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Nadav Amit decided to dig into why some small kernel functions were not being inlined by GCC; the result is a detailed investigation into how these things can go wrong. "Ignoring the assembly shenanigans that this code uses, we can see that in practice it generates a single ud2 instruction. However, the compiler considers this code to be 'big' and consequently oftentimes does not inline functions that use WARN() or similar functions. The reason turns to be the newline characters (marked as '\n' above). The kernel compiler, GCC, is unaware to the code size that will be generated by the inline assembly. It therefore tries to estimate its size based on newline characters and statement separators (';' on x86)."
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Even though the Mac line has grown less repairable over time, fixers have still managed to develop techniques for performing essential screen and battery repairs  until now. According to an internal Apple service document, any Mac with an Apple T2 chip now requires the proprietary 'Apple Service Toolkit 2 (AST 2) System Configuration Suite' (whew, that's a mouthful!) to complete certain repairs. This issue has received extensive coverage, but we wanted to perform some lab testing before we took our shot. Let's break down what all this means first. This is inevitable  Macs have becoming ever more closed and less repairable for years now. This sucks  but at the same time, nobody is forcing you to buy a Mac. There are countless premium Windows and Linux laptops out there that are just as good, and even many nonpremium Windows laptops are more than good enough replacements.
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tedlistens writes: Police officers wearing new cameras by Axon, the U.S.'s largest body camera supplier, will soon be able to send live video from their cameras back to base and elsewhere, potentially expanding police surveillance. Another feature of the new device  set to be released next year  triggers the camera to start recording and alerts command staff once an officer has fired their weapon, a possible corrective to the problem of officers forgetting to switch them on. (The initial price of $699 doesn't include other costs, like a subscription to Axon's Evidence.com data management system.) But adding new technologies to body camera video introduces new privacy concerns, say legal experts, who have cautioned that a network of livestreaming cameras risks turning officers into roving sentinels for a giant panopticonlike surveillance system. Harlan Yu, the executive director of Upturn, a Washington nonprofit consultancy that has studied body cameras, says that livestreaming c
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China has issued a new regulation setting out wideranging police powers to inspect internet service providers and users, as the government tightens its grip on the country's heavily restricted cyberspace. Local media reports: Under the new rule, effective from November 1, central and local public security authorities can enter the premises of all companies and entities that provide internet services and look up and copy information considered relevant to cybersecurity. The regulation was issued by the Ministry of Public Security last month and released on its website on Sunday. It comes more than a year after a controversial cybersecurity law was introduced that has caused widespread concern among foreign companies operating in China. Despite its broad scope, the legislation gives few details about implementation, making it all the more difficult for companies trying to avoid its repercussions. Analysts said the new regulation sheds some light on how the law will be implemented. "That
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Hackaday's Tom Nardi writes about the Federal Aviation Administration's push to repeal Section 336, which states that small remotecontrolled aircraft as used for hobby and educational purposes aren't under FAA jurisdiction. "Despite assurances that the FAA will work towards implementing waivers for hobbyists, critics worry that in the worst case the repeal of Section 336 might mean that remote control pilots and their craft may be held to the same standards as their humancarrying counterparts," writes Nardi. From the report: Section 336 has already been used to shoot down the FAA's illconceived attempt to get RC pilots to register themselves and their craft, so it's little surprise they're eager to get rid of it. But they aren't alone. The Commercial Drone Alliance, a nonprofit association dedicated to supporting enterprise use of Unmanned Aerial Systems (UAS), expressed their support for repealing Section 336 in a June press release: "Basic 'rules of the road' are needed to manage
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Citing corporate values, Google has decided not to compete for the Pentagon's $10 billion cloudcomputing contract. Bloomberg reports: The project, known as the Joint Enterprise Defense Infrastructure cloud, or JEDI, involves transitioning massive amounts of Defense Department data to a commercially operated cloud system. Companies are due to submit bids for the contract, which could last as long as 10 years, on October 12th. Google's announcement on Monday came just months after the company decided not to renew its contract with a Pentagon artificial intelligence program, after extensive protests from employees of the internet giant about working with the military. The company then released a set of principles designed to evaluate what kind of artificial intelligence projects it would pursue. "We are not bidding on the JEDI contract because first, we couldn't be assured that it would align with our AI Principles," a Google spokesman said in a statement. "And second, we determined that
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Google exposed the private data of hundreds of thousands of users of the Google+ social network and then opted not to disclose the issue this past spring, in part because of fears that doing so would draw regulatory scrutiny and cause reputational damage, WSJ reported Monday, citing people briefed on the incident and documents. From the report: As part of its response to the incident, the Alphabet unit plans to announce a sweeping set of data privacy measures that include permanently shutting down all consumer functionality of Google+, the people said. The move effectively puts the final nail in the coffin of a product that was launched in 2011 to challenge Facebook and is widely seen as one of Google's biggest failures. A software glitch in the social site gave outside developers potential access to private Google+ profile data between 2015 and March 2018, [Editor's note: the link may be paywalled; alternative source] when internal investigators discovered and fixed the issue, accordi
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Google exposed the private data of hundreds of thousands of users of the Google+ social network and then opted not to disclose the issue this past spring, in part because of fears that doing so would draw regulatory scrutiny and cause reputational damage, according to people briefed on the incident and documents reviewed by The Wall Street Journal. [...] A software glitch in the social site gave outside developers potential access to private Google+ profile data between 2015 and March 2018, when internal investigators discovered and fixed the issue, according to the documents and people briefed on the incident. A memo reviewed by the Journal prepared by Google's legal and policy staff and shared with senior executives warned that disclosing the incident would likely trigger "immediate regulatory interest" and invite comparisons to Facebook's leak of user information to data firm Cambridge Analytica. Data leaks and breaches happen. They are a fact of life we're pretty much
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The Hubble Space Telescope is operating with only essential functions after it lost one of the gyroscopes needed to point the spacecraft. From a report: The observatory, described as one of the most important scientific instruments ever created, was placed in "safe mode" over the weekend, while scientists try to fix the problem. Hubble had been operating with four of its six gyroscopes when one of them failed on Friday. The telescope was launched in 1990. After the gyro failure at the weekend, controllers tried to switch on a different one, but that was found to be malfunctioning. That leaves Hubble with only two fully functional gyros. At any given time, Hubble needs three of its gyroscopes to work for optimal efficiency.
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An anonymous reader shares a report: Four years ago, IBM announced that it was investing $3 billion over the next five years into the future of nanoelectronics with a broad project it dubbed "7nm and Beyond." With at least one major chipmaker, GlobalFoundries, hitting the wall at the 7nm node, IBM is forging ahead, using graphene to deposit nanomaterials in predefined locations without chemical contamination. In research described in the journal Nature Communications, the IBM researchers for the first time electrified graphene so that it helps to deposit nanomaterials with 97% accuracy. "As this method works for a wide variety of nanomaterials, we envision integrated devices with functionalities that represent the unique physical properties of the nanomaterial," said Mathias Steiner, manager at IBM ResearchBrazil. "We also can envision onchip light detectors and emitters operating within a distinct wavelength range determined by the optical properties of the nanomaterial." As an exa
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Intel unveiled its 9thgeneration Core desktop chips, with the notable omission of a key feature: HyperThreading, at least on all but the most exclusive Core i99900K for mainstream PCs. HyperThreading has also been reserved for a new iteration of Intel's Xseries processors, which includes up to 18 cores and 36 threads. From a report: In a livestream Monday morning from its Fall Launch Event in New York, the company announced just a single Core i9 chip, the $488 Core i99900K. Later, the company privately revealed two others in the Core i7 and Core i5 families. Intel also announced a new series of Xclass chips, ranging from 8 cores and 16 threads through 18 cores and 36 threads. Prices will range from $589 to $1,979. It's certainly fair to say that Intel surprised us all with the unexpected shift of its upcoming 28core chip to the Xeon family, as well as the announcement of the Xseries chips, too. And what's the deal with hyperthreading? Intel's announcement certainly adds some n
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Among many of Intel's announcements today, a key one for a lot of users will be the launch of Intel's 9th Generation Core desktop processors, offering up to 8cores on Intel's mainstream consumer platform. These processors are dropin compatible with current Coffee Lake and Z370 platforms, but are accompanied by a new Z390 chipset and associated motherboards as well. The highlights from this launch is the 8core Core i9 parts, which include a 5.0 GHz turbo Core i99900K, rated at a 95W TDP. Biggest news for me is that Intel unveiled that these new processors will switch from a cheap paste as thermal interface material between the die and the IHS to a layer of solder. This should greatly aid in cooling.
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An anonymous reader quotes a report from Ars Technica: Lawyers representing a Southern California limousine company that sued Uber last month over state unfair competition allegations have now filed a motion for partial summary judgement. If the filing is granted by the judge, the motion would substantially streamline the case and answer the vexing question: are Uber drivers employees or not? The proposed classaction lawsuit, known as Diva Limousine v. Uber, relies on a recently decided California Supreme Court decision that makes it more difficult for companies to unilaterally declare their workers as contractors, which effectively deprives them of benefits that they would otherwise receive as employees. In the California Supreme Court case, known as Dynamex, that court came up with a threepart test, known as the ABC test, to figure out whether companies can assert contractor status or not: "(A) that the worker is free from the control and direction of the hiring entity in connectio
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Earlier this month, The New York Times ran a story which looks at the ways a network of illegal radio stations changed British music, and wonders where young people are going to make culture now, now that the internet is killing off the pirate radio. An excerpt from the story: Ofcom, the British communications regulator, estimated there are now just 50 pirate stations in London, down from about 100 a decade ago, and hundreds in the 1990s, when stations were constantly starting up and shutting down. Ofcom considers this good news, because illegal broadcasters could interfere with radio frequencies used by emergency services and air traffic control, a spokesman said. But pirate radio stations also offered public services, of a different sort: They gave immigrant communities programming in their native languages, ran charity drives and created the first radio specifically for black Britons. Pirate radio was also the site of some of Britain's most important musical innovations, introducing
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Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$filtrations when $p$ is not too small. Moreover, we give applications of this theorem to $p^n$filtrations for $n > 1$, to modules containing the Steinberg module as a tensor factor, and to the Donkin conjecture on modules having $p$filtrations.
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A graph $G$ is called edgemagic if there is a bijective function $f$ from the set of vertices and edges to the set $\{1,2,\ldots,V(G)+E(G)\}$ such that the sum $f(x)+f(xy)+f(y)$ for any $xy$ in $E(G)$ is constant. Such a function is called an edgemagic labelling of G and the constant is called the valence of $f$. An edgemagic labelling with the extra property that $f(V(G))= \{1,2,\ldots,V(G)\}$ is called super edgemagic. In this paper, we establish a relationship between the valences of (super) edgemagic labelings of certain types of bipartite graphs and the existence of a particular type of decompositions of such graphs.
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A convex optimizationbased method is proposed to numerically solve dynamic programs in continuous state and action spaces. This approach using a discretization of the state space has the following salient features. First, by introducing an auxiliary optimization variable that assigns the contribution of each grid point, it does not require an interpolation in solving an associated Bellman equation and constructing a control policy. Second, the proposed method allows us to solve the Bellman equation with a desired level of precision via convex programming in the case of linear systems and convex costs. We can also construct a control policy of which performance converges to the optimum as the grid resolution becomes finer in this case. Third, when a nonlinear controlaffine system is considered, the convex optimization approach provides an approximate control policy with a provable suboptimality bound. Fourth, for general cases, the proposed convex formulation of dynamic programming op
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FirstFit is a greedy algorithm for partitioning the elements of a poset into chains. Let $\textrm{FF}(w,Q)$ be the maximum number of chains that FirstFit uses on a $Q$free poset of width $w$. A result due to Bosek, Krawczyk, and Matecki states that $\textrm{FF}(w,Q)$ is finite when $Q$ has width at most $2$. We describe a family of posets $\mathcal{Q}$ and show that the following dichotomy holds: if $Q\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \le 2^{c(\log w)^2}$ for some constant $c$ depending only on $Q$, and if $Q\not\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \ge 2^w  1$.
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Computationally efficient numerical methods for highorder approximations of convolution integrals involving weakly singular kernels find many practical applications including those in the development of fast quadrature methods for numerical solution of integral equations. Most fast techniques in this direction utilize uniform grid discretizations of the integral that facilitate the use of FFT for $O(n\log n)$ computations on a grid of size $n$. In general, however, the resulting error converges slowly with increasing $n$ when the integrand does not have a smooth periodic extension. Such extensions, in fact, are often discontinuous and, therefore, their approximations by truncated Fourier series suffer from Gibb's oscillations. In this paper, we present and analyze an $O(n\log n)$ scheme, based on a Fourier extension approach for removing such unwanted oscillations, that not only converges with highorder but is also relatively simple to implement. We include a theoretical error analys
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We present a hybrid spectral elementFourier spectral method for solving the coupled system of NavierStokes and CahnHilliard equations to simulate wallbounded twophase flows in a threedimensional domain which is homogeneous in at least one direction. Fourier spectral expansions are employed along the homogeneous direction and $C^0$ highorder spectral element expansions are employed in the other directions. A critical component of the method is a strategy we developed in a previous work for dealing with the variable density/viscosity of the twophase mixture, which makes the efficient use of Fourier expansions in the current work possible for twophase flows with different densities and viscosities for the two fluids. The attractive feature of the presented method lies in that the twophase computations in the threedimensional space are transformed into a set of decoupled twodimensional computations in the planes of the nonhomogeneous directions. The overall scheme consists of
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In this paper we study class $\mathcal{S}^+$ of univalent functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients. For such functions we give estimates of the FeketeSzeg\H{o} functional and sharp estimates of their initial coefficients and logarithmic coefficients. Also, we present necessary and sufficient conditions for $f\in \mathcal{S}^+$ to be starlike of order $1/2$.
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Computation biology helps to understand all processes in organisms from interaction of molecules to complex functions of whole organs. Therefore, there is a need for mathematical methods and models that deliver logical explanations in a reasonable time. For the last few years there has been a growing interest in biological theory connected to finite fields: the algebraic modeling tools used up to now are based on Gr\"obner bases or Boolean group. Let $n$ variables representing gene products, changing over the time on $p$ values. A Polynomial dynamical system (PDS) is a function which has several components, each one is a polynom with $n$ variables and coefficient in the finite field $Z/pZ$ that model the evolution of gene products. We propose herein a method using algebraic separators, which are special polynomials abundantly studied in effective Galois theory. This approach avoids heavy calculations and provides a first Polynomial model in linear time.
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Linear Hamiltonian systems with timedependent coefficients are of importance to nonlinear Hamiltonian systems, accelerator physics, plasma physics, and quantum physics. It is shown that the solution map of a linear Hamiltonian system with timedependent coefficients can be parameterized by an envelope matrix $w(t)$, which has a clear physical meaning and satisfies a nonlinear envelope matrix equation. It is proved that a linear Hamiltonian system with periodic coefficients is stable iff the envelope matrix equation admits a solution with periodic $\sqrt{w^{\dagger}w}$ and a suitable initial condition. The mathematical devices utilized in this theoretical development with significant physical implications are timedependent canonical transformations, normal forms for stable symplectic matrices, and horizontal polar decomposition of symplectic matrices. These tools systematically decompose the dynamics of linear Hamiltonian systems with timedependent coefficients, and are expected to b
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While the definition of a fractional integral may be codified by Riemann and Liouville, an agreedupon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants of integration in their results. An elimination of constants of integration opens the door to an operator that reconciles all known fractional derivatives and shows surprising results in areas unobserved before, including the appearance of the Riemann Zeta Function and fractional Laplace and Fourier Transforms. A new class of functions, known as Zero Functions and closely related to the Dirac Delta Function, are necessary for one to perform elementary operations of functions without using constants. The operator also allows for a generalization of the Volterra integral equation, and provides a method of solving for Riemann's "complimentary" function introduced during his research on fractional derivatives.
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In The Delta Conjecture (arxiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values $(q,t,z,w) = (1,1,0,0)$ is equal to the Catalan number $\frac{1}{n+1}\binom{2n}{n}$. We prove the compositional version of this conjecture (which implies the noncompositional version) that states that the coefficient of $s_{r,1^{nr}}$ in the expression $\Delta_{h_\ell} \nabla C_\alpha$ is equal to a weighted sum over decorated Dyck paths.
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In this paper we study a subFinsler geometric problem on the freenilpotent group of rank 2 and step 3. Such a group is also called Cartan group and has a natural structure of Carnot group, which we metrize considering the $\ell_\infty$ norm on its first layer. We adopt the point of view of timeoptimal control theory. We characterize extremal curves via Pontryagin maximum principle. We describe abnormal and singular arcs, and construct the bangbang flow.
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We consider the dynamic multichannel access problem, which can be formulated as a partially observable Markov decision process (POMDP). We first propose a modelfree actorcritic deep reinforcement learning based framework to explore the sensing policy. To evaluate the performance of the proposed sensing policy and the framework's tolerance against uncertainty, we test the framework in scenarios with different channel switching patterns and consider different switching probabilities. Then, we consider a timevarying environment to identify the adaptive ability of the proposed framework. Additionally, we provide comparisons with the DeepQ network (DQN) based framework proposed in [1], in terms of both average reward and the time efficiency.
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Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the meanfield limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregationdiffusion equations. Over the past fifteen years, aggregationdiffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating wellposedness from finite time blowup. In the present work, we review known analytical results for aggregationdiffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, as well as localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also revie
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In this paper we focus on a certain selfdistributive multiplication on coalgebras, which leads to socalled rack bialgebra. We construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra. Our motivation is deformation quantization of Leibniz algebras in the sense of [6]. Namely, the canonical rack bialgebras we have constructed for any Leibniz algebra lead to a simple explicit formula of the rackstarproduct on the dual of a Leibniz algebra recently constructed by Dherin and Wagemann in [6]. We clarify this framework setting up a general deformation theory for rack bialgebras and show that the rackstarproduct turns out to be a deformation of the trivial rack bialgebra product.
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The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are primes, there always exists a subnormal series: $\langle {e} \rangle = G_{o} < G_{1} < \dots < G_{n} = G$ such that $G_{i}/G_{i1}$ is isomorphic to a cyclic group of order $p_{i}$, $i = 1,2,\dots,n$. Associated with this series, there exists a system of generators consisting $n$ elements $x_{1}, x_{2}, \dots, x_{n}$ (say), such that $G_{i} = \langle x_{1}, x_{2}, \dots, x_{i} \rangle$, $i = 1,2,\dots,n$, which is called a "long system of generators". In terms of this system of generators and conjugacy class sum of $x_{i}$ in $G_{i}$, $i = 1,2, \dots, n$, we present an algorithm for constructing the irreducible matrix representations of $G$ over $\mathbb{C}$ within the group algebra $\mathbb{C}[G]$. This algorithmic construction needs the knowled
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Equipping a nonequivariant topological $E_\infty$operad with the trivial $G$action gives an operad in $G$spaces. For a $G$spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called na\"{i}vecommutative ring $G$spectra. In this paper we take $G=SO(2)$ and we show that commutative algebras in the algebraic model for rational $SO(2)$spectra model rational na\"{i}vecommutative ring $SO(2)$spectra. In particular, this applies to show that the $SO(2)$equivariant cohomology associated to an elliptic curve $C$ from previous work of the second author is represented by an $E_\infty$ring spectrum. Moreover, the category of modules over that $E_\infty$ring spectrum is equivalent to the derived category of sheaves over the elliptic curve $C$ with the Zariski torsion point topology.
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We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schr\"{o}dinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary. The uniqueness proof relies on proving a suitable Carleman estimate for functions which vanish only on a part of boundary and constructing complex geometric optics solutions which vanish on a part of the boundary.
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We review and apply CheegerGromov theory on $l^2$cohomology of infinite coverings of complete manifolds with bounded curvature and finite volume. Applications focus on $l^2$cohomology of (pullback of) harmonic Higgs bundles on some covering of Zariski open sets of K\"ahler manifolds. The $l^2$Dolbeault to DeRham spectral sequence of these Higgs bundles is seen to degenerate at $E_2$.
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We introduce approximation schemes for a type of countablyinfinitedimensional linear programs (CILPs) whose feasible points are unsigned measures and whose optimal values are bounds on the averages of these measures. In particular, we explain how to approximate the program's optimal value, optimal points, and minimal point (should one exist) by solving finitedimensional linear programs. We show that the approximations converge to the CILP's optimal value, optimal points, and minimal point as the size of the finitedimensional program approaches that of the CILP. Inbuilt in our schemes is a degree of error control: they yield lower and upper bounds on the optimal values and we give a simple bound on the approximation error of the minimal point. To motivate our work, we discuss applications of our schemes taken from the Markov chain literature: stationary distributions, occupation measures, and exit distributions.
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We provide an atomic decomposition of the product Hardy spaces $H^p(\widetilde{X})$ which were recently developed by Han, Li, and Ward in the setting of product spaces of homogeneous type $\widetilde{X} = X_1 \times X_2$. Here each factor $(X_i,d_i,\mu_i)$, for $i = 1$, $2$, is a space of homogeneous type in the sense of Coifman and Weiss. These Hardy spaces make use of the orthogonal wavelet bases of Auscher and Hyt\"onen and their underlying reference dyadic grids. However, no additional assumptions on the quasimetric or on the doubling measure for each factor space are made. To carry out this program, we introduce product $(p,q)$atoms on $\widetilde{X}$ and product atomic Hardy spaces $H^{p,q}_{{\rm at}}(\widetilde{X})$. As consequences of the atomic decomposition of $H^p(\widetilde{X})$, we show that for all $q > 1$ the product atomic Hardy spaces coincide with the product Hardy spaces, and we show that the product Hardy spaces are independent of the particular choices of both
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In this article, we study the enumeration by length of several walk models on the square lattice. We obtain bijections between walks in the upper halfplane returning to the $x$axis and walks in the quarter plane. A recent work by Bostan, Chyzak, and Mahboubi has given a bijection for models using small north, west, and southeast steps. We adapt and generalize it to a bijection between halfplane walks using those three steps in two colours and a quarterplane model over the symmetrized step set consisting of north, northwest, west, south, southeast, and east. We then generalize our bijections to certain models with large steps: for given $p\geq1$, a bijection is given between the halfplane and quarterplane models obtained by keeping the small southeast step and replacing the two steps north and west of length 1 by the $p+1$ steps of length $p$ in directions between north and west. This model is close to, but distinct from, the model of generalized tandem walks studied by Bousqu
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We show that if $h(x,y)=ax^2+bxy+cy^2\in \mathbb{Z}[x,y]$ satisfies $b^2\neq 4ac$, then any subset of $\{1,2,\dots,N\}$ with no nonzero differences in the image of $h$ has size at most a constant depending on $h$ times $N\exp(c\sqrt{\log N})$, where $c=c(h)>0$. We achieve this goal by adapting an $L^2$ density increment strategy previously used to establish analogous results for sums of one or more singlevariable polynomials. Our exposition is thorough and selfcontained, in order to serve as an accessible gateway for readers who are unfamiliar with previous implementations of these techniques.
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The cycle joining method and the crossjoin pairing are two main construction techniques for de Bruijn sequences. This work shows how to combine Zech's logarithms and each of the two techniques to efficiently construct binary de Bruijn sequences of large orders. A basic implementation is supplied as a proofofconcept. In the cycle joining method, the cycles are generated by an LFSR with a chosen period. We prove that determining Zech's logarithms is equivalent to identifying conjugate pairs shared by any pair of cycles. The approach quickly finds enough number of conjugate pairs between any two cycles to ensure the existence of trees containing all vertices in the adjacency graph of the LFSR. When the characteristic polynomial of the LFSR is a product of distinct irreducible polynomials, the approach via Zech's logarithms combines nicely with a recently proposed method to determine the conjugate pairs. This allows us to efficiently generate de Bruijn sequences with larger orders. Alon
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We show that fixed dimensional klt weak Fano pairs with alphainvariants and volumes bounded away from $0$ and the coefficients of the boundaries belong to the set of hyperstandard multiplicities $\Phi(\mathscr{R})$ associated to a fixed finite set $\mathscr{R}$ form a bounded family. We also show $\alpha(X,B)^{d1}\mathrm{vol}((K_X+B))$ are bounded from above for all klt weak Fano pairs $(X,B)$ of a fixed dimension $d$.
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A paper of U. First & Z. Reichstein proves that if $R$ is a commutative ring of dimension $d$, then any Azumaya algebra $A$ over $R$ can be generated as an algebra by $d+2$ elements, by constructing such a generating set, but they do not prove that this number of generators is required, or even that for an arbitrarily large $r$ that there exists an Azumaya algebra requiring $r$ generators. In this paper, for any given fixed $n\ge 2$, we produce examples of a base ring $R$ of dimension $d$ and an Azumaya algebra of degree $n$ over $R$ that requires $r(d,n) = \lfloor \frac{d}{2n2} \rfloor + 2$ generators. While $r(d,n) < d+2$ in general, we at least show that there is no uniform upper bound on the number of generators required for Azumaya algebras. The method of proof is to consider certain varieties $B^r_n$ that are universal varieties for degree$n$ Azumaya algebras equipped with a set of $r$ generators, and specifically we show that a natural map on Chow group $CH^{(r1)(n1)}
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In 1968, H. Brezis introduced a notion of operator pseudomonotonicity which provides a unified approach to monotone and nonmonotone variational inequalities (VIs). A closely related notion is that of KyFan hemicontinuity, a continuity property which arises if the famous KyFan minimax inequality is applied to the VI framework. It is clear from the corresponding definitions that KyFan hemicontinuity implies Brezis pseudomonotonicity, but quite surprisingly, a recent publication by Sadeqi and Paydar (J. Optim. Theory Appl., 165(2):344358, 2015) claims the equivalence of the two properties. The purpose of the present note is to show that this equivalence is false; this is achieved by providing a concrete example of a nonlinear operator which is Brezis pseudomonotone but not KyFan hemicontinuous.
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We study Stochastic Gradient Descent (SGD) with diminishing step sizes for convex objective functions. We introduce a definitional framework and theory that defines and characterizes a core property, called curvature, of convex objective functions. In terms of curvature we can derive a new inequality that can be used to compute an optimal sequence of diminishing step sizes by solving a differential equation. Our exact solutions confirm known results in literature and allows us to fully characterize a new regularizer with its corresponding expected convergence rates.
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Millimeterwave communications rely on narrowbeam transmissions to cope with the strong signal attenuation at these frequencies, thus demanding precise alignment between transmitter and receiver. However, the beamalignment procedure may entail a huge overhead and its performance may be degraded by detection errors. This paper proposes a coded energyefficient beamalignment scheme, robust against detection errors. Specifically, the beamalignment sequence is designed such that the errorfree feedback sequences are generated from a codebook with the desired error correction capabilities. Therefore, in the presence of detection errors, the errorfree feedback sequences can be recovered with high probability. The assignment of beams to codewords is designed to optimize energy efficiency, and a waterfilling solution is proved. The numerical results with analog beams depict up to 4dB and 8dB gains over exhaustive and uncoded beamalignment schemes, respectively.
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We derive a nonlinear integrodifferential transport equation describing collective evolution of weights under gradient descent in largewidth neuralnetworklike models. We characterize stationary points of the evolution and analyze several scenarios where the transport equation can be solved approximately. We test our general method in the special case of linear freeknot splines, and find good agreement between theory and experiment in observations of global optima, stability of stationary points, and convergence rates.
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