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This paper generalizes the borderedalgebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe effective computations of the resulting invariant.
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In this work, we derive upper bounds on the cardinality of tandem and palindromic duplication correcting codes by deriving the generalized sphere packing bound for these error types. We first prove that an upper bound for tandem or palindromic deletions is also an upper bound for inserting the respective type of duplications. Therefore, we derive the bounds based on these special deletions as this results in tighter bounds. We determine the spheres for tandem and palindromic duplications/deletions and the number of words with a specific sphere size. Our upper bounds on the cardinality directly imply lower bounds on the redundancy which we compare with the redundancy of the best known construction correcting arbitrary burst errors. Our results indicate that the correction of palindromic duplications requires more redundancy than the correction of tandem duplications. Further, there is a significant gap between the minimum redundancy of duplication correcting codes and burst insertion co
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Given a compact Riemann surface $X$ and a complex reductive Lie group $G$ equipped with real structures, we define antiholomorphic involutions on the moduli space of $G$Higgs bundles over $X$. We investigate how the various components of the fixed point locus match up, as one passes from $G$ to its Langlands dual $^LG$. Special attention is given to the case $G=SL(2,\C)$ and $^LG=PGL(2,\C)$.
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In this paper, we extend the geometric concept of linearity to betweenness structures, suitable combinatorial abstractions of finite metric spaces. As our main result, we prove that every linear betweenness structure is induced by either a path (of length $n$) or a cycle of length 4.
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A Riemannian symmetric space is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. On such spaces, we introduce the notion of a distributional lattice, generalizing the notion of lattice. Distributional lattices exist in any Riemannian symmetric space: specifically the Voronoi tessellation of an stationary Poisson point process is an example. We show that for an appropriate notion of amenability, the amenability of a distributional lattice is equivalent to the amenability of the ambient space. Using this equivalence, we show that the simple random walk on any distributional lattice has positive embedded speed. For nonpositively curved, simply connected spaces, we show that the simple random walk on a PoissonVoronoi tessellation has positive graph speed by developing some additional structure for PoissonVoronoi tessellations.
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In this paper, distributed convex optimization problem over nondirected dynamical networks is studied. Here, networked agents with singleintegrator dynamics are supposed to rendezvous at a point that is the solution of a global convex optimization problem with some local inequality constraints. To this end, all agents shall cooperate with their neighbors to seek the optimum point of the networks global objective function. A distributed optimization algorithm based on the interiorpoint method is proposed, which combines an optimization algorithm with a nonlinear consensus protocol to find the optimum value of the global objective function. We tackle this problem by addressing its subproblems, namely a consensus problem and a convex optimization problem. Firstly, we propose a saturation protocol for the consensus subproblem. Then to solve the distributed optimization part, we implement a centralized control law, which yields the optimum value of the global objective function, in a d
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We study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the $SL(2,\mathbb{C})$ version of Casson invariants. As applications, we discuss nonexistence for chirally cosmetic surgeries on almost positive knots, and we give a complete classification of chirally cosmetic surgeries on two bridge knots of genus one.
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We show that, on a complete and possibly noncompact Riemannian manifold of dimension at least 2 without close conjugate points at infinity, the existence of a closed geodesic with local homology in maximal degree and maximal index growth under iteration forces the existence of infinitely many closed geodesics. For closed manifolds, this was a theorem due to Hingston.
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The rise of machinetomachine communications has rekindled the interest in random access protocols as a support for a massive number of uncoordinatedly transmitting devices. The legacy ALOHA approach is developed under a collision model, where slots containing collided packets are considered as waste. However, if the common receiver (e.g., base station) is capable to store the collision slots and use them in a transmission recovery process based on successive interference cancellation, the design space for access protocols is radically expanded. We present the paradigm of coded random access, in which the structure of the access protocol can be mapped to a structure of an erasurecorrecting code defined on graph. This opens the possibility to use coding theory and tools for designing efficient random access protocols, offering markedly better performance than ALOHA. Several instances of coded random access protocols are described, as well as a case study on how to upgrade a legacy ALO
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In this article, we propose new Bayesian methods for selecting and estimating a sparse coefficient vector for skewed heteroscedastic response. Our novel Bayesian procedures effectively estimate the median and other quantile functions, accommodate nonlocal prior for regression effects without compromising ease of implementation via sampling based tools, and asymptotically select the true set of predictors even when the number of covariates increases in the same order of the sample size. We also extend our method to deal with some observations with very large errors. Via simulation studies and a reanalysis of a medical cost study with large number of potential predictors, we illustrate the ease of implementation and other practical advantages of our approach compared to existing methods for such studies.
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Our aim is to establish relations between Drazin inverses of the pesudoblock matrix $(P,Q,R,S)$ and the block matrix composed of $P,R,S,Q$, where $R^2=S^2=0$. Based on the relations, we give representations for Drazin inverses of the sum $P+Q+R+S$ under weaker restrictions. As its applications, several expressions for Drazin inverses of a $2\times 2$ block matrix are presented under some assumptions. Our results generalize several results in the literature.
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The correlations of multiple renewable power plants (RPPs) should be fully considered in the power system with very high penetration renewable power integration. This paper models the uncertainties, spatial correlation of multiple RPPs based on Copula theory and actual probability historical histograms by onedimension distributions for economic dispatch (ED) problem. An efficient dynamic renewable power scenario generation method based on Gibbs sampling is proposed to generate renewable power scenarios considering the uncertainties, spatial correlation and variability (temporal correlation) of multiple RPPs, in which the sampling space complexity do not increase with the number of RPPs. Distributionbased and scenariobased methods are proposed and compared to solve the realtime ED problem with multiple RPPs. Results show that the proposed dynamic scenario generation method is much more consist with the actual renewable power. The proposed ED methods show better understanding for the
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This document briefly describes the BlackBox MultiObjective Optimization Benchmarking (BMOBench) platform. It presents the test problems, evaluation procedure, and experimental setup. To this end, the BMOBench is demonstrated by comparing recent multiobjective solvers from the literature, namely SMSEMOA, DMS, and MOSOO.
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In dynamic imaging, a key challenge is to reconstruct image sequences with high temporal resolution from strong undersampling projections due to a relatively slow data acquisition speed. In this paper, we propose a variational model using the infimal convolution of Bregman distance with respect to total variation to model edge dependence of sequential frames. The proposed model is solved via an alternating iterative scheme, for which each subproblem is convex and can be solved by existing algorithms. The proposed model is formulated under both Gaussian and Poisson noise assumption and the simulation on two sets of dynamic images shows the advantage of the proposed method compared to previous methods.
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A mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element edge functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension, in order to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl. This allows for the exact conservation of first order moments (mass, vorticity), as well as quadratic moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exac
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Every partial colouring of a Hamming graph is uniquely related to a partial Latin hyperrectangle. In this paper we introduce the $\Theta$stabilized $(a,b)$colouring game for Hamming graphs, a variant of the $(a,b)$colouring game so that each move must respect a given autotopism $\Theta$ of the resulting partial Latin hyperrectangle. We examine the complexity of this variant by means of its chromatic number. We focus in particular on the bidimensional case, for which the game is played on the Cartesian product of two complete graphs, and also on the hypercube case.
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In this paper, convergence of a characteristicsbased hybrid method recently introduced in Daripa & Dutta (J. Comput. Phys., 335:249282, 2017) has been proved. This method which combines a discontinuous finite element method and a modified method of characteristics (MMOC) has been successfully applied to solve a coupled, nonlinear system of elliptic and transport equations that arise in multicomponent twophase porous media flows. The novelty in this paper is the convergence analysis of the MMOC procedure for a nonlinear system of transport equations. For this purpose, an analogous singlecomponent system of transport equations has been considered and possible extension to multicomponent systems has been discussed. Error estimates have been obtained and these estimates have also been validated by realistic numerical simulations of flows arising in enhanced oil recovery processes.
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That a monotone, stable, and consistent scheme converges to the viscosity solution of a fully nonlinear second order local equation satisfying a comparison principle is a seminal result in the viscosity theory. We extend these ideas in a very general manner to weakly nonlocal equations and apply our results to obtain convergent schemes for finite and infinitehorizon equations arising from impulse control, including a new "stochastic semiLagrangian" scheme that is fully explicit, unconditionally stable, trivially monotone in higher dimensions, and embarrassingly parallel.
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Convolutional dictionary learning (CDL or sparsifying CDL) has many applications in image processing and computer vision. There has been growing interest in developing efficient algorithms for CDL, mostly relying on the augmented Lagrangian (AL) method or the variant alternating direction method of multipliers (ADMM). When their parameters are properly tuned, AL methods have shown fast convergence in CDL. However, the parameter tuning process is not trivial due to its data dependence and, in practice, the convergence of AL methods depends on the AL parameters for nonconvex CDL problems. To moderate these problems, this paper proposes a new practically feasible and convergent Block Proximal Gradient method using a Majorizer (BPGM) for CDL. The BPGMbased CDL is investigated with different block updating schemes and majorization matrix designs, and further accelerated by incorporating some momentum coefficient formulas and restarting techniques. All of the methods investigated incorpor
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In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is $\frak{g} = A \otimes \frak{k}$, where $\frak{k}$ is a compact simple Lie superalgebra and $A$ is a supercommutative associative (super)algebra; the crucial case is when $A = \Lambda_s(\mathbb{R})$ is a Gra\ss{}mann algebra. Since we are interested in projective representations, the first step consists of determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if $\frak{k}$ is a simple compact Lie superalgebra with $\frak{k}_1\neq \{0\}$, then each (projective) unitary representation of $\Lambda_s(\mathbb{R})\otimes \frak{k}$ factors through a (projective) unitary representation of $\frak{k}$ itself, and these are known by Jakobsen's classification. If $\frak{k}_1 = \{0\}$, then we likewise reduce the classification problem to semidirect products of compact Lie groups $K$ with a CliffordLie su
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We introduce three new cut tree structures of graphs $G$ in which the vertex set of the tree is a partition of $V(G)$ and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup \cite{Kawarabayashi2015a} presented the first deterministic nearlinear edgeconnectivity recognition algorithm. A crucial step in this algorithm uses the existence of vertex subsets of a simple graph $G$ whose contractions leave a graph with $\tilde{O}(n/\delta)$ vertices and $\tilde{O}(n)$ edges ($n := V(G)$) such that all nontrivial mincuts of $G$ are preserved. We improve this result by eliminating the polylogarithmic factors, that is, we show a contractionbased sparsification that leaves $O(n/\delta)$ vertices and $O(n)$ edges and preserves all nontrivial mincuts. We complement this result by giving a sparsification that leaves $O(n/\delta)$ vertices and $O(n)$ edges such that all (possibly not minimum) cuts of size l
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We study the behavior of the spectrum of the Dirac operator together with a symmetric $W^{1, \infty}$potential on spin manifolds under a collapse of codimension one with bounded sectional curvature and diameter. If there is an induced spin structure on the limit space $N$ then there are convergent eigenvalues which converge to the spectrum of a first order differential operator $D$ on $N$ together with a symmetric $W^{1,\infty}$potential. If the dimension of the limit space is even then $D$ is the Dirac operator $D^N$ on $N$ and if the dimension of the limit space is odd, then $D = D^N \oplus D^N$.
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Recently, deep learning approaches have achieved significant performance improvement in various imaging problems. However, it is still unclear why these deep learning architectures work. Moreover, the link between the deep learning and the classical signal processing approaches such as wavelet, nonlocal processing, compressed sensing, etc, is still not well understood, which often makes signal processors in deep troubles. To address these issues, here we show that the longsearchedfor missing link is the convolutional framelets for representing a signal by convolving local and nonlocal bases. The convolutional framelets was originally developed to generalize the recent theory of lowrank Hankel matrix approaches, and this paper significantly extends the idea to derive a deep neural network using multilayer convolutional framelets with perfect reconstruction (PR) under rectified linear unit (ReLU). Our analysis also shows that the popular deep network components such as residual blo
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In this paper, we continue our understanding of the stable process from the perspective of the theory of selfsimilar Markov processes in the spirit of the recent papers of Kyprianou (2016) and Kyprianou et al. (2017). In particular, we turn our attention to the case of $d$dimensional isotropic stable process, for $d\geq 2$. Using a completely new approach we consider the distribution of the point of closest reach. This leads us to a number of other substantial new results for this class of stable processes. We engage with a new radial excursion theory, never before used, from which we develop the classical BlumenthalGetoorRay identities for first entry/exit into a ball, cf. Blumenthal et al. (1961), to the setting of $n$tuple laws. We identify explicitly the stationary distribution of the stable process when reflected in its running radial supremum. Moreover, we provide a representation of the WienerHopf factorisation of the MAP that underlies the stable process through the La
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Let p be a prime. For any finite pgroup G, the deep transfers T(H,G'):H/H' > G'/G'' from the maximal subgroups H of index (G:H)=p in G to the derived subgroup G' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels kappa_d(G)=(ker(T(H,G')))_{(G:H)=p}. For all finite 3groups G of coclass cc(G)=1, the family kappa_d(G) is determined explicitly. The results are applied to the Galois groups G=Gal(F_3^\infty/F) of the Hilbert 3class towers of all real quadratic fields F=Q(d^1/2) with fundamental discriminants d>1, 3class group Cl_3(F)~C_3*C_3, and total 3principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1<d<5*10^6, and a few exceptional cases are pointed out for 1<d<64*10^6.
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The exact recovery property of Basis pursuit (BP) and Orthogonal Matching Pursuit (OMP) has a relation with the coherence of the underlying frame. A frame with low coherence provides better guarantees for exact recovery. In particular, Incoherent Unit Norm Tight Frames (IUNTFs) play a significant role in sparse representations. IUNTFs with special structure, in particular those given by a union of several orthonormal bases, are known to satisfy better theoretical guarantees for recovering sparse signals. In the present work, we propose to construct structured IUNTFs consisting of large number of orthonormal bases. For a given $r, k, m$ with $k$ being less than or equal to the smallest prime power factor of $m$ and $r<k,$ we construct a CS matrix of size $mk \times (mk\times m^{r})$ with coherence at most $\frac{r}{k},$ which consists of $m^{r}$ number of orthonormal bases and with density $\frac{1}{m}$. We also present numerical results of recovery performance of union of orthonorma
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In this paper we focus on the finitedimensional approximation of quasistatic evolutions of critical points of the phasefield model of brittle fracture. In a space discretized setting, we first discuss an alternating minimization scheme which, together with the usual timediscretization procedure, allows us to construct such finitedimensional evolutions. Then, passing to the limit as the space discretization becomes finer and finer, we prove that any limit of a sequence of finitedimensional evolutions is itself a quasistatic evolution of the phasefield model of fracture. In particular, our proof shows for the first time the consistency of numerical schemes related to the study of fracture mechanics and image processing.
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Given two distinct newforms with real Fourier coefficients, we show that the set of primes where the Hecke eigenvalues of one of them dominate the Hecke eigenvalues of the other has density at least 1/16. Furthermore, if the two newforms do not have complex multiplication, and neither is a quadratic twist of the other, we also prove a similar result for the squares of their Hecke eigenvalues.
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The introduction of a fractional differential operator defined in terms of the RiemannLiouville derivative makes it possible to generalize the kinetic equations used to model relaxation in dielectrics. In this context such fractional equations are called fractional kinetic relaxation equations and their solutions, called fractional kinetic relaxation functions, are given in terms of MittagLeffler functions. These fractional kinetic relaxation functions generalize the kinetic relaxation functions associated with the Debye, ColeCole, ColeDavidson and HavriliakNegami models, as the latter functions become particular cases of the fractional solutions, obtained for specific values of the parameter specifying the order of the derivative. The aim of this work is to analyse the behavior of these fractional functions in the time variable. As theoretical tools we use the theorem by Bernstein on the complete monotonicity of functions together with Titchmarsh's inversion formula. The last par
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A method is proposed for solving equality constrained nonlinear optimization problems involving twice continuously differentiable functions. The method employs a trust funnel approach consisting of two phases: a first phase to locate an $\epsilon$feasible point and a second phase to seek optimality while maintaining at least $\epsilon$feasibility. A twophase approach of this kind based on a cubic regularization methodology was recently proposed along with a supporting worstcase iteration complexity analysis. Unfortunately, however, in that approach, the objective function is completely ignored in the first phase when $\epsilon$feasibility is sought. The main contribution of the method proposed in this paper is that the same worstcase iteration complexity is achieved, but with a first phase that also accounts for improvements in the objective function. As such, the method typically requires fewer iterations in the second phase, as the results of numerical experiments demonstrate.
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In the paper we study the models of timechanged Poisson and Skellamtype processes, where the role of time is played by compound PoissonGamma subordinators and their inverse (or first passage time) processes. We obtain explicitly the probability distributions of considered timechanged processes and discuss their properties.
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We consider the remote vector source coding problem in which a vector Gaussian source is to be estimated from noisy linear measurements. For this problem, we derive the performance of the compressandestimate (CE) coding scheme and compare it to the optimal performance. In the CE coding scheme, the remote encoder compresses the noisy source observations so as to minimize the local distortion measure, independent from the joint distribution between the source and the observations. In reconstruction, the decoder estimates the original source realization from the lossycompressed noisy observations. For the CE coding in the Gaussian vector case, we show that, if the code rate is less than a threshold, then the CE coding scheme attains the same performance as the optimal coding scheme. We also introduce lower and upper bounds for the performance gap above this threshold. In addition, an example with two observations and two sources is studied to illustrate the behavior of the performance
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Massive MTC support is an important future market segment, but not yet efficiently supported in cellular systems. In this paper we followup on recent concepts combining advanced MAC protocols with Compressed Sensing (CS) based multiuser detection. Specifically, we introduce a concept for sparse joint activity, channel and data detection in the context of the Coded ALOHA (FDMA) protocol. We will argue that a simple sparse activity and data detection is not sufficient (as many papers do) because control resources are in the order of the data. In addition, we will improve on the performance of such protocols in terms of the reduction of resources required for the user activity, channel estimation and data detection. We will mathematically analyze the system accordingly and provide expressions for the capture probabilities of the underlying sparse multiuser detector. Finally, we will provide structured CS algorithms for the joint estimation scheme and evaluate its performance.
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The computation of the tropical prevariety is the first step in the application of polyhedral methods to compute positive dimensional solution sets of polynomial systems. In particular, pretropisms are candidate leading exponents for the power series developments of the solutions. The computation of the power series may start as soon as one pretropism is available, so our parallel computation of the tropical prevariety has an application in a pipelined solver. We present a parallel implementation of dynamic enumeration. Our first distributed memory implementation with forked processes achieved good speedups, but quite often resulted in large variations in the execution times of the processes. The shared memory multithreaded version applies work stealing to reduce the variability of the run time. Our implementation applies the thread safe Parma Polyhedral Library (PPL), in exact arithmetic with the GNU Multiprecision Arithmetic Library (GMP), aided by the fast memory allocations of TCMa
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We study the sets that are computable from both halves of some (MartinL\"of) random sequence, which we call \emph{$1/2$bases}. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e.\ elements. It is a proper subideal of the $K$trivial sets. We characterise $1/2$bases as the sets computable from both halves of Chaitin's $\Omega$, and as the sets that obey the cost function $\mathbf c(x,s) = \sqrt{\Omega_s  \Omega_x}$. Generalising these results yields a dense hierarchy of subideals in the $K$trivial degrees: For $k< n$, let $B_{k/n}$ be the collection of sets that are below any $k$ out of $n$ columns of some random sequence. As before, this is an ideal generated by its c.e.\ elements and the random sequence in the definition can always be taken to be $\Omega$. Furthermore, the corresponding cost function characterisation reveals that $B_{k/n}$ is independent of the particular representation of the rational $k/n$, and that $B_
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L.J. Savage once hoped to show that "the superficially incompatible systems of ideas associated on the one hand with [subjective Bayesianism] and on the other hand with [classical statistics]...lend each other mutual support and clarification." By 1972, however, he had largely "lost faith in the devices" of classical statistics. One aspect of those "devices" that he found objectionable is that differences among the "stopping rules" that are used to decide when to end an experiment which are "noninformative" from a Bayesian perspective can affect decisions made using a classical approach. Two experiments that produce the same data using different stopping rules seem to differ only in the intentions of the experimenters regarding whether or not they would have carried on if the data had been different, which seem irrelevant to the evidential import of the data and thus to facts about what actions the data warrant. I argue that classical and Bayesian ideas about stopping rules do in fact
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schwit1 shares an article from ZDNet: A new analysis of documents leaked by whistleblower Edward Snowden details a highly classified technique that allows the National Security Agency to "deliberately divert" U.S. internet traffic, normally safeguarded by constitutional protections, overseas in order to conduct unrestrained data collection on Americans. According to the new analysis, the NSA has clandestine means of "diverting portions of the river of internet traffic that travels on global communications cables," which allows it to bypass protections put into place by Congress to prevent domestic surveillance on Americans. The new findings follow a 2014 paper by researchers Axel Arnbak and Sharon Goldberg, published on sistersite CBS News, which theorized that the NSA, whose job it is to produce intelligence from overseas targets, was using a "traffic shaping" technique to route US internet data overseas so that it could be incidentally collected under the authority of a largely unk
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As trust and familiarity with sharing economy services continues to grow, so too will the number of users, estimates marketing research firm eMarketer. From the article: Over a quarter (26.0%) of US adult internet users  or 56.5 million people  will use a sharing economy service at least once in 2017. This is a higher figure than previously projected due to strongerthanexpected uptake of both ride and homesharing services. This year, 16.9% of US adult internet users are expected to use their Airbnb account at least once, equating to 36.8 million people.
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"If you don't feel rejuvenated and keen to face Monday after two workfree days, there might be a reason: You're doing your weekend wrong," an anonymous reader writes, citing a Quartz article. From the article: According to University of Calgary sociologist Robert Stebbins, most leisure falls into two categories: casual and serious. Casual leisure pursuits are short lived, immediately gratifying, and often passive; they include activities like drinking, online shopping, and bingewatching. These diversions provide instant hedonic pleasure  quite literally, actually, as all these pastimes cause the brain to release dopamine and provide instant soothing comfort. In a culture where many people exist all week in an ampedup, overworked state, casual weekend leisure easily becomes the default for quick decompression. But serious leisure is a far more beneficial pursuit. Serious leisure activities provide deeper fulfillment, and  to invoke a fuzzy '70s word  "selfactualization." Self
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An anonymous reader writes: Friday technology investor Dave McClure tweeted a link to a statement from the new CEO of the startup incubator he cofounded which announces his demotion after engaging in "inappropriate interactions with women in the tech community." The new CEO of 500 Startups says "I sincerely apologize for the choices he made and the pain and stress they've caused people. But apologies aren't enough without meaningful actions and change. Because of this, we made the decision a few months ago to change the leadership structure at 500." Meanwhile, McClure will attend counseling "to work on changing his perspectives and preventing his previous unacceptable behavior... As much as we want to be part of the solution, we clearly have also been part of the problem." The same day more than two dozen female entrepreneurs told the New York Times about incidents of sexual harassment in the startup industry, "often providing corroborating messages and emails." Several women told t
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