Solidot 公告
请在发布文章时用HTML代码加上至少一条新闻来源的链接；原创性消息，可加入相关信息（如涉及公司的网址）的链接。有任何问题，邮件至：he.fang#zhiding.cn
ken：feigaobox@gmail.com
注意：收到邮件乱码的用户请修改客户端的默认字体编码，从"简体中文（GB2312）"修改为"Unicode（UTF8）"。
投 票
信息流

Advances in mobile computing technologies have made it possible to monitor and apply datadriven interventions across complex systems in real time. Markov decision processes (MDPs) are the primary model for sequential decision problems with a large or indefinite time horizon. Choosing a representation of the underlying decision process that is both Markov and lowdimensional is nontrivial. We propose a method for constructing a lowdimensional representation of the original decision process for which: 1. the MDP model holds; 2. a decision strategy that maximizes mean utility when applied to the lowdimensional representation also maximizes mean utility when applied to the original process. We use a deep neural network to define a class of potential process representations and estimate the process of lowest dimension within this class. The method is illustrated using data from a mobile study on heavy drinking and smoking among college students.
收起

Let $\mathcal{B}$ denote a set of bicolorings of $[n]$, where each bicoloring is a mapping of the points in $[n]$ to $\{1,+1\}$. For each $B \in \mathcal{B}$, let $Y_B=(B(1),\ldots,B(n))$. For each $A \subseteq [n]$, let $X_A \in \{0,1\}^n$ denote the incidence vector of $A$. A nonempty set $A$ is said to be an `unbiased representative' for a bicoloring $B \in \mathcal{B}$ if $\left\langle X_A,Y_B\right\rangle =0$. Given a set $\mathcal{B}$ of bicolorings, we study the minimum cardinality of a family $\mathcal{A}$ consisting of subsets of $[n]$ such that every bicoloring in $\mathcal{B}$ has an unbiased representative in $\mathcal{A}$.
收起

We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a $3$dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined spacetime paraproducts on parabolic H\"older spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.
收起

The BHK interpretation interprets propositional statements as descriptions of the world of proofs; a world which is hierarchical in nature. It consists of different layers of the concept of proof; the proofs, the proofs about proofs and so on. To describe this hierarchical world, one approach is the Russellian approach in which we use a typed language to reflect this hierarchical nature in the syntax level. In this case, since the connective responsible for this hierarchical behavior is implication, we will use a typed language equipped with a hierarchy of implications, $\{\rightarrow_n\}_{n=0}^{\infty}$. In fact, using this typed propositional language, we will introduce the hierarchical counterparts of the logics $\mathbf{BPC}$, $\mathbf{EBPC}$, $\mathbf{IPC}$ and $\mathbf{FPL}$ and then by proving their corresponding soundnesscompleteness theorems with respect to their natural BHK interpretations, we will show how these different logics describe different worlds of proofs embodying
收起

Let $Q$ be a nondegenerate quadratic form on a vector space $V$ of even dimension $n$ over a number field $F$. Via the circle method or automorphic methods one can give good estimates for smoothed sums over the number of zeros of the quadratic form whose coordinates are of size at most $X$ (properly interpreted). For example, when $F=\mathbb{Q}$ and $\dim V>4$ HeathBrown has given an asymptotic of the form \begin{align} \label{HB:esti} c_1X^{n2}+O_{Q,\varepsilon,f}(X^{n/2+\varepsilon}) \end{align} for any $\varepsilon>0$. Here $c_1 \in \mathbb{C}$ and $f \in \mathcal{S}(V(\mathbb{R}))$ is a smoothing function. We refine HeathBrown's work to give an asymptotic of the form $$ c_1X^{n2}+c_2X^{n/2}+O_{Q,\varepsilon,f}(X^{n/2+\varepsilon1}) $$ over any number field. Here $c_2 \in \mathbb{C}$. Interestingly the secondary term $c_2$ is the sum of a rapidly decreasing function on $V(\mathbb{R})$ over the zeros of $Q^{\vee}$, the form whose matrix is inverse to the matrix of $Q$. We
收起

In this paper, we investigate secure transmission over the largescale multipleantenna wiretap channel with finite alphabet inputs. First, we investigate the case where instantaneous channel state information (CSI) of the eavesdropper is known at the transmitter. We show analytically that a generalized singular value decomposition (GSVD) based design, which is optimal for Gaussian inputs, may exhibit a severe performance loss for finite alphabet inputs in the high signaltonoise ratio (SNR) regime. In light of this, we propose a novel PerGroupGSVD (PGGSVD) design which can effectively compensate the performance loss caused by the GSVD design. More importantly, the computational complexity of the PGGSVD design is by orders of magnitude lower than that of the existing design for finite alphabet inputs in [1] while the resulting performance loss is minimal. Then, we extend the PGGSVD design to the case where only statistical CSI of the eavesdropper is available at the transmitter.
收起

Security updates have been issued by Debian (activemq, libav, minicom, mysql5.5, tiff3, and xen), Fedora (ansible, collectd, icu, and pcre), openSUSE (chromium and firefox), Red Hat (chromiumbrowser and kernel), Slackware (firefox), and Ubuntu (kernel, linux, linuxaws, linuxgke, linuxraspi2, linuxsnapdragon, linux, linuxraspi2, linuxhwe, linuxltstrusty, linuxltsxenial, qemu, and samba).
收起

The existence of a countably compact group without nontrivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without nontrivial convergent sequences having the following "selective" compactness property: For each free ultrafilter p on N and every sequence {U_n:n in N} of nonempty open subsets of G one can choose a point x_n in U_n for all n in such a way that the resulting sequence {x_n:n in N} has a plimit in G, that is, {n in N: x_n in V} belongs to p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of DorantesAldama and the first author. As a byproduct, we show that the free precompact Boolean group over any disjoint sum of maximal countable spaces contains no infinite compact subsets.
收起

This paper is concerned with semiconcavity of viscosity solutions for a class of degenerate elliptic integrodifferential equations in $\mathbb R^n$. This class of equations includes Bellman equations containing operators of L\'evyIt\^o type. H\"{o}lder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integrodifferential equations are also provided.
收起

Principal component analysis (PCA) is a fundamental dimension reduction tool in statistics and machine learning. For large and highdimensional data, computing the PCA (i.e., the singular vectors corresponding to a number of dominant singular values of the data matrix) becomes a challenging task. In this work, a singlepass randomized algorithm is proposed to compute PCA with only one pass over the data. It is suitable for processing extremely large and highdimensional data stored in slow memory (hard disk) or the data generated in a streaming fashion. Experiments with synthetic and real data validate the algorithm's accuracy, which has orders of magnitude smaller error than an existing singlepass algorithm. For a set of highdimensional data stored as a 150 GB file, the proposed algorithm is able to compute the first 50 principal components in just 24 minutes on a typical 24core computer, with less than 1 GB memory cost.
收起

Let $H$ be a subgroup of $\pi_{1}(X,x_{0})$. In this paper, we extend the concept of $X$ being SLT space to $H$SLT space at $x_0$. First, we show that the fibers of the endpoint projection $p_{H}:\tilde{X}_{H}\rightarrow X$ are topological group when $X$ is $H$SLT space at $x_0$ and $H$ is a normal subgroup. Also, we show that under these conditions the concepts of homotopically path Hausdorff relative to $H$ and homotopically Hausdorff relative to $H$ coincide. Moreover, among other things, we show that the endpoint projection map $p_{H}$ has the unique path lifting property if and only if $H$ is a closed normal subgroup of $\pi_{1}^{qtop}(X,x_{0})$ when $X$ is SLT at $x_{0}$. Second, we present conditions under which the whisker topology is agree with the quotient of compactopen topology on $\tilde{X}_{H}$. Also, we study the relationship between open subsets of $\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{qtop}(X,x_{0})$.
收起

We consider pure SU(2) YangMills theory on fourdimensional de Sitter space dS$_4$ and construct a smooth and spatially homogeneous magnetic solution to the YangMills equations. Slicing dS$_4$ as ${\mathbb R}\times S^3$, via an SU(2)equivariant ansatz we reduce the YangMills equations to ordinary matrix differential equations and further to Newtonian dynamics in a doublewell potential. Its local maximum yields a YangMills solution whose colormagnetic field at time $\tau\in{\mathbb R}$ is given by $\tilde{B}_a=\frac12 I_a/(R^2\cosh^2\!\tau)$, where $I_a$ for $a=1,2,3$ are the SU(2) generators and $R$ is the de Sitter radius. At any moment, this spatially homogeneous configuration has finite energy, but its action is also finite and of the value $\frac34\pi^3$. Similarly, the doublewell bounce produces a family of homogeneous finiteaction electricmagnetic solutions with the same energy. There is a continuum of other solutions whose energy and action extend down to zero.
收起

We give a unified approach to constructing quaternary sequences with even period and low autocorrelation from pairs of binary sequences with even period and optimal autocorrelation. We obtain several families of balanced and almost balanced quaternary sequences with low autocorrelation using this approach, as well as investigate their linear complexity.
收起

The Ehrhart polynomial and the reciprocity theorems by Ehrhart \& Macdonald are extended to tensor valuations on lattice polytopes. A complete classification is established of tensor valuations of rank up to eight that are equivariant with respect to the special linear group over the integers and translation covariant. Every such valuation is a linear combination of the Ehrhart tensors which is shown to no longer hold true for rank nine.
收起

Deflation techniques for Krylov subspace methods have seen a lot of attention in recent years. They provide means to improve the convergence speed of these methods by enriching the Krylov subspace with a deflation subspace. The most common approach for the construction of deflation subspaces is to use (approximate) eigenvectors, but also more general subspaces are applicable. In this paper we discuss two results concerning the accuracy requirements within the deflated CG method. First we show that the effective condition number which bounds the convergence rate of the deflated conjugate gradient method depends asymptotically linearly on the size of the perturbations in the deflation subspace. Second, we discuss the accuracy required in calculating the deflating projection. This is crucial concerning the overall convergence of the method, and also allows to save some computational work. To show these results, we use the fact that as a projection approach deflation has many similarities
收起

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of halfintegral weight not divisible by some prime $\ell$. Among the applications of this we show that there are $\gg \sqrt{X}/\log \log X$ integers $n \leq X$ for which the partition function $p(n)$ is not divisible by $\ell$, and that there are $\gg \sqrt{X}/\log \log X$ values of $n \leq X$ for which $c(n)$, the $n$th Fourier coefficient of the $j$invariant, is odd.
收起

A causal scenario is a graph that describes the cause and effect relationships between all relevant variables in an experiment. A scenario is deemed `not interesting' if there is no deviceindependent way to distinguish the predictions of classical physics from any generalised probabilistic theory (including quantum mechanics). Conversely, an interesting scenario is one in which there exists a gap between the predictions of different operational probabilistic theories, as occurs for example in Belltype experiments. Henson, Lal and Pusey (HLP) recently proposed a sufficient condition for a causal scenario to not be interesting. In this paper we supplement their analysis with some new techniques and results. We first show that existing graphical techniques due to Evans can be used to confirm by inspection that many graphs are interesting without having to explicitly search for inequality violations. For three exceptional cases  the graphs numbered 15,16,20 in HLP  we show that there
收起

Wireless surveillance is becoming increasingly important to protect the public security by legitimately eavesdropping suspicious wireless communications. This paper studies the wireless surveillance of a twohop suspicious communication link by a halfduplex legitimate monitor. By exploring the suspicious link's twohop nature, the monitor can adaptively choose among the following three eavesdropping modes to improve the eavesdropping performance: (I) \emph{passive eavesdropping} to intercept both hops to decode the message collectively, (II) \emph{proactive eavesdropping} via {\emph{noise jamming}} over the first hop, and (III) \emph{proactive eavesdropping} via {\emph{hybrid jamming}} over the second hop. In both proactive eavesdropping modes, the (noise/hybrid) jamming over one hop is for the purpose of reducing the endtoend communication rate of the suspicious link and accordingly making the interception more easily over the other hop. Under this setup, we maximize the eavesdropp
收起

In this article we describe the life and work of Wolf Barth who died on 30th December 2016. Wolf Barth's contributions to algebraic variety span a wide range of subjects. His achievements range from what is now called the BarthLefschetz theorems to his fundamental contributions to the theory of algebraic surfaces and moduli of vector bundles, and include his later work on algebraic surfaces with many singularities, culminating in the famous Barth sextic.
收起

The multiqueue block layer subsystem, introduced in 2013, was a necessary step for the kernel to scale to the fastest storage devices on large systems. The implementation in current kernels is incomplete, though, in that it lacks an I/O scheduler designed to work with multiqueue devices. That gap is currently set to be closed in the 4.12 development cycle when the kernel will probably get not just one, but two new multiqueue I/O schedulers.
收起

We propose two nonconforming finite elements to approximate a boundary value problem arising from strain gradient elasticity, which is a higherorder perturbation of the linearized elastic system. Our elements are H$^2$nonconforming while H$^1$conforming. We show both elements converges in the energy norm uniformly with respect to the perturbation parameter.
收起

In this work we study the following three space problem for operator spaces: if X is an operator space with base space isomorphic to a Hilbert space and X contains a completely isomorphic copy of the operator Hilbert space OH with respective quotient also completely isomorphic to OH, must X be completely isomorphic to OH? This problem leads us to the study of short exact sequences of operator spaces, more specifically those induced by complex interpolation, and their splitting. We show that the answer to the three space problem is negative, giving two different solutions.
收起

In this paper, we study the "minimal requirement" on the incoming radiation that guarantees a trapped surface to form in vacuum. First, we extend the region of existence in Christodoulou's theorem on the formation of trapped surfaces and consequently show that the lower bound required to form a trapped surface can be relaxed. Second, we demonstrate that trapped surfaces form dynamically from a class of initial data which are large merely in a scalingcritical norm. This result is motivated in part by the scaling in Christodoulou's formation of trapped surfaces theorem for the Einsteinscalar field system in spherical symmetry.
收起

The KunduEckhaus equation is a nonlinear partial differential equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. In spite of its importance, exact solution to this nonlinear equation are rarely found in literature. In this work, we solve this equation and present a new approach to obtain the solution by means of the combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). Besides, we compare the behaviour of the solutions obtained with the only exact solutions given in the literature through fractional calculus. Moreover, it is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.
收起

We show that the Grothendieck group associated to integral polytopes in $\mathbb{R}^n$ is freeabelian by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sending a polytope to its induced seminorm on the dual of $\mathbb{R}^n$.
收起

We construct a faithful categorical representation of an infinite TemperleyLieb algebra on the periplectic analogue of Deligne's category. We use the corresponding combinatorics to classify thick tensor ideals in this periplectic Deligne category. This allows to determine the kernel of the tensor functor going to the module category of the periplectic Lie supergroup, which in turn yields a description of the tensor powers of the natural representation.
收起

We obtain an important generalization of the inverse weighted FermatTorricelli problem for tetrahedra in R^3 by assigning at the corresponding weighted FermatTorricelli point a remaining positive number (residual weight). As a consequence, we derive a new plasticity principle of weighted FermatTorricellitrees of degree five for boundary closed hexahedra in R^3 by applying a geometric plasticity principle which lead to the plasticity of mass transportation networks of degree five in R^3. We also derive a complete solution for an important generalization of the inverse weighted FermatTorricelli problem for three noncollinear points and a new plasticity principle of mass networks of degree four for boundary convex quadrilaterals in R^2. The plasticity of mass transportation networks provides some first evidence in a creation of a new field that we may call in the future Mathematical Botany.
收起

Square ice is a statistical mechanics model for twodimensional ice, widely believed to have a conformally invariant scaling limit. We associate a Peano (space filling) curve to a square ice configuration, and more generally to a socalled 6vertex model configuration, and argue that its scaling limit is a spacefilling version of the random fractal curve SLE$_\kappa$, SchrammLoewner evolution with parameter $\kappa$, where $4<\kappa\leq 12+8\sqrt{2}$. For square ice, $\kappa=12$. At the "freefermion point" of the 6vertex model, $\kappa=8+4\sqrt{3}$. These unusual values lie outside the classical interval $2\le \kappa\le 8$.
收起

Many realworld control systems, such as the smart grid and human sensorimotor control systems, have decentralized components that react quickly using local information and centralized components that react slowly using a more global view. This paper seeks to provide a theoretical framework for how to design controllers that are decomposed across timescales in this way. The framework is analogous to how the network utility maximization framework uses optimization decomposition to distribute a global control problem across independent controllers, each of which solves a local problem; except our goal is to decompose a global problem temporally, extracting a timescale separation. Our results highlight that decomposition of a multitimescale controller into a fast timescale, reactive controller and a slow timescale, predictive controller can be nearoptimal in a strong sense. In particular, we exhibit such a design, named Multitimescale Reflexive Predictive Control (MRPC), which maintain
收起

Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hardcore model and monomerdimer model are the independence and matching polynomials respectively. We show how stability results follow naturally from the recently developed occupancy method for maximizing and minimizing physical observables over classes of regular graphs, and then show these stability results can be used to obtain tight extremal bounds on the individual coefficients of the corresponding partition functions. As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markstr\"om and a conjecture of Kahn on independent sets for a wide range of parameters. Additi
收起

OutputFeedback Stochastic Model Predictive Control based on Stochastic Optimal Control for nonlinear systems is computationally intractable because of the need to solve a Finite Horizon Stochastic Optimal Control Problem. However, solving this problem leads to an optimal probing nature of the resulting control law, called dual control, which trades off benefits of exploration and exploitation. In practice, intractability of Stochastic Model Predictive Control is typically overcome by replacement of the underlying Stochastic Optimal Control problem by more amenable approximate surrogate problems, which however come at a loss of the optimal probing nature of the control signals. While probing can be superimposed in some approaches, this is done suboptimally. In this paper, we examine approximation of the system dynamics by a Partially Observable Markov Decision Process with its own Finite Horizon Stochastic Optimal Control Problem, which can be solved for an optimal control policy, imp
收起

In this paper, we study some order theoretic properties of $M$ideals in order smooth $\infty$normed spaces. We obtain an order theoretic version of the AlfsenEfffros' cone decomposition theorem \cite[Theorem 2.9]{AE} for order smooth $1$normed spaces satisfying condition $(OS.1.2)$. As an application of this result, we sharpen a result on the extension of bounded positive linear functionals on subspaces of order smooth $\infty$normed spaces. We also give two different characterizations for \emph{Mideals} of order smooth $\infty$normed spaces. Finally, we characterize approximate order unit spaces as those order smooth $\infty$normed spaces $V$ that are $M$ideals in $\tilde{V}$. Here $\tilde{V}$ is the order unit space obtained by adjoining an order unit to $V$. We obtain this result by realising a complete order smooth $\infty$normed space $V$ as $A_0(Q(V))$, the space of continuous affine functions on $Q(V)$ vanishing at $0$. Here $Q(V)$ is the set of quasistates of $V$.
收起

It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in $q$variation resp. $1/q$H\"{o}lder type metrics on the space of rough paths, for any regularity $1/q \in (0,1]$. We extend this to a new class of BesovNikolskiitype metrics, with arbitrary regularity $1/q\in (0,1]$ and integrability $p\in [ q,\infty ]$, where the case $p\in \{ q,\infty \} $ corresponds to the known cases. Interestingly, the result is obtained as consequence of known $q$variation rough path estimates.
收起

If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.
收起

We begin the study of how one can define means on infinite sets. We investigate many definitions, their properties and their relations to each other. One method is based on sequences of ideals, other deal with accumulation/isolated points, other with sequences of symmetric approximating sets, other with limit of average on surroundings and one deals with evenly distributed samples. We also present some result that is based on the correspondent version of Riemann's Rearrangement Theorem for Cesaro summation.
收起

For $n\ge5$, it is well known that the moduli space $\mathfrak{M_{0,\:n}}$ of unordered $n$ points on the Riemann sphere is a quotient space of the Zariski open set $K_n$ of $\mathbb C^{n3}$ by an $S_n$ action. The stabilizers of this $S_n$ action at certain points of this Zariski open set $K_n$ correspond to the groups fixing the sets of $n$ points on the Riemann sphere. Let $\alpha$ be a subset of $n$ distinct points on the Riemann sphere. We call the group of all linear fractional transformations leaving $\alpha$ invariant the stabilizer of $\alpha$, which is finite by observation. For each nontrivial finite subgroup $G$ of the group ${\rm PSL}(2,{\Bbb C})$ of linear fractional transformations, we give the necessary and sufficient condition for finite subsets of the Riemann sphere under which the stabilizers of them are conjugate to $G$. We also prove that there does exist some finite subset of the Riemann sphere whose stabilizer coincides with $G$. Next we obtain the irreducible
收起

In this paper we study the moduli space of properly Alexandrovembedded, minimal annuli in $\mathbb{H}^2 \times \mathbb{R}$ with horizontal ends. We say that the ends are horizontal when they are graphs of $\mathcal{C}^{2, \alpha}$ functions over $\partial_\infty \mathbb{H}^2$. Contrary to expectation, we show that one can not fully prescribe the two boundary curves at infinity, but rather, one can prescribe the bottom curve, but the top curve only up to a translation and a tilt, along with the position of the neck and the vertical flux of the annulus. We also prove general existence theorems for minimal annuli with discrete groups of symmetries.
收起

In this paper, we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions $n=2,3$, which yields a conforming and strongly symmetric approximation for stress. Applying $\mathcal{P}_{k+1}\mathcal{P}_k$ as the local approximation for the stress and displacement, the mixed methods achieve the optimal order of convergence for both the stress and displacement when $k \ge n$. For the lower order case $(n2\le k<n)$, the stability and convergence still hold on some special grids. The proposed mixed methods are efficiently implemented by hybridization, which imposes the interelement normal continuity of the stress by a Lagrange multiplier. Then, we develop and analyze multilevel solvers for the Schur complement of the hybridized system in the two dimensional case. Provided that no nearly singular vertex on the grids, the proposed solvers are proved to be uniformly convergent with respect to both the grid size and Poisson's ratio. Numerical exp
收起

In this paper, we study the problem of the nonlinear interaction of impulsive gravitational waves for the Einstein vacuum equations. The problem is studied in the context of a characteristic initial value problem with data given on two null hypersurfaces and containing curvature delta singularities. We establish an existence and uniqueness result for the spacetime arising from such data and show that the resulting spacetime represents the interaction of two impulsive gravitational waves germinating from the initial singularities. In the spacetime, the curvature delta singularities propagate along 3dimensional null hypersurfaces intersecting to the future of the data. To the past of the intersection, the spacetime can be thought of as containing two independent, noninteracting impulsive gravitational waves and the intersection represents the first instance of their nonlinear interaction. Our analysis extends to the region past their first interaction and shows that the spacetime still
收起

These notes have been prepared for the Workshop on "(Non)existence of complex structures on $\mathbb{S}^6$", to be celebrated in Marburg in March, 2017. The material is not intended to be original. It contains a survey about the smallest of the exceptional Lie groups: $G_2$, its definition and different characterizations joint with its relationship with $\mathbb{S}^6$ and with $\mathbb{S}^7$. With the exception of the summary of the KillingCartan classification, this survey is selfcontained, and all the proofs are given, mainly following linear algebra arguments. Although these proofs are wellknown, they are spread and some of them are difficult to find. The approach is algebraical, working at the Lie algebra level most of times. We analyze the complex Lie algebra (and group) of type $G_2$ as well as the two real Lie algebras of type $G_2$, the split and the compact one. Octonions will appear, but it is not the starting point. Also, 3forms approach and spinorial approach are viewe
收起

Let $f_1,..., f_k:X\to N$ be maps from a complex $X$ to a compact manifold $N$, $k\ge 2$. In previous works \cite{BLM,MS}, a Lefschetz type theorem was established so that the nonvanishing of a Lefschetz type coincidence class $L(f_1,...,f_k)$ implies the existence of a coincidence $x\in X$ such that $f_1(x)=...=f_k(x)$. In this paper, we investigate the converse of the Lefschetz coincidence theorem for multiple maps. In particular, we study the obstruction to deforming the maps $f_1,...,f_k$ to be coincidence free. We construct an example of two maps $f_1,f_2:M\to T$ from a sympletic $4$manifold $M$ to the $2$torus $T$ such that $f_1$ and $f_2$ cannot be homotopic to coincidence free maps but for {\it any} $f:M\to T$, the maps $f_1,f_2,f$ are deformable to be coincidence free.
收起

In this note we investigate the $p$degree function of elliptic curves over the field $\mathbb{Q}_p$ of $p$adic numbers. The $p$degree measures the least complexity of a nonzero $p$torsion point on an elliptic curve. We prove some properties of this function and compute it explicitly in some special cases.
收起

We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots, f_k(t))=\bigsqcup^{a1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots, f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic to $\mathbb{Z}[t^{\pm1}]/J$ with an explicit ideal $J$. By using this, we see how a quandle is decomposed into connected components for some Alexander quandles. We introduce a decomposition of a quandle into the disjoint union of maximal connected subquandles. In some cases, this decomposition is obtained by iterating a connected component decomposition. We also discuss the maximal connected submultiple conjugation quandle decomposition.
收起

Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various timedependent settings, including the heat equation on a K\"{a}hler manifold, Ricci flow, K\"{a}hlerRicci flow, and mean curvature flow, to name a few. As an elliptic analogue, Colding proved a sharp gradient estimate for the Green function on a manifold with nonnegative Ricci curvature. In this paper we prove a related matrix inequality on manifolds with suitable curvature and volume growth assumptions.
收起

John Deere has turned itself into the posterchild for the DMCA, fighting farmers who say they want to fix their own tractors and access their data by saying that doing so violates the 1998 law's prohibition on bypassing copyright locks. Deere's just reiterated that position to a US Copyright Office inquiry on the future of the law, joined by auto manufacturers (but not Tesla) and many other giant corporations, all of them arguing that since the gadgets you buy have software, and since that software is licensed, not sold, you don't really own any of that stuff. You are a licensee, and you have to use the gadget according to the license terms, which spell out where you have to buy your service, parts, consumables, apps, and so on. This is finally a moment where without a doubt I can be all smug and entirely unhelpful by saying I do not use any stuff made by John Deere.
收起

In this article a framework is presented for the joint optimization of the analog transmit and receive filter with respect to a channel estimation problem. At the receiver, conventional signal processing systems restrict the bandwidth of the analog prefilter $B$ to the rate of the analogtodigital converter $f_s$ in order to comply with the wellknown Nyquist sampling theorem. In contrast, here we consider a transceiver that by design violates the common paradigm $B\leq f_s$. To this end, at the receiver we allow for a higher prefilter bandwidth $B>f_s$ and study the achievable channel estimation accuracy under a fixed sampling rate when the transmit and receive filter are jointly optimized with respect to the Bayesian Cram\'{e}rRao lower bound. For the case of a channel with unknown delayDoppler shift we show how to approximate the required Fisher information matrix and solve the transceiver design problem by an alternating optimization algorithm. The presented approach allows
收起

We continue our investigation of integral spans of tight frames in Euclidean spaces. In a previous paper, we considered the case of an equiangular tight frame (ETF), proving that if its integral span is a lattice then the frame must be rational, but overlooking a simple argument in the reverse direction. Thus our first result here is that the integral span of an ETF is a lattice if and only if the frame is rational. Further, we discuss conditions under which such lattices are eutactic and perfect and, consequently, are local maxima of the packing density function in the dimension of their span. In particular, the unit (276, 23) equiangular tight frame is shown to be eutactic and perfect. More general tight frames and their normforms are considered as well, and definitive results are obtained in dimensions two and three.
收起

A generalisation of the Lie symmetry method is applied to classify a coupled system of reactiondiffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasisteady but the other not. A complete Lie symmetry classification, including a number of the cases characterised being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific movingboundary problems motivated by biological applications tumour growth). Graphical representations of the solutions are provided and biological interpretation addressed briefly. The results are generalised on multidimensional case under assumption of radially symmetrical shape of the tumour.
收起

Transversality is a simple and effective method for implementing quantum computation faulttolerantly. However, no quantum errorcorrecting code (QECC) can transversally implement a quantum universal gate set (Eastin and Knill, Phys. Rev. Lett., 102, 110502). Since reversible classical computation is often a dominating part of useful quantum computation, whether or not it can be implemented transversally is an important open problem. We show that, other than a small set of nonadditive codes that we cannot rule out, no binary QECC can transversally implement a classical reversible universal gate set. In particular, no such QECC can implement the Toffoli gate transversally. We prove our result by constructing an information theoretically secure (but inefficient) quantum homomorphic encryption (ITSQHE) scheme inspired by Ouyang et al. (arXiv:1508.00938). Homomorphic encryption allows the implementation of certain functions directly on encrypted data, i.e. homomorphically. Our scheme bui
收起

We prove the $W^{2s,p}_{\textrm{loc}}$ local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of $\mathbb{R}^N$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cutoff, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.
收起

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with Dirichlet boundary condition on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. The key tool consists in combining classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic equation.
收起

We give intrinsic characterisations for the uniformly localized versions of the Besov spaces $B^{s}_{p,q}({\mathbb R}^n)$, where $p,q\in [1,+\infty]$, and of the LizorkinTriebel spaces $F^{s}_{p,q}({\mathbb R}^n)$, where $q\in [1,+\infty]$ and $p\in [1,+\infty[$, whatever be the real number $s>0$.
收起

This paper considers the design of the beamformers for a multipleinput singleoutput (MISO) downlink system that seeks to mitigate the impact of the imperfections in the channel state information (CSI) that is available at the base station (BS). The goal of the design is to minimize the outage probability of specified signaltointerferenceandnoise ratio (SINR) targets, while satisfying perantenna power constraints (PAPCs), and to do so at a low computational cost. Based on insights from the offset maximization technique for robust beamforming, and observations regarding the structure of the optimality conditions, lowcomplexity iterative algorithms that involve the evaluation of closedform expressions are developed. To further reduce the computational cost, algorithms are developed for perantenna powerconstrained variants of the zeroforcing (ZF) and maximum ratio transmission (MRT) beamforming directions. In the MRT case, our lowcomplexity version for systems with a large num
收起

The paper discusses stably trivial torsors for spin and orthogonal groups over smooth affine schemes over infinite perfect fields of characteristic unequal to 2. We give a complete description of all the invariants relevant for the classification of such objects over schemes of dimension at most $3$, along with many examples. The results are based on the $\mathbb{A}^1$representability theorem for torsors and transfer of known computations of $\mathbb{A}^1$homotopy sheaves along the sporadic isomorphisms to spin groups.
收起

We expose some concepts concerning the channel impulse response (CIR) of linear timevarying (LTV) channels to give a proper characterization of the mobiletomobile underwater channel. We find different connections between the linear timeinvariant (LTI) CIR of the static channel and two definitions of LTV CIRs of the dynamic mobiletomobile channel. These connections are useful to design a dynamic channel simulator from the static channel models available in the literature. Such feature is particularly interesting for overspread channels, which are hard to characterize by a measuring campaign. Specifically, the shallow water acoustic (SWA) channel is potentially overspread due the signal low velocity of propagation which prompt long delay spread responses and great Doppler effect. Furthermore, from these connections between the LTI static CIRs and the LTV dynamic CIRS, we find that the SWA mobiletomobile CIR does not only depend on the relative velocity between transceivers, but a
收起

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matou\v{s}ek and T. Tokuyama introduced "implicit computational geometry" in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative meth
收起

The branch of provability logic investigates the provabilitybased behavior of the mathematical theories. In a more precise way, it studies the relation between a mathematical theory $T$ and a modal logic $L$ via the provability interpretation which interprets the modality as the provability predicate of $T$. In this paper we will extend this relation to investigate the provabilitybased behavior of a hierarchy of theories. More precisely, using the modal language with infinitely many modalities, $\{\Box_n\}_{n=0}^{\infty}$, we will define the hierarchical counterparts of some of the classical modal theories such as $\mathbf{K4}$, $\mathbf{KD4}$, $\mathbf{GL}$ and $\mathbf{S4}$. Then we will define their canonical provability interpretations and their corresponding soundnesscompleteness theorems.
收起

We study periodic windtree models, billiards in the plane endowed with $\mathbb{Z}^2$periodically located identical connected symmetric rightangled obstacles. We exhibit effective asymptotic formulas for the number of (isotopy classes of) periodic billiard trajectories (up to $\mathbb{Z}^2$translations) on Veech windtree billiards, that is, windtree billiards whose underlying compact translation surfaces are Veech surfaces. We show that the error term depends on spectral properties of the Veech group and give explicit estimates in the case when obstacles are squares of side length $1/2$.
收起

We reexamine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the BeilinsonBernstein localization theorem, a theory of HarishChandra bimodules and their relationship to convolution operators on cohomology, and a discrete group action on the derived category of representations, generalizing the braid group action on category O via twisting functors. Our primary goal is to apply these results to other quantized symplectic resolutions, including quiver varieties and hypertoric varieties. This provides a new context for known results about Lie algebras, Cherednik algebras, finite Walgebras, and hypertoric enveloping algebras, while also pointing to the study of new algebras arising from
收起

These are the notes for a minicourse held in Odessa (2016) and Belo Horizonte (2017). My aim was to provide a short introduction to basic notions of category theory and representation theory of finitedimensional algebras. We learnt the concept of adjoint functors and showed that the construction "quiver"<> "algebra" can be interpreted as a pair of adjoint functors between certain categories. Lectures almost do not contain the proofs, theoretical part is accompanied with examples, and sometimes introduced in the form of exercises.
收起

From the classic work of Gohberg and Krein (1958), it is well known that the set of partial indices of a nonsingular matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the larger and the smaller partial indices is larger than unity then, in any neighborhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a nonstable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing matrix functions with nonstable sets of partial indices, and instead focuses on determining
收起

One of the most intriguing problems, in $q$analogs of designs, is the existence question of an infinite family of $q$analog of Steiner systems, known also as $q$Steiner systems, (spreads not included) in general, and the existence question for the $q$analog of the Fano plane, known also as the $q$Fano plane, in particular. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$Steiner systems were found for one set of parameters. In this paper, a definition for the $q$analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the $q$analog properties. The existence of a design with the parameters of the residual $q$Steiner system in general and the residual $q$Fano plane in particular are examined. We prove the existence of the residual $q$Fano plane for all $q$, where $q$ is a pri
收起

I started to reverse engineer APFS and want to share what I found out so far. Notice: I created a test image with macOS Sierra 10.12.3 (16D32). All results are guesses and the reverse engineering is work in progress. Also newer versions of APFS might change structures. The information below is neither complete nor proven to be correct.
收起