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An anonymous reader quotes a report from Bleeping Computer: Cryptocurrency miners are now being distributed by a new campaign pretending to be Adobe Flash Player installers. While this is not new, this particular campaign is going the extra mile to appear legitimate by not only installing a miner, but also updating Flash Player as well. In a new malware campaign discovered by Palo Alto Unit 42 researcher Brad Duncan, it was found that a fake Flash Player Trojan not only installed a XMRig miner, but it also automatically updated his installed Flash Player. This real Flash installer was downloaded by the Trojan from Adobe's site. By actually performing an upgrade of the desired program, it makes the user less suspicious and adds further legitimacy that the Trojan was a real Adobe installer for Adobe Flash Player. While Flash Player is now updated, what the victim does not know is that a coinminer was silently installed on the computer and started. Once started, this sample would connect
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Slate's Rachel Withers argues that "tech companies that profit from Wikipedia's extensive database owe Wikimedia a much greater debt." Amazon's Alexa, for example, uses Wikipedia "without credit, contribution, or compensation." The Google Assistant also sources Wikipedia, but they credit the encyclopedia  and other sites  when it uses it as a resource. From the report: Amazon recently donated $1 million to the Wikimedia Endowment, a fund that keeps Wikipedia running, as "part of Amazon's and CEO Jeff Bezos' growing work in philanthropy," according to CNET. It's being framed as a "gift," one that  as Amazon puts it  recognizes their shared vision to "make it easier to share knowledge globally." Obviously, and as alluded to by CNET, $1 million is hardly a magnanimous donation from Amazon and Bezos, the former a trilliondollar company and the latter a man with a net worth of more than $160 billion. But it's not just the fact that this donation is, in the scheme of things, paltry.
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Yesterday, at a routine vote on regulations for selfdriving cars, members of the European Peoples' Party voted down a clause that would protect a vehicle's telemetry so that it couldn't become someone's property. The clause affirmed that "data generated by autonomous transport are automatically generated and are by nature not creative, thus making copyright protection or the right on databases inapplicable." Boing Boing reports: This is data that we will need to evaluate the safety of autonomous vehicles, to finetune their performance, to ensure that they are working as the manufacturer claims  data that will not be public domain (as copyright law dictates), but will instead be someone's exclusive purview, to release or withhold as they see fit. Who will own this data? It's unlikely that it will be the owners of the vehicles. It's already the case that most auto manufacturers use license agreements and DRM to lock up your car so that you can't fix it yourself or take it to an inde
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Facebook is purging hundreds of accounts and pages in the U.S., many of which spread political misinformation, for breaking the company's terms against "inauthentic" content and spam. The Verge reports: The company said in a blog post that 559 pages and 251 accounts would be removed. While the accounts used "sensational political content," Facebook did not say that was the reason for the purge. Instead, the accounts and pages will be taken down after they had "consistently broken" the company's rules against gaming its platform. Facebook noted that many used strategies like posting on fake or multiple accounts to generate traffic, or to inflate their popularity. Still, Facebook noted the proximity to the U.S. midterm elections, and said that networks like the ones it removed today are "increasingly" promoting political content that is "often indistinguishable from legitimate political debate." The company said this was the reason it has turned to "behavior" instead of "content" when se
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We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$, assuming uniform volume bounds and $L^{n/2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are precompact in the GromovHausdorff topology. Our study is based on the use of $A_\infty$weights from harmonic analysis, and provides geometric controls on the limit spaces thus obtained. Our techniques also show that any conformal deformation of the Euclidean metric on $R^n$ with infinite volume and finite $L^{n/2}$ norm of the scalar curvature satisfies the Euclidean isoperimetric inequality.
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3Dprinted plastic objects can track their own use without any electronics
1011 MIT Technology 6157We propose a formula for the enumeration of closed lattice random walks of length $n$ enclosing a given algebraic area. The information is contained in the Kreft coefficients which encode, in the commensurate case, the Hofstadter secular equation for a quantum particle hopping on a lattice coupled to a perpendicular magnetic field. The algebraic area enumeration is possible because it is split in $2^{n/21}$ pieces, each tractable in terms of explicit combinatorial expressions.
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An anonymous reader quotes a report from Bloomberg: Amazon is sweetening the pay for some of its longtime warehouse workers after employees criticized the loss of bonuses and stock awards as part of the company's pledge to boost all wages to at least $15 an hour. The world's largest online retailer grabbed headlines last week with its minimumpay pledge  followed by concerns from veteran workers who feared their compensation would actually decline because the company also eliminated bonuses and stock awards. Amazon said any workers already earning $15 would get raises of $1 per hour. Now, some of those employees are learning their hourly raises will actually be $1.25 an hour. Additionally, Amazon is introducing a new cash bonus of $1,500 to $3,000 for tenure milestones at five, 10, 15 and 20 years. Workers with good attendance in the month of December will also get a $100 bonus, according to the company. "All hourly Operations and Customer Service employees will see an increase in th
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Jeffrey Dastin, reporting for Reuters: Amazon's machinelearning specialists uncovered a big problem: their new recruiting engine did not like women. The team had been building computer programs since 2014 to review job applicants' resumes with the aim of mechanizing the search for top talent, five people familiar with the effort told Reuters. Automation has been key to Amazon's ecommerce dominance, be it inside warehouses or driving pricing decisions. The company's experimental hiring tool used artificial intelligence to give job candidates scores ranging from one to five stars  much like shoppers rate products on Amazon, some of the people said. "Everyone wanted this holy grail," one of the people said. "They literally wanted it to be an engine where I'm going to give you 100 resumes, it will spit out the top five, and we'll hire those." But by 2015, the company realized its new system was not rating candidates for software developer jobs and other technical posts in a genderneut
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Apple is planning a new digital video service that will provide original content free to its device owners, CNBC reported Wednesday. From the report: Apple is preparing a new digital video service that will marry original content and subscription services from legacy media companies, according to people familiar with the matter. Owners of Apple devices, such as the iPhone, iPad and Apple TV will find the stillintheworks service in the preinstalled "TV" application, said the people, who asked not to be named because the details of the project are private. The product will include Appleowned content, which will be free to Apple device owners, and subscription "channels" which will allow customers to sign up for onlineonly services, such as those from HBO and Starz. Apple plans to debut the revamped app early next year, the people said. As Bloomberg reported in May, the subscription channels will essentially copy Amazon's Prime Video Channel Subscriptions. Customers will be able to
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From a report: Within the past four years, Apple has managed to "dramatically reduce" the rate of iPhonerelated repair fraud in its retail stores in China, according to The Information's Wayne Ma. The report is based on interviews with more than a dozen former Apple employees who spoke on condition of anonymity. In 2013, Apple is said to have discovered a highly sophisticated fraud scheme in which organized thieves would buy or steal iPhones, remove valuable components like the processor or logic board, swap in fake components, and return the "broken" iPhones to receive replacements they could resell. From the report: "Thieves would stand outside stores with suitcases full of iPhones with some of the original components stripped out and replaced with inferior parts, two of the people said. The fraudsters would hire people to pretend to be customers to return them, each taking a device to stand in line at the Genius Bar, the people said. Once the phones were swapped, the actors would p
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Japanese retailer Uniqlo in Tokyo's Ariake district has managed to cut 90% of its staff and replace them with robots that are capable of inspecting and sorting the clothing housed there. The automation also allows them to operate 24 hours a day. Quartz reports: The company recently remodeled the existing warehouse with an automated system created in partnership with Daifuku, a provider of material handling systems. Now that the system is running, the company revealed during a walkthrough of the new facility, Uniqlo has been able to cut staff at the warehouse by 90%. The Japan News described how the automation works: "The robotic system is designed to transfer products delivered to the warehouse by truck, read electronic tags attached to the products and confirm their stock numbers and other information. When shipping, the system wraps products placed on a conveyor belt in cardboard and attaches labels to them. Only a small portion of work at the warehouse needs to be done by employees,
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The BBC had to replace live broadcasts with recorded material on its TV news channels for about an hour on Wednesday following a technical glitch. BBC News reports: The News at Six was also presented from the BBC's Millbank studio instead of its usual home of New Broadcasting House. The issue affected OpenMedia, a new computer system rolled out across BBC News outlets over the past six months. OpenMedia supplier Annova has been helping to investigate the fault. Engineers believe they have now addressed the problem. BBC News Home Editor Mark Easton shared on social media that he was rushing across London to the Millbank studio.
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The Android Developers Blog describes the controlflow integrity work that is shipping on the Pixel 3 handset. "LLVM's CFI implementation adds a check before each indirect branch to confirm that the target address points to a valid function with a correct signature. This prevents an indirect branch from jumping to an arbitrary code location and even limits the functions that can be called. As C compilers do not enforce similar restrictions on indirect branches, there were several CFI violations due to function type declaration mismatches even in the core kernel that we have addressed in our CFI patch sets for kernels 4.9 and 4.14."
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During a hearing in front of the Senate Homeland Security Committee on Wednesday, FBI Director Christopher Wray told senators to "be careful what you read," when asked about a recent story involving spy chips from China being secretly embedded into servers owned by Apple, Amazon and other big companies. From a report: Senator Ron Johnson, RWis., chairman of the committee, asked Wray when his agency found out about the chips that server manufacturer Super Micro implanted into server hardware, as reported last week by Bloomberg Businessweek. "I would say to the newspaper article or, I mean, the magazine article, I would say be careful what you read," Wray replied. "Especially in this context." Johnson called on Wray to speak to the accuracy of the story, telling the FBI director that, "We don't want false information out there." Wray said he couldn't offer much detail because the agency has a policy of not confirming or denying that an investigation is underway. "I do want to be careful
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We discuss the constant $\sigma_{2}$ problem for conic 4spheres. Based on earlier works of ChangHanYang and HanLiTeixeira, we are able to find a necessary condition for the existence problem. In particular, when the condition is sharp, we have the uniqueness result similar to that of Troyanov in dimension 2. It indicates that the boundary of the moduli of all conic 4spheres with constant $\sigma_{2}$ metrics consists of conic spheres with 2 conic points and rotational symmetry.
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Let $S$ be a smooth, totally real, compact immersion in $\mathbb{C}^n$ of real dimension $m \leq n$, which is locally polynomially convex and it has finitely many points where it selfintersects finitely many times, transversely or nontransversely. We prove that $S$ is rationally convex if and only if it is isotropic with respect to a "degenerate" K\"ahler form in $\mathbb{C}^n$.
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This paper presents a theoretical analysis of numerical integration based on interpolation with a Stein kernel. In particular, the case of integrals with respect to a posterior distribution supported on a general Riemannian manifold is considered and the asymptotic convergence of the estimator in this context is established. Our results are considerably stronger than those previously reported, in that the optimal rate of convergence is established under a basic Sobolevtype assumption on the integrand. The theoretical results are empirically verified on $\mathbb{S}^2$.
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We introduce a direct, linear sampling approach to imaging in an acoustic waveguide with sound hard walls. The waveguide terminates at one end and has unknown geometry due to compactly supported wall deformations. The goal of imaging is to determine these deformations and to identify localized scatterers in the waveguide, using a remote array of sensors that emits time harmonic probing waves and records the echoes. We present a theoretical analysis of the imaging approach and illustrate its performance with numerical simulations.
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In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are wellknown Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants.
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This paper concerns a FokkerPlanck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a nonlocal order parameter related to the excess mass of the system. The first main result concerns the wellposedness and regularity of the Cauchy problem. The wellposedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield H\"older regularity of the nonlocal order parameter, which is improved by an iteration argument. The asymptotic behavior of solutions depends on some order parameter $\rho$ depending on the initial data. The system shows different behavior depending on a value $\rho_s>0$, determined from the potentials and diffusion coefficient. For $\rho \leq \rho_s$, there exists an equilibrium solution $c^{\text{eq}}_{(\rho)}$. If $\rho\le\rho_s$ the solution converges strongly to $c^{\text{eq}}
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The purpose of this note is to prove the $G$equivariant Sarkisov program for a connected algebraic group $G$ following the proof of the Sarkisov program by Hacon and McKernan. As a consequence, we obtain a characterisation of connected subgroups of $Bir(Z)$ acting rationally on $Z$.
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For any rational prime $p$, we define a certain $p$stabilization of holomorphic Siegel Eisenstein series for the symplectic group ${\rm Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for the Fourier coefficients and conclude their $p$adic interpolation problems. Consequently, for any odd prime $p$, we deduce the existence of a $\Lambda$adic form (in the sense of A. Wiles, H. Hida and R.L. Taylor) such that after taking a suitable constant multiple, it interpolates $p$adic analytic families of the abovementioned $p$stabilized Siegel Eisenstein series with nebentypus characters locally trivial at $p$ and Siegel Eisenstein series with nebentypus characters locally nontrivial at $p$ simultaneously. This can be viewed as a quite natural generalization of the ordinary $\Lambda$adic Eisenstein series for ${\rm GL}(2)$.
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We study a basic linear elliptic equation on a lower dimensional rectifiable set $S$ in $\mathbb{R}^N$ with the Neumann boundary data. Set $S$ is a support of a finite Borel measure $\mu$. We will use the measure theoretic tools to interpret the equation and the Neumann boundary condition. For this purpose we recall the Sobolevtype space dependent on the measure $\mu$. We establish existence and uniqueness of weak solutions provided that an appropriate source term is given.
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We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of wellknown fractional Sobolev spaces. As an application, we obtain apriori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$Laplacian.
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The analyses of cellular network performance based on stochastic geometry generally ignore the traffic dynamics in the network. This restricts the proper evaluation and dimensioning of the network from the perspective of a mobile operator. To address the effect of dynamic traffic, recently, the mean cell approach has been introduced, which approximates the average network load by the zero cell load. However, this is not a realistic characterization of the network load, since a zero cell is statistically larger than a random cell drawn from the population of cells, i.e., a typical cell. In this paper, we analyze the load of a noiselimited network characterized by high signal to noise ratio (SNR). The noiselimited assumption can be applied to a variety of scenarios, e.g., millimeter wave networks with efficient interference management mechanisms. First, we provide an analytical framework to obtain the cumulative density function of the load of the typical cell. Then, we obtain two appr
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Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer's velocity. These restrictions define certain `manoeuvres' of the saucer, which we call `attacking,' `landing,' or `G2 mode' manoeuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manoeuvre corresponds to the flat Legendrean contact structure, the landing manoeuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1,1), and the most complicated G2 manoeuvre corresponds to the contact Engel structure with split real form of the exceptional Lie group G2 as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5dimensional parabolic G2 geometries is used to construct a `G2 joystic
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We identify various structures on the configuration space C of a flying saucer, moving in a threedimensional smooth manifold M. Always C is a fivedimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of `parabolic geometry') and we provide examples when this geometry is `flat,' meaning that its symmetries comprise the split form of the exceptional Lie algebra G2.
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Generation expansion planning (GEP) models have been useful aids for longterm planning. Recent growth in intermittent renewable generation has increased the need to represent the capability for nonrenewables to respond to rapid changes in daily loads, leading research to bring unit commitment (UC) features into GEPs. Such GEP+UC models usually contain discrete variables which, along with many details, make computation times impractically long for analysts who need to develop, debug, modify and use the GEP for many alternative runs. We propose a GEP with generation aggregated by technology type, and with the minimal UC content necessary to represent the limitations on generation to respond to rapid changes in demand, i.e., rampup and rampdown constraints, with ramp limits estimated from historical data on maximum rates of change of each generation type. We illustrate with data for the province of Ontario in Canada; the GEP is a large scale linear program that solves in less than one
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Given a rational map $f:\overline{\mathbb C}\to \overline{\mathbb C}$ and a finite graph $G\subset \overline{\mathbb C}$ such that $f(G)\subset G$ and $f$ is expanding on some neighbourhood of $G$, we show that there is another finite graph $G'\subset \bigcup _{n\ge 0}f^{n}(G)$ in an arbitrarily small neighbourhood of $G$ such that $f^N(G')\subset G'$ for some integer $N$ but $\bigcup _{i=0}^{N1}f^{i}(G')$ contains accumulating {\em{plaits}} and {\em{nests}}
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In this article, we analyse a stabilised equalorder finite element approximation for the Stokes equations on anisotropic meshes. In particular, we allow arbitrary anisotropies in a subdomain, for example along the boundary of the domain, with the only condition that a maximum angle is fulfilled in each element.This discretisation is motivated by applications on moving domains as arising e.g. in fluidstructure interaction or multiphaseflow problems. To deal with the anisotropies, we define a modification of the original Continuous Interior Penalty stabilisation approach. We show analytically the discrete stability of the method and convergence of order ${\cal O}(h^{3/2})$ in the energy norm and ${\cal O}(h^{5/2})$ in the $L^2$norm of the velocities. We present numerical examples for a linear Stokes problem and for a nonlinear fluidstructure interaction problem, that substantiate the analytical results and show the capabilities of the approach.
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In noisy evolutionary optimization, sampling is a common strategy to deal with noise. By the sampling strategy, the fitness of a solution is evaluated multiple times (called \emph{sample size}) independently, and its true fitness is then approximated by the average of these evaluations. Previous studies on sampling are mainly empirical. In this paper, we first investigate the effect of sample size from a theoretical perspective. By analyzing the (1+1)EA on the noisy LeadingOnes problem, we show that as the sample size increases, the running time can reduce from exponential to polynomial, but then return to exponential. This suggests that a proper sample size is crucial in practice. Then, we investigate what strategies can work when sampling with any fixed sample size fails. By two illustrative examples, we prove that using parent or offspring populations can be better. Finally, we construct an artificial noisy example to show that when using neither sampling nor populations is effecti
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We give normal forms for generic kdimensional parametric families $(Z_\varepsilon)_\varepsilon$ of germs of holomorphic vector fields near $0\in\mathbb{C}^2$ unfolding a saddlenode singularity $Z_0$, under the condition that there exists a family of invariant analytic curves unfolding the weak separatrix of $Z_0$. These normal forms provide a moduli space for these parametric families. In our former 2008 paper, a modulus of a family was given as the unfolding of the MartinetRamis modulus, but the realization part was missing. We solve the realization problem in that partial case and show the equivalence between the two presentations of the moduli space. Finally, we completely characterize the families which have a modulus depending analytically on the parameter. We provide an application of the result in the field of nonlinear, parameterized differential Galois theory.
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In this paper, an Artificial Neural Network (ANN) technique is developed to find solution of celebrated Fractional order Differential Equations (FDE). Compared to integer order differential equation, FDE has the advantage that it can better describe sometimes various real world application problems of physical systems. Here we have employed multilayer feed forward neural architecture and error back propagation algorithm with unsupervised learning for minimizing the error function and modification of the parameters (weights and biases). Combining the initial conditions with the ANN output gives us a suitable approximate solution of FDE. To prove the applicability of the concept, some illustrative examples are provided to demonstrate the precision and effectiveness of this method. Comparison of the present results with other available results by traditional methods shows a close match which establishes its correctness and accuracy of this method.
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We give an introduction into some aspects of the emerging mathematical theory of manybody localization (MBL) for disordered quantum spin chains. In particular, we discuss manifestations of MBL such as zerovelocity LiebRobinson bounds, quasilocality of the time evolution of local observables, as well as exponential clustering and low entanglement of eigenstates. Explicit models where such properties have recently been verified are the XY and XXZ spin chain, in each case with disorder introduced in the form of a random exterior field. We introduce these models, state many of the available results and try to provide some general context. We discuss methods and ideas which enter the proofs and, in a few illustrative examples, include more detailed arguments. Finally, we also mention some directions for future mathematical work on MBL.
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We prove upper bounds for the average size of the $\ell$torsion $\Cl_K[\ell]$ of the class group of $K$, as $K$ runs through certain natural families of number fields and $\ell$ is a positive integer. We refine a key argument, used in almost all results of this type, which links upper bounds for $\Cl_K[\ell]$ to the existence of many primes splitting completely in $K$ that are small compared to the discriminant of $K$. Our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument, in conjunction with new counting results for these invariants. This leads to significantly improved upper bounds for the average and sometimes even higher moments of $\Cl_K[\ell]$ for many families of number fields $K$ considered in the literature, for example, for the families of all degree$d$fields for $d\in\{2,3,4,5\}$ (and non$D_4$ if $d=4$). As an application of the case $d=2$ we obtain the best upper bou
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In this paper, bicomplex kFibonacci quaternions are defined. Also, some algebraic properties of bicomplex kFibonacci quaternions which are connected with bicomplex numbers and kFibonacci numbers are investigated. Furthermore, the Honsberger identity, the d'Ocagne's identity, Binet's formula, Cassini's identity, Catalan's identity for these quaternions are given.
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We prove that nearthreshold negative energy solutions to the 2D cubic ($L^2$critical) focusing ZakharovKuznetsov (ZK) equation blowup in finite or infinite time. The proof consists of several steps. First, we show that if the blowup conclusion is false, there are negative energy solutions arbitrarily close to the threshold that are globally bounded in $H^1$ and are spatially localized, uniformly in time. In the second step, we show that such solutions must in fact be exact remodulations of the ground state, and hence, have zero energy, which is a contradiction. This second step, a nonlinear Liouville theorem, is proved by contradiction, with a limiting argument producing a nontrivial solution to a (linear) linearized ZK equation obeying uniformintime spatial localization. Such nontrivial linear solutions are excluded by a localviral spacetime estimate. The general framework of the argument is modeled on Merle [29] and Martel & Merle [24], who treated the 1D problem of the
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We study strong types and Galois groups in model theory from a topological and descriptivesettheoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems related to cardinality and Borel cardinality of strong types, quotients of definable groups and related objecets, generalising (and often improving) essentially all hitherto known results in this area. In particular, we show that under reasonable assumptions, strong type spaces are "locally" quotients of compact Polish groups. It follows that they are smooth if and only if they are typedefinable, and that a quotient of a typedefinable group by an analytic subgroup is either finite or of cardinality at least continuum.
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In this article we prove a new central limit theorem (CLT) for coupled particle filters (CPFs). CPFs are used for the sequential estimation of the difference of expectations w.r.t. filters which are in some sense close. Examples include the estimation of the filtering distribution associated to different parameters (finite difference estimation) and filters associated to partially observed discretized diffusion processes (PODDP) and the implementation of the multilevel Monte Carlo (MLMC) identity. We develop new theory for CPFs and based upon several results, we propose a new CPF which approximates the maximal coupling (MCPF) of a pair of predictor distributions. In the context of ML estimation associated to PODDP with discretization $\Delta_l$ we show that the MCPF and the approach in Jasra et al. (2018) have, under assumptions, an asymptotic variance that is upperbounded by an expression that is (almost) $\mathcal{O}(\Delta_l)$, uniformly in time. The $\mathcal{O}(\Delta_l)$ rate pr
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We propose the following definition of topological quantum phases valid for mixed states: two states are in the same phase if there exists a time independent, fast and local Lindbladian evolution driving one state into the other. The underlying idea, motivated by Koenig and Pastawski in 2013, is that it takes time to create new topological correlations, even with the use of dissipation. We show that it is a good definition in the following sense: (1) It divides the set of states into equivalent classes and it establishes a partial order between those according to their level of "topological complexity". (2) It provides a path between any two states belonging to the same phase where observables behave smoothly. We then focus on pure states to relate the new definition in this particular case with the usual definition for quantum phases of closed systems in terms of the existence of a gapped path of Hamiltonians connecting both states in the corresponding ground state path. We show first
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Despite the fact that many important problems (including clustering) can be described using hypergraphs, theoretical foundations as well as practical algorithms using hypergraphs are not well developed yet. In this paper, we propose a hypergraph modularity function that generalizes its well established and widely used graph counterpart measure of how clustered a network is. In order to define it properly, we generalize the ChungLu model for graphs to hypergraphs. We then provide the theoretical foundations to search for an optimal solution with respect to our hypergraph modularity function. Two simple heuristic algorithms are described and applied to a few small illustrative examples. We show that using a strict version of our proposed modularity function often leads to a solution where a smaller number of hyperedges get cut as compared to optimizing modularity of 2section graph of a hypergraph.
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In this paper the authors obtain a new equivalent norms of the Besov spaces of variable smoothness and integrability. Our main tools are the continuous version of Calderon reproducing formula, maximal inequalities and variable exponent technique, but allowing the parameters to vary from point to point will raise extra difficulties which, in general, are overcome by imposing regularity assumptions on these exponents.
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We address the question of convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Samesign particles repel each other, and oppositesign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blowup in finite time. We remedy this situation by regularising the interaction potential at a lengthscale $\delta_n>0$, which converges to zero as the number of particles $n$ tends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of
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Under $\mathfrak{p} = \mathfrak{c}$, we answer Question 24 of \cite{dikranjan&shakhmatov3} for cardinality ${\mathfrak c}$ , by showing that if a nontorsion Abelian group of size continuum admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with nontrivial convergent sequences.
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We study wavespackets in nonlinear periodic media in arbitrary ($d$) spatial dimension, modeled by the cubic GrossPitaevskii equation. In the asymptotic setting of small and broad wavespackets with $N\in \mathbb{N}$ carrier Bloch waves the effective equations for the envelopes are first order coupled mode equations (CMEs). We provide a rigorous justification of the effective equations. The estimate of the asymptotic error is carried out in an $L^1$norm in the Bloch variables. This translates to a supremum norm estimate in the physical variables. In order to investigate the existence of gap solitons of the $d$dimensional CMEs, we discuss spectral gaps of the CMEs. For $N=4$ and $d=2$ a family of time harmonic gap solitons is constructed formally asymptotically and numerically. Moving gap solitons have not been found for $d>1$ and for the considered values of $N$ due to the absence of a spectral gap in the standard moving frame variables.
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Throughout history, recreational mathematics has always played a prominent role in advancing research. Following in this tradition, in this paper we extend some recent work with crazy sequential representations of numbers equations made of sequences of one through nine (or nine through one) that evaluate to a number. All previous work on this type of puzzle has focused only on base ten numbers and whether a solution existed. We generalize this concept and examine how this extends to arbitrary bases, the ranges of possible numbers, the combinatorial challenge of finding the numbers, efficient algorithms, and some interesting patterns across any base. For the analysis, we focus on bases three through ten. Further, we outline several interesting mathematical and algorithmic complexity problems related to this area that have yet to be considered.
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We give a generalization of Dorronsoro's Theorem on critical $\mathrm{L}^p$Taylor expansions for $\mathrm{BV}^k$maps on $\mathrm{R}^n$, i.e., we characterize homogeneous linear differential operators $\mathbb{A}$ of $k$th order such that $D^{kj}u$ has $j$th order $\mathrm{L}^{n/(nj)}$Taylor expansion a.e. for all $u\in\mathrm{BV}^\mathbb{A}_{\text{loc}}$ (here $j=1,\ldots, k$, with an appropriate convention if $j\geq n$). The space $\mathrm{BV}^\mathbb{A}_{\text{loc}}$ consists of those locally integrable maps $u$ such that $\mathbb{A} u$ is a Radon measure on $\mathbb{R}^n$. A new $\mathrm{L}^\infty$Sobolev inequality is established to cover higher order expansions. Lorentz refinements are also considered. The main results can be seen as pointwise regularity statements for linear elliptic systems with measuredata.
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We consider random multimodal $C^3$ maps with negative Schwarzian derivative, defined on a finite union of closed intervals in $[0,1]$, onto the interval $[0,1]$ with the base space $\Omega$ and a base invertible ergodic map $\theta:\Omega\to\Omega$ preserving a probability measure $m$ on $\Omega$. We denote the corresponding skew product map by $T$ and call it a critically finite random map of an interval. We prove that there exists a subset $AA(T)$ of $[0,1]$ with the following properties: (1) For each $t\in AA(T)$ a $t$conformal random measure $\nu_t$ exists. We denote by $\lambda_{t,\nu_t,\omega}$ the corresponding generalized eigenvalues of the corresponding dual operators $\mathcal{L}_{t,\omega}^*$, $\omega\in\Omega$. (2) Given $t\ge 0$ any two $t$conformal random measures are equivalent. (3) The expected topological pressure of the parameter $t$: $$\mathcal{E}P(t):=\int_{\Omega}\log\lambda_{t,\nu,\omega}dm(\omega) $$ is independent of the choice of a $t$conformal random measu
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