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In [APS], the authors characterize the partitions of $n$ whose corresponding representations of $S_n$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups $W$. Namely, given a nontrivial multiplicative character $\omega$ of $W$, we give a closed formula for the number of irreducible representations of $W$ with determinant $\omega$. For Coxeter groups of type $B_n$ and $D_n$, this is accomplished by characterizing the bipartitions associated to such representations.
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We consider a family of dense initializations for limitedmemory quasiNewton methods. The proposed initialization uses two parameters to approximate the curvature of the Hessian in two complementary subspaces. This family of dense initializations is proposed in the context of a limitedmemory BroydenFletcherGoldfarbShanno (LBFGS) trustregion method that makes use of a shapechanging norm to define each subproblem. As with LBFGS methods that traditionally use diagonal initialization, the dense initialization and the sequence of generated quasiNewton matrices are never explicitly formed. Numerical experiments on the CUTEst test set suggest that this initialization together with the shapechanging trustregion method outperforms other LBFGS methods for solving general nonconvex unconstrained optimization problems. While this dense initialization is proposed in the context of a special trustregion method, it has broad applications for more general quasiNewton trustregion and li
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The Debian 9.2 point release is available; it includes fixes for a long list of problems. "As a special case for this point release, those using the 'aptget' tool to perform the upgrade will need to ensure that the 'distupgrade' command is used, in order to update to the latest kernel packages."
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In this paper, we propose a new distributed energy beamforming and information transfer (DEBIT) scheme for realizing simultaneous wireless information and power transfer (SWIPT) in multiway relay networks (MWRNs), where multiple singleantenna users exchange information via an energyconstrained singleantenna relay node. We investigate the optimal transceiver designs to maximize the achievable sumrate or the harvested power. The resultant sumrate maximization problem is nonconvex and the global optimal solution can be obtained through a threedimensional search in combination with conventional convex optimization. To reduce the computation complexity, a suboptimal DEBIT scheme is also proposed, for which the optimization problem becomes linear programming. The achievable sumrate performance is analyzed and a closedform lower bound is derived for the MWRN with a large number of users. Furthermore, we consider the harvestedpower maximization problem under a target sumrate constra
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We introduce a new notion of depth of low computational complexity for high dimensional data. For small dimensions, it behaves similarly to simplicial and halfspace depths. We show that this new depth notion shares the set of desirable properties that were introduced in Serfling (1980). For the empirical version of this depth function both uniform consistency and the asymptotic distribution are studied. Our new notion is extended to the important setup of data in Riemannian manifolds. The behaviour of the depth is illustrated through several examples.
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We prove that if a closed polarized complex manifold admits a conformally K\"ahler, EinsteinMaxwell metric, or more generally, a K\"ahler metric of constant $(\xi, a, p)$scalar curvature, then it minimizes the $(\xi,a,p)$Mabuchi functional. Our method of proof extends the approach introduced by Donaldson, via finite dimensional approximations and generalized balanced metrics. As an application of our result and the recent construction of KocaT{\o}nnesenFriedman, we describe the K\"ahler classes on a geometrically ruled complex surface of genus greater than 2, which admit conformally K\"ahler, EinsteinMaxwell metrics.
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A new system model reflecting the clustered structure of distributed storage is suggested to investigate bandwidth requirements for repairing failed storage nodes. Large data centers with multiple racks/disks or local networks of storage devices (e.g. sensor network) are good applications of the suggested clustered model. In realistic scenarios involving clustered storage structures, repairing storage nodes using intact nodes residing in other clusters is more bandwidthconsuming than restoring nodes based on information from intracluster nodes. Therefore, it is important to differentiate between intracluster repair bandwidth and crosscluster repair bandwidth in modeling distributed storage. Capacity of the suggested model is obtained as a function of fundamental resources of distributed storage systems, namely, node storage capacity, intracluster repair bandwidth and crosscluster repair bandwidth. The capacity is shown to be asymptotically equivalent to a monotonic decreasing fun
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In recent work, generalized persistence modules have proved useful in distinguishing noise from the legitimate topological features of a data set. Algebraically, generalized persistence modules can be viewed as representations for the poset algebra. The interplay between various metrics on persistence modules has been of wide interest, most notably, the isometry theorem of Bauer and Lesnick for (onedimensional) persistence modules. The interleaving metric of Bubenik, de Silva and Scott endows the collection of representations of a poset with values in any category with the structure of a metric space. This metric makes sense for any poset, and has the advantage that postcomposition by any functor is a contraction. In this paper, we prove an isometry theorem using this interleaving metric on a full subcategory of generalized persistence modules for a large class of posets.
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We prove that if $\mathcal{X}$ is a smooth Artin stack with stable good moduli space $\mathcal{X} \to X$, then there is a canonical sequence of birational morphisms of smooth Artin stacks $\mathcal{X}_n \to \mathcal{X}_{n1} \to \ldots \to \mathcal{X}_0 = \mathcal{X}$ with the following properties: (1) the maximum dimension of a stabilizer of a point of $\mathcal{X}_{k+1}$ is strictly smaller than the maximum dimension of a stabilizer of $\mathcal{X}_k$ and the final stack $\mathcal{X}_n$ has constant stabilizer dimension; (2) the morphisms $\mathcal{X}_{k+1} \to \mathcal{X}_k$ induce proper and birational morphisms of good moduli spaces $X_{k+1} \to X_k$; and (3) the algebraic space $X_n$ has tame quotient singularities and is a partial desingularization of the good moduli space $X$. Combining our result with D. Bergh's recent destackification theorem for tame stacks, we obtain a full desingularization of $X$.
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In statistical mechanics, for a system with fixed number of particles, e.g., a finitesize system, strictly speaking, the thermodynamic quantity needs to be calculated in the canonical ensemble. Nevertheless, the calculation of the canonical partition function is difficult; even the canonical partition function of ideal Bose and Fermi gases cannot be obtained exactly. In this paper, based on the mathematical theory of the symmetric function and the Bell polynomial, we suggest a method for the calculation of the canonical partition function of ideal quantum gases, including ideal Bose, Fermi, and Gentile gases. Moreover, we also reveal that the canonical partition functions of interacting classical and quantum gases given by the classical and quantum cluster expansion methods are indeed the Bell polynomial in mathematics. The virial coefficients of ideal Bose, Fermi, and Gentile gases is calculated from the exact canonical partition function. The virial coefficients of interacting class
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We establish a 3parameter family of integral identities to be used on a class of theories possessing solitons with spherical symmetry in $d$ spatial dimensions. The construction provides five boundary charges that are related to certain integrals of the profile functions of the solitons in question. The framework is quite generic and we give examples of both topological defects (like vortices and monopoles) and topological textures (like Skyrmions) in 2 and 3 dimensions. The class of theories considered here is based on a kinetic term and three functionals often encountered in reduced Lagrangians for solitons. One particularly interesting case provides a generalization of the wellknown Pohozaev identity. Our construction, however, is fundamentally different from scaling arguments behind Derrick's theorem and virial relations. For BPS vortices, we find interestingly an infinity of integrals simply related to the topological winding number.
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We conjecture that a countable group $G$ admits a nonsingular Bernoulli action of type III$_1$ if and only if the first $L^2$cohomology of $G$ is nonzero. We prove this conjecture for all groups that admit at least one element of infinite order. We also give numerous explicit examples of type III$_1$ Bernoulli actions of the group of integers and the free groups, with different degrees of ergodicity.
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We study the growth rate of the inclusion length of an almost periodic function. For a given a. p. function such growth rate depends on the algebraic structure of Fourier exponents, i. e. on how good they can be approximated by rational numbers. In additional, as appears from the definition, the inclusion length carries some information about the translation numbers (almost periods). Our result is a lower bound of the growth rate of the inclusion interval of a quasiperiodic function (theorem 3). Here we use methods from dimension theory. We do not assume anything about exponents, but rationally independence. This suggest an idea that this lower bound can be reached (in asymptotic sense) for some "bad" exponents. Koichiro Naito in his papers on estimates of the fractal dimension of almost periodic attractors proved an upper bound of the inclusion length for some class of a.p. functions, using simultaneous Diophantine approximations. For the special case of badly approximable exponents w
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This paper is concerned with the Cauchy problem of heatconductive ideal gas without viscosity. We show that, for the nonviscous case, if the strengths of the wave patterns and the initial perturbation are suitably small, the unique globalintime solution exists and asymptotically tends toward the corresponding the viscous contact wave or the composition of a viscous contact wave with rarefaction waves determined by the initial condition, which extended the results by HuangLiMatsumura[13], where they treated the viscous and heatconductive ideal gas.
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A problem in representation theory of $p$adic groups is the computation of the \textit{Casselman basis} of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials which are deformations of the KazhdanLusztig Rpolynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.
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We prove a general normal approximation theorem for local graph statistics in the configuration model, together with an explicit bound on the error in the approximation with respect to the Wasserstein metric. Such statistics take the form $T := \sum_{v \in V} H_v$, where $V$ is the vertex set, and $H_v$ depends on a neighbourhood in the graph around $v$ of size at most $\ell$. The error bound is expressed in terms of $\ell$, $V$, an almost sure bound on $H_v$, the maximum vertex degree $d_{\rm max}$, and the variance of $T$. Under suitable assumptions on the convergence of the empirical degree distributions to a limiting distribution, we deduce that the size of the giant component in the configuration model has asymptotically Gaussian fluctuations.
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Let $G$ be a simple undirected graph. For $\alpha \in [0,1]$, let \begin{equation*} A_{\alpha}\left( G\right) =\alpha D\left( G\right) +(1\alpha)A\left( G\right) , \end{equation*} where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the diagonal matrix of the degrees of $G$. In particular, $A_{0}(G)=A(G)$ and $A_{\frac{1}{2}}(G)=\frac{1}{2}Q(G)$ where $Q(G)$ is the signless Laplacian matrix of $G$. A bug $B_{p,q,r}$ is a graph obtained from a complete graph $K_{p}$ by deleting an edge and attaching paths $P_{q}$ and $P_{r}$ to its ends. In \cite{HaSt08}, Hansen and Stevanovi\'{c} proved that, among the graphs $G$ of order $n$ and diameter $d$, the largest spectral radius of $A(G)$ is attained by the bug $B_{nd+2,\lfloor d/2\rfloor,\lceil d/2\rceil}$. In \cite{LiLu14}, Liu and Lu proved the same result for the spectral radius of $Q(G)$. Let $\rho_{\alpha}(G)$ be the spectral radius of $A_{\alpha}(G)$. In this note, for a bug $B$ of order $n$ and diameter $d$, it is shown that $(n
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We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This confirms a conjecture of Albenque and Knauer and implies that there is a polynomial time algorithm to compute the convex hullnumber of a graph, when all its convex subgraphs are given as input. We then show that computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNPcomplete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear extension graph, that was conjectured to be in P. The intent to show that the latter problem is LOGSNPcomplete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $S=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is
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The notion of unboundedly order converges has been recieved recently a particular attention by several authors. The main result of the present paper shows that the notion is efficient and deserves that care. It states that a vector lattice is universally complete if and only if it is unboundedly order complete. Another notion of completeness will be treated is the notion of supcompletion introduced by Donner.
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We propose a slightly revised MillerHagberg (MH) algorithm that efficiently generates a random network from a given expected degree sequence. The revision was to replace the approximated edge probability between a pair of nodes with a combinatorically calculated edge probability that better captures the likelihood of edge presence especially where edges are dense. The computational complexity of this combinatorial MH algorithm is still in the same order as the original one. We evaluated the proposed algorithm through several numerical experiments. The results demonstrated that the proposed algorithm was particularly good at accurately representing highdegree nodes in dense, heterogeneous networks. This algorithm may be a useful alternative of other more established network randomization methods, given that the data are increasingly becoming larger and denser in today's network science research.
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We study quasiisometry types of expanders that come from a warped cone construction over group actions on homogeneous spaces. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if the group has no abelian factors, then two such warped cones are quasiisometric if and only if the actions are conjugate in finite covers. As a consequence, we produce a continuum of nonquasiisometric expanders and superexpanders. The proof relies on the use of coarse topology for warped cones, such as a computation of their coarse fundamental group.
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We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady's classifying space for braid groups and the SteinFarley complexes used for various relatives of Thompson's groups. It recovers the fact that Garside groups have finite classifying spaces. We describe the categories and Garside structures underlying certain Thompson groups. The ZappaSz\'ep product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our methods we introduce the group Braided T and show that it is of type $F_\infty$.
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We define spatial $L^p$ AF algebras for $p \in [1, \infty) \setminus \{ 2 \}$, and prove the following analog of the Elliott AF algebra classification theorem. If $A$ and $B$ are spatial $L^p$ AF algebras, then the following are equivalent: 1) $A$ and $B$ have isomorphic scaled preordered $K_0$groups. 2) $A \cong B$ as rings. 3) $A \cong B$ (not necessarily isometrically) as Banach algebras. 4) $A$ is isometrically isomorphic to $B$ as Banach algebras. 5) $A$ is completely isometrically isomorphic to $B$ as matrix normed Banach algebra. As background, we develop the theory of matrix normed $L^p$ operator algebras, and show that there is a unique way to make a spatial $L^p$ AF algebra into a matrix normed $L^p$ operator algebra. We also show that any countable scaled Riesz group can be realized as the scaled preordered $K_0$group of a spatial $L^p$ AF algebra.
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We use physical principles to derive a water wheel model under the assumption of an asymmetric water wheel for which the water inflow rate is in general unsteady (modeled by an arbitrary function of time). Our model allows one to recover the asymmetric water wheel with steady flow rate, as well as the symmetric water wheel, as special cases. Under physically reasonable assumptions we then reduce the underlying model into a nonautonomous nonlinear system. In order to determine parameter regimes giving chaotic dynamics in this nonautonomous nonlinear system, we consider an application of competitive modes analysis. In order to apply this method to a nonautonomous system, we are required to generalize the competitive modes analysis so that it is applicable to nonautonomous systems. The nonautonomous nonlinear water wheel model is shown to satisfy competitive modes conditions for chaos in certain parameter regimes, and we employ the obtained parameter regimes to construct the chaotic
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turkeydance shares a report from The Telegraph: If you struggle to get a tan, consider yourself a night owl or are plagued with arthritis, then your Neanderthal ancestors could be to blame, a new genetic study has shown. Although Neanderthals are often portrayed in drawings as swarthy, in fact they arrived in Northern Europe thousands of years before modern humans, giving time for their skin to become paler as their bodies struggled to soak up enough sun. When they interbred with modern humans those pale genes were passed on. Likewise, genetic mutations which predispose people to arthritis also came from our Neanderthal ancestors, as did the propensity to be a night owl rather than a lark, as northern latitudes altered their body clocks. A raft of new papers published in the journals Science and the American Journal of Human Genetics has shed light on just how many traits we owe to our Neanderthal ancestors. Scientists also now think that differences in hair color, mood and whether som
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Mark Wilson writes: Bitcoin has been in the news for some time now as its value climbs and drops, but most recently interest turned to mining code embedded in websites. The Pirate Bay was one of the first sites to be seen using Coinhive code to secretly mine using visitors' CPU time, and then we saw similar activity from the SafeBrowse extension for Chrome. The discovery of the code was a little distressing for visitors to the affected sites, and internet security and content delivery network (CDN) firm Cloudflare is taking action to clamp down on what it is describing as malware. Torrent proxy site ProxyBunker.online has contacted TorrentFreak to say that Cloudflare has dropped it as a customer. The reason given for ProxyBunker's suspension is that the site has been using Coinhive code on several of the domains it owns.
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Today we're marking Slashdot's 20th birthday. 20 years is a long time on the internet. Many websites have come and gone over that time, and many that stuck around haven't had any interest in preserving their older content. Fortunately, as Slashdot approaches its 163,000th story, we've managed to keep track of almost all our old postings  all but the first 2^10, or so. In addition to that, we've held onto user comments, the lifeblood of the site, from 1999 onward. As we celebrate Slashdot's 20th anniversary this month, we thought we'd take a moment to highlight a few of the notable or interesting stories and discussions that have happened here in the past decade and a half. This is part of our 20year anniversary celebration, and we've set up a page to coordinate user meetups. We'll be continuing to run some special pieces throughout the month, so keep an eye out for those. Read on for a trip down memory lane. Update: Slashdot founder CmdrTaco has taken to Medium with some of his own
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Longtime package delivery companies UPS and FedEx may have some new competition from Amazon. The company is experimenting with a new delivery service of its own intended to make more products available for free twoday delivery and relieve overcrowding in its warehouses. Bloomberg reports: The service began two years ago in India, and Amazon has been slowly marketing it to U.S. merchants in preparation for a national expansion, said the people, who asked not to be identified because the U.S. pilot project is confidential. Amazon is calling the project Seller Flex, one person said. The service began on a trial basis this year in West Coast states with a broader rollout planned in 2018, the people said. Amazon will oversee pickup of packages from warehouses of thirdparty merchants selling goods on Amazon.com and their delivery to customers' homes, the people said  work that is now often handled by UPS and FedEx. Amazon could still use these couriers for delivery, but the company will
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It's official: Facebook will be investing $1 billion in a new data center in Henrico County, which is just outside Richmond, Virginia. According to CNNMoney, Facebook is putting $750 million into construction and $250 million to multiple solar facilities that will power the data center. From the report: The investment is expected to create 100 fulltime jobs. Facebook will receive about $19 million in state tax exemptions through 2035, according to the Virginia Economic Development Partnership. Facebook already has data centers in Oregon, North Carolina and Iowa. Centers in Fort Worth, Texas; Los Lunas, New Mexico; and New Albany, Ohio are currently under construction. "One of the many important factors in our search for a new data center location is being able to source clean and renewable energy. We also look for great partnerships within the local community, robust infrastructure ... and a strong pool of local talent," Rachel Peterson, Facebook's director of data center strategy, sa
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An anonymous reader shares a report: Amazon, disrupter of industries from book selling to grocery shopping, has found its latest sector to upend  recruiting at the nation's elite business schools. The Seattlebased retail giant is now the top recruiter at the business schools of Carnegie Mellon University, Duke University and University of California, Berkeley. It is the biggest internship destination for firstyear M.B.A.s at the University of Michigan, Massachusetts Institute of Technology, Dartmouth College and Duke. Amazon took in more interns from the University of Chicago's Booth School of Business than either Bain & Co. or McKinsey & Co., which were until recently among the school's top hirers of interns, according to Madhav Rajan, Booth's dean. All told, Amazon has hired some 1,000 M.B.A.s in the past year, according to Miriam Park, Amazon's director of university programs  a drop in the bucket for a company that plans to add 50,000 software developers in the next y
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Nilay Patel on the further disappearance of the headphone jack, and its replacement, Bluetooth: To improve Bluetooth, platform vendors like Apple and Google are riffing on top of it, and that means theyâre building custom solutions. And building custom solutions means theyâre taking the opportunity to prioritize their own products, because that is a fair and rational thing for platform vendors to do. Unfortunately, what is fair and rational for platform vendors isnât always great for markets, competition, or consumers. And at the end of this road, we will have taken a simple, universal thing that enabled a vibrant market with tons of options for every consumer, and turned it into yet another limited market defined by ecosystem lockin. This is exactly what's happening, and it is turning something simple and straightforward  get headphones, plug it in literally every single piece of headphonesenabled audio equipment made in the last 100 years, and have it work  into an inco
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An anonymous reader quotes a report from Reuters: A Seattlearea startup, backed by the venture capital arms of Boeing and JetBlue announced plans on Thursday to bring a small hybridelectric commuter aircraft to market by 2022. The small airliner is the first of several planes planned by Zunum Aero, which said it would seat up to 12 passengers and be powered by two electric motors, dramatically reducing the travel time and cost of trips under 1,000 miles (1,600 km). Zunum's plans and timetable underscore a rush to develop small electric aircraft based on rapidly evolving battery technology and artificial intelligence systems that avoid obstacles on a road or in the sky. In a separate but related development, Boeing said on Thursday it plans to acquire a company that specializes in electric and autonomous flight to help its own efforts to develop such aircraft. Zunum's planes would fly from thousands of small airports around big cities to cut regional travel times and costs.
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Joe Rossignol, writing for MacRumours: Brazilian software developer Matheus Mariano appears to have discovered a significant Disk Utility bug that exposes the passwords of encrypted Apple File System volumes in plain text on macOS High Sierra. Mariano added a new encrypted APFS volume to a container, set a password and hint, and unmounted and remounted the container in order to force a password prompt for demonstration purposes. Then, he clicked the "Show Hint" button, which revealed the full password in plain text rather than the hint. [...] Apple has addressed this bug by releasing a macOS High Sierra 10.13 Supplemental Update, available from the Updates tab in the Mac App Store.
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It has long been known that the differential operator $D$ represents a typical examples of unbounded operators in many Banach spaces including the classical Fock spaces, the FockSobolev spaces, and the generalized Fock spaces where the weight decays faster than the Gaussian weight. In this note we identify Fock type spaces where the operator admits some basic spectral structures including compactness and membership in the Schatten $\mathcal{S}_p$ classes. We also showed that its nontrivial spectrum while acting on such spaces is precisely the closed unit disk $\D$ in the complex plane.
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We consider an elliptic boundary problem over a bounded region $\Omega$ in $\mathbb{R}^n$ and acting on the generalized Sobolev space $W^{0,\chi}_p(\Omega)$ for $1 < p < \infty$. We note that similar problems for $\Omega$ either a bounded region in $\mathbb{R}^n$ or a closed manifold acting on $W^{0,\chi}_2(\Omega)$, called H\"{o}rmander space, have been the subject of investigation by various authors. Then in this paper we will, under the assumption of parameterellipticity, establish results pertaining to the existence and uniqueness of solutions of the boundary problem. Furthermore, under the further assumption that the boundary conditions are null, we will establish results pertaining to the spectral properties of the Banach space operator induced by the boundary problem, and in particular, to the angular and asymptotic distribution of its eigenvalues.
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We show that any accordion complex associated to a dissection of a convex polygon is isomorphic to the support $\tau$tilting simplicial complex of an explicit finite dimensional algebra. To this end, we prove a property of some induced subcomplexes of support $\tau$tilting simplicial complexes of finite dimensional algebras.
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The aim of this paper is to prove the strong law of large numbers (SLLN) as well as the central limit theorem (CLT) for a class of vectorvalued stochastic processes which arise as solutions of the stochastic evolution inclusion \begin{align*} \eta(t,z) N_{\Theta}(dt \otimes z)\in dX(t)+\mathcal{A} X(t)dt, \end{align*} where $\mathcal{A}$ is a multivalued operator and $N_{\Theta}$ is the counting measure induced by a point process $\Theta$. The SLLN and the CLT will be proven not only for realvalued, but also for vectorvalued functionals and the applicability of these results to the (weighted) $p$Laplacian evolution equation (for "small" $p$) will be demonstrated. The key assumption needed in this paper is that the nonlinear semigroup arising from the multivalued operator $\mathcal{A}$ extincts in finite time.
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The purpose of this paper is to study stochastic evolution inclusions of the form \begin{align*} \eta(t,z) N_{\Theta}(dt \otimes z)\in dX(t)+\mathcal{A} X(t)dt, \end{align*} where $\mathcal{A}$ is a multivalued operator acting on a separable Banach space and $N_{\Theta}$ is the counting measure induced by a point process $\Theta$. Firstly, we will set up the concepts of strong and mild solutions; then we will derive existence as well as uniqueness criteria for these kinds of solutions and give a representation formula for the solutions. The results will be formulated by means of nonlinear semigroup theory and except for separability, no assumptions on the underlying Banach space are required.
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We present a criterion of local Normal Embedding of a semialgebraic (or definable in an ominimal structure) contained in $R^n$ in terms orders of contact of arcs. Namely, we prove that a semialgebraic set is normally embedded at a point x if and only if for any pair of arcs, coming to this point the inner order of contact is equal to the outer order of contact.
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Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $\ell \not= p$, the Frobenius automorphism (unique up to conjugation) acts on the $\ell$adic Tate module of $E$. The characteristic polynomial of the Frobenius automorphism is defined over $\mathbb{Z}$ and is independent of $\ell$. Its splitting field is called the Frobenius field of $E$ at $p$. Let $E_1$ and $E_2$ be two elliptic curves defined over $\bar{Q}$ that are nonisogenous and both without complex multiplication over $\overline{\mathbb{Q}}$. Motivated by the LangTrotter conjecture, it is natural to consider the asymptotic behaviour of the function that counts the number of primes $p \le x$ such that the Frobenius fields of $E_1$ and $E_2$ at $p$ coincide. In this short note, using HeathBrown's square sieve, we provide both conditional (upon GRH) and unconditional upper bounds.
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We prove that special functions of bounded deformation with small jump set are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness of the jump set for local minimizers of the Griffith energy.
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We discuss a general approach permitting the identification of a broad class of sets of Poissoncommuting Hamiltonians, which are integrable in the sense of Liouville. It is shown that all such Hamiltonians can be solved explicitly by a separation of variables {\em Ansatz}. The method leads in particular to a proof that the socalled "goldfish" Hamiltonian is maximally superintegrable, and leads to an elementary identification of a full set of integrals of motion. The Hamiltonians in involution with the "goldfish" Hamiltonian are also explicitly integrated. New integrable Hamiltonians are identified, among which some have the property of being isochronous, that is, that all their orbits have the same period. Finally, a peculiar structure is identified in the Poisson brackets between the elementary symmetric functions and the set of Hamiltonians commuting with the "goldfish" Hamiltonian: these can be expressed as products between elementary symmetric functions and Hamiltonians. The stru
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We prove an Erd\H{o}sKac type of theorem for the set $S(x,y)=\{n\leq x: pn \Rightarrow p\leq y \}$. If $\omega (n)$ is the number of prime factors of $n$, we prove that the distribution of $\omega(n)$ for $n \in S(x,y)$ is Gaussian for a certain range of $y$ using method of moments. The advantage of the present approach is that it recovers classical results for the range $u=o(\log \log x )$ where $u=\frac{\log x}{\log y}$, with a much simpler proof.
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In this manuscript, we discuss the use of describing functions as a systematic approach to the analysis and design of oscillators. Describing functions are traditionally used to study the stability of nonlinear control systems, and have been adapted for analyzing LC oscillators. We show that they can be applied to other categories of oscillators too, including relaxation and ring oscillators. With the help of several examples of oscillators from various physical domains, we illustrate the techniques involved, and also demonstrate the effectiveness and limitations of describing functions for oscillator analysis.
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As we explain, when a positive integer $n$ is not squarefree, even over $\mathbb{C}$ the moduli stack that parametrizes generalized elliptic curves equipped with an ample cyclic subgroup of order $n$ does not agree at the cusps with the $\Gamma_0(n)$level modular stack $\mathscr{X}_0(n)$ defined by Deligne and Rapoport via normalization. Following a suggestion of Deligne, we present a refined moduli stack of ample cyclic subgroups of order $n$ that does recover $\mathscr{X}_0(n)$ over $\mathbb{Z}$ for all $n$. The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: $\mathscr{X}_0(n)$ is also regular at the cusps. We also prove such regularity for $\mathscr{X}_1(n)$ and several other modular stacks, some of which have been treated by Conrad by a different method. For the proofs we introduce a tower of compactifications $\overline{Ell}_m$ of the stack $Ell$ that parametrizes elliptic curvesthe ability to vary $m$ in the tower permits robust re
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We study the minimal gap statistic for fractional parts of sequences of the form $\mathcal A^\alpha = \{\alpha a(n)\}$ where $\mathcal A = \{a(n)\}$ is a sequence of distinct of integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\alpha$, the minimal gap $\delta_{\min}^\alpha(N)=\min\{\alpha a(m)\alpha a(n)\bmod 1: 1\leq m\neq n\leq N\}$ is close to that of a random sequence.
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People with depression are more likely to feel bad in response to perceived inequality, according to a study published this week in Nature Human Behaviour. From a report: Simply, in experiments where participants were tasked with playing a game with a strong element of unfairness, those participants with higher levels of brain activity in depressionlinked brain regions  as recorded via fMRI scans  were more likely to later demonstrate signs of clinical depression. This is a new test of an old idea, one that's been demonstrated in previous research. People with depression commonly demonstrate increased concern for others, or for the perspectives of others. More precisely, prosocial attitudes predict depression, which is in contrast to individualist attitudes. Individualist here basically just means selfish, or relatively selfish. The researchers behind the current study hypothesized that they would be able to observe these tendencies at the level of actual brain activity. Fortunate
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An anonymous reader quotes a report from Reuters: The U.S. Senate Commerce Committee on Wednesday unanimously approved a bill to speed selfdriving cars to market without human controls and bar states from imposing regulatory road blocks. The bill still must be approved by the full Senate. The U.S. House passed a similar version last month unanimously. General Motors Co, Alphabet Inc, Ford Motor Co and others have lobbied for the landmark legislation. Despite some complaints from Republicans, the Senate bill does not speed approval of selfdriving technology for large commercial trucks after labor unions raised safety and employment concerns. The measure, the first significant federal legislation aimed at speeding selfdriving cars to market, would allow automakers to win exemptions from current safety rules that prohibit vehicles without human controls. States could still set rules on registration, licensing, liability, insurance and safety inspections, but not performance standards.
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AnalogDiehard writes: Congress created the Inter Partes Review (IPR) in 2012 within the U.S. Patent Office Patent Trials and Appeals Board (PTAB) as a faster and cheaper way to challenge and invalidate bad patents. The IPR expense is a fraction of the cost of a multimillion dollar patent court trial; it is loved by patent challengers and hated by patent owners. The pharmaceutical company Allergen has exploited a novel tactic to evade the IPR process: they hand them to a Native American Indian tribe for safekeeping. Under the arrangement, the tribes earn millions in royalties as long as the patents are valid, they license them back to Allergan, and the patents under the tribes' ownership is immune from lawsuits via sovereign immunity. Under the colonialera concept of "sovereign immunity" which is codified in the 11th amendment, certain groups like states, universities, and tribes are immune from lawsuits, thus the drug patents are shielded from the IPR process leaving only a full blown
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