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  • We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander--Reiten theory. More precisely, for each Nakayama algebra $A$ and each positive integer $d$, we construct a finite dimensional algebra $A^{(d)}$ having a distinguished $d$-cluster-tilting $A^{(d)}$-module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of $A$. We also construct higher dimensional analogues of the mesh category of type $\mathbb{ZA}_\infty$ and the tubes.

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  • Radio frequency energy harvesting (RFEH) is a promising technology to charge unattended Internet of Things (IoT) low-power devices remotely. To enable this, in future IoT system, besides the traditional data access points (DAPs) for collecting data, energy access points (EAPs) should be deployed to charge IoT devices to maintain their sustainable operations. Practically, the DAPs and EAPs may be operated by different operators, and the DAPs thus need to provide effective incentives to motivate the surrounding EAPs to charge their associated IoT devices. Different from existing incentive schemes, we consider a practical scenario with asymmetric information, where the DAP is not aware of the channel conditions and energy costs of the EAPs. We first extend the existing Stackelberg game-based approach with complete information to the asymmetric information scenario, where the expected utility of the DAP is defined and maximized. To deal with asymmetric information more efficiently, we then

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  • The fifth generation (5G) wireless communications brag both high spectrum efficiency and high energy efficiency. To meet the requirements, various new techniques have been proposed. Among these, the recently-emerging index modulation has attracted significant interests. By judiciously activating a subset of certain communication {building blocks, such as} antenna, subcarrier and time slot, index modulation is claimed to have the potential to meet the challenging 5G needs. In this article, we will discuss index modulation and its general and specific representations, enhancements, and potential applications in various 5G scenarios. The objective is to reveal whether, and how, index modulation may strive for more performance gains with less medium resource occupation.

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  • Weakly interacting quantum fluids allow for a natural kinetic theory description which takes into account the fermionic or bosonic nature of the interacting particles. In the simplest cases, one arrives at the Boltzmann-Nordheim equations for the reduced density matrix of the fluid. We discuss here two related topics: the kinetic theory of the fermionic Hubbard model, in which conservation of total spin results in an additional Vlasov type term in the Boltzmann equation, and the relation between kinetic theory and thermalization.

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  • We extend the Koszul calculus defined on quadratic algebras by Berger, Lambre and Solotar, to N-homogeneous algebras. When N>2, the Koszul cup and cap products are defined by specific expressions, and they are compatible with the Koszul differentials, providing associative products on (co)homology classes. The N-Koszul calculus is calculated for the truncated polynomial algebras.

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  • Let $M$ be a Liouville 6-manifold which is the smooth fiber of a Lefschetz fibration on $\mathbb{C}^4$ constructed by suspending a Lefschetz fibration on $\mathbb{C}^3$. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category $\mathcal{F}(M)$ and the wrapped Fukaya category $\mathcal{W}(M)$ are related through $A_\infty$-Koszul duality, by identifying them with cyclic and Calabi-Yau completions of the same quiver algebra. This implies the split-generation of the compact Fukaya category $\mathcal{F}(M)$ by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi-dilations in the sense of Seidel-Solomon are obtained.

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  • We study a topologically exact, negative Schwarzian unimodal map whose critical point is non-recurrent and flat. Assuming the critical order is either logarithmic or polynomial, we establish the Large Deviation Principle and give a partial description of the zeros of the corresponding rate functions. We apply our main results to a certain parametrized family of unimodal maps in the same topological conjugacy class, and give a complete description of the zeros of the rate functions. We observe a qualitative change at a transition parameter, and show that the sets of zeros depend continuously on the parameter even at the transition.

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  • Let $n$ be a positive integer. For each $0\leq j \leq n-1$ we let $C_n^j$ denote the Cayley graph of the cyclic group $\mathbb{Z}_n$ with respect to the subset $\{1,j\}$. Utilizing the Smith Normal Form process, we give an explicit description of the Grothendieck group of each of the Leavitt path algebras $L_K(C_n^j)$ for any field $K$. Our general method significantly streamlines the approach that was used in previous work to establish this description in the specific case $j=2$. Along the way, we give necessary and sufficient conditions on the pairs $(j,n)$ which yield that this group is infinite. We subsequently focus on the case $j = 3$, where the structure of this group turns out to be related to a Fibonacci-like sequence, called the Narayana's Cows sequence.

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  • We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. We utilize -- and reprove -- D. Anick's fundamental result on approximation of polynomial automorphisms, adapt it to the case of symplectomorphisms, and formulate the lifting problem. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory.

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  • In this paper, we propose a second-order energy-conserving approximation procedure for Hamiltonian systems with holonomic constraints. The derivation of the procedure relies on the use of the so-called line integral framework. We provide numerical experiments to illustrate theoretical findings.

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  • In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Ne\v{s}et\v{r}il and Pudl\'ak in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. In just the last year, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width, and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms o

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  • We study local and global trace formulae for $\mathcal{C}^{\infty}$ ($a\,priori$ non-analytic) hyperbolic diffeomorphisms, that is we compare the sum of its Ruelle resonances with the flat trace of its transfer operator. We explicit the link between trace formulae and dynamical determinants and give examples of systems with explicit dynamical determinants exhibiting interesting behaviours (in particular, trace formulae do not always hold). We also establish a bound on the growth of the dynamical determinant for Gevrey dynamics from which we deduce a control on the number of resonances outside of a disc centred at $0$.

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  • The aim of this paper is to classify compact, simply connected K\"ahler manifolds which admit J-invariant Killing tensor with two eigenvalues of multiplicity 2 and n-2 and with constant eigenvalue corresponding to 2-dimensional eigendistribution.

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  • In this expository paper we review some recent results about representations of Kac-Moody groups. We sketch the construction of these groups. If practical, we present the ideas behind the proofs of theorems. At the end we pose open questions.

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  • In this paper, we introduce an inertial version of the Proximal Incremental Aggregated Gradient method (PIAG) for minimizing the sum of smooth convex component functions and a possibly nonsmooth convex regularization function. Theoretically, we show that the inertial Proximal Incremental Aggregated Gradiend (iPIAG) method enjoys a global linear convergence under a quadratic growth condition, which is strictly weaker than strong convexity, provided that the stepsize is not larger than a constant. Moreover, we present two numerical expreiments which demonstrate that iPIAG outperforms the original PIAG.

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  • We consider a compressed sensing problem in which both the measurement and the sparsifying systems are assumed to be frames (not necessarily tight) of the underlying Hilbert space of signals, which may be finite or infinite dimensional. The main result gives explicit bounds on the number of measurements in order to achieve stable recovery, which depends on the mutual coherence of the two systems. As a simple corollary, we prove the efficiency of nonuniform sampling strategies in cases when the two systems are not incoherent, but only asymptotically incoherent, as with the recovery of wavelet coefficients from Fourier samples. This general framework finds applications to inverse problems in partial differential equations, where the standard assumptions of compressed sensing are often not satisfied. Several examples are discussed, with a special focus on electrical impedance tomography.

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  • It is well-known that every derivation of a semisimple Lie algebra $L$ over an algebraically closed field $F$ with characteristic zero is inner. The aim of this paper is to show what happens if the characteristic of $F$ is prime with $L$ an exceptional Lie algebra. We prove that if $L$ is a Chevalley Lie algebra of type $\{G_2,F_4,E_6,E_7,E_8\}$ over a field of characteristic $p$ then the derivations of $L$ are inner except in the cases $G_2$ with $p=2$, $E_6$ with $p=3$ and $E_7$ with $p=2$.

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  • Motivated by the need to detect an underground cavity within the procedure of an On-Site-Inspection (OSI), of the Comprehensive Nuclear Test Ban Treaty Organization, the aim of this paper is to present results on the comparison of our numerical simulations with an analytic solution. The accurate numerical modeling can facilitate the development of proper analysis techniques to detect the remnants of an underground nuclear test. The larger goal is to help set a rigorous scientific base of OSI and to contribute to bringing the Treaty into force. For our 3D numerical simulations, we use the discontinuous Galerkin Spectral Element Code SPEED jointly developed at MOX (The Laboratory for Modeling and Scientific Computing, Department of Mathematics) and at DICA (Department of Civil and Environmental Engineering) of the Politecnico di Milano.

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  • We construct an obstruction for the existence of embeddings of homology $3$-sphere into homology $S^3\times S^1$ under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer cohomology due to R.Fintushel-R.Stern. We make use of the $\mathbb{Z}$-fold covering space of homology $S^3\times S^1$ and the instantons on it.

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  • We study the effective action for the integrable $\lambda$-deformation of the $G_{k_1} \times G_{k_2}/G_{k_1+k_2}$ coset CFTs. For unequal levels theses models do not fall into the general discussion of $\lambda$-deformations of CFTs corresponding to symmetric spaces and have many attractive features. We show that the perturbation is driven by parafermion bilinears and we revisit the derivation of their algebra. We uncover a non-trivial symmetry of these models parametric space, which has not encountered before in the literature. Using field theoretical methods and the effective action we compute the exact in the deformation parameter $\beta$-function and explicitly demonstrate the existence of a fixed point in the IR corresponding to the $G_{k_1-k_2} \times G_{k_2}/G_{k_1}$ coset CFTs. The same result is verified using gravitational methods for $G=SU(2)$. We examine various limiting cases previously considered in the literature and found agreement.

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  • Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid, is the quotient of a finite dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an "Almeida-Molino" integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a "de Rham" integrable lift for any given transitive Abelian Lie algebroid.

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  • In this paper, we develop an extremum seeking control method integrated with iterative learning control to track a time-varying optimizer within finite time. The behavior of the extremum seeking system is analyzed via an approximating system - the modified Lie bracket system. The modified Lie bracket system is essentially an online integral-type iterative learning control law. The paper contributes to two fields, namely, iterative learning control and extremum seeking. First, an online integral type iterative learning control with a forgetting factor is proposed. Its convergence is analyzed via $k$-dependent (iteration- dependent) contraction mapping in a Banach space equipped with $\lambda$-norm. Second, the iterative learning extremum seeking system can be regarded as an iterative learning control with "control input disturbance." The tracking error of its modified Lie bracket system can be shown uniformly bounded in terms of iterations by selecting a sufficiently large frequency. Fu

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  • This work is devoted to solve the problem of estimating the carrier frequency offset, phase offset, amplitude, and SNR between two mmWave transceivers. The Cram\'{e}r-Rao Lower Bound (CRLB) for the different parameters is provided first, as well as the condition for the CRLB to exist, known as Regularity Condition. Thereafter, the problem of finding suitable estimators for the parameters is adressed, for which the proposed solution is the Maximum Likelihood estimator (ML).

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  • We prove the local mixing theorem for geodesic flows on abelian covers finite volume hyperbolic surfaces with cusps, which is a continuation of the work of Oh-Pan. We also describe applications to counting problems and the prime geodesic theorem.

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  • We study the exact controllability for spatially periodic water waves with surface tension, by localized exterior pressures applied to free surfaces. We prove that in any dimension, the exact controllability holds within arbitrarily short time, for sufficiently small and regular data, provided that the region of control satisfies the geometric control condition. This result was previously obtained by Alazard, Baldi, and Han-Kwan for 2-D water waves. Our proof combines an iterative scheme, that reduces the controllability of the original quasi-linear equation to that of a sequence of linear equations, with a semiclassical approach for the linear control problems.

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  • Distance multivariance was recently introduced as a measure of multivariate dependence. Here we discuss several new aspects and present a guide to its use. In particular, $m$-multivariance is defined, which is a new dependence measure yielding tests for pairwise independence and independence of higher order. These tests are computational feasible and they are consistent against all alternatives. Based on distance multivariance we also propose a visualization scheme for higher order dependence which fits into/extends the framework of probabilistic graphical models. Finally, it is indicated by several simulation studies that distance multivariance and the new measures match or outperform any other recently introduced multivariate independence measure. Many examples are included. All functions for the use of distance multivariance in applications are published in the R-package 'multivariance'.

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  • Work of Jean Renault shows that, for topologically principal \'etale groupoids, a diagonal-preserving isomorphism of reduced $C^*$-algebras yields an isomorphism of groupoids. Several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses. In this paper, we extend the known results by allowing more general coefficient rings and by weakening the hypotheses on the groupoids. Our approach has the additional feature that we only need to impose conditions on one of the two groupoids. Applications are given to Leavitt path algebras.

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  • A ranked poset is called strongly Sperner if the size of $k$-family cannot exceed the sum of $k$-largest Whitney numbers. In a sense of function ordering, a function $f$ is (weakly) majorized by $g$ if the the sum of $k$-largest values in $f$ cannot exceed the sum of $k$-largest values in $g$. Two definitions arise from different contexts, but each share a strong similarity with each other. Furthermore, the product of two weighted posets assumes a structural similarity with a convolution of two functions. Elements in the product of weighted posets with ranks capture underlying structures of the building blocks in the convolution. Combining all together, we are able to derive several types of entropy inequalities including discrete entropy power inequalities, and discuss more applications.

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  • We propose a conjectural construction of various slices for double affine Grassmannians as Coulomb branches of 3-dimensional N=4 supersymmetric affine quiver gauge theories. It generalizes the known construction for the usual affine Grassmannian, and makes sense for arbitrary symmetric Kac-Moody algebras.

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  • We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreover, we show that our abstract approach unifies existing results which characterize hypercyclicity, resp. topological mixing, of (weighted) composition operators on the space of holomorphic functions on a simply connected domain in the complex plane, on the space of smooth functions on an open subset of $\mathbb{R}^d$, as well as results characterizing topological transitiviy of such operators on

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  • We consider the inverse problem in geophysics of imaging the subsurface of the Earth in cases where a region below the surface is known to be formed by strata of different materials and the depths and thicknesses of the strata and the (possibly anisotropic) conductivity of each of them need to be identified simultaneously. This problem is treated as a special case of the inverse problem of determining a family of nested inclusions in a medium $\Omega\subset\mathbb{R}^n$, $n \geq 3$.

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  • We enumerate the state diagrams of the twist knot shadow which consist of the disjoint union of two trivial knots. The result coincides with the maximal number of regions into which the plane is divided by a given number of circles. We then establish a bijection between the state enumeration and this particular partition of the plane by means of binary words.

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  • For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in $M$. This gives a quantitative version of the main result of \cite{irie-marques-neves}, by Irie and the first two authors, that established denseness of minimal hypersurfaces for generic metrics. As in \cite{irie-marques-neves}, the main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich and the first two authors in \cite{liokumovich-marques-neves}.

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  • We discuss the error analysis of the lowest degree Crouzeix-Raviart and Raviart-Thomas finite element methods applied to a two-dimensional Poisson equation. To obtain error estimations, we use the techniques developed by Babu\v{s}ka-Aziz and the authors. We present error estimates in terms of the circumradius and the diameter of triangles in which the constants are independent of the geometric properties of the triangulations. Numerical experiments confirm the obtained results.

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  • This paper studies cache-aided interference networks with arbitrary number of transmitters and receivers, whereby each transmitter has a cache memory of finite size. Each transmitter fills its cache memory from a content library of files in the placement phase. In the subsequent delivery phase, each receiver requests one of the library files, and the transmitters are responsible for delivering the requested files from their caches to the receivers. The objective is to design schemes for the placement and delivery phases to maximize the sum degrees of freedom (sum-DoF) which expresses the capacity of the interference network at the high signal-to-noise ratio regime. Our work mainly focuses on a commonly used uncoded placement strategy. We provide an information-theoretic bound on the sum-DoF for this placement strategy. We demonstrate by an example that the derived bound is tighter than the bounds existing in the literature for small cache sizes. We propose a novel delivery scheme with

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  • Analogues of 1-shuffle elements for complex reflection groups of type $G(m,1,n)$ are introduced. A geometric interpretation for $G(m,1,n)$ in terms of rotational permutations of polygonal cards is given. We compute the eigenvalues, and their multiplicities, of the 1-shuffle element in the algebra of the group $G(m,1,n)$. Considering shuffling as a random walk on the group $G(m,1,n)$, we estimate the rate of convergence to randomness of the corresponding Markov chain. We report on the spectrum of the 1-shuffle analogue in the cyclotomic Hecke algebra $H(m,1,n)$ for $m=2$ and small $n$.

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  • This paper is concerned with a class of controlled singular Volterra integral equations, which could be used to describe problems involving memories. The well-known fractional order ordinary differential equations of the Riemann--Liouville or Caputo types are strictly special cases of the equations studied in this paper. Well-posedness and some regularity results in proper spaces are established for such kind of questions. For the associated optimal control problem, by using a Liapounoff's type theorem and the spike variation technique, we establish a Pontryagin's type maximum principle for optimal controls. Different from the existing literature, our method enables us to deal with the problem without assuming regularity conditions on the controls, the convexity condition on the control domain, and some additional unnecessary conditions on the nonlinear terms of the integral equation and the cost functional.

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  • We develop tools to construct Lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped Euler system under the influence of external and interaction potential forces with respect to the 2-Wasserstein distance. We also discuss the overdamped limit to a nonlocal equation used in the modelling of granular media with respect to the 2-Wasserstein distance, and provide rigorous proofs for particular examples in one spatial dimension.

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  • This paper studies the use of convex Lipschitz continuous functions to approximate the value functions in Markov decision processes containing a finite number of possible actions. Compact convergence is proved under various sampling schemes for the driving state disturbance. Under some assumptions, these approximations give a non-decreasing sequence of lower bounding or a non-increasing sequence of upper bounding functions. Numerical experiments involving piecewise linear approximations for a Bermudan put option demonstrate that tight bounding functions for its fair price over the entire state space can be obtained with excellent speed (fractions of a cpu second).

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  • In this work, we introduce and develop a theory of convex drawings of the complete graph $K_n$ in the sphere. A drawing $D$ of $K_n$ is convex if, for every 3-cycle $T$ of $K_n$, there is a closed disc $\Delta_T$ bounded by $D[T]$ such that, for any two vertices $u,v$ with $D[u]$ and $D[v]$ both in $\Delta_T$, the entire edge $D[uv]$ is also contained in $\Delta_T$. As one application of this perspective, we consider drawings containing a non-convex $K_5$ that has restrictions on its extensions to drawings of $K_7$. For each such drawing, we use convexity to produce a new drawing with fewer crossings. This is the first example of local considerations providing sufficient conditions for suboptimality. In particular, we do not compare the number of crossings {with the number of crossings in} any known drawings. This result sheds light on Aichholzer's computer proof (personal communication) showing that, for $n\le 12$, every optimal drawing of $K_n$ is convex. Convex drawings are characte

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  • Empirical observations show that natural communities of species with mutualistic interactions such microbes, can have a number of coexisting species of the order of hundreds. However standard models in population dynamics predict that ecosystem stability should decrease as the number of species in the community increases and that cooperative systems are less stable than communities with only competitive and/or exploitative interactions. Here we propose a stochastic model which is appropriate for species communities with mutualistic/commensalistic and exploitative interactions and that can be exactly solved at the leading order in the system size. We obtain results for relevant macro-ecological patterns, such as species abundance distributions and correlation functions. We also find that, in the large system size limit, any number of species can coexist for a very general class of interaction networks. For pure mutualistic/commensalistic interactions we analytically find the topological

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  • We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales. At the same time, we introduce the concept of i.o. subuniformity and relate this concept to recursive measure theory. We prove that there are classes closed downwards under Turing reducibility that have recursive measure zero and that are not i.o. subuniform. This shows that there are examples of classes that cannot be covered with methods other than probabilistic ones. It is easily seen that every set of hyperimmune degree can cover the recursive sets. We prove that there are both examples of hyperimmune-free degree that can and that cannot compute such a cover.

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  • In this paper, we study the weak asymptotic in the plane of some wave functions resulting from the WKB techniques applied to a Shrodinger equation with quartic oscillator and having some boundary condition. In first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions .Especially , we investigate the properties of the Cauchy transform of the root counting measure of a re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form $\mathcal{C}^{2}\left( z\right) +r\left( z\right) \mathcal{C}\left( z\right) +s\left( z\right) =0$, where $r,s$ are also polynomials. In second step, we discuss the existence of solutions (as Cauchy transform of a signed measures) of this algebraic equation.This problem remains to describe the critical graph of a related 4-degree polynomial quadratic differential $-p\left( z\right) dz^{2}$. In particular, we discuss the existence(and their number

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  • We construct and study curves with low H-constants on abelian and K3 surfaces. Using the Kummer $(16_{6})$-configurations on Jacobian surfaces and some $(16_{10})$-configurations of curves on $(1,3)$-polarized Abelian surfaces, we obtain examples of configurations of curves of genus $>1$ on a generic Jacobian K3 surface with H-constants $<-4$.

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  • We discuss relations between several known (some false, some open) conjectures on 3-edge-connected, cubic graphs and how they all fit into the same framework related to cuts in matchings. We then provide a construction of 3-edge-connected digraphs satisfying the property that for every even subgraph $E$, the graph obtained by contracting the edges of $E$ is not strongly connected. This disproves a recent conjecture of Hochst\"attler [A flow theory for the dichromatic number. European Journal of Combinatorics, 66, 160--167, 2017]. Furthermore, we show that an open conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices.

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  • We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an arbitrary centrally-symmetric convex body in R^n to the class of unit balls of n-dimensional subspaces of L_p-spaces. Finally, we prove a result of the Busemann-Petty type for these moments.

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  • Let $R$ be a smooth affine domain of dimension $d\geq 2$ over an infinite perfect field $k$ such that $2R=R$. We establish a morphism from the Euler class group $E^d(R)$ to $Um_{d+1}(R)/E_{d+1}(R)$, the group of elementary orbits of unimodular rows.

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  • The regular icosahedron is connected to many exceptional objects in mathematics. Here we describe two constructions of the $\mathrm{E}_8$ lattice from the icosahedron. One uses a subring of the quaternions called the "icosians", while the other uses du Val's work on the resolution of Kleinian singularities. Together they link the golden ratio, the quaternions, the quintic equation, the 600-cell, and the Poincare homology 3-sphere. We leave it as a challenge to the reader to find the connection between these two constructions.

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  • In this article, we put forward a new approach to electrodynamics of materials. Based on the identification of induced electromagnetic fields as the microscopic counterparts of polarization and magnetization, we systematically employ the mutual functional dependencies of induced, external and total field quantities. This allows for a unified, relativistic description of the electromagnetic response without assuming the material to be composed of electric or magnetic dipoles. Using this approach, we derive universal (material-independent) relations between electromagnetic response functions such as the dielectric tensor, the magnetic susceptibility and the microscopic conductivity tensor. Our formulae can be reduced to well-known identities in special cases, but more generally include the effects of inhomogeneity, anisotropy, magnetoelectric coupling and relativistic retardation. If combined with the Kubo formalism, they would also lend themselves to the ab initio calculation of all lin

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  • In this paper, a new class of energy-preserving integrators is proposed and analysed for Poisson systems by using functionally-fitted technology. The integrators exactly preserve energy and have arbitrarily high order. It is shown that the proposed approach allows us to obtain the energy-preserving methods derived in BIT 51 (2011) by Cohen and Hairer and in J. Comput. Appl. Math. 236 (2012) by Brugnano et al. for Poisson systems. Furthermore, we study the sufficient conditions that ensure the existence of a unique solution and discuss the order of the new energy-preserving integrators.

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  • Generalized B\"acklund-Darboux transformations (GBDTs) of discrete skew-selfadjoint Dirac systems have been successfully used for explicit solving of direct and inverse problems of Weyl-Titchmarsh theory. During explicit solving of the direct and inverse problems, we considered GBDTs of the trivial initial systems. However, GBDTs of arbitrary discrete skew-selfadjoint Dirac systems are important as well and we introduce these transformations in the present paper. The obtained results are applied to the construction of explicit solutions of the interesting related non-stationary systems.

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  • For every nonconstant polynomial $f\in\mathbb Q[x]$, let $\Phi_{4,f}$ denote the fourth dynatomic polynomial of $f$. We determine here the structure of the Galois group and the degrees of the irreducible factors of $\Phi_{4,f}$ for every quadratic polynomial $f$. As an application we prove new results related to a uniform boundedness conjecture of Morton and Silverman. In particular we show that if $f$ is a quadratic polynomial, then, for more than $39\%$ of all primes $p$, $f$ does not have a point of period four in $\mathbb Q_p$.

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  • The Wishart probability distribution on symmetricmatrices has been initially defined by mean of the multivariateGaussian distribution as an of the chi-square distribution. A moregeneral definition is given using results for harmonic analysis.Recently a probability distribution on symmetric matrices called theRiesz distribution has been defined by its Laplace transform as ageneralization of the Wishart distribution. The aim of the presentpaper is to show that some Riesz probability distributions which arenot necessarily Wishart may also be presented by mean of theGaussian distribution using Gaussian samples with missing data.

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  • T\^ete-\`a-t\^ete graphs and relative t\^ete-\`a-t\^ete graphs were introduced by N. A'Campo in 2010 to model monodromies of isolated plane curves. By recent workof Fdez de Bobadilla, Pe Pereira and the author, they provide a way of modeling the periodic mapping classes that leave some boundary component invariant. In this work we introduce the notion of general t\^ete-\`a-t\^ete graph and prove that they model all periodic mapping classes. We also describe algorithms that take a Seifert manifold and a horizontal surface and return a t\^ete-\`a-t\^ete graph and vice versa.

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  • Let $S$ be a surface of genus $g$ at least $2$. A representation $\rho:\pi_1S\longrightarrow \text{PSL}_2\Bbb R$ is said to be purely hyperbolic if its image consists only of hyperbolic elements other than the identity. We may wonder under which conditions such representations arise as holonomy of a hyperbolic cone-structure on $S$. In this work we will characterize them completely, giving necessary and sufficient conditions.

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  • The time dependent Eikonal equation is a Hamilton-Jacobi equation with Hamiltonian $H(P)=|P|$, which is not strictly convex nor smooth. The regularizing effect of Hamiltonian for the Eikonal equation is much weaker than that of strictly convex Hamiltonians, therefore leading to new phenomena such as the appearance of "contact discontinuity". In this paper, we study the set of singularity points of solutions in the upper half space for $C^1$ or $C^2$ initial data, with emphasis on the countability of connected components of the set. The regularity of solutions in the complement of the set of singularity points is also obtained.

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  • Several infinite products are studied along with their finite counterparts. For certain values of the parameters these infinite products reduce to modular forms. The finite product formulas give an elementary proof of a particular modular transformation.

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  • We present a new solution for fundamental problems in nonlinear dynamical systems: finding, verifying, and stabilizing cycles. The solution we propose consists of a new control method based on mixing previous states of the system (or the functions of these states). This approach allows us to locally stabilize and to find a priori unknown cycles of a given length. Our method generalizes and improves on the existing one dimensional space solutions to multi-dimensional space while using the geometric complex functions theory rather than a linear algebra approach. Several numerical examples are considered. All statements and formulas are given in final form. The formulas derivation and reasoning may be found in the cited references. The article focuses on practical applications of methods and algorithms.

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  • In this paper, we investigate the effect of the filter for the hyperbolic moment equations(HME) [15] of the Vlasov-Poisson equations and propose a novel quasi time- consistent filter to suppress the numerical recurrence effect. By taking properties of HME into consideration, the filter preserves a lot of physical properties of HME, including Galilean invariance and the conservation of mass, momentum and energy. We present two viewpoints, collisional viewpoint and dissipative viewpoint, to dissect the filter, and show that the filtered hyperbolic moment method can be treated as a solver of Vlasov equation. Numerical simulations of the linear Landau damping and two stream instability are tested to demonstrate the effectiveness of the filter in restraining recurrence arising from particle streaming. Both the analysis and the numerical results indicate that the filtered HME can capture the evolution of the Vlasov equation, even when phase mixing and filamentation are dominant.

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  • In this paper, applying the Bethe ansatz method, we investigate the Schr\"odinger equation for the three quasi-exactly solvable double-well potentials, namely the generalized Manning potential, the Razavy bistable potential and the hyperbolic Shifman potential. General exact expressions for the energies and the associated wave functions are obtained in terms of the roots of a set of algebraic equations. Also, we solve the same problems using the Lie algebraic approach of quasi-exact solvability through the sl(2) algebraization and show that the results are the same. The numerical evaluation of the energy spectrum is reported to display explicitly the energy levels splitting.

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  • Piateskii-Shapiro defined local $L$-factors $L^{PS}(s,\Pi,\Lambda)$ attached to irreducible admissible representations $\Pi$ of the group $GSp(4)$ over local fields and Bessel models of $\Pi$ attached to Bessel data $\Lambda$. These local $L$-factors decompose into a product $L^{PS}(s,\Pi,\Lambda)= L^{PS}_{ex}(s,\Pi,\Lambda) L^{PS}_{reg}(s,\Pi,\Lambda)$ of an exceptional and a regular $L$-factor. In this paper we compute the exceptional factors $L^{PS}_{ex}(s,\Pi,\Lambda)$ for split Bessel models of $\Pi$.

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  • In this article we prove local-in-time existence and uniqueness of solution to a non-isothermal cross-diffusion system with Maxwell-Stefan structure.

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  • We study the non-linear minimization problem on $H^1_0(\Omega)\subset L^q$ with $q=\frac{2n}{n-2}$, $\alpha>0$ and $n\geq4$~: \[\inf_{\substack{u\in H^1_0(\Omega) \|u\|_{L^q}=1}}\int_\Omega a(x,u)|\nabla u|^2 - \lambda \int_{\Omega} |u|^2.\] where $a(x,s)$ presents a global minimum $\alpha$ at $(x_0,0)$ with $x_0\in\Omega$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behaviour of $a(x,s)$ with respect to $s$. The model case is \[\inf_{\substack{u\in H^1_0(\Omega) \|u\|_{L^q}=1}}\int_\Omega (\alpha+|x|^\beta |u|^k)|\nabla u|^2 - \lambda \int_{\Omega} |u|^2.\] In a previous paper dedicated to the same problem with $\lambda=0$, we showed that minimizers exist only in the range $\beta<kn/q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\beta\geq kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda\leq \alpha\lambda_1(\Omega)$, $0\leq k\leq q-2$ and

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  • In this paper we introduce exotic twisted $\mathbb T$-equivariant K-theory of loop space $LZ$ depending on the (typically non-flat) holonomy line bundle ${\mathcal L}^B$ on $LZ$ induced from a gerbe with connection $B$ on $Z$. We also define exotic twisted $\mathbb T$-equivariant Chern character that maps the exotic twisted $\mathbb T$-equivariant K-theory of $LZ$ into the exotic twisted $\mathbb T$-equivariant cohomology as defined in an earlier paper of ours, and which localises to twisted cohomology of $Z$.

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  • In this paper, we present FPT-algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems' formulations are near square. The parameter is the maximum absolute value of rank minors of the corresponding matrices. Additionally, we present FPT-algorithms with respect to the same parameter for the problems, when the matrices have no singular rank sub-matrices.

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  • Locally acyclic cluster algebras are Krull domains. Hence their factorization theory is determined by their (divisor) class group and the set of classes containing height-1 prime ideals. Motivated by this, we investigate class groups of cluster algebras. We show that any cluster algebra that is a Krull domain has a finitely generated free abelian class group, and that every class contains infinitely many height-$1$ prime ideals. For a cluster algebra associated to an acyclic seed, we give an explicit description of the class group in terms of the initial exchange matrix. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type. In the acyclic case, we prove the sufficiency of necessary conditions for factoriality given by Geiss--Leclerc--Schr\"oer.

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  • We prove upper and lower bounds on the ground-state energy of the ideal two-dimensional anyon gas. Our bounds are extensive in the particle number, as for fermions, and linear in the statistics parameter $\alpha$. The lower bounds extend to Lieb-Thirring inequalities for all anyons except bosons.

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  • We show that the field $\Q(x,y)$, generated by two singular moduli~$x$ and~$y$, is generated by their sum ${x+y}$, unless~$x$ and~$y$ are conjugate over~$\Q$, in which case ${x+y}$ generates a subfield of degree at most~$2$. We obtain a similar result for the product of two singular moduli.

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  • We consider the consistency properties of a regularised estimator for the simultaneous identification of both changepoints and graphical dependency structure in multivariate time-series. Traditionally, estimation of Gaussian Graphical Models (GGM) is performed in an i.i.d setting. More recently, such models have been extended to allow for changes in the distribution, but only where changepoints are known a-priori. In this work, we study the Group-Fused Graphical Lasso (GFGL) which penalises partial-correlations with an L1 penalty while simultaneously inducing block-wise smoothness over time to detect multiple changepoints. We present a proof of consistency for the estimator, both in terms of changepoints, and the structure of the graphical models in each segment.

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  • We prove upper bounds for the probability that a radial SLE$_{\kappa}$ curve, $\kappa\in(0,8)$, comes within specified radii of $n$ different points in the unit disc. Using this estimate, we then prove a similar upper bound for a whole-plane SLE$_{\kappa}$ curve. We then use these estimates to show that the lower Minkowski content of both the radial and whole-plane SLE$_{\kappa}$ traces restricted in a bounded region have finite moments of any order.

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  • We have derived some new results for the Mellin transform formulas, as well as for the Gauss hypergeometric function. Also, we have found the connection between the Legendre functions of the second kind. Some of the results obtained we used in quantum mechanics of two charged particles of a continuous spectrum.

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  • In this article, we establish a truncated non-integrated defect relation for algebraically nondegenerate meromorphic mappings from an $m$-dimensional complete K\"{a}hler manifold into a subvariety $V$ of $k-$dimension in $P^n(C)$ intersecting $q$ hypersurfaces $Q_1,...,Q_q$ in $N$-subgeneral position of degree $d_i$ with respect to $V$, i.e., the intersection of any $N+1$ hypersurfaces and $V$ is empty. In our result, the truncation level of the counting functions is explicitly estimated. Our result generalizes and improves previous results.

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  • In this paper, we provide various formulas for partial sums of weighted averages over regular integers modulo $n$ of the $\gcd$-sum function with any arithmetical function involving the Bernoulli polynomials and the Gamma function. Many interesting applications of the results are also given.

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  • In this paper, we study joint functional calculus for commuting $n$-tuple of Ritt operators. We provide an equivalent characterisation of boundedness for joint functional calculus for Ritt operators on $L^p$-space.

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  • We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1\over 2}$ on $\Gamma_0(4).$

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  • Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singularity of $\mathsf m$ or $\{\infty\}$. The pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ has conical singularities of angles $4\pi$ at the critical points of $f$ and other conical singularities that are the preimages of those of $\mathsf m$. We study the $\zeta$-regularized determinant $\operatorname{Det}' \Delta_F$ of the (Friedrichs extension of) Laplace-Beltrami operator on $(X,f^*\mathsf m)$ as a functional on the moduli space of pairs $(X, f)$ and obtain an explicit formula for $\operatorname{Det}' \Delta_F$.

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  • Norway switches off FM radio, but a station is defying government order

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  • In order to preserve net neutrality and the free and open internet, we must end our reliance on monopolistic corporations and build something fundamentally different: internet infrastructure that is locally owned and operated and is dedicated to serving the people who connect to it, writes Jason Koebler, editor-in-chief of Vice's Motherboard news outlet. He writes: The good news is a better internet infrastructure is possible: Small communities, nonprofits, and startup companies around the United States have built networks that rival those built by big companies. Because these networks are built to serve their communities rather than their owners, they are privacy-focused and respect net neutrality ideals. These networks are proofs-of-concept around the country that a better internet is possible. This week, Motherboard and VICE Media are committing to be part of the change we'd like to see. We will build a community network based at our Brooklyn headquarters that will provide internet

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  • After a year of teases, puzzles, and feints, the bad guys are here—and they are pissed.

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  • Google is preparing to use light beams to bring rural areas of the planet online after it announced to a planned rollout in India. From a report: The firm is working with a telecom operator in Indian state Andhra Pradesh, home to over 50 million people, to use Free Space Optical Communications (FSOC), a technology that uses beams of light to deliver high-speed, high-capacity connectivity over long distances. Now partner AP State FiberNet will introduce 2,000 FSOC links starting from January to add additional support to its network backbone in the state. The Google project is aimed at "critical gaps to major access points, like cell-towers and WiFi hotspots, that support thousands of people," Google said. The initiative ties into a government initiative to connect 12 million households to the internet by 2019, the U.S. firm added.

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