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Generalizing previous constructions, we present a dual pair of decompositions of the complement of a link L into bipyramids, given any multicrossing projection of L. When L is hyperbolic, this gives new upper bounds on the volume of L given its multicrossing projection. These bounds are realized by three closely related infinite tiling weaves.
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Soot trapped in the feathers of songbirds over the past 100 years is causing scientists to revise their records of air pollution. From a report: US researchers measured the black carbon found on 1,300 larks, woodpeckers and sparrows over the past century. They've produced the most complete picture to date of historic air quality over industrial parts of the US. The study also boosts our understanding of historic climate change. [...] This new study takes an unusual approach to working out the scale of soot coming from this part of the US over the last 100 years. The scientists trawled through natural history collections in museums in the region and measured evidence of black carbon, trapped in the feathers and wings of songbirds as they flew through the smoky air. The researchers were able to accurately estimate the amount of soot on each bird by photographing them and measuring the amount of light reflected off them. "We went into natural history collections and saw that birds from 10
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The paper introduces a variable exponent space $X$ which has in common with $L^{\infty}([0,1])$ the property that the space $C([0,1])$ of continuous functions on $[0,1]$ is a closed linear subspace in it. The associate space of $X$ contains both the Kolmogorov and the Marcinkiewicz examples of functions in $L^{1}$ with a.e. divergent Fourier series.
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Let $X\rightarrow {\mathbb P}^1$ be an elliptically fibered $K3$ surface with a section, admitting a sequence of Ricciflat metrics collapsing the fibers. Let $\mathcal E$ be a generic, holomoprhic $SU(n)$ bundle over $X$ such that the restriction of $\mathcal E$ to each fiber is semistable. Given a sequence $\Xi_i$ of HermitianYangMills connections on $\mathcal E$ corresponding to this degeneration, we prove that, if $E$ is a given fiber away from a finite set, the restricted sequence $\Xi_i_{E}$ converges to a flat connection uniquely determined by the holomorphic structure on $\mathcal E$.
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We consider 1D dissipative transport equations with nonlocal velocity field: \[ \theta_t+u\theta_x+\delta u_{x} \theta+\Lambda^{\gamma}\theta=0, \quad u=\mathcal{N}(\theta), \] where $\mathcal{N}$ is a nonlocal operator given by a Fourier multiplier. Especially we consider two types of nonlocal operators: $\mathcal{N}=\mathcal{H}$, the Hilbert transform, $\mathcal{N}=(1\partial_{xx} )^{\alpha}$. In this paper, we show several global existence of weak solutions depending on the range of $\gamma$ and $\delta$. When $0<\gamma<1$, we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variables), which have infinite energy, when $\gamma \in (0,2)$.
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By using the action of a Galois group on complex irreducible characters and conjugacy classes, we define the Galois characters and Galois classes. We will introduce a set of Galois characters, called Galois irreducible characters, that each Galois character is a positive linear combination of the Galois irreducible characters. It is shown that whenever the complex characters of the groups of a tower produce a positive selfdual Hopf algebra (PSH), Galois characters of the groups of the tower also produce a PSH. Then we will classify the Galois characters and Galois classes of the general linear groups over finite fields. In the end, we will precisely indicate the isomorphism between the PSH of Galois characters and a certain tensor product of Hopf algebras isomorphic to symmetric functions.
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We construct differential algebras in which spaces of (onedimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with the pointwise multiplication of ordinary functions. In particular, we embed the space of hyperfunctions on the unit circle into a differential algebra in such a way that the multiplication of real analytic functions on the unit circle coincides with their pointwise multiplication. Furthermore, we introduce a notion of regularity in our newly defined algebras and show that an embedded ultradistribution is regular if and only if it is an ultradifferentiable function.
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We have studied Pad\'e interpolation problems on an additive grid, related to additive difference ($d$) Painlev\'e equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. By solving those problems, we can derive time evolution equations, Lax pairs of scalar type and determinant formulae of special solutions, for the corresponding $d$Painlev\'e equations. We show how the special solutions are expressed as determinant formulae by using terminating (generalized) hypergeometric functions.
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In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of $\nu$Tamari lattices. In our framework, the main role of "Catalan objects" is played by $(I,\overline{J})$trees: bipartite trees associated to a pair $(I,\overline{J})$ of finite index sets that stand in simple bijection with lattice paths weakly above a lattice path $\nu=\nu(I,\overline{J})$. Such trees label the maximal simplices of a triangulation whose dual polyhedral complex gives a geometric realization of the $\nu$Tamari lattice introduced by Pr\'evileRatelle and Viennot. In particular, we obtain geometric realizations of $m$Tamari lattices as polyhedral subdivisions of associahedra induced by an arrangement of tropical hyperplanes, giving a positive answer to an open question of F.~Bergeron. The simplicial complex underlying our triangulation endows the $\nu$Tamari lattice with a full simplicial complex struct
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We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ {\Delta}u+ u=\big(I_\alpha*u^{\frac{\alpha}{N}+1}\big)u^{\frac{\alpha}{N}1}u+f(x,u)\qquad \text{ in } \mathbb{R}^N \] where $N\geq 1$, $I_\alpha$ is the Riesz potential of order $\alpha \in (0, N)$, the exponent $\frac{\alpha}{N}+1$ is critical with respect to the HardyLittlewoodSobolev inequality and the nonlinear perturbation $f$ satisfies suitable growth and structural assumptions.
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We study the Derivative Nonlinear Schr\"odinger (DNLS). equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities corresponding to algebraic solitons). We show that the set of such initial data is open and dense in a weighted Sobolev space, and includes data of arbitrarily large $L^2$norm. We prove global wellposedness on this open and dense set. In a subsequent paper, we will use these results and a steepest descent analysis to prove the soliton resolution conjecture for the DNLS equation with the initial data considered here and asymptotic stability of $N$soliton solutions.
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We introduce homassociative Ore extensions as nonassociative, nonunital Ore extensions with a homassociative multiplication, as well as give some necessary and sufficient conditions when such exist. Within this framework, we also construct a family of homassociative Weyl algebras as generalizations of the classical analogue, and prove that they are simple.
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In this paper, we provide a general framework to study general class of linear and nonlinear kinetic equations with random uncertainties from the initial data or collision kernels, and their stochastic Galerkin approximations, in both incompressible NavierStokes and Euler (acoustic) regimes. First, we show that the general framework put forth in [C. Mouhot and L. Neumann, Nonlinearity, 19, 969998, 2006, M. Briant, J. Diff. Eqn., 259, 60726141, 2005] based on hypocoercivity for the deterministic kinetic equations can be easily adopted for sensitivity analysis for random kinetic equations, which gives rise to an exponential convergence of the random solution toward the (deterministic) global equilibrium, under suitable conditions on the collision kernel. Then we use such theory to study the stochastic Galerkin (SG) methods for the equations, establish hypocoercivity of the SG system and regularity of its solution, and spectral accuracy and exponential decay of the numerical error of t
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The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $\beta$ is allowed to vary with the matrix size $n$. In particular, we show that as $n \to \infty$ with $n\beta \to \infty$, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.
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When dilute charged particles are confined in a bounded domain, boundary effects are crucial in the dynamics of particles. We construct a unique globalintime solution to the VlasovPoissonBoltzmann system in convex domains with the diffuse boundary condition. The construction is based on $L^{2}$$L^{\infty}$ framework with a new weighted $W^{1,p}$estimate of distribution function and $C^{2}$estimate of the selfconsistent electric potential. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.
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In this paper we study an experimentallyobserved connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call "grid peeling") is the convexlayer decomposition of subsets $G\subset \mathbb Z^2$ of the integer grid, previously studied for the particular case $G=\{1,\ldots,m\}^2$ by HarPeled and Lidick\'y (2013). The second one is the affine curveshortening flow (ACSF), first studied by Alvarez et al. (1993) and Sapiro and Tannenbaum (1993). We present empirical evidence that, in a certain welldefined sense, grid peeling behaves at the limit like ACSF on convex curves. We offer some theoretical arguments in favor of this conjecture. We also pay closer attention to the simple case where $G=\mathbb N^2$ is a quarterinfinite grid. This case corresponds to ACSF starting with an infinite Lshaped curve, which when transformed using the ACSF becomes a hyperbola for all times $t>0$. We prove
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In the space of cubic polynomials, Milnor defined a notable curve $\mathcal S_p$, consisting of cubic polynomials with a periodic critical point, whose period is exactly $p$. In this paper, we show that for any integer $p\geq 1$, any bounded hyperbolic component on $\mathcal{S}_p$ is a Jordan disk.
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Vector perturbation (VP) precoding is a promising technique for multiuser communication systems operating in the downlink. In this work, we introduce a hybrid framework to improve the performance of lattice reduction (LR) aided (LRA) VP. Firstly, we perform a simple precoding using zero forcing (ZF) or successive interference cancellation (SIC) based on a reduced lattice basis. Since the signal space after LRZF or LRSIC precoding can be shown to be bounded to a small range, then along with sufficient orthogonality of the lattice basis guaranteed by LR, they collectively pave the way for the subsequent application of an approximate message passing (AMP) algorithm, which further boosts the performance of any suboptimal precoder. Our work shows that the AMP algorithm in compressed sensing can be beneficial for a lattice decoding problem whose signal constraint lies in \mathbb{Z} and entries of the input lattice basis not necessarily being i.i.d. Gaussian. Numerical results show that the
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Describing and simulating acoustic wave propagation can be difficult and time consuming; especially when modeling threedimensional (3D) problems. As the propagating waves exit the computational domain, the amplitude needs to be sufficiently small otherwise reflections can occur from the boundary influencing the numerical solution. This paper will attempt to quantify what is meant by `sufficiently small' and investigate whether the geometry of the computational boundary can be manipulated to reduce reflections at the outer walls. The 3D compressible Euler equations were solved using the discontinuous Galerkin method on a graphical processing unit. A pressure pulse with an amplitude equivalent to 10% of atmospheric pressure was simulated through a modified trumpet within seven different geometries. The numerical results indicate that if the amplitude of the pulse is less than 0.5% of atmospheric pressure, reflections are minimal and do not significantly influence the solution in the dom
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We discuss the Hochschild cohomology of the category of Dmodules associated to an algebraic stack. In particular we describe the Hochschild cohomology of the category of torusequivariant Dmodules as the cohomology of a Dmodule on the loop space of the quotient stack. Finally, we give an approach for understanding the Hochschild cohomology of Dmodules on general stacks via a relative compactification of the diagonal. This work is motivated by a desire to understand the support theory (in the sense of BensonIyengarKrause) of Dmodules on stacks.
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We describe a new cognitive ability, i.e., functional conceptual substratum, used implicitly in the generation of several mathematical proofs and definitions. Furthermore, we present an initial (firstorder) formalization of this mechanism together with its relation to classic notions like primitive positive definability and Diophantiveness. Additionally, we analyze the semantic variability of functional conceptual substratum when small syntactic modifications are done. Finally, we describe mathematically natural inference rules for definitions inspired by functional conceptual substratum and show that they are sound and complete w.r.t. standard calculi.
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The distribution of descents in a fixed conjugacy class of $S_n$ is studied, and it is shown that its moments have an interesting property. A particular conjugacy class that is of interest is the class of matchings (also known as fixed point free involutions). This paper provides a bijective proof of the symmetry of the descents and major indices of matchings and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.
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Scientists often seek simplified representations of complex systems to facilitate prediction and understanding. If the factors comprising a representation allow us to make accurate predictions about our system, but obscuring any subset of the factors destroys our ability to make predictions, we say that the representation exhibits informational synergy. We argue that synergy is an undesirable feature in learned representations and that explicitly minimizing synergy can help disentangle the true factors of variation underlying data. We explore different ways of quantifying synergy, deriving new closedform expressions in some cases, and then show how to modify learning to produce representations that are minimally synergistic. We introduce a benchmark task to disentangle separate characters from images of words. We demonstrate that Minimally Synergistic (MinSyn) representations correctly disentangle characters while methods relying on statistical independence fail.
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In this paper, we introduce artificial boundary conditions for the linearized GreenNaghdi system of equations. The derivation of such continuous (respectively discrete) boundary conditions include the inversion of Laplace transform (respectively Ztransform) and these boundary conditions are in turn non local in time. In the case of continuous boundary conditions, the inversion is done explicitly. We consider two spatial discretisations of the initial system either on a staggered grid or on a collocated grids, both of interest from the practical point of view. We use a Crank Nicolson time discretization. The proposed numerical scheme with the staggered grid permits explicit Ztransform inversion whereas the collocated grid discretization do not. A stable numerical procedure is proposed for this latter inversion. We test numerically the accuracy of the described method with standard Gaussian initial data and wave packet initial data which are more convenient to explore the dispersive p
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A detachment of a hypergraph $\scr F$ is a hypergraph obtained from $\scr F$ by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph $\scr G$ can be thought of as taking $\scr G$, partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph $\scr F$. In this paper we use NashWilliams lemma on laminar families to prove a detachment theorem for amalgamated 3uniform hypergraphs, which yields a substantial generalization of previous amalgamation theorems by Hilton, Rodger and NashWilliams. To demonstrate the power of our detachment theorem, we show that the complete 3uniform $n$partite multihypergraph $\lambda K_{m_1,\ldots,m_n}^{3}$ can be expressed as the union $\scr G_1\cup \ldots \cup\scr G_k$ of $k$ edgedisjoint factors, where for $i=1,\ldots, k$, $\scr G_i$ is $r_i$regular, if and only if (i) $m_i=m_j:=m$ for all $1\leq i,j\leq k$, (ii) $3$ divides $r_imn$ f
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An anonymous reader shares a report: Software produced by Microsoft has been acquired by state organizations and firms in Russia and Crimea despite sanctions barring U.Sbased companies from doing business with them, official documents show. The acquisitions, registered on the Russian state procurement database, show the limitations in the way foreign governments and firms enforce the U.S. sanctions, imposed on Russia over its annexation of the Crimea peninsula from Ukraine in 2014. Some of the users gave Microsoft fictitious data about their identity, people involved in the transactions told Reuters, exploiting a gap in the U.S. company's ability to keep its products out of their hands. The products in each case were sold via third parties and Reuters has no evidence that Microsoft sold products directly to entities hit by the sanctions. "Microsoft has a strong commitment to complying with legal requirements and we have been looking into this matter in recent weeks," a Microsoft repre
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Under certain general conditions, an explicit formula to compute the greatest deltaepsilon function of a continuous function is given. From this formula, a new way to analyze the uniform continuity of a continuous function is given. Several examples illustrating the theory are discussed.
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We consider multiagent stochastic optimization problems over reproducing kernel Hilbert spaces (RKHS). In this setting, a network of interconnected agents aims to learn decision functions, i.e., nonlinear statistical models, that are optimal in terms of a global convex functional that aggregates data across the network, with only access to locally and sequentially observed samples. We propose solving this problem by allowing each agent to learn a local regression function while enforcing consensus constraints. We use a penalized variant of functional stochastic gradient descent operating simultaneously with lowdimensional subspace projections. These subspaces are constructed greedily by applying orthogonal matching pursuit to the sequence of kernel dictionaries and weights. By tuning the projectioninduced bias, we propose an algorithm that allows for each individual agent to learn, based upon its locally observed data stream and message passing with its neighbors only, a regression
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In this paper we consider a statistical estimation problem known as atomic deconvolution. Introduced in reliability, this model has a direct application when considering biological data produced by flow cytometers. In these experiments, biologists measure the fluorescence emission of treated cells and compare them with their natural emission to study the presence of specific molecules on the cells' surface. They observe a signal which is composed of a noise (the natural fluorescence) plus some additional signal related to the quantity of molecule present on the surface if any. From a statistical point of view, we aim at inferring the percentage of cells expressing the selected molecule and the probability distribution function associated with its fluorescence emission. We propose here an adaptive estimation procedure based on a previous deconvolution procedure introduced by [vEGS08, GvES11]. For both estimating the mixing parameter and the mixing density automatically, we use the Leps
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We consider the perturbation of an electric potential due to an insulating inclusion with corners. This perturbation is known to admit a multipole expansion whose coefficients are linear combinations of generalized polarization tensors. We define new geometric factors of a simple planar domain in terms of a conformal mapping associated with the domain. The geometric factors share properties of the generalized polarization tensors and are the Fourier series coefficients of a kind of generalized external angle of the inclusion boundary. Since the generalized external angle contains the Dirac delta singularity at corner points, we can determine the criterion for the existence of corner points on the inclusion boundary in terms of the geometric factors. We illustrate and validate our results with numerical examples computed to a high degree of precision using integral equation techniques, Nystr\"om discretization, and recursively compressed inverse preconditioning.
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Geometric aspects play an important role in the construction and analysis of structurepreserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical lowrank approximation of timedependent large matrices and tensors, and its use in numerical integrators for Hamiltonian tensor network approximations in quantum dynamics.
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This paper introduces a new and unified bit error rates performance analysis of spacetime block codes (STBC) deployed in wireless systems with spatial diversity in generalized shadowed fading and noise scenarios. Specifically, we derive a simple and a very accurate approximate expressions for the average error rates of coherent modulation schemes in generalized $\eta\mu$ and $\kappa\mu$ shadowed fading channels with multiple input multiple output (MIMO) systems. The noise in the network is assumed to be modeled using the additive white generalized Gaussian noise (AWGGN), which encompasses the classical Laplacian and the Gaussian noise environments as special cases. The derived results obviate the need to rederive the error rates for MIMO STBC systems under many multipath fading and noise conditions while avoiding any special functions with high computational complexity. Published results from the literature, as well as numerical evaluations, corroborate the accuracy of our derived
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We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$trivial surface. In the first half of this article, we define the notion of limit tilt stability, which is closely related to Bayer's polynomial stability. We show that the FourierMukai transform of a slope stable torsionfree sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit stable object. We also show that the inverse FourierMukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsionfree sheaf, up to modification in codimension 2. In the second half of this article, we define a limit stability for complexes that vanish on the generic fiber of the fibration. We show that onedimensional stable sheaves with positive twisted third Chern character correspond to such limit stable complexes under a FourierMukai transform. When the elliptic fibration has a numerically $
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We study an effective model of microscopic facet formation for low temperature three dimensional microscopic Wulff crystals above the droplet condensation threshold. The model we consider is a 2+1 solid on solid surface coupled with high and low density bulk Bernoulli fields. At equilibrium the surface stays flat. Imposing a canonical constraint on excess number of particles forces the surface to "grow" through the sequence of spontaneous creations of macroscopic size monolayers. We prove that at all sufficiently low temperatures, as the excess particle constraint is tuned, the model undergoes an infinite sequence of first order transitions, which traces an infinite sequence of first order transitions in the underlying variational problem. Away from transition values of canonical constraint we prove sharp concentration results for the rescaled level lines around solutions of the limiting variational problem.
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The PoissonBoltzmann equation (PBE) is a nonlinear elliptic PDE that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges immersed in an ionic solution. Efficient numerical computation of the PBE yields a high number of degrees of freedom in the resultant algebraic system of equations. Coupled with the fact that in most cases the PBE requires to be solved multiple times for a large number of system configurations, this poses great computational challenges to conventional numerical techniques. To accelerate such computations, we here present the reduced basis method (RBM) which greatly reduces this computational complexity by constructing a reduced order model of typically low dimension. In this study, we employ a simple version of the PBE for proof of concept and discretize the linearized PBE (LPBE) with a centered finite difference scheme. The resultant linear system is s
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Uncertainty Quantification through stochastic spectral methods is rising in popularity. We derive a modification of the classical stochastic Galerkin method, that ensures the hyperbolicity of the underlying hyperbolic system of partial differential equations. The modification is done using a suitable "slope" limiter, based on similar ideas in the context of kinetic moment models. We apply the resulting modified stochastic Galerkin method to the compressible Euler equations and the $M_1$ model of radiative transfer. Our numerical results show that it can compete with other UQ methods like the intrusive polynomial moment method while being computationally inexpensive and easy to implement.
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Quantum particle bound in an infinite, onedimensional square potential well is one of the problems in Quantum Mechanics (QM) that most of the textbooks start from. There, calculating an allowed energy spectrum for an arbitrary wave function often involves Riemann zeta function resulting in a $\pi$ series. In this work, two "$\pi$ formulas" are derived when calculating a spectrum of possible outcomes of the momentum measurement for a particle confined in such a well, the series, $\frac{\pi^2}{8} = \sum_{k=1}^{k=\infty} \frac{1}{(2k1)^2}$, and the integral $\int_{\infty}^{\infty} \frac{sin^2 x}{x^2} dx =\pi$. The spectrum of the momentum operator appears to peak on classically allowed momentum values only for the states with even quantum number. The present article is inspired by another quantum mechanical derivation of $\pi$ formula in \cite{wallys}.
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We introduce an extended KeplerCoulomb quantum model in spherical coordinates. The Schr\"{o}dinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higherorder, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.
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We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of timechanges of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the timechanges of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive certain spectral results for skew products over translations and Furstenberg transformations.
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We consider a family of fractional porous media equations, recently studied by Caffarelli and V\'azquez. We show the construction of a weak solution as Wasserstein gradient flow of a square fractional Sobolev norm. Energy dissipation inequality, regularizing effect and decay estimates for the $L^p$ norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case.
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We calculate model theoretic ranks of Painlev\'e equations in this article, showing in particular, that any equation in any of the Painlev\'e families has Morley rank one, extending results of Nagloo and Pillay (2011). We show that the type of the generic solution of any equation in the second Painlev\'e family is geometrically trivial, extending a result of Nagloo (2015). We also establish the orthogonality of various pairs of equations in the Painlev\'e families, showing at least generically, that all instances of nonorthogonality between equations in the same Painlev\'e family come from classically studied B{\"a}cklund transformations. For instance, we show that if at least one of $\alpha, \beta$ is transcendental, then $P_{II} (\alpha)$ is nonorthogonal to $P_{II} ( \beta )$ if and only if $\alpha+ \beta \in \mathbb Z$ or $\alpha  \beta \in \mathbb Z$. Our results have concrete interpretations in terms of characterizing the algebraic relations between solutions of Painlev\'e equat
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A simple framework for Dirac spinors is developed that parametrizes admissible quantum dynamics and also analytically constructs electromagnetic fields, obeying Maxwell's equations, which yield a desired evolution. In particular, we show how to achieve dispersionless rotation and translation of wave packets. Additionally, this formalism can handle control interactions beyond electromagnetic. This work reveals unexpected flexibility of the Dirac equation for control applications, which may open new prospects for quantum technologies.
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Segregated direct boundarydomain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable nonsmooth (or limitedsmoothness) coefficient on Lipschitz domains are formulated. The PDE right hand sides belong to the Sobolev (Besselpotential) space $H^{s2}(\Omega)$ or $\widetilde H^{s2}(\Omega)$, $1/2<s<3/2$, when neither strong classical nor weak canonical conormal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/nonuniqueness, as well as Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, however some finitedimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.
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In this article, We investigate an inverse problem of determining the timedependent source factor in parabolic integrodifferential equations from boundary data. We establish the uniqueness and the conditional stability estimate of H\"older type for the inverse source problem in a cylindrical shaped domain. Our methodology is based on the BukhgeimKlibanov method by means of the Carleman estimate. Here we also derive the "timelike" Carleman estimate for parabolic integrodifferential equations.
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The main problem considered in the present paper is to single out classes of convex sets, whose convexity property is preserved under nonlinear smooth transformations. Extending an approach due to B.T. Polyak, the present study focusses on the class of uniformly convex subsets of Banach spaces. As a main result, a quantitative condition linking the modulus of convexity of such kind of set, the regularity behaviour around a point of a nonlinear mapping and the Lipschitz continuity of its derivative is established, which ensures the images of uniformly convex sets to remain uniformly convex. Applications of the resulting convexity principle to the existence of solutions, their characterization and to the Lagrangian duality theory in constrained nonconvex optimization are then discussed.
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J. R. Wilton obtained a formula for the product of two Riemann zeta functions. This formula played a crucial role to find an approximate functional equation for the product of two Riemann zeta functions in the critical region. We find an analogous formula for the product of two Dedekind zeta functions and then use this formula to find the values of Dedekind zeta function attached to an arbitrary imaginary quadratic field at any positive integer. We finally compute some values of Dedekind zeta functions attached to certain imaginary quadratic fields.
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In this article we combine two developments in polynomial optimization. On the one hand, we consider nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the second and the third author. On the other hand, we investigate geometric programming methods for constrained polynomial optimization problems, which were recently developed by Ghasemi and Marshall. We show that the combination of both results yields a new method to solve a huge class of constrained polynomial optimization problems, particularly for high degree polynomials. Experimentally, the resulting method is significantly faster than semidefinite programming as we demonstrate in various examples.
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The detection of landmarks or patterns is of interest for extracting features in biological images. Hence, algorithms for finding these keypoints have been extensively investigated in the literature, and their localization and detection properties are well known. In this paper, we study the complementary topic of local orientation estimation, which has not received similar attention. Simply stated, the problem that we address is the following: estimate the angle of rotation of a pattern with steerable filters centered at the same location, where the image is corrupted by colored isotropic Gaussian noise. For this problem, we use a statistical framework based on the Cram\'{e}rRao lower bound (CRLB) that sets a fundamental limit on the accuracy of the corresponding class of estimators. We propose a scheme to measure the performance of estimators based on steerable filters (as a lower bound), while considering the connection to maximum likelihood estimation. Beyond the general results, w
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For the pair $\{\Delta, \Delta\alpha\delta_\mathcal{C}\}$ of selfadjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent) DirichlettoNeumann maps. Here $\delta_{\cal{C}}$ denotes a singular $\delta$potential which is supported on a smooth compact hypersurface $\mathcal{C}\subset\mathbb{R}^n$ and $\alpha$ is a realvalued function on $\mathcal{C}$.
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Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. L\"{u}, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+\lambda\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^\lambda$ contain an interval for all $x\in(0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of nonmatching parameters studied by the first author and Kalle (2017).
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We propose a variational approximation to Bayesian posterior distributions, called $\alpha$VB, with provable statistical guarantees for models with and without latent variables. The standard variational approximation is a special case of $\alpha$VB with $\alpha=1$. When $\alpha \in(0,1)$, a novel class of variational inequalities are developed for linking the Bayes risk under the variational approximation to the objective function in the variational optimization problem, implying that maximizing the evidence lower bound in variational inference has the effect of minimizing the Bayes risk within the variational density family. Operating in a frequentist setup, the variational inequalities imply that point estimates constructed from the $\alpha$VB procedure converge at an optimal rate to the true parameter in a wide range of problems. We illustrate our general theory with a number of examples, including the meanfield variational approximation to (low)highdimensional Bayesian linear
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Assume that $X_{\Sigma}\in\mathbb{R}^{n}$ is a random vector following a multivariate normal distribution with zero mean and positive definite covariance matrix $\Sigma$. Let $g:\mathbb{R}^{n}\to\mathbb{C}$ be measurable and of moderate growth, e.g., $g(x) \lesssim (1+x)^{N}$. We show that the map $\Sigma\mapsto\mathbb{E}\left[g(X_{\Sigma})\right]$ is smooth, and we derive convenient expressions for its partial derivatives, in terms of certain expectations $\mathbb{E}\left[(\partial^{\alpha}g)(X_{\Sigma})\right]$ of partial (distributional) derivatives of $g$. As we discuss, this result can be used to derive bounds for the expectation $\mathbb{E}\left[g(X_{\Sigma})\right]$ of a nonlinear function $g(X_{\Sigma})$ of a Gaussian random vector $X_{\Sigma}$ with possibly correlated entries. For the case when $g(x) =g_{1}(x_{1})\cdots g_{n}(x_{n})$ has tensorproduct structure, the above result is known in the engineering literature as Price's theorem, originally published in 1958. For d
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We use the SU(2) 't HooftPolyakov monopole configuration, and its BPS version, to test the integral equations of the YangMills theory. Those integral equations involve two (complex) parameters which do not appear in the differential YangMills equations, and if they are considered to be arbitrary it then implies that nonabelian gauge theories (but not abelian ones) possess an infinity of integral equations. For static monopole configurations only one of those parameters is relevant. We expand the integral YangMills equation in a power series of that parameter and show that the 't HooftPolyakov monopole and its BPS version satisfy the integral equations obtained in first and second order of that expansion. Our results points to the importance of exploring the physical consequences of such an infinity of integral equations on the global properties of the YangMills theory.
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An anonymous reader quotes a report from The New York Times (Warning: source may be paywalled; alternative source: The Trump administration announced Monday that it would take formal steps to repeal President Barack Obama's signature policy to curb greenhouse gas emissions from power plants, setting up a bitter fight over the future of America's efforts to tackle global warming. At an event in eastern Kentucky, Scott Pruitt, the head of the Environmental Protection Agency, said that his predecessors had departed from regulatory norms in crafting the Clean Power Plan, which was finalized in 2015 and would have pushed states to move away from coal in favor of sources of electricity that produce fewer carbon emissions. The repeal proposal, which will be filed in the Federal Register on Tuesday, fulfills a promise President Trump made to eradicate his predecessor's environmental legacy. Eliminating the Clean Power Plan makes it less likely the United States can fulfill its promise as part
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Alphabet's Waymo and Intel announced plans today to sponsor ads about selfdriving cars. "Alphabet's Waymo is launching a public education campaign today called "Let's Talk SelfDriving" aimed at addressing the skepticism many people have about autonomous technology," reports The Verge. Meanwhile, "Intel said it would be airing its commercial starring LeBron James in the runup to the NBA season opener on October 17th. From the report: The ad campaign will launch first in Arizona, before spreading to other states. Waymo is preparing to launch its first commercial ridehailing service powered by its selfdriving Chrysler Pacifica minivans, according to a recent report in The Information. This public education campaign would appear to be a prelude to inviting ordinary people to take a ride in a driverless vehicle. Both companies recognize that in order to make lots of money, there will need to be a robust effort to persuade people that autonomous vehicles are as safe, if not safer, than
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This paper is a sequel to [He11] and [GH17]. In [He11] a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_\mu^{mar}$ for marked singularities in one $\mu$homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichm\"uller space. It comes together with a $\mu$constant monodromy group $G^{mar}\subset G_{\mathbb{Z}}$. Here $G_{\mathbb{Z}}$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_\mu^{mar}$ is connected. This is equivalent to $G^{mar}= G_{\mathbb{Z}}$. Also Torelli type conjectures were formulated. In [He11] and [GH17] $M_\mu^{mar}, G_{\mathbb{Z}}$ and $G^{mar}$ were determined and all conjectures were proved for the simple, the unimodal and the exceptional bimodal singularities. In this paper the quadrangle singularities and the bimodal series are treated. The Torelli type conjectures are true. But the conjecture $G^{mar}= G_{\mathbb{Z}}
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We study $321$avoiding affine permutations, and prove a formula for their enumeration with respect to the inversion number by using a combinatorial approach. This is done in two different ways, both related to Viennot's theory of heaps. First, we encode these permutations using certain heaps of monomers and dimers. This method specializes to the case of affine involutions. For the second proof, we introduce periodic parallelogram polyominoes, which are new combinatorial objects of independent interest. We enumerate them by extending the approach of BousquetM\'elou and Viennot used for classical parallelogram polyominoes. We finally establish a connection between these new objects and $321$avoiding affine permutations.
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A datadriven formulation of the optimal transport problem is presented and solved using adaptively refined meshes to decompose the problem into a sequence of finite linear programming problems. Both the marginal distributions and their unknown optimal coupling are approximated through mixtures, which decouples the problem into the the optimal transport between the individual components of the mixtures and a classical assignment problem linking them all. A factorization of the components into products of singlevariable distributions makes the first subproblem solvable in closed form. The size of the assignment problem is addressed through an adaptive procedure: a sequence of linear programming problems which utilize at each level the solution from the previous coarser mesh to restrict the size of the function space where solutions are sought. The linear programming approach for pairwise optimal transportation, combined with an iterative scheme, gives a data driven algorithm for the W
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A comparison theorem is proved for a pair of solutions that satisfy in a weak sense opposite differential inequalities with nonlinearity of the form $f (u)$ with $f$ belonging to the class $L^p_{loc}$. The solutions are assumed to have nonvanishing gradients in the domain, where the inequalities are considered. The comparison theorem is applied to the problem describing steady, periodic water waves with vorticity in the case of arbitrary freesurface profiles including overhanging ones. Bounds for these profiles as well as streamfunctions and admissible values of the total head are obtained.
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