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A lot was discussed and presented in the three hours allotted to the Testing and Fuzzing microconference at this year's Linux Plumbers Conference (LPC), but some spilled out of that slot. We have already looked at some discussions on kernel testing that occurred both before and during the microconference. Much of the rest of the discussion is summarized in the article from this week's edition, which subscribers can access from the link below.
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In this paper we compute the $RO(G)$graded homotopy Mackey functor for $H\underline{\mathbb{Z}}$, the EilenbergMac Lane spectrum of constant Mackey functor for $G = C_{p^2}$, and give some computation for larger $G$. As an application, we use it to give some computation of homological algebra of $\underline{\mathbb{Z}}$modules.
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Email is such a pain in the butt. We've been doing everything in our power to fight the influence it has on our lives, to minimize the spam, the marketing, the burden. That burden leads lots of folks to fruitlessly hunt for the perfect email client like I hunt for the perfect word processor. Others have followed the path of least resistance: Either Gmail or Outlook. But there was a time when we didn't feel this way, when getting email was actually exciting. The email client Eudora, named for Eudora Welty, was designed to capture this excitement  the idea that mailboxes were no longer tethered to physical space. But even as the diehards held on, it couldn't. Tonight's Tedium ponders the demise of Eudora, and whether we lost something great. I don't have a lot of experience with Eudora personally, but I know it had quite the enthusiastic and fervent fanbase back then.
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When sales of a product are affected by randomness in demand, retailers can use dynamic pricing strategies to maximise their profits. In this article the pricing problem is formulated as a stochastic optimal control problem, where the optimal policy can be found by solving the associated Bellman equation. The aim is to investigate Approximate Dynamic Programming algorithms for this problem. For realistic retail applications, modelling the problem and solving it to optimality is intractable. Thus practitioners make simplifying assumptions and design suboptimal policies, but a thorough investigation of the relative performance of these policies is lacking. To better understand such assumptions, we simulate the performance of two algorithms on a oneproduct system. It is found that for more than half of the realisations of the random disturbance, the oftenused, but approximate, Certainty Equivalent Control policy yields larger profits than an optimal, maximum expectedvalue policy. This
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Given an $\alpha > 1$ and a $\theta$ with unbounded continued fraction entries, we characterise new relations between Sturmian subshifts with slope $\theta$ with respect to (i) an $\alpha$H\"oder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of $\theta$, and (iii) complexity notions which we call $\alpha$repetitive, $\alpha$repulsive and $\alpha$finite; generalisations of the properties known as linearly repetitive, repulsive and power free, respectively. We show that the level sets relate naturally to (exact) Jarn\'{\i}k sets and prove that their Hausdorff dimension is $2/(\alpha + 1)$.
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We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width at most three, then the flow categories are move equivalent. The process we describe is essentially algorithmic and can often be performed by hand, without the aid of a computer program.
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We present an identity management scheme built into the Bitcoin blockchain, allowing for identities that are as indelible as the blockchain itself. Moreover, we take advantage of Bitcoin's decentralized nature to facilitate a shared control between users and identity providers, allowing users to directly manage their own identities, fluidly coordinating identities from different providers, even as identity providers can revoke identities and impose controls.
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In the second paper of the series we introduce the concept of the global extended exactness of penalty and augmented Lagrangian functions, and derive the localization principle in the extended form. The main idea behind the extended exactness consists in the extension of the original constrained optimization problem by adding some extra variables, and then the construction of a penalty/augmented Lagrangian function for the extended problem. This approach allows one to design extended penalty/augmented Lagrangian functions having some useful properties (such as smoothness), which their counterparts for the original problem might not possess. In turn, the global exactness of such extended functions can be easily proved with the use of the localization principle presented in this paper, which reduces the study of global exactness to a local analysis of sufficient optimality conditions and constraint qualifications. We utilize the localization principle in order to obtain simple necessary
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We present a new formulation based on the classical DirichletNeumann formulation for interface coupling problems in linearized elasticity. By using Taylor series expansions, we derive a new set of interface conditions that allow our formulation to pass the linear consistency test. In addition, we propose an iterative method to determine the solution of our formulation. We demonstrate in our numerical results that we may achieve the desired piecewise linear finite element error bounds for both nonoverlapping domain decomposition problems as well as for interface coupling problems where the Lam\'e parameters of the structures differ.
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A graph $G$ is \emph{$(a,b)$choosable} if given any list assignment $L$ with $L(v)=a$ for each $v\in V(G)$ there exists a function $\varphi$ such that $\varphi(v)\in L(v)$ and $\varphi(v)=a$ for all $v\in V(G)$, and whenever vertices $x$ and $y$ are adjacent $\varphi(x)\cap \varphi(y)=\emptyset$. Meng, Puleo, and Zhu conjectured a characterization of (4,2)choosable graphs. We prove their conjecture.
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We develop a theory of $p$adic automorphic forms on unitary groups that allows $p$adic interpolation in families and holds for all primes $p$ that do not ramify in the reflex field $E$ of the associated unitary Shimura variety. If the ordinary locus is nonempty (a condition only met if $p$ splits completely in $E$), we recover Hida's theory of $p$adic automorphic forms, which is defined over the ordinary locus. More generally, we work over the $\mu$ordinary locus, which is open and dense. By eliminating the splitting condition on $p$, our framework should allow many results employing Hida's theory to extend to infinitely many more primes. We also provide a construction of $p$adic families of automorphic forms that uses differential operators constructed in the paper. Our approach is to adapt the methods of Hida and Katz to the more general $\mu$ordinary setting, while also building on papers of each author. Along the way, we encounter some unexpected challenges and subtleties tha
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In this paper, we generalize a mathematical definition of GopakumarVafa (GV) invariants on CalabiYau 3folds introduced by Maulik and the author, using an analogue of BPS sheaves introduced by DavisonMeinhardt on the coarse moduli spaces of one dimensional twisted semistable sheaves with arbitrary holomorphic Euler characteristics. We show that our generalized GV invariants are independent of twisted stability conditions, and conjecture that they are also independent of holomorphic Euler characteristics, so that they define the same GV invariants. As an application, we will show the flop transformation formula of GV invariants.
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In 1964, Massey introduced a class of codes with complementary duals which are called Linear Complimentary Dual (LCD for short) codes. He showed that LCD codes have applications in communication system, sidechannel attack (SCA) and so on. LCD codes have been extensively studied in literature. On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice. The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. We exemplify this construction and obtain several classes of explicit LCD MDS codes for the odd characteristic case.
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In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} \Delta(g(\nabla u_2^2) u + u^r) = a u + b u^p& \mbox{in}~\Omega, u>0& \mbox{in}~\Omega, u= 0& \mbox{on}~\partial\Omega, \end{array} \right. \end{equation*} in a bounded and smooth domain $\Omega$ in ${\rm I}\hskip 0.85mm{\rm R}$. By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters $a,b,p,r$ and the nonlinearity $g$, the existence of positive solutions.
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Delay and sum (DAS) is the most common beamforming algorithm in lineararray photoacoustic imaging (PAI) as a result of its simple implementation. However, it leads to a low resolution and high sidelobes. Delay multiply and sum (DMAS) was used to address the incapabilities of DAS, providing a higher image quality. However, the resolution improvement is not well enough compared to eigenspacebased minimum variance (EIBMV). In this paper, the EIBMV beamformer has been combined with DMAS algebra, called EIBMVDMAS, using the expansion of DMAS algorithm. The proposed method is used as the reconstruction algorithm in lineararray PAI. EIBMVDMAS is experimentally evaluated where the quantitative and qualitative results show that it outperforms DAS, DMAS and EIBMV. The proposed method degrades the sidelobes for about 365 %, 221 % and 40 %, compared to DAS, DMAS and EIBMV, respectively. Moreover, EIBMVDMAS improves the SNR about 158 %, 63 % and 20 %, respectively.
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Minimum storage regenerating codes have minimum storage of data in each node and therefore are maximal distance separable (MDS for short) codes. Thus, the number of nodes is upper bounded by $2^{\fb}$, where $\fb$ is the bits of data stored in each node. From both theoretical and practical points of view (see the details in Section 1), it is natural to consider regenerating codes that nearly have minimum storage of data, and meanwhile the number of nodes is unbounded. One of the candidates for such regenerating codes is an algebraic geometry code. In this paper, we generalize the repairing algorithm of ReedSolomon codes given in \cite[STOC2016]{GW16} to algebraic geometry codes and present an efficient repairing algorithm for arbitrary onepoint algebraic geometry codes. By applying our repairing algorithm to the onepoint algebraic geometry codes based on the GarciaStichtenoth tower, one can repair a code of rate $1\Ge$ and length $n$ over $\F_{q}$ with bandwidth $(n1)(1\Gt)\log
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The observational census of transNeptunian objects with semimajor axes greater than ~250 AU exhibits unexpected orbital structure that is most readily attributed to gravitational perturbations induced by a yetundetected, massive planet. Although the capacity of this planet to (i) reproduce the observed clustering of distant orbits in physical space, (ii) facilitate dynamical detachment of their perihelia from Neptune, and (iii) excite a population of longperiod centaurs to extreme inclinations is well established through numerical experiments, a coherent theoretical description of the dynamical mechanisms responsible for these effects remains elusive. In this work, we characterize the dynamical processes at play, from semianalytic grounds. We begin by considering a purely secular model of orbital evolution induced by Planet Nine, and show that it is at odds with the ensuing stability of distant objects. Instead, the longterm survival of the clustered population of longperiod KBO
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Applying the method of the paper [CT], we perform a quantum version of the DrinfeldSokolov reduction in Reflection Equation algebras and braided Yangians, associated with involutive and Hecke symmetries of general forms. This reduction is based on the CayleyHamilton identity valid for the generating matrices of these algebras.
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A matroid is Ingleton if all quadruples of subsets of its ground set satisfy Ingleton's inequality. In particular, representable matroids are Ingleton. We show that the number of Ingleton matroids on ground set $[n]$ is doubly exponential in $n$; it follows that almost all Ingleton matroids are nonrepresentable.
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This paper investigates the problem of distributed medium access control in a time slotted wireless multiple access network with an unknown finite number of homogeneous users. Assume that each user has a single transmission option. In each time slot, a user chooses either to idle or to transmit a packet. Under a general channel model, a distributed medium access control framework is proposed to adapt transmission probabilities of all users to a value that is near optimal with respect to a predetermined symmetric network utility. Probability target of each user in the proposed algorithm is calculated based upon a channel contention measure, which is defined as the conditional success probability of a virtual packet. It is shown that the proposed algorithm falls into the classical stochastic approximation framework with guaranteed convergence when the contention measure can be directly obtained from the receiver. On the other hand, computer simulations show that, when the contention meas
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We introduce a category of stochastic maps (certain Markov kernels) on compact Hausdorff spaces, construct a stochastic analogue of the Gelfand spectrum functor, and prove a stochastic version of the commutative GelfandNaimark Theorem. This relates concepts from algebra and operator theory to concepts from topology and probability theory. For completeness, we review stochastic matrices, their relationship to positive maps on commutative $C^*$algebras, and the GelfandNaimark Theorem. No knowledge of probability theory nor $C^*$algebras is assumed and several examples are drawn from physics.
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Online enrichment is the extension of a reduced solution space based on the solution of the reduced model. Procedures for online enrichment were published for many localized model order reduction techniques. We show that residual based online enrichment on overlapping domains converges exponentially. Furthermore, we present an optimal enrichment strategy which couples the global reduced space with a local fine space. Numerical experiments on the two dimensional stationary heat equation with high contrast and channels confirm and illustrate the results.
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We explicitly find the rate of exponential longterm convergence for the ruin probability in a leveldependent L\'evydriven risk model, as time goes to infinity. Siegmund duality allows to reduce the problem to longterm convergence of a reflected jumpdiffusion to its stationary distribution, which is handled via Lyapunov functions.
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The feedback vertex number $\tau(G)$ of a graph $G$ is the minimum number of vertices that can be deleted from $G$ such that the resultant graph does not contain a cycle. We show that $\tau(S_p^n)=p^{n1}(p2)$ for the Sierpi\'{n}ski graph $S_p^n$ with $p\geq 2$ and $n\geq 1$. The generalized Sierpi\'{n}ski triangle graph $\hat{S_p^n}$ is obtained by contracting all nonclique edges from the Sierpi\'{n}ski graph $S_p^{n+1}$. We prove that $\tau(\hat{S}_3^n)=\frac {3^n+1} 2=\frac{V(\hat{S}_3^n)} 3$, and give an upper bound for $\tau(\hat{S}_p^n)$ for the case when $p\geq 4$.
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We consider a mechanical system consisting of $n$ penduli and a $d$dimensional generalized rotator subject to a timedependent perturbation. The perturbation is not assumed to be either Hamiltonian, or periodic or quasiperiodic. The strength of the perturbation is given by a parameter $\epsilon\in\mathbb{R}$. For all $\epsilon$ sufficiently small, the augmented flow has a $(2d + 1)$dimensional normally hyperbolic locally invariant manifold $\tilde\Lambda_\epsilon$. We define a Melnikov vector, which gives the first order expansion of the displacement of the stable and unstable manifolds of $\tilde\Lambda_0$ under the perturbation. We provide an explicit formula for the Melnikov vector in terms of convergent improper integrals of the perturbation along homoclinic orbits of the unperturbed system. We show that if the perturbation satisfies some explicit nondegeneracy conditions, then the stable and unstable manifolds of $\tilde\Lambda_\epsilon$, $W^s(\tilde\Lambda_\epsilon)$ and $W
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We study the $r$th generalized minimum distance function (gmd function for short) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. If $\mathbb{X}$ is a set of projective points over a finite field and $I(\mathbb{X})$ is its vanishing ideal, we show that the gmd function and the Vasconcelos function of $I(\mathbb{X})$ are equal to the $r$th generalized Hamming weight of the corresponding ReedMullertype code $C_\mathbb{X}(d)$. We show that the $r$th generalized footprint function of $I(\mathbb{X})$ is a lower bound for the $r$th generalized Hamming weight of $C_\mathbb{X}(d)$. As an application to coding theory we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine cartesian code.
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We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of the 2sphere, the Euclidean plane or the hyperbolic plane. This requires generalizing the angle structures program of Casson and Rivin for triangulations with a mixture of finite, ideal, and truncated (i.e. ultraideal) vertices. A consequence of this decomposition is that the volumes of spherical tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description. In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces S_g x I with genus g at least 2. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for links in S_g x I, ranging over all g at least 2, is a dense subset of the interval [0, 2v_{oct}], where v_{oct}, approximately 3.66386, is the volu
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We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and planeoriented random trees, as the size of the tree goes to infinity. All these models can be seen as special cases of the onesplit branching random walk for which we also provide an Edgeworth expansion. These expansions lead to new results on mode, width and occupation numbers of the trees, settling several open problems raised in Devroye and Hwang [Ann. Appl. Probab. 16(2): 886918, 2006], Fuchs, Hwang and Neininger [Algorithmica, 46 (34): 367407, 2006], and Drmota and Hwang [Adv. in Appl. Probab., 37 (2): 321341, 2005]. The aforementioned results are special cases and corollaries of a general theorem: an Edgeworth expansion for an arbitrary sequence of random or deterministic functions $\mathbb L_n:\mathbb Z\to\mathbb R$ which converges in the mod$\phi$sense. Applications to Stirling numbers of the first kind will be given in a sepa
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This article is a generalisation of results of Ciliberto and Sernesi. For a general canonically embedded curve $C$ of genus $g\geq 5$, let $d\le g1$ be an integer such that the BrillNoether number $\rho(g,d,1)=g2(gd+1)\geq 1$. We study the family of $d$secant $\mathbb{P}^{d2}$'s to $C$ induced by the smooth locus of the BrillNoether locus $W^1_d(C)$. Using the theory of foci and a structure theorem for the rank one locus of special $1$generic matrices by Eisenbud and Harris, we prove a Torellitype theorem for general curves by reconstructing the curve from its BrillNoether loci $W^1_d(C)$ of dimension at least $1$.
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In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical YangBaxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of Lie groups between the group of affine transformations of a flat affine and simply connected manifold and the classical group of affine transformations of $\mathbb{R}^n$. In the last section, for each flat left invariant affine symplectic connection on the group of affine transformations of the real line, describe for MedinaSaldarriagaGiraldo, we determine the affine symplectomorphisms. Finally we exhibit the Hess connection, associated to a Lagrangian bifoliation, which is flat left invariant affine.
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Let $X$ be an algebraic variety with quotient field $K(X)$. Let $s$ be the highest multiplicity of $X$ and $F_s(X)$ the set of points of multiplicity $s$. A sequence of blow ups at regular centers $Y_i \subset F_s(X_i)$, $X \leftarrow X_1 \leftarrow \dotsb \leftarrow X_n$, is a reduction of the multiplicity if $F_s(X_n) $ is empty. In characteristic zero there is an algorithm which assigns to each $X$ a unique reduction of the multiplicity. Fix $K(X) \subset L$ a finite extension of fields of degree $r$. For a finite map $\beta : X' \to X$ with $K(X') = L$ the highest multiplicity of $X'$ is at most $r \cdot s$. When the bound is achieved we say that $\beta$ is transversal. We will see that, if $\beta$ is transversal, then $F_{rs}(X')$ is homeomorphic to its image and $\beta(F_{rs}(X'))\subset F_{s}(X)$. We will see that a blow up $X' \leftarrow X'_1$ along a regular center $Y'_1 \subset F_{rs}(X'_1)$ induces a blow up $X \leftarrow X_1$ along a center $Y_1 \subset F_s(X_1)$ and a fini
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Identification of the parameters of stable linear dynamical systems is a wellstudied problem in the literature, both in the low and highdimensional settings. However, there are hardly any results for the unstable case, especially regarding finite time bounds. For this setting, classical results on leastsquares estimation of the dynamics parameters are not applicable and therefore new concepts and technical approaches need to be developed to address the issue. Unstable linear systems reflect key real applications in control theory, econometrics, and finance. This study establishes finite time bounds for the identification error of the leastsquares estimates for a fairly large class of heavytailed noise distributions, and transition matrices of such systems. The results relate the time length required as a function of the problem dimension and key characteristics of the true underlying transition matrix and the noise distribution. To obtain them, appropriate concentration inequaliti
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This work presents a novel diffusion based dualphase molecular communication system where the source leverages multiple cooperating nanomachines to improve the endtoend reliability of communication. The NeymanPearson (NP) Likelihood Ratio Tests (LRT) are derived for each of the cooperative as well as the destination nanomachines in the presence of multiuser interference (MUI). Further, to characterize the performance of the aforementioned system, closed form expressions are derived for the probabilities of detection, false alarm at the individual cooperative, destination nanomachines, as well as the overall endtoend probability of error. Simulation results demonstrate a significant improvement in the endtoend performance of the proposed cooperative framework in comparison to multipleinput singleoutput (MISO) and singleinput singleoutput (SIMO) molecular communication scenarios in the existing literature.
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Players will have the chance to explore Assassin's Creed Origins' virtual Egypt free of combat and story constraints in a new "Discovery Tour" gamemode, developerpublisher Ubisoft announced today. Discovery Tour turns Origins' map, as the company puts it, into a "combatfree living museum, with guided tours that let players delve into its history firsthand." Given the lengths Ubisoft went to creating a largescale asaccurateaspossible map of the country, hiring historians and Egyptologists as consultants, this is a chance for the developer to showcase its map and the functioning virtual world it's created, rather than it simply existing as a backdrop for action. This is a great move, as it turns what is normally 'just' a game into a tool that can be used for education and learning, or something more casual as just walking around in a beautiful environment without having to worry about being attacked or killed or whatever.
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We review the main results of the theory of rankmetric codes, with emphasis on their combinatorial properties. We study their duality theory and MacWilliams identities, comparing in particular rankmetric codes in vector and matrix representation. We then investigate the combinatorial structure of MRD codes and optimal anticodes in the rank metric, describing how they relate to each other.
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A problem of classification of local field potentials (LFPs), recorded from the prefrontal cortex of a macaque monkey, is considered. An adult macaque monkey is trained to perform a memory based saccade. The objective is to decode the eye movement goals from the LFP collected during a memory period. The LFP classification problem is modeled as that of classification of smooth functions embedded in Gaussian noise. It is then argued that using minimax function estimators as features would lead to consistent LFP classifiers. The theory of Gaussian sequence models allows us to represent minimax estimators as finite dimensional objects. The LFP classifier resulting from this mathematical endeavor is a spectrum based technique, where Fourier series coefficients of the LFP data, followed by appropriate shrinkage and thresholding, are used as features in a linear discriminant classifier. The classifier is then applied to the LFP data to achieve high decoding accuracy. The function classificati
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Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$module. By a result of Glauberman it has been known for a long time that if $G$ is nilpotent of class 2, then $G < V$. In this paper we generalize this result as follows. Assuming $G$ to be solvable, we show that the order of the maximal class 2 quotient of $G$ is strictly bounded above by $V$.
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Using the metric duality theory developed by Vaisala, we characterize generalized John domains in terms of higher dimensional homological bounded turning for its complement under mild assumptions. Simple examples indicate that our assumptions for such a characterization are optimal. Furthermore, we show that similar results in terms of higher dimensional homotopic bounded turning do not hold in three dimension.
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The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We investigate dual formulations for this approach and leverage these dual programs for developing new analysis tools for the JSR. We show that the dual of this convex problem searches for the occupations measures of trajectories with high asymptotic growth rate. We both show how to generate a sequence of guaranteed high asymptotic growth rate and how to detect cases where we can provide lower bounds to the JSR. We deduce from it a new guarantee for the upper bound provided by the sum of squares lyapunov program. We end this paper with a method to reduce the computation of the JSR of low rank matrices to th
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We revisit the construction of signature classes in C*algebra Ktheory, and develop a variation that allows us to prove equality of signature classes in some situations involving homotopy equivalences of noncompact manifolds that are only defined outside of a compact set. As an application, we prove a counterpart for signature classes of a codimension two vanishing theorem for the index of the Dirac operator on spin manifolds (the latter is due to Hanke, Pape and Schick, and is a development of wellknown work of Gromov and Lawson).
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We prove that every smoothly embedded surface in a 4manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized bridge trisection}, extends the authors' definition of bridge trisections for surfaces in $S^4$. Using this new construction, we give diagrammatic representations called \emph{shadow diagrams} for knotted surfaces in 4manifolds. We also provide a lowcomplexity classification for these structures and describe several examples, including the important case of complex curves inside $\mathbb{CP}^2$. Using these examples, we prove that there exist exotic 4manifolds with $(g,0)$trisections for certain values of $g$. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shad
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We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian curvature measure.
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We generalise the exponential AxSchanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by J. Kirby and C. Crampin we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised AxSchanuel inequalities are adequate for them.
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We study the signed Bernoulli convolution $$\nu_\beta^{(n)}=*_{j=1}^n \left (\frac12\delta_{\beta^{j}}\frac12\delta_{\beta^{j}}\right ),\ n\ge 1$$ where $\beta>1$ satisfies $$\beta^m=\beta^{m1}+\cdots+\beta+1$$ for some integer $m\ge 2$. When $m$ is odd, we show that the variation $\nu_\beta^{(n)}$ coincides the unsigned Bernoulli convolution $$\mu_\beta^{(n)}=*_{j=1}^n \left (\frac12\delta_{\beta^{j}}+\frac12\delta_{\beta^{j}}\right ).$$ When $m$ is even, we obtain the exact asymptotic of the total variation $\\nu_\beta^{(n)}\$ as $n\rightarrow\infty$.
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We study the compressible and incompressible twophase flows separated by a sharp interface with a phase transition and a surface tension. In particular, we consider the problem in $\mathbb{R}^N$, and the NavierStokesKorteweg equations is used in the upper domain and the NavierStokes equations is used in the lower domain. We prove the existence of $\mathcal{R}$bounded solution operator families for a resolvent problem arising from its model problem. According to Shibata \cite{GS2014}, the regularity of $\rho_+$ is $W^1_q$ in space, but to solve the kinetic equation: $\mathbf{u}_\Gamma\cdot\mathbf{n}_t = [[\rho\mathbf{u}]]\cdot\mathbf{n}_t /[[\rho]]$ on $\Gamma_t$ we need $W^{21/q}_q$ regularity of $\rho_+$ on $\Gamma_t$, which means the regularity loss. Since the regularity of $\rho_+$ dominated by the NavierStokesKorteweg equations is $W^3_q$ in space, we eliminate the problem by using the NavierStokesKorteweg equations instead of the compressible NavierStokes equations.
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In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include abelian varieties that have both prime dimension and nontrivial endomorphism ring. The proof is based on a crystalline analogue of Faltings' isogeny theorem due to Bost and the known cases of the MumfordTate conjecture.
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We analyse the convergence of numerical schemes in the GDMELLAM (Gradient Discretisation MethodEulerian Lagrangian Localised Adjoint Method) framework for a strongly coupled ellipticparabolic PDE which models miscible displacement in porous media. These schemes include, but are not limited to Mixed Finite ElementELLAM and Hybrid Mimetic MixedELLAM schemes. A complete convergence analysis is presented on the coupled model, using only weak regularity assumptions on the solution (which are satisfied in practical applications), and not relying on $L^\infty$ bounds (which are impossible to ensure at the discrete level given the anisotropic diffusion tensors and the general grids used in applications).
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A family of invariants of smooth, oriented fourdimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parameterised by a pivotal functor from a spherical fusion category into a ribbon fusion category. A state sum formula for the invariant is constructed via the chainmail procedure, so a large class of topological state sum models can be expressed as link invariants. Most prominently, the CraneYetter state sum over an arbitrary ribbon fusion category is recovered, including the nonmodular case. It is shown that the CraneYetter invariant for nonmodular categories is stronger than signature and Euler invariant. A special case is the fourdimensional untwisted DijkgraafWitten model. Derivations of state space dimensions of TQFTs arising from the state sum model agree with recent calculations of ground state degeneracies in WalkerWang models. Relations to different approaches to quantum gravity such as Cartan geome
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Generalized permutohedra are deformations of regular permutohedra, and arise in many different fields of mathematics. One important characterization of generalized permutohedra is the Submodular Theorem, which is related to the deformation cone of the Braid fan. We lay out general techniques for determining deformation cones of a fixed polytope and apply it to the Braid fan to obtain a natural combinatorial proof for the Submodular Theorem. We also consider a refinement of the Braid fan, called the nested Braid fan, and construct usual (respectively, generalized) nested permutohedra which have the nested Braid fan as (respectively, refining) their normal fan. We extend many results on generalized permutohedra to this new family of polytopes, including a onetoone correspondence between faces of nested permutohedra and chains in ordered partition posets, and a theorem analogous to the Submodular Theorem.
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Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $k$automorphisms defined as $h(x_g)=x_{hg}$ for any $g,h\in G$. We denote the fixed field $k(x_g : g\in G)^G$ by $k(G)$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is wellknown that if $\bC(G)$ is stably rational over $\bC$, then all the unramified cohomology groups $H_{\rm nr}^i(\bC(G),\bQ/\bZ)=0$ for $i \ge 2$. Hoshi, Kang and Kunyavskii [HKK] showed that, for a $p$group of order $p^5$ ($p$: an odd prime number), $H_{\rm nr}^2(\bC(G),\bQ/\bZ)\neq 0$ if and only if $G$ belongs to the isoclinism family $\Phi_{10}$. When $p$ is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY] exhibit some $p$groups $G$ which are of the form of a central extension of certain elementary abelian $p$group by another one with $H_{\rm nr}^2(\bC(G),\bQ/\bZ)= 0$ and $H_{\rm nr}^3(\bC(G),\bQ/\bZ) \neq 0$. However, it is difficult to tell whe
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The nonLeibniz formalism is introduced in this article. The formalism is based on the generalized differentiation operator (kappaoperator) with a nonzero Leibniz defect. The Leibniz defect of the introduced operator linearly depends on one scaling parameter. In a special case, if the Leibniz defect vanishes, the generalized differentiation operator reduces to the common differentiation operator. The kappaoperator allows the formulation of the variational principles and corresponding Lagrange and Hamiltonian equations. The solutions of some generalized dynamical equations are provided closed form.With a positive Leibniz defect the amplitude of free vibration remains constant with time with the fading frequency (<<red shift>>). The negative Leibniz defect leads the opposite behavior, demonstrating the growing frequency (<<blue shift>>). However, the Hamiltonian remains constant in time in both cases. Thus the introduction of nonzero Leibniz defect leads to an
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We construct a selfsimilar local regular Dirichlet form on the Sierpi\'nski gasket using $\Gamma$convergence of stablelike nonlocal closed forms. Such a Dirichlet form was constructed previously by Kigami \cite{Kig01}, but our construction has the advantage that it is a realization of a more general method of construction of a local regular Dirichlet form that works also on the Sierpi\'nski carpet \cite{GY17}. A direct consequence of this construction is the fact that the domain of the local Dirichlet form is some Besov space.
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We study the constrained minimum energy problem with an external field relative to the $\alpha$Riesz kernel $xy^{\alphan}$ of order $\alpha\in(0,n)$ for a condenser $\mathbf A=(A_i)_{i\in I}$ in $\mathbb R^n$, $n\geqslant 3$, whose oppositely charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint are both varied, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our arguments are particularly based on the simultaneous use of the vague topology and a suitable semimetric structure on a set of vector measures associated with $\mathbf A$, and the establishment of completeness theorems for proper semimetric spaces. The results obtained remain valid
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We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of an element in a generated group. While one form of connectivity essentially reflects the connectivity of the poset diagram, the other two are a bit more involved: Hurwitzconnectivity has its origins in algebraic geometry, and shellability in topology. We propose a framework to study these connectivity properties in a uniform way. Our main tool is a certain total order of the generators that is compatible with the chosen element.
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According to Kolmogorov complexity, every finite binary string is compressible to a shortest code  its information content  from which it is effectively recoverable. We investigate the extend to which this holds for infinite binary sequences (streams). We devise a coding method which uniformly codes every stream $X$ into an algorithmically stream $Y$, in such a way that the first $n$ bits of $X$ are recoverable from the first $I(X\upharpoonright_n)$ bits of $X$, where $I$ is any information content measure which is either computable or satisfies a certain property. As a consequence, if $g$ is any computable upper bound on the initial segment \pf complexity of $X$, then $X$ is computable from an algorithmically random $Y$ with oracleuse at most $g$.If no such computable upper bound is available, the oracleuse is bounded above by $K(X\upharpoonright_n)+K(n)$.
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In this paper we prove that for any simplicial set $B$, there is a Quillen equivalence between the covariant model structure on $\mathbf{S}/B$ and a certain localization of the projective model structure on the category of simplicial presheaves on the simplex category $\Delta/B$ of $B$. We extend this result to give a new Quillen equivalence between this covariant model structure and the projective model structure on the category of simplicial presheaves on the simplicial category $\mathfrak{C}[B]$. We study the relationship with Lurie's straightening theorem. Along the way we prove some results on localizations of simplicial categories and quasicategories.
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The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a hyperplane arrangement can be expressed in terms of its Tutte polynomial. We also show that, even if one is only interested in computing the Tutte polynomial of a graph or a matroid, the theory of hyperplane arrangements provides a powerful finite field method for this computation.
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We derive basic properties of TriebelLizorkinLorentz spaces important in the treatment of PDE. For instance, we prove TriebelLizorkinLorentz spaces to be of class $\mathcal{HT}$, to have property $(\alpha)$, and to admit a multiplier result of Mikhlin type. By utilizing these properties we prove the Laplace and the Stokes operator to admit a bounded $H^\infty$calculus. This is finally applied to derive local strong wellposedness for the NavierStokes equations on corresponding TriebelLizorkinLorentz ground spaces.
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Hawkes processes are a class of simple point processes that are selfexciting and have clustering effect, with wide applications in finance, social networks and many other fields. This paper considers a selfexciting Hawkes process where the baseline intensity is timedependent, the exciting function is a general function and the jump sizes of the intensity process are independent and identically distributed nonnegative random variables. This Hawkes model is nonMarkovian in general. We obtain closedform formulas for the Laplace transform, moments and the distribution of the Hawkes process. To illustrate the applications of our results, we use the Hawkes process to model the clustered arrival of trades in a dark pool and analyze various performance metrics including timetofirstfill, timetocompletefill and the expected fill rate of a resting dark order.
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We analyze the relation of the notion of a pluriLagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, for any Lagrangian system with $m$ commuting variational symmetries, one can construct a pluriLagrangian 1form in the $(m+1)$dimensional time, whose multitime EulerLagrange equations coincide with the original system supplied with $m$ commuting evolutionary flows corresponding to the variational symmetries. We also give a Hamiltonian counterpart of this construction, leading, for any system of commuting Hamiltonian flows, to a pluriLagrangian 1form with coefficients depending on functions in the phase space.
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We consider a coupled PDEODE system describing the motion of the rigid body in a container filled with the incompressible, viscous fluid. The fluid and the rigid body are coupled via Navier slip boundary condition. We prove that the local in time strong solution is unique in the larger class of weak solutions on the interval of its existence. This is the first weakstrong uniqueness result in the area of fluidstructure interaction.
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In this paper, we establish the boundedness of the commutators of multilinear Hausdorff operators on the product of some weighted MorreyHerz type spaces with variable exponent with their symbols belong to both Lipschitz space and central BMO space. By these, we generalize and strengthen some previous known results.
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Odoo is, according to Wikipedia, "the most popular open source ERP system." Thus, any survey of opensource accounting systems must certainly take a look in that direction. This episode in the ongoing search for a suitable accounting system for LWN examines the accounting features of Odoo; unfortunately, it comes up a bit short.
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We explore a web of connections between quantum entanglement and knot theory by examining how topological entanglement entropy probes the braiding data of quasiparticles in ChernSimons theory, mainly using $SU(2)$ gauge group as our working example. The problem of determining the Renyi entropy is mapped to computing the expectation value of an auxiliary Wilson loop in $S^3$ for each braid. We study various properties of this auxiliary Wilson loop for some 2strand and 3strand braids, and demonstrate how they reflect some geometrical properties of the underlying braids.
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While the adoption of OpenPGP by the general population is marginal at best, it is a critical component for the security community and particularly for Linux distributions. For example, every package uploaded into Debian is verified by the central repository using the maintainer's OpenPGP keys and the repository itself is, in turn, signed using a separate key. If upstream packages also use such signatures, this creates a complete trust path from the original upstream developer to users. Beyond that, pull requests for the Linux kernel are verified using signatures as well. Therefore, the stakes are high: a compromise of the release key, or even of a single maintainer's key, could enable devastating attacks against many machines.
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Jens Axboe is the maintainer of the block layer of the kernel. In this capacity, he spoke at Kernel Recipes 2017 on what's new in the storage world for Linux, with a particular focus on the new blockmultiqueue subsystem: the degree to which it's been adopted, a number of optimizations that have recently been made, and a bit of speculation about how it will further improve in the future. Subscribers can click below for a report from the Kernel Recipes talk by guest author Tom Yates.
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