamma function. Although we characterize compactness conditions on those spaces using the main tools of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is $\ell _{\infty }$. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.
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Geometric quantization of 2+1 dimensional pure YangMills theory is studied with focusing on finite large scales. It is previously shown that (Yildirim, 2015, Int. J. Mod. Phys A, 30(7), 1550034), topologically massive YangMills theory exhibits a ChernSimons splitting behavior at large scales, similar to the topologically massive AdS gravity model in its near ChernSimons limit. This splitting occurs as two ChernSimons parts with levels $+k/2$ each, where $k$ is the original ChernSimons level in the Lagrangian. In the pure ChernSimons limit, split parts combine to give the original ChernSimons level. The opposite limit of the gravitational analogue gives an EinsteinHilbert term with a negative cosmological constant, which can be written as a two half ChernSimons terms with opposite signs. With this motivation, the gauge theory analogue of this limit is investigated, showing that at finite large distances pure YangMills theory acts like a topological theory that consists of spl
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The Chow ring of a matroid (or more generally, atomic latice) is an invariant whose importance was demonstrated by Adiprasito, Huh and Katz, who used it to resolve the longstanding HeronRotaWelsh conjecture. Here, we make a detailed study of the Chow rings of uniform matroids and of matroids of finite vector spaces. In particular, we express the Hilbert series of such matroids in terms of permutation statistics; in the full rank case, our formula yields the majexc $q$Eulerian polynomials of Shareshian and Wachs. We also provide a formula for the CharneyDavis quantities of such matroids, which can be expressed in terms of either determinants or $q$secant numbers.
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Inspired by the AxKochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the $E(n,p)$local categories over any nonprinicipal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. This shows that chromatic homotopy theory at a fixed height is asymptotically algebraic.
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The integration of a massive number of largescale wind turbines brought about urgent technical challenge to power transmission network operators in terms of secure power supply and energy dispatching optimization. In this paper, an optimal framework is proposed for transmission expansion planning in a deregulated power market environment. The level of congestion in the network is utilized as the driving signal for the need of network expansion. A compromise between the congestion cost and the investment cost is used to determine the optimal expansion scheme. The longterm network expansion problem is formed as the decoupled combination of: 1) the master problem (minimization of investment costs subject to investment constraints and the Benders cuts generated by the operational problem (power pool) and 2) the operational problem, whose solution provides congestion details and associated multipliers. A proper powerpool model is developed and solved for congestion cost, congestion reven
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We study a generalization of genuszero $r$spin theory in which exactly one insertion has a negativeone twist, which we refer to as the "closed extended" theory, and which is closely related to the open $r$spin theory of Riemann surfaces with boundary. We prove that the generating function of genuszero closed extended intersection numbers coincides with the genuszero part of a special solution to the system of differential equations for the wave function of the $r$th GelfandDickey hierarchy. This parallels an analogous result for the open $r$spin generating function in the companion paper to this work.
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For each of the groups $G = O(2), SU(2), U(2)$, we compute the integral and $\mathbb{F}_2$cohomology rings of $B_\text{com} G$ (the classifying space for commutativity of $G$), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of $E_\text{com} G$ (the homotopy fiber of the inclusion $B_\text{com} G \to BG$), and some lowdimensional homotopy groups of $B_\text{com} G$.
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The FockBargmannHartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\w\^2<e^{\mu\z\^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded nonhyperbolic domain in $\mathbb{C}^{n+m}$. Recently, TuWang obtained the rigidity result that proper holomorphic selfmappings of $D_{n,m}(\mu)$ are automorphisms for $m\geq 2$, and found a counterexample to show that the rigidity result isn't true for $D_{n,1}(\mu)$. In this article, we obtain a classification of proper holomorphic mappings between $D_{n,1}(\mu)$ and $D_{N,1}(\mu)$ with $N<2n$.
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One proves that the $n$D stochastic controlled equation $dX+AXdt=\sigma(X)dW+Bu\,dt$, where $\sigma\in\mbox{Lip}((\R^n,\L(\R^d,\R^n))$ and the pair $A\in\L(\R^n)$, $B\in\L(\R^m,\R^n)$ satisfies the Kalman rank condition, is exactly controllable in each $y\in\R^n$, $\sigma(y)=0$ on each finite interval $(0,T)$. An application to approximate controllability to stochastic heat equation is given.
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In this article, we present new general results on existence of augmented Lagrange multipliers. We define a penalty function associated with an augmented Lagrangian, and prove that, under a certain growth assumption on the augmenting function, an augmented Lagrange multiplier exists if and only if this penalty function is exact. We also develop a new general approach to the study of augmented Lagrange multipliers called the localization principle. The localization principle allows one to study the local behaviour of the augmented Lagrangian near globally optimal solutions of the initial optimization problem in order to prove the existence of augmented Lagrange multipliers.
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How big is the risk that a few initial failures of nodes in a network amplify to large cascades that span a substantial share of all nodes? Predicting the final cascade size is critical to ensure the functioning of a system as a whole. Yet, this task is hampered by uncertain or changing parameters and missing information. In infinitely large networks, the average cascade size can often be well estimated by established approaches building on local tree approximations and mean field approximations. Yet, as we demonstrate, in finite networks, this average does not even need to be a likely outcome. Instead, we find broad and even bimodal cascade size distributions. This phenomenon persists for system sizes up to $10^{7}$ and different cascade models, i.e. it is relevant for most real systems. To show this, we derive explicit closedform solutions for the full probability distribution of the final cascade size. We focus on two topological limit cases, the complete network representing a den
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