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For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$. We answer this question for $p=2$ when $G$ is a proper double cover of the symmetric group. Our techniques involve constructing part of the decomposition matrix for a Rouquier block of a double cover, restricting to subgroups using the Brundan--Kleshchev modular branching rules and comparing the dimensions of irreducible representations via the bar-length formula.
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In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $\mathcal{D}^{sg}(X)$ of a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $\mathrm{K}_0(\mathcal{D}^{sg}(X))$ is finite torsion, and that $\mathrm{K}_1(\mathcal{D}^{sg}(X)) = 0$. One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincare duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups.
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We give a detailed, self-contained proof of Geoffrey Martin's normal form theorem for Lagrangian submanifolds of standard multisymplectic manifolds (that generalises Alan Weinstein's famous normal form theorem in symplectic geometry), providing also complete proofs for the necessary results in foliated differential topology, i.e., a foliated tubular neighborhood theorem and a foliated relative Poincar\'e lemma.
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The guesswork problem was originally studied by Massey to quantify the number of guesses needed to ascertain a discrete random variable. It has been shown that for a large class of random processes the rescaled logarithm of the guesswork satisfies the large deviation principle and this has been extended to the case where $k$ out $m$ sequences are guessed. The study of conditional guesswork, where guessing of a sequence is aided by the observation of another one, was initiated by Ar{\i}kan in his simple derivation of the upper bound of the cutoff rate for sequential decoding. In this note, we extend these large deviation results to the setting of conditional guesswork.
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For each fixed integer $\ell\ge 3$ and $u>0$ we establish the leading order of the exponential rate function for the probability that the number of cycles of length $\ell$ in the Erd\H{o}s--R\'enyi graph $G(N,p)$ exceeds its expectation by a factor $1+u$, assuming $N^{-1/2}\ll p\ll 1$ (up to log corrections) when $\ell\ge 4$, and $N^{-1/3}\ll p\ll 1$ in the case of triangles. We additionally obtain the sharp upper tail for Schatten norms of the adjacency matrix, and for general subgraph counts in a narrower range of $p$, as well as the sharp lower tail for counts of seminorming graphs. As in recent works of Chatterjee--Dembo and Eldan on the emerging theory of \emph{nonlinear large deviations}, our general approach applies to "low complexity" functions on product spaces, though the notion of complexity used here is somewhat different. For the application to random graphs, our argument can be viewed as a quantitative refinement of the spectral approach to Szemer\'edi's regularity lem
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The Frobenius test exponent $\operatorname{Fte}(R)$ of a local ring $(R,\mathfrak{m})$ of prime characteristic $p > 0$ is the smallest $e_0 \in \mathbb{N}$ such that for every ideal $\mathfrak{q}$ generated by a (full) system of parameters, the Frobenius closure $\mathfrak{q}^F$ has $(\mathfrak{q}^F)^{[p^{e_0}]} = \mathfrak{q}^{[p^{e_0}]}$. We establish a suffcient condition for $\operatorname{Fte}(R)<\infty$ and use it to show that if $R$ is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e. $H^j_{\mathfrak{m}}(R) / 0^F_{H^j_{\mathfrak{m}}(R)}$ is finite length for $0 \le j < \dim(R)$, then $\operatorname{Fte}(R)<\infty$.
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Google launched today a new set of services for enterprise customers of VirusTotal, a website that lets users test suspicious files and URLs against an aggregate of multiple antivirus scanning engines at the same time. From a report: This collection of new tools is part of the new VirusTotal Enterprise service, which Google described as "the most significant upgrade in VirusTotal's 14-year history." As the name implies, this new service is specifically aimed at enterprise customers and is an expansion of VirusTotal's current Premium Services. Google says VirusTotal Enterprise consists of existing VirusTotal capabilities, but also new functionality, such as improved threat detection and a faster search system that uses a brand new interface that unifies capabilities in VirusTotal's free and paid sites. "VirusTotal Enterprise allows users to search for malware samples (using VT Intelligence), hunt for future malware samples (using VT Hunt with YARA), analyze malware relationships (using
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An SEC suit has accused Elon Musk of misleading investors, deepening his le
09-28 MIT Technology 2942The Frobenius test exponent $\operatorname{Fte}(R)$ of a local ring $(R,\mathfrak{m})$ of prime characteristic $p > 0$ is the smallest $e_0 \in \mathbb{N}$ such that for every ideal $\mathfrak{q}$ generated by a (full) system of parameters, the Frobenius closure $\mathfrak{q}^F$ has $(\mathfrak{q}^F)^{[p^{e_0}]} = \mathfrak{q}^{[p^{e_0}]}$. We establish a suffcient condition for $\operatorname{Fte}(R)<\infty$ and use it to show that if $R$ is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e. $H^j_{\mathfrak{m}}(R) / 0^F_{H^j_{\mathfrak{m}}(R)}$ is finite length for $0 \le j < \dim(R)$, then $\operatorname{Fte}(R)<\infty$.
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One of the most appealing new features of the Apple Watch Series 4 is its electrocardiogram (ECG) sensor that measures the electrical activity of your heartbeat, providing you with a heart rhythm classification that can be shared with your doctor. While the feature will be available later this year in the United States, 9to5Mac reports that it could take years for it to be approved in the United Kingdom. From a report: The Medicines and Healthcare products Regulatory Agency (MHRA) explains that the process starts by examining Apple's documentation surrounding the ECG feature and performing an audit of the quality assurance system. While this step doesn't appear to be lengthy, the proceeding steps could make the process longer. MHRA says it would require Apple to perform a new clinical investigation to judge the effectiveness of the ECG on Apple Watch, but Apple would likely not be able to use any of the data from the studies it's already completed because MHRA requires companies to not
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Mark Wilson shares a report from BetaNews: Security researchers have discovered an issue with the Device Enrollment Program used by Apple to allow organizations to manage their MacBooks and iPhones. Duo Security says that using nothing more than a serial number, it is possible to gain access to sensitive data about enrolled devices and their owners. It is even possible to enroll new devices that can then access Wi-Fi passwords, VPN configurations and more. Apple was alerted to the issue way back in May, but has not done anything about it as the company does not regard it as a vulnerability. James Barclay from Duo Security, and Rich Smith from Duo Labs share their findings in a paper entitled MDM Me Maybe: Device Enrollment Program Security. They point out that while there are various easy ways to obtain devices' serial numbers, the researchers have been able to create a simple serial generator that can be used to search for information. In regard to the serial generator, Smith told CNE
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It used to happen sporadically but now it is a daily experience. As I am browsing the net I click on a link (usually a newspaper website). The page starts to load. Then I wait. And I wait. And I wait. It takes several seconds. Once loaded, my patience is not rewarded since my MacBook Air mid-2011 seems to barely be able to keep up. Videos start playing left and right. Sound is not even turned off by default anymore. This shitshow festival of lights and sounds is discouraging but I am committed to learn about world news. I continue. I have the silly idea to scroll down (searching for the meaty citations located between double quotes) and the framerate drops to 15 frames per second. Later, for no apparent reason, all fans will start running at full speed. The air exhaust will expel burning hot air. MacOS X's ActivityMonitor.app reveals countless "Helpers" processes which are not helping at all. I wonder if the machine is going to die on my lap, or take off like a jet and fly awa
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An anonymous reader writes: For those unfamiliar with the project, Stage 9 is a beautiful virtual recreation of the Enterprise ship from Star Trek: The Next Generation for Windows, Mac and Linux. More experience than game, Stage 9 was built by fans over two years in the Unreal Engine. "There were two things that we were always pretty careful with," says project leader 'Scragnog'. "We made it as clear as we possibly could that this was NOT an officially licensed project. We had no affiliation with CBS or Paramount and the IP we were trying our hardest to treat with respect was not our own. We were fans, just creating fan art." In an announcement this week, Scragnog reminded fans that no one involved in the project was in it for any financial reason and everyone was well aware that throwing money into the mix could be a problem. However, the team says it has always known that they could be shut down at any time on the whim of a license holder because in this world, that's what can happen
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During a talk at the Oculus Connect conference today, Oculus' CTO, John Carmack, compared the company's newly announced Oculus Quest headset to the Xbox 360 and PS3 in terms of power. Ars Technica reports: That doesn't mean the Quest, which is powered by a Qualcomm Snapdragon 835 SoC, can generate VR scenes comparable to those seen in Xbox 360 or PS3 games, though. As Carmack pointed out, most games of that generation targeted a 1280x720 resolution at 30 frames per second. On Quest, the display target involves two 1280x1280 images per frame at 72fps. That's 8.5 times as many pixels per second, with additional high-end anti-aliasing effects needed for VR as well. "It is not possible to take a game that was done at a high-quality level [on the Xbox 360 or PS3] and expect it to look good in VR," Carmack said. Expecting Rift-level performance from a self-contained mobile headset like the quest isn't realistic, Carmack said, partly for simple electrical reasons. While a high-end gaming PC o
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After pulling its recommendation in August, Consumer Reports announced that it is once again recommending Microsoft's Surface laptops. "Microsoft's reliability is now on-par with most other laptop brands," says Martin Lachter, a senior research associate at Consumer Reports. The Verge reports: Consumer Reports originally revoked its recommendations after a survey of 90,000 laptop and tablet owners found that 25 percent of Surface users reported having problems by the end of their second year owning the device. Its latest survey concluded that that's no longer the case (although the recommendation site didn't disclose the exact numbers for this year's polling). The newly re-gained recommendation applies to most of Microsoft's Surface lineup, including the Surface Pro, Surface Laptop, and Surface Book 2. Notably missing is the recently released 10-inch Surface Go, although that isn't getting the recommendation badge due to poor performance in Consumer Reports' lab testing, not reliabilit
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Homeland Security's Inspector General has issued a report warning that its airport face scanning system is struggling with "technical and operational challenges." The report says that Customs and Border Protection "could only use the technology with 85 percent of passengers due to staff shortages, network problems and hastened boarding times during flight delays," reports Engadget. "The system did catch 1,300 people overstaying their allowed time in the U.S., but it might have caught more -- and there were problems 'consistently' matching people from specific age groups and countries." From the report: The watchdog also pointed out uncertainty about help from airlines, such as requiring them buy the cameras needed for taking passengers' photos. That represents a "significant point failure" for the face scanning system, the Inspector General said. As a result, the oversight body warned that Homeland Security might not make its target of having the face scanning system completely ready f
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Jann Horn describes CVE-2018-17182, a locally exploitable memory-management bug in the kernel, in great detail. "Fundamentally, this bug can be triggered by any process that can run for a sufficiently long time to overflow the reference counter (about an hour if MAP_FIXED is usable) and has the ability to use mmap()/munmap() (to manage memory mappings) and clone() (to create a thread). These syscalls do not require any privileges, and they are often permitted even in seccomp-sandboxed contexts, such as the Chrome renderer sandbox (mmap, munmap, clone), the sandbox of the main gVisor host component, and Docker's seccomp policy."
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Amazon is opening a new store in New York City that sells items from its website rated 4 stars or above. The company appears to be making a bigger push into brick-and-mortar retail following its acquisition of Whole Foods and the gradual rollout of its cashier-less convenience stores. CNBC reports: The store is set to open this Thursday in New York's SoHo neighborhood, on 72 Spring Street. The company said Amazon Prime members will pay "the Amazon.com price" when they shop there. An Amazon spokeswoman said the store is permanent, not a pop-up location, as some of its shops -- including one that sold Calvin Klein merchandise in SoHo last year -- have been before. Amazon said customers will find items like a mini skillet, card games and many of the company's own electronic devices including the Echo Spot and Fire TV Stick, in the new store, called Amazon 4-star. It said it will also include some "Trending Around NYC" items there, which Amazon is able to track on its website. Amazon said
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An anonymous reader quotes a report from Gizmodo: Amazon, the country's second-largest employer, has so far remained immune to any attempts by U.S. workers to form a union. With rumblings of employee organization at Whole Foods -- which Amazon bought for $13.7 billion last year -- a 45-minute union-busting training video produced by the company was sent to Team Leaders of the grocery chain last week, according to sources with knowledge of the store's activities. Recordings of that video, obtained by Gizmodo, provide valuable insight into the company's thinking and tactics. Each of the video's six sections, which the narrator states are "specifically designed to give you the tools that you need for success when it comes to labor organizing," take place in an animated simulacrum of a Fulfillment Center. The video's narrators are clad in the reflective vests typical of the real-world setting. "We are not anti-union, but we are not neutral either," the video states, drawing a distinction t
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A former Google employee has warned of the firm's "disturbing" plans in China, in a letter to US lawmakers. BBC: Jack Poulson, who had been a senior researcher at the company until resigning in August, wrote that he was fearful of Google's ambitions. His letter alleges Google's work on a Chinese product -- codenamed Dragonfly -- would aid Beijing's efforts to censor and monitor its citizens online. Google has said its work in China to date has been "exploratory." Ben Gomes, Google's head of search, told the BBC earlier this week: "Right now all we've done is some exploration, but since we don't have any plans to launch something there's nothing much I can say about it." A report by news site The Intercept last week alleged Google had demanded employees delete an internal memo that discussed the plans. Google has not commented on the staff row, but said: "We've been investing for many years to help Chinese users, from developing Android, through mobile apps such as Google Translate and
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Facebook's Oculus has announced its new $399 standalone virtual-reality headset that's scheduled to launch in the spring of 2019. Facebook CEO Mark Zuckerberg says that "with Oculus Quest, we will complete our first generation of Oculus products," adding that the Oculus Quest combines "the key attributes of the ideal VR system" -- a wireless design, virtual hand controllers, and full positional tracking. The Verge reports: The Oculus Quest is a consumer version of what was previously known as Project Santa Cruz. It uses motion controllers similar to Oculus Touch, and four wide-angle cameras provide positional tracking that lets people walk through virtual space. It's supposed to support "Rift-quality" experiences, with a starting catalog of over 50 titles, including well-known existing games like climbing simulator The Climb and adventure-puzzle game Moss. Oculus Quest essentially combines the high-end, tethered Oculus Rift headset with the relatively cheap, standalone Oculus Go device
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Facebook's David Marcus, who until recently ran the Facebook Messenger before starting the blockchain group earlier this year, is defending the company and CEO Mark Zuckerberg after a WhatsApp founder spoke critically of his experience at the company. Marcus: [...] On the business model. I was present in a lot of these meetings. Again, Mark protected WhatsApp for a very long period of time. And you have to put this in the context of a large organization with businesses knocking on our door to have the ability to engage and communicate with their customers on WhatsApp the same way they were doing it on Messenger. During this time, it became pretty clear that while advocating for business messaging, and being given the opportunity to build and deliver on that promise, Brian actively slow-played the execution, and never truly went for it. In my view, if you're passionate about a certain path -- in this case, letting businesses message people and charging for it -- and if you have internal
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Kashmir Hill, reporting for Gizmodo: Last week, I ran an ad on Facebook targeted at a computer science professor named Alan Mislove. Mislove studies how privacy works on social networks and had a theory that Facebook is letting advertisers reach users with contact information collected in surprising ways. I was helping him test the theory by targeting him in a way Facebook had previously told me wouldn't work. I directed the ad to display to a Facebook account connected to the landline number for Alan Mislove's office, a number Mislove has never provided to Facebook. He saw the ad within hours. One of the many ways that ads get in front of your eyeballs on Facebook and Instagram is that the social networking giant lets an advertiser upload a list of phone numbers or email addresses it has on file; it will then put an ad in front of accounts associated with that contact information. A clothing retailer can put an ad for a dress in the Instagram feeds of women who have purchased from the
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A heterochromatic (or rainbow) graph is an edge-colored graph whose edges have distinct colors, that is, where each color appears at most once. In this paper, I propose a $(g,f)$-chromatic graph as an edge-colored graph where each color $c$ appears at least $g(c)$ times and at most $f(c)$ times. I also present a necessary and sufficient condition for edge-colored graphs (not necessary to be proper) to have a $(g,f)$-chromatic spanning tree. Using this criterion, I show that an edge-colored complete graph $G$ has a spanning tree with a color probability distribution `similar' to that of $G$. Moreover, I conjecture that an edge-colored complete graph $G$ of order $2n$ $(n \ge 3)$ can be partitioned into $n$ edge-disjoint spanning trees such that each has a color probability distribution `similar' to that of $G$.
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If $\mathbb{F}_{q}$ is a finite field, $C$ is a vector subspace of $\mathbb{F}_{q}^{n}$ (linear code), and $G$ is a subgroup of the group of linear automorphisms of $\mathbb{F}_{q}^{n}$, $C$ is said to be $G$-invariant if $g(C)=C$ for all $g\in G$. A solution to the problem of computing all the $G$-invariant linear codes $C$ of $\mathbb{F}_{q}^{n}$ is offered. This will be referred as the invariance problem. When $n=|G|t$, we determine conditions for the existence of an isomorphism of $\mathbb{F}_{q}[G]$-modules between $\mathbb{F}_{q}^{n}$ and $\mathbb{F}_{q}[G]\times \cdots \times \mathbb{F}_{q}[G]$ ($t$-times), that preserves the Hamming weight. This reduces the invariance problem to the determination of the $\mathbb{F}_{q}[G]$-submodules of $\mathbb{F}_{q}[G]\times \cdots \times \mathbb{F}_{q}[G]$ ($t$-times). The concept of Gaussian binomial coefficient for semisimple $\mathbb{F}_{q}[G]$-modules, which is useful for counting $G$-invariant codes, is introduced. Finally, a systemati
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We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs.
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In this mainly expository note, we state a criterion for when a left Kan extension of a lax monoidal functor along a strong monoidal functor can itself be equipped with a lax monoidal structure, in a way that results in a left Kan extension in MonCat. This belongs to the general theory of algebraic Kan extensions, as developed by Melli\`es-Tabareau, Koudenburg and Weber, and is very close to an instance of a theorem of Koudenburg. We find this special case particularly important due to its connections with the theory of graded monads.
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The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions, including compressible, turbulent, as well as flows involving further physics such as non-equilibrium internal energy exchange and chemical reactions. Despite its wide applicability, deterministic solution of the Boltzmann equation presents a huge computational challenge, and often the collision operator is simplified for practical reasons. In this work, we introduce a highly accurate deterministic method for the full Boltzmann equation which couples the Runge-Kutta discontinuous Galerkin (RKDG) discretization in time and physical space (Su et al., Comp. Fluids, 109 pp. 123-136, 2015) and the recently developed fast Fourier spectral method in velocity space (Gamba et al., SIAM J. Sci. Comput., 39 pp.~B658--B674, 2017). The novelty of this approach encompasses three aspects: firs
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We study the numerical approximation of linear-quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed. Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.
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Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis.
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Having the right assortment of shipping boxes in the fulfillment warehouse to pack and ship customer's online orders is an indispensable and integral part of nowadays eCommerce business, as it will not only help maintain a profitable business but also create great experiences for customers. However, it is an extremely challenging operations task to strategically select the best combination of tens of box sizes from thousands of feasible ones to be responsible for hundreds of thousands of orders daily placed on millions of inventory products. In this paper, we present a machine learning approach to tackle the task by formulating the box design problem prescriptively as a generalized version of weighted $k$-medoids clustering problem, where the parameters are estimated through a variety of descriptive analytics. We test this machine learning approach on fulfillment data collected from Walmart U.S. eCommerce, and our approach is shown to be capable of improving the box utilization rate by
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Optimally encoding classical information in a quantum system is one of the oldest and most fundamental challenges of quantum information theory. Holevo's bound places a hard upper limit on such encodings, while the Holevo-Schumacher-Westmoreland (HSW) theorem addresses the question of how many classical messages can be "packed" into a given quantum system. In this article, we use Sen's recent quantum joint typicality results to prove a one-shot multiparty quantum packing lemma generalizing the HSW theorem. The lemma is designed to be easily applicable in many network communication scenarios. As an illustration, we use it to straightforwardly obtain quantum generalizations of well-known classical coding schemes for the relay channel: multihop, coherent multihop, decode-forward, and partial decode-forward. We provide both finite blocklength and asymptotic results, the latter matching existing formulas. Given the key role of the classical packing lemma in network information theory, our p
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We present a new and efficient method for deriving finite-size effects in statistical physics models solvable by Bethe Ansatz. It is based on the study of the functional that maps a function to the sum of its evaluations over the Bethe roots. A simple and powerful constraint is derived when applying this functional to infinitely derivable test functions with compact support, that generalizes then to more general test functions. The method is presented in the context of the simple spin-$1/2$ XXZ chain for which we derive the finite-size corrections to leading eigenvalues of the Hamiltonian for any configuration of Bethe numbers with real Bethe roots. The expected results for the central charge and conformal dimensions are recovered.
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We study the Cauchy problem for non-linear non-local operators that may be degenerate. Our general framework includes cases where the jump intensity is allowed to depend on the values of the solution itself, e.g. the porous medium equation with the fractional Laplacian and the parabolic fractional $p$-Laplacian. We show the existence, uniqueness of bounded solutions and study their further properties. Several new examples of non-local, non-linear operators are provided.
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We give a list of statements on the geometry of elliptic threefolds phrased only in the language of topology and homological algebra. Using only notions from topology and homological algebra, we recover existing results and prove new results on torsion pairs in the category of coherent sheaves on an elliptic threefold.
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Fleischner introduced the idea of splitting a vertex of degree at least three in a connected graph and used the operation to characterize Eulerian graphs. Raghunathan et. al. extended the splitting operation from graphs to binary matroids. It has been studied that splitting operation, in general, may not preserve the connectedness of the binary matroid. Interestingly, it is true that the splitting matroid of a disconnected matroid may be connected. In this paper, we characterize the binary disconnected matroids whose splitting matroid is connected.
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We devise a simplified parameter estimator for a second order stochastic differential equation by a first order system based on the Smoluchowski-Kramers approximation. We establish the consistency of the estimator by using {\Gamma}-convergence theory. We further illus- trate our estimation method by an experimentally studied movement model of a colloidal particle immersed in water under conservative force and constant diffusion.
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The Stokes-Brinkman equations model flow in heterogeneous porous media by combining the Stokes and Darcy models of flow into a single system of equations. With suitable parameters, the equations can model either flow without detailed knowledge of the interface between the two regions. Thus, the Stokes-Brinkman equations provide an alternative to coupled Darcy-Stokes models. After a brief review of the Stokes-Brinkman problem and its discretization using Taylor-Hood finite elements, we present a residual-based a posteriori error estimate and use it to drive an adaptive mesh refinement process. We compare several strategies for the mesh refinement, and demonstrate its effectiveness by numerical experiments in both 2D and 3D.
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We analyze a reaction coefficient identification problem for the spectral fractional powers of a symmetric, coercive, linear, elliptic, second-order operator in a bounded domain $\Omega$. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $\Omega \times (0,\infty)$. We thus consider an equivalent coefficient identification problem, where the coefficient to be identified appears explicitly. We derive existence of local solutions, optimality conditions, and a rapid decay of solutions on the extended domain $(0,\infty)$. The latter property suggests a truncation that is suitable for numerical approximation. We thus propose and analyze a fully discrete scheme that discretizes the set of admissible coefficients with piecewise constant functions. The discretization of the state equation relies on the tensorization of a first-degree FEM in $\Omega$ with a suitable $hp$-FEM in the extended dimension. We derive
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Over one year ago, a very long preprint posted on arXiv [arXiv:1709.03771] and HAL announced a proof of Lehmer's Conjecture (and of other related results). Unfortunately, as was remarked by several specialists, this proof contains a (at least one) fatal error. The aim of this very short note is to inform the mathematical community, which could be aware of this.
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Fully interpreted languages whose interpretations conform to classical logic have fully interpreted extensions which have truth predicates. A theory of truth can be formulated for the extended languages so that it satisfies the norms presented in Hannes Leitgeb's paper 'What Theories of Truth Should be Like (but Cannot be)'.
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In this paper, we associate to two given continuous maps $f,g: X\rightarrow Z$, on a path connected space $X$, the relative topological complexity $TC^{(f, g, Z)}(X):=TC_X(X\times _ZX)$ of their fiber space $X\times _ZX$. When $g=f$ we obtain a variant of the topological complexity $TC(f)$ of $f: X\longrightarrow Z$ generalizing Farber's topological complexity $TC(X)$ in the sens that $TC(X)=TC(cst_{x_0})$; being $cst_{x_0}$ the constant map on $X$. Moreover, we prove that $TC(f)$ is a fiberwise homotopy equivalence invariant. When $(X,x_0)$ is a pointed space, we prove that $TC^{(f, cst_{x_0}, Z)}(X)$ interpolates $cat(X)$ and $TC(X)$ for any continuous map $f$.
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Passengers board a fully booked airplane in order. The first passenger picks one of the seats at random. Each subsequent passenger takes his or her assigned seat if available, otherwise takes one of the remaining seats at random. It is well known that the last passenger obtains her own seat with probability $1/2$. We study the distribution of the number of incorrectly seated passengers, and we also discuss the case of several absent-minded passengers.
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We consider surfaces of geometric genus $3$ with the property that their transcendental cohomology splits into $3$ pieces, each piece coming from a $K3$ surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).
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Automatic algorithms attempt to provide approximate solutions that differ from exact solutions by no more than a user-specified error tolerance. This paper describes an automatic, adaptive algorithm for approximating the solution to a general linear problem on Hilbert spaces. The algorithm employs continuous linear functionals of the input function, specifically Fourier coefficients. We assume that the Fourier coefficients of the solution decay sufficiently fast, but do not require the decay rate to be known a priori. We also assume that the Fourier coefficients decay steadily, although not necessarily monotonically. Under these assumptions, our adaptive algorithm is shown to produce an approximate solution satisfying the desired error tolerance, without prior knowledge of the norm of the function to be approximated. Moreover, the computational cost of our algorithm is shown to be essentially no worse than that of the optimal algorithm. We provide a numerical experiment to illustrate o
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In this study, we focus on the market clearing problem of Turkish day-ahead electricity market. We propose a mathematical model by extending the variety of bid types for different price regions. The commercial solvers may not find any feasible solution for the proposed problem in some instances within the given time limits. Hence, we design an adaptive tabu search (ATS) algorithm to solve the problem. ATS discretizes continuous search space arising from the flow variables. Our method has adaptive radius and it achieves backtracking by a commercial solver. Then, we compare the performance of ATS with a heuristic decomposition method from the literature by using synthetic data sets. We evaluate the performances of the algorithms with respect to their solution times and surplus differences. ATS performs better in most of the sets.
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In this paper we describe an algorithm for the computation of canonical forms of finite subsets of $\mathbb{Z}^d$, up to affinities over $\mathbb{Z}$. For fixed dimension $d$, this algorithm has worst-case asymptotic complexity $O(n \log^2 n \, s\,\mu(s))$, where $n$ is the number of points in the given subset, $s$ is an upper bound to the size of the binary representation of any of the $n$ points, and $\mu(s)$ is an upper bound to the number of operations required to multiply two $s$-bit numbers. In particular, the problem is fixed-parameter tractable with respect to the dimension $d$. This problem arises e.g. in the context of computation of invariants of finitely presented groups with abelianized group isomorphic to $\mathbb{Z}^d$. In that context one needs to decide whether two Laurent polynomials in $d$ indeterminates, considered as elements of the group ring over the abelianized group, are equivalent with respect to a change of basis.
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We point out that Chubanov's oracle-based algorithm for linear programming [5] can be applied almost as it is to linear semi-infinite programming (LSIP). In this note, we describe the details and prove the polynomial complexity of the algorithm based on the real computation model proposed by Blum, Shub and Smale (the BSS model) which is more suitable for floating point computation in modern computers. The adoption of the BBS model makes our description and analysis much simpler than the original one by Chubanov [5]. Then we reformulate semidefinite programming (SDP) and second-order cone programming (SOCP) into LSIP, and apply our algorithm to obtain new complexity results for computing interior feasible solutions of homogeneous SDP and SOCP.
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