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We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.
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We study a class of discretetime random dynamical systems with compact phase space. Assuming that the deterministic counterpart of the system in question possesses a dissipation property, its linearisation is approximately controllable, and the driving noise has a decomposable structure, we prove that the corresponding family of Markov processes has a unique stationary measure, which is exponentially mixing in the dualLipschitz metric. The abstract result is applicable to nonlinear dissipative PDEs perturbed by a random force which affects only a few Fourier modes and belongs to a certain class of random processes. We assume that the nonlinear PDE in question is well posed, its nonlinearity is nondegenerate in the sense of the control theory, and the random force is a regular and bounded function of time which satisfies some decomposability and observability hypotheses. This class of forces includes random Haar series, where coefficients for high Haar modes decay sufficiently fast.
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Exact lower and upper bounds on the best possible misclassification probability for a finite number of classes are obtained in terms of the total variation norms of the differences between the subdistributions over the classes. These bounds are compared with the exact bounds in terms of the conditional entropy obtained by Feder and Merhav.
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The equilibration of a finite Bose system is modelled using a gradient expansion of the collision integral that leads to a nonlinear transport equation. For constant transport coefficients, it is solved in closed form through a nonlinear transformation. Using schematic initial conditions, the exact solution and the equilibration time are derived and compared to the corresponding case for fermions. Applications to the equilibration of the gluon system created initially in relativistic heavyion collisions, and to cold quantum gases are envisaged.
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We analyze complexity in spatial network ensembles through the lens of graph entropy. Mathematically, we model a spatial network as a soft random geometric graph, i.e., a graph with two sources of randomness, namely nodes located randomly in space and links formed independently between pairs of nodes with probability given by a specified function (the "pair connection function") of their mutual distance. We consider the general case where randomness arises in node positions as well as pairwise connections (i.e., for a given pair distance, the corresponding edge state is a random variable). Classical random geometric graph and exponential graph models can be recovered in certain limits. We derive a simple bound for the entropy of a spatial network ensemble and calculate the conditional entropy of an ensemble given the node location distribution for hard and soft (probabilistic) pair connection functions. Under this formalism, we derive the connection function that yields maximum entropy
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To make better use of file diversity provided by random caching and improve the successful transmission probability (STP) of a file, we consider retransmissions with random discontinuous transmission (DTX) in a largescale cacheenabled heterogeneous wireless network (HetNet) employing random caching. We analyze and optimize the STP in two mobility scenarios, i.e., the high mobility scenario and the static scenario. First, in each scenario, by using tools from stochastic geometry, we obtain the closedform expressions for the STP in the general and low signaltointerference ratio (SIR) threshold regimes, respectively. The analysis shows that a larger caching probability corresponds to a higher STP in both scenarios; random DTX can improve the STP in the static scenario and its benefit gradually diminishes when mobility increases. Then, in each scenario, we consider the maximization of the STP with respect to the caching probability and the BS activity probability, which is a challengi
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The dimension of a graph $G$ is the smallest $d$ for which its vertices can be embedded in $d$dimensional Euclidean space in the sense that the distances between endpoints of edges equal $1$ (but there may be other unit distances). Answering a question of Erd\H{o}s and Simonovits [Ars Combin. 9 (1980) 229246], we show that any graph with less than $\binom{d+2}{2}$ edges has dimension at most $d$. Improving their result, we prove that that the dimension of a graph with maximum degree $d$ is at most $d$. We show the following Ramsey result: if each edge of the complete graph on $2d$ vertices is coloured red or blue, then either the red graph or the blue graph can be embedded in Euclidean $d$space. We also derive analogous results for embeddings of graphs into the $(d1)$dimensional sphere of radius $1/\sqrt{2}$.
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We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvaturedimension inequality $CD(0,\infty)$. This estimate is independent of the size of the graph and provides a general method to obtain higher order spectral estimates. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted graphs satisfying $CD(0,\infty)$. We also discuss a higher order Cheeger constant ratio estimate and related topics about expanders.
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Following a previous article we continue our study on nonterminating hypergeometric series with one free parameter, which aims to find arithmetical constraints for a given hypergeometric series to admit a gamma product formula. In this article we exploit the concepts of duality and reciprocity not only to extend already obtained results to a larger region but also to strengthen themselves substantially. Among other things we are able to settle the rationality and finiteness conjectures posed in the previous article.
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One of the main challenges in highspeed mobile communications is the presence of large Doppler spreads. Thus, accurate estimation of maximum Doppler spread (MDS) plays an important role in improving the performance of the communication link. In this paper, we derive the dataaided (DA) and nondataaided (NDA) CramerRao lower bounds (CRLBs) and maximum likelihood estimators (MLEs) for the MDS in multipleinput multipleoutput (MIMO) frequencyselective fading channel. Moreover, a lowcomplexity NDAmomentbased estimator (MBE) is proposed. The proposed NDAMBE relies on the second and fourthorder moments of the received signal, which are employed to estimate the normalized squared autocorrelation function of the fading channel. Then, the problem of MDS estimation is formulated as a nonlinear regression problem, and the leastsquares curvefitting optimization technique is applied to determine the estimate of the MDS. This is the first time in the literature when DAand NDAMDS estima
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In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes $A_{\vec{p}}$, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calder\'onZygmund operators. Furthermore, we go beyond the classes $A_{\vec{p}}$ and extrapolate within the classes $A_{\vec{p},\vec{r}}$ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several applications which can be easily obtained using extrapolation. First, for the bilinear Hilbert transform one can extrapolate from the recent result of Culiuc et al. who considered the Banach range and extend the estimates to the quasiBanach range. Also, we obtain for free vectorvalued inequalities as those proved by Benea and Muscalu. We also extend recent results of Carando et al. on Marcinkie
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We investigate integral formulations and fast algorithms for the steadystate radiative transfer equation with isotropic and anisotropic scattering. When the scattering term is a smooth convolution on the unit sphere, a model reduction step in the angular domain using the Fourier transformation in 2D and the spherical harmonic transformation in 3D significantly reduces the number of degrees of freedoms. The resulting Fourier coefficients or spherical harmonic coefficients satisfy a Fredholm integral equation of the second kind. We study the uniqueness of the equation and proved an a priori estimate. For a homogeneous medium, the integral equation can be solved efficiently using the FFT and iterative methods. For an inhomogeneous medium, the recursive skeletonization factorization method is applied instead. Numerical simulations demonstrate the efficiency of the proposed algorithms in both homogeneous and inhomogeneous cases and for both transport and diffusion regimes.
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We consider a class of singular ordinary differential equations describing analytic systems of arbitrary finite dimension, subject to a quasiperiodic forcing term and in the presence of dissipation. We study the existence of response solutions, i.e. quasiperiodic solutions with the same frequency vector as the forcing term, in the case of large dissipation. We assume the system to be conservative in the absence dissipation, so that the forcing term is  up to the sign  the gradient of a potential energy, and both the mass and damping matrices to be symmetric and positive definite. Further, we assume a nondegeneracy condition on the forcing term, essentially that the timeaverage of the potential energy has a strict local minimum. On the contrary, no condition is assumed on the forcing frequency; in particular we do not require any Diophantine condition. We prove that, under the assumptions above, a response solution always exist provided the dissipation is strong enough. This e
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We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given.
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In light of the rich results of expansiveness in the dynamics of diffeomorphisms, it is natural to consider another notions of expansiveness such as countablyexpansive, measure expansive, $N$expansive and so on. In this paper, we introduce the notion of $N$expansiveness for flows on a $C^{\infty}$ compact connected Riemannian manifold by using the kinematic expansiveness which is extension of the $N$expansive diffeomorphisms. And we prove that a vector field $X$ on $M$ is $C^1$ robustly kinematic $N$expansive then $X$ satisfies quasiAnosov. Furthermore, we consider the hyperbolicity of local dynamical systems with kinematic $N$expansiveness.
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We analyze an iterative coupling of mixed and discontinuous Galerkin methods for numerical modelling of coupled flow and mechanical deformation in porous media. The iteration is based on an optimized fixedstress split along with a discontinuous variational time discretization. For the spatial discretization of the subproblem of flow mixed finite element techniques are applied. The discretization of the subproblem of mechanical deformation uses discontinuous Galerkin methods. They have shown their ability to eliminate locking that sometimes arises in numerical algorithms for poroelasticity and causes nonphysical pressure oscillations.
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We study $\infty$modulus on general metric spaces and establish its relation to shortest lengths of paths. This connection was already known for modulus on graphs, but the formulation in metric measure spaces requires more attention to exceptional families. We use this to define a metric that we call the essential metric, and show how this recovers a metric that had already been advanced in the literature by De Cecco and Palmieri.
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Detecting the presence of an active random wireless source with minimum latency utilizing array signal processing is considered. The problem is studied under the constraint that the analogtodigital conversion at each radio sensor is restricted to the reading of the analog receive signal sign. We formulate the digital signal processing task as a sequential hypothesis test in simple form. To circumvent the intractable loglikelihood ratio of the resulting multivariate binary array data, a reduced model representation within the exponential family is employed. This approach allows us to design a sequential test and to analyze its analytic performance along classical arguments. In the context of wireless spectrum monitoring for satellitebased navigation and synchronization systems, we study the achievable processing latency, characterized by the average sample number, as a function of the antennas in use. The practical feasibility and potential of the discussed lowcomplexity sensing an
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Following ideas of Brunnbauer and Hanke, we construct a notion of infinite Karea for homology classes of simplicial complexes with finitely generated fundamental groups. As in Brunnbauer's and Hanke's work, the results concerning these homology classes will imply that Gromov's property ofhaving infinite Karea depends only on the image of the fundamental class under the classifying map of the universal cover. As acorollary we obtain another proof of a theorem of Fukumoto, that the property of infinite Karea is invariant under $p$surgery with $p\neq 1$. As a result of the proof of our main theorem, we will clarify a point left open in a paper of Mishchenko and Teleman about extensions of almost flat bundles.
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We study a naturally occurring $E_{\infty}$subalgebra of the full $E_2$Hochschild cochain complex arising from coherent cochains. For group rings and certain category algebras, these cochains detect $H^*(B {\cal{C}})$, the simplicial cohomology of the classifying space of the underlying group or category, $\cal{C}$. In this setting the simplicial cup product of cochains on $B{\cal{C}}$ agrees with the Gerstenhaber product and Steenrod's cupone product of cochains agrees with the preLie product. We extend the idea of coherent cochains to algebras more general than category algebras and dub the resulting cochains autopoietic. Coefficients are from a commutative ring $k$ with unit, not necessarily a field.
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We generalize problems in Wasan geometry which involve no folded figures but are related to Haga's fold in origami. Using the tangent circles appeared in those problems we give a parametric representation of the generalized Haga's fold given in the first part of this twopart paper.
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We show two FreidlinWentzell type Large Deviations Principles in path space topologies (uniform and H\"older) for the solution process of McKeanVlasov Stochastic Differential Equations (MVSDEs) using techniques addressing the presence of the law in the coefficients directly and avoiding altogether decoupling arguments or limits of particle systems. We provide existence and uniqueness results along with several properties for a class of MVSDEs having random coefficients and drifts of superlinear growth . As an application of our results, we establish a Functional Strassen type result (Law of Iterated Logarithm) for the solution process of an MVSDEs.
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We propose a deformed version of the generalized Heisenberg algebra by using techniques borrowed from the theory of pseudobosons. In particular, this analysis is relevant when non selfadjoint Hamiltonians are needed to describe a given physical system. We also discuss relations with nonlinear pseudobosons. Several examples are discussed.
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We introduce a notion of global weak solution to the NavierStokes equations in three dimensions with initial values in the critical homogeneous Besov spaces $\dot B^{1+\frac{3}{p}}_{p,\infty}$, $p > 3$. Our solutions satisfy certain stability properties with respect to the weak$\ast$ convergence of initial data. We provide applications to blowup criteria, minimal blowup initial data, and forward selfsimilar solutions.
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In an effort to force websites to better protect their users, the Chrome web browser will label all sites not encrypted traffic as "Not secure" in the web address bar, Google announced Thursday. From a report: Encrypted traffic allows users to access data on a website without allowing potential eavesdroppers to see anything the users visit. HTTPS also prevents meddlers from changing information in transit. During normal web browsing, Google currently displays a "Not secure" warning in the next to a site's URL if it forgoes HTTPS encryption and a user enters data. Now the browser will label all sites without HTTPS encryption this way.
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Alphabet is close to acquiring the iconic Chelsea Market in Midtown Manhattan for over $2 billion. The market totals 1.2 million square feet and sits across the street from the company's New York headquarters, a 2.9 millionsquarefoot building that it bought for $1.8 billion in 2010. Quartz reports: Google is already the Market's largest tenant, having steadily expanded its footprint to about 400,000 square feet. The tech giant hasn't revealed plans for the property, but according to The Real Deal, the company is expected to maintain the status quo. Alphabet's aggressive expansion in New York follows a growing trend of tech giants taking over cities. With their outsized share of the economy, tech companies are exerting increasing influence over urban infrastructure and development.
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schwit1 shares a report from Ars Technica, highlighting the problems the Germany Navy is facing right now. It has no working submarines due to a chronic repair parts shortage, and its newest ships face problems so severe that the first of the class failed its sea trials and was returned to the shipbuilders in December. From the report: The BadenWurttemberg class frigates were ordered to replace the 1980sera Bremen class ships, all but two of which have been retired already. At 149 meters (488 feet) long with a displacement of 7,200 metric tons (about 7,900 U.S. tons), the BadenWurttembergs are about the size of destroyers and are intended to reduce the size of the crew required to operate them. Like the Zumwalt, the frigates are intended to have improved land attack capabilities  a mission capability largely missing from the Deutsche Marine's other postunification ships. The new frigate was supposed to be a master of all trades  carrying Marines to deploy to fight ashore, provi
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We consider the Dirac equation with cubic Hartreetpye nonlinearity derived by uncoupling the MaxwellDirac or DiracKleinGordon systems. We prove small data scattering result. Main ingredients of the proof are the localized strichartz estimates and improved bilnear estimates thanks to nullstructure hidden in Dirac operator. We apply the projection operator and get system of equations which we work on. This result is shown to be optimal by proving iteration method based on Duhamel's formula of the system over superciritical range fails.
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Let $G$ be a locally compact group with left regular representation $\lambda_{G}.$ We say that $G$ admits a frame of translates if there exist a countable set $\Gamma\subset G$ and $\varphi\in L^{2}(G)$ such that $(\lambda_{G}(x) \varphi)_{x \in\Gamma}$ is a frame for $L^{2}(G).$ The present work aims to characterize locally compact groups having frames of translates, and to this end, we derive necessary and/or sufficient conditions for the existence of such frames. Additionally, we exhibit surprisingly large classes of Lie groups admitting frames of translates.
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The paper deals with learning the probability distribution of the observed data by artificial neural networks. We suggest a socalled gradient conjugate prior (GCP) update appropriate for neural networks, which is a modification of the classical Bayesian update for conjugate priors. We establish a connection between the gradient conjugate prior update and the maximization of the loglikelihood of the predictive distribution. Unlike for the Bayesian neural networks, we do not impose a prior on the weights of the neural networks, but rather assume that the ground truth distribution is normal with unknown mean and variance and learn by neural networks the parameters of a prior (normalgamma distribution) for these unknown mean and variance. The update of the parameters is done, using the gradient that, at each step, directs towards minimizing the KullbackLeibler divergence from the prior to the posterior distribution (both being normalgamma). We obtain a corresponding dynamical system
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A homogenizable structure $\mathcal{M}$ is a structure where we may add a finite amount of new relational symbols to represent some $\emptyset$definable relations in order to make the structure homogeneous. In this article we will divide the homogenizable structures into different classes which categorize many known examples and show what makes each class important. We will show that model completeness is vital for the relation between a structure and the amalgamation bases of its age and give a necessary and sufficient condition for an $\omega$categorical modelcomplete structure to be homogenizable.
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A Gadget, more precisely a scalar Gadget, is defined as a mathematical calculation acting over a domain of one or more adinkra graphs and whose range is a real number. A 2010 work on the subject of automorphisms of adinkra graphs, implied the existence of multiple numbers of Gadgets depending on the number of colors under consideration. For four colors, this number is two. In this work, we verify the existence of a second such Gadget and calculate (both analytically and via explicit computerenabled algorithms) its 1,358,954,496 matrix elements over 36,864 minimal valise adinkras related to the Coxeter Group BC4.
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In this paper we provide the existence of classical solutions to stationary mean field game systems in the whole space $\mathbb{R}^N$, with coercive potential, aggregating local coupling, and under general conditions on the Hamiltonian, completing the analysis started in the companion paper [6]. The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian. This result is obtained using a variational approach based on the analysis of the nonconvex energy associated to the system. Finally, we show that in the vanishing viscosity limit mass concentrates around the flattest minima of the potential, and that the asymptotic shape of the solutions in a suitable rescaled setting converges to a ground state, i.e. a classical solution to a mean field game system without potential.
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These notes are the English version of the paper "Hyperbolicit\'e du graphe des rayons et quasimorphismes sur un gros groupe modulaire". The mapping class group Gamma of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of Gamma on this graph to find an explicit non trivial quasimorphism on Gamma and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a hyperbolic element of Gamma with vanishing stable commutator length. This carries out a program proposed by Danny Calegari.
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To reinforce the analogy between the mapping class group and the Cremona group of rank 2 over an algebraic closed field, we look for a graph analoguous to the curve graph and such that the Cremona group acts on it nontrivially. The first candidate is a graph introduced by D. Wright. However, we demonstrate that it's not Gromovhyperbolic. Then, we construct two other graphs associated to the Vorono\"i tesselation. We show that one is quasiisometric to the Wright's graph and so it's not Gromovhyperbolic. We prove that the other one is Gromovhyperbolic.
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Nextcloud 13 has been released. "This release brings improvements to the core File Sync and Share like easier moving of files and a tech preview of our endtoend encryption for the ultimate protection of your data. It also introduces collaboration and communication capabilities, like autocomplete of comments and integrated realtime chat and video communication. Last but not least, Nextcloud was optimized and tuned to deliver up to 80% faster LDAP, much faster object storage and Windows Network Drive performance and a smoother user interface."
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We compute the onelevel density for the family of cubic Dirichlet $L$functions when the support of the Fourier transform of a test function is in $(1,1)$. We also establish the Ratios conjecture prediction for the onelevel density for this family, and confirm that it agrees with the onelevel density we obtain.
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In this paper, we study the asymptotic behaviour of minimizing solutions of a GinzburgLandau type functional with potential having a zero at 1 of infinite order and we estimate the energy. We generalize in this case a lower bound for the energy of unit vector field given by BrezisMerleRivi\`ere.
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This paper shows a practical way of modelling nonlinear dynamic systems with one input using the discrete form of the integral operator of the Urysohn type. A new iterative procedure for the identification of the Uryshon operator is proposed. It represents a sequence of elementary computational steps providing iterative improvements for the Urysohn model using observable inputoutput data. The controllability of the system is not required. The method is applicable as a real time identification process. The main advantages of the suggested identification technique are simplicity (the procedure can be implemented in $13$ lines of Matlab code) and accuracy. It can also be used for the identification of one input Hammerstein and linear systems, since the Urysohn model is the generic form of these models. The efficiency of the algorithm is demonstrated using one example of a nonlinear mechanical system with precise and noisy inputoutput data.
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We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergencefree velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in $W^{r,p}$, with $r>1$, and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space $W^{r,p}$ does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper "Exponential selfsimilar mixing by incompressible flows" (Preprint arXiv:1605.02090), and have been announced in "Exponential selfsimilar mixing and loss of regularity for continuity equations" (C. R. Math. Ac
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We study the set of logcanonical thresholds (or critical integrability indices) of holomorphic (resp. real analytic) function germs in $\mathbb{C}^2$ (resp. $\mathbb{R}^2$). In particular, we prove that the ascending chain condition holds, and that the positive accumulation points of decreasing sequences are precisely the integrability indices of holomorphic (resp. real analytic) functions in dimension $1$. This gives a new proof of a theorem of PhongSturm.
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For a general measure space $(X, \sL, \l)$ the pointwise nature of weak convergence in $\Li$ is investigated using singular functionals analogous to $\d$functions in the theory of continuous functions on topological spaces. The implications for pointwise behaviour in $X$ of weakly convergent sequences in $\Li$ are inferred and the composition mapping $u \mapsto F(u)$ is shown to be sequentially weakly continuous on $\Li$ when $F:\RR \to \RR$ is continuous. When $\sB$ is the Borel $\sigma$algebra of a locally compact Hausdorff topological space $(X,\varrho)$ and $f \in L_\infty(X, \sB, \l)^*$ is arbitrary, let $\nu$ be the finitely additive measure in the integral representation of $f$ on $L_\infty(X, \sB, \l)$, and let $\hat \nu$ be the Borel measure in the integral representation of $f$ restricted to $C_0(X,\varrho)$. From a minimax formula for $\hat \nu$ in terms $\nu$ it emerges that when $(X,\varrho)$ is not compact, $\hat\nu$ may be zero when $\nu$ is not, and the set of $\nu$ f
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We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{c r}$ with $c>0$. By considering suitable parameter spaces for $m$periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$critical for all $c$ in terms of weighted spherical $2$designs contained in the set. Especially for $2$periodic sets like the family $\mathsf{D}^+_n$ we obtain expressions for the hessian of the energy function, allowing to certify $f_c$optimality in certain cases. For odd integers $n\geq 9$ we can hereby in particular show that $\mathsf{D}^+_n$ is locally $f_c$optimal among periodic sets for all sufficiently large~$c$.
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We prove LiebRobinson bounds for a general class of lattice fermion systems. By making use of a suitable conditional expectation onto subalgebras of the CAR algebra, we can apply the LiebRobinson bounds much in the same way as for quantum spin systems. We preview how to obtain the spectral flow automorphisms and to prove stability of the spectral gap for frustrationfree gapped systems satisfying a Local Topological Quantum Order condition.
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Random walk in a dynamic i.i.d. beta random environment, conditioned to escape at an atypical velocity, converges to a Doob transform of the original walk. The Doobtransformed environment is correlated in time, i.i.d. in space, and its marginal density function is a product of a beta density and a hypergeometric function. Under its averaged distribution the transformed walk obeys the wandering exponent 2/3 that agrees with KardarParisiZhang universality. The harmonic function in the Doob transform comes from a Busemanntype limit and appears as an extremal in a variational problem for the quenched large deviation rate function.
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Let $X$ be a set of $K$rational points in $P^1 \times P^1$ over a field $K$ of characteristic zero, let $Y$ be a fat point scheme supported at $ X$, and let $R_Y$ be the bihomogeneus coordinate ring of $Y$. In this paper we investigate the module of Kaehler differentials $\Omega^1_{R_Y/K}$. We describe this bigraded $R_Y$module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support $X$ is a complete intersection or an almost complete intersection in $P^1 \times P^1$. Moreover, we introduce a Kaehler different for $Y$ and use it to characterize reduced fat point schemes in $P^1 \times P^1$ having the CayleyBacharach property.
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Recently S. Patrikis, J.F. Voloch and Y. Zarhin have proven, assuming several well known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, under some technical restrictions. Our approach is possible since abelian varieties and K3s are quite well described by `Hodgetheoretical' results. In particular the theorem we present can be interpreted as follows: a family of $\ell$adic representations that \emph{looks like} the one induced by the transcendental part of the $\ell$adic cohomology of a K3 surface (defined over a number field) determines a Hodge structure which in turn determines a K3 surface (which may be defined over a number field).
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We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the Gequivariant sphere spectrum, including their HillHopkinsRavenel norms, where G is any finite group. Our results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, our analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, we obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category.
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Let $A$ be a finite subset of a field $\mathbb{F}$ and $D_n(A)$ be a set of all matrices with entries in $A$, namely $$ D_n(A)=\{D\in \mathbb{F}\ \ \exists a_{ij}\in A, 1 \le i,j \le n, \det\bigl((a_{ij})\bigr)=D\}, $$ where the symbol $(a_{ij})$ defines the matrix with elements $a_{ij}$. How big is the size of the set $D_n(A)$ comparing to the size of the set $A$?
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Equilibrium shapes of twodimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed.
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In this note, we aim to establish a number of embeddings between various function spaces that are frequently considered in the theory of Fourier series. More specifically, we give sufficient conditions for the embeddings $\Phi V[h]\subseteq \Lambda\text{BV}^{(p_n\uparrow p)}$, $\Lambda V[h_1]^{(p)}\subseteq\Gamma V[h_2]^{(q)}$ and $\Lambda\text{BV}^{(p_n\uparrow p)}\subseteq\Gamma\text{BV}^{(q_n\uparrow q)}$. Our results are new even for the wellknown spaces that have been studied in the literature. In particular, a number of results due to M. Avdispahi\'{c}, that describe relationships between the classes $\Lambda\text{BV}$ and $V[h]$, are derived as special cases.
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The appearance of primes in a family of linear recurrence sequences labelled by a positive integer $n$ is considered. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in $n$) dilated versions of the socalled Chebyshev polynomials of the fourth kind, also known as airfoil polynomials. It is proved that when the value of $n$ is given by a dilated Chebyshev polynomial of the first kind evaluated at a suitable integer, either the sequence contains a single prime, or no term is prime. For all other values of $n$, it is conjectured that the sequence contains infinitely many primes, whose distribution has analogous properties to the distribution of Mersenne primes among the Mersenne numbers. Similar results are obtained for the sequences associated with negative integers $n$, which correspond to Chebyshev polynomials of the third kind, and to another family of Lehmer numbers.
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Peter Cameron introduced the concept of the circular altitude of graphs; a parameter which was shown by Bamberg et al. that provides a lower bound on the circular chromatic number. In this note, we investigate this parameter and show that the circular altitude of a graph is equal to the maximum of circular altitudes of its blocks. Also, we show that homomorphically equivalent graphs have the same circular altitudes. Finally, we prove that the circular altitude of the Cartesian product of two graphs is equal to the maximum of circular altitudes of its factors.
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We consider constant mean curvature 1 surfaces in $\mathbb{R}^3$ arising via the DPW method from a holomorphic perturbation of the standard Delaunay potential on the punctured disk. Kilian, Rossman and Schmitt have proven that such a surface is asymptotic to a Delaunay surface. We consider families of such potentials parametrised by the necksize of the model Delaunay surface and prove the existence of a uniform disk on which the surfaces are close to the model Delaunay surface and are embedded in the unduloid case.
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In this paper, we study a class of nonweight modules over two kinds of algebras related to the Virasoro algebra, i.e., the loopVirasoro algebras $\mathfrak{L}$ and a class of Block type Lie algebras $\mathfrak{B(q)}$, where $q$ is a nonzero complex number. We determine those modules whose restriction to the Cartan subalgebra (modulo center) are free of rank one. We also provide a sufficient and necessary condition for such modules to be simple, and determine their isomorphism classes. Moreover, we obtain the simplicity of modules over loopVirasoro algebras by taking tensor products of some irreducible modules mentioned above with irreducible highest weight modules or Whittaker modules.
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Nonequilibrium steady states (NESS) of Boson systems with different phases are investigated with the aid of the C^*algebraic method. The system consists of sample and several free or purehopping bosonic reservoirs coupled with each other. Initially, the sample and reservoirs are uncoupled and in equilibrium with temperatures, local densities, and phases. At t=0, the reservoirs couple with each other through the sample. NESS are constructed as time limit of the composition of initial state and time evolution of coupled system. In NESS, Josephson currents and the mean entropy production rates are calculated. The mean entropy production rate does not depend on phase differences. As consequence, the currents provided to phase differences produces no entropy.
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Fullduplex systems require very strong selfinterference cancellation in order to operate correctly and a significant part of the selfinterference signal is due to nonlinear effects created by various transceiver impairments. As such, linear cancellation alone is usually not sufficient and sophisticated nonlinear cancellation algorithms have been proposed in the literature. In this work, we investigate the use of a neural network as an alternative to the traditional nonlinear cancellation method that is based on polynomial basis functions. Measurement results from a fullduplex testbed demonstrate that a small and simple feedforward neural network canceller works exceptionally well, as it can match the performance of the polynomial nonlinear canceller with significantly lower computational complexity.
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We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as $$ F(x,u,Du,D^2u) =\lambda c(x)u+\langle M(x)D u, D u \rangle +h(x) $$ in a bounded domain with a Dirichlet boundary condition; here $\lambda \in\mathbb{R}$, $c,\, h \in L^p(\Omega)$, $p>n\geq 1$, $c\gneqq 0$ and the matrix $M$ satisfies $0<\mu_1 I\leq M\leq \mu_2 I$. Recently this problem was studied in the "coercive" case $c\le0$, where uniqueness of solutions can be expected. It was conjectured that the solution set is more complex for noncoercive equations. This conjecture was recently verified by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, where the integral formulation of the problem was exploited. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use novel techniques based on the maximum principle. We develop a method to obtain the crucial uniform a priori bounds, which permit
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We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a NadelNakano type vanishing theorem of a vector bundle with a singular Hermitian metric. We do not use an approximation technique of a singular Hermitian metric. We apply these theorems to a singular Hermitian metric induced by holomorphic sections and a big vector bundle, and we obtain a generalization of Griffiths' vanishing theorem. Finally, we show a generalization of Ohsawa's vanishing theorem.
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We consider detection based on deep learning, and show it is possible to train detectors that perform well, without any knowledge of the underlying channel models. Moreover, when the channel model is known, we demonstrate that it is possible to train detectors that do not require channel state information (CSI). In particular, a technique we call sliding bidirectional recurrent neural network (SBRNN) is proposed for detection where, after training, the detector estimates the data in realtime as the signal stream arrives at the receiver. We evaluate this algorithm, as well as other neural network (NN) architectures, using the Poisson channel model, which is applicable to both optical and chemical communication systems. In addition, we also evaluate the performance of this detection method applied to data sent over a chemical communication platform, where the channel model is difficult to model analytically. We show that SBRNN is computationally efficient, and can perform detection unde
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