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A function $U:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ is called $\left( 1,\omega_{1}\right) $\emph{weakly universal }if for every function $F:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ there is an injective function $h:\omega_{1}\longrightarrow\omega_{1}$ and a function $e:\omega \longrightarrow\omega$ such that $F\left( \alpha,\beta\right) =e\left( U\left( h\left( \alpha\right) ,h\left( \beta\right) \right) \right) $ for every $\alpha,\beta\in\omega_{1}$. We will prove that it is consistent that there are no $\left( 1,\omega_{1}\right) $\emph{}weakly universal functions, this answers a question of Shelah and Stepr\={a}ns. In fact, we will prove that there are no $\left( 1,\omega_{1}\right) $\emph{}weakly universal functions in the Cohen model and after adding $\omega_{2}$ Sacks reals sidebyside. However, we show that there are $\left( 1,\omega _{1}\right) $\emph{}weakly universal functions in the Sacks model. In particular, the existence of such graphs is co
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The aim of this paper is to present the global bounds for renormalized solutions to the following quasilinear elliptic problem: \begin{align*} \begin{cases} \div(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ u &=0 \quad \text{on} \ \ \partial \Omega, \end{cases} \end{align*} in LorentzMorrey spaces, where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$), $\mu$ is a finite Radon measure, $A$ is a monotone Carath\'eodory vector valued function defined on $W^{1,p}_0(\Omega)$ and the $p$capacity uniform thickness condition is imposed on our domain. There have been research activities on the gradient estimates in LorentzMorrey spaces with various hypotheses. For instance, in \cite{55Ph1} Nguyen Cong Phuc proposed the Morrey global bounds of solution to this equation, but for the regular case $2\frac{1}{n}<p\le n$, in \cite{MP2018}, our first result provides us with the good$\lambda$ bounds of solution in Lorentz space for $\frac{3n2}{2n1}<p \le 2  \frac{1}{n}$; and in
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This paper derives noncentral asymptotic results for nonlinear integral functionals of homogeneous isotropic Gaussian random fields defined on hypersurfaces in $\mathbb{R}^d$. We obtain the rate of convergence for these functionals. The results extend recent findings for solid figures. We apply the obtained results to the case of sojourn measures and demonstrate different limit situations.
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In this article we lay out the details of Fukaya's A $\infty$structure of the Morse complexe of a manifold possibily with boundary. We show that this A $\infty$structure is homotopically independent of the made choices. We emphasize the transversality arguments that some fiber product constructions make valid.
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We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic kclique of each color, then G has at least 4(k1) vertices. This confirms a conjecture of Bucic, Lidicky, Long, and Wagner (arXiv:1805.11278[math.CO]) and thereby solves the 2dimensional case of their problem about partitions of discrete boxes with the kpiercing property. We also characterize the case of equality in our result.
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Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon's conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.
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We obtain an improved version of a recent result concerning the existence of nonnegative nonradial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left x\right ^{\alpha }dx)$ to the equation \[ \triangle u+\displaystyle\frac{A}{\left x\right ^{\alpha }}u=f\left( u\right) \quad \text{in }\mathbb{R}^{N},\quad N\geq 3,\quad A,\alpha >0, \] where $f$ is a continuous nonlinearity satisfying $f\left( 0\right) =0$.
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In this paper, we develop a new computational approach which is based on minimizing the difference of two convex functionals (DC) to solve a broader class of phase retrieval problems. The approach splits a standard nonlinear least squares minimizing function associated with the phase retrieval problem into the difference of two convex functions and then solves a sequence of convex minimization subproblems. For each subproblem, the Nesterov's accelerated gradient descent algorithm or the BarzilaiBorwein (BB) algorithm is used. In the setting of sparse phase retrieval, a standard $\ell_1$ norm term is added into the minimization mentioned above. The subproblem is approximated by a proximal gradient method which is solved by the shrinkagethreshold technique directly without iterations. In addition, a modified AttouchPeypouquet technique is used to accelerate the iterative computation. These lead to more effective algorithms than the Wirtinger flow (WF) algorithm and the GaussNewton (
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In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian in bounded domains of a Euclidean space which strengthens the wellknown AshbaughBeguria inequality about the ratio of the first two Dirichlet eigenvalues of the same domains and supports strongly a conjecture of AshbaughBenguria.
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Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We prove the convergence of this microscopic distance, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean firstpassage percolation for nonhomogeneous Poisson point processes.
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We introduce first the largecardinal notion of $\Sigma_n$supercompactness as a higherlevel analog of the wellknown Magidor's characterization of supercompact cardinals, and show that a cardinal is $C^{(n)}$extendible if and only if it is $\Sigma_{n+1}$supercompact. This yields a new characterization of $C^{(n)}$extendible cardinals which underlines their role as natural milestones in the region of the largecardinal hierarchy between the first supercompact cardinal and Vop\v{e}nka's Principle ($\rm{VP}$). We then develop a general setting for the preservation of $\Sigma_n$supercompact cardinals under class forcing iterations. As a result we obtain new proofs of the consistency of the GCH with $C^{(n)}$extendible cardinals (cf.~\cite{Tsa13}) and the consistency of $\rm{VP}$ with the GCH (cf.~\cite{Broo}). Further, we show that $C^{(n)}$extendible cardinals are preserved after forcing with standard Easton class forcing iterations for any $\Pi_1$definable possible behaviour of
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The dichromatic number of a digraph $D$, denoted by $\chi_A(D)$, is the minimum $k$ such that $D$ admits a $k$coloring of its vertex set in such a way that each color class is acyclic. In 1976, Bondy proved that the chromatic number of a digraph $D$ is at most its circumference, the length of a longest cycle. In this paper we will construct three graphs from $D$ whose chromatic numbers will bound $\chi_A(D)$. Moreover, we prove: i) for integers $k\geq 2$, $s\geq 1$ and $r_1, \ldots, r_s$ with $k\geq r_i\geq 0$ and $r_i\neq 1$ for each $i\in[s]$, that if all cycles in $D$ have length $r$ modulo $k$ for some $r\in\{r_1,\ldots,r_s\}$, then $\chi_A(D)\leq 2s+1;$ ii) if $D$ has girth $g$, the length of a shortest cycle, and circumference $c$, then $\chi_A(D)\leq \lceil \frac{c1}{g1} \rceil +1$, which improves, substantially, the bound proposed by Bondy; iii) if $D$ has girth $g$ and there are integers $k$ and $p,$ with $k\geq g1\geq p\geq 1$ such that $D$ contains no cycle of length $r$
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Nontransversal intersection of the free and fixed boundary is shown to hold and a classification of blowup solutions is given for obstacle problems generated by fully nonlinear uniformly elliptic operators in two dimensions which appear in the meanfield theory of superconducting vortices.
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For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasicoherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all quasiprojective schemes over commutative noetherian rings. The main tools of the proof are the GabrielPopescu embedding and Roos' characterization of Grothendieck categories satisfying Ab6 and Ab4*.
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This paper is the first step in the project of categorifying the bialgebra structure on the half of quantum group $U_{q}(\mathfrak{g})$ by using geometry and Hall algebras. We equip the category of Dmodules on the moduli stack of objects of the category $Rep_{\mathbb{C}}(Q)$ of representations of a quiver with the structure of an algebra object in the category of stable $\infty$categories. The data for this construction is provided by an extension of the Waldhausen construction for the category $Rep_{\mathbb{C}}(Q)$. We discuss the connection to the KhovanovLaudaRouquier categorification of half of the quantum group $U_{q}(\mathfrak{g})$ associated to the quiver $Q$ and outline our approach to the categorification of the bialgebra structure.
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In the early days of what ultimately became Waymo, Google's selfdriving car division (known at the time as "Project Chauffeur"), there were "more than a dozen accidents, at least three of which were serious," according to a new article in The New Yorker . From a report: The magazine profiled Anthony Levandowski, the former Google engineer who was at the center of the Waymo v. Uber trade secrets lawsuit. According to the article, back in 2011, Levandowski also modified the autonomous software to take the prototype Priuses on "otherwise forbidden routes." Citing an anonymous source, The New Yorker reports that Levandowski sat behind the wheel as the safety driver, along with Isaac Taylor, a Google executive. But while they were in the car, the Prius "accidentally boxed in another vehicle," a Camry. As The New Yorker wrote: "A human driver could easily have handled the situation by slowing down and letting the Camry merge into traffic, but Google's software wasn't prepared for this scena
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Gabrielle Hamilton, April Bloomfield, and the Problem with Leaving Women to
1018 THE NEW YORKER 189 
An anonymous reader quotes a report from Bloomberg: Apple on Wednesday began allowing users in the U.S. to download a copy of all of the data that they have stored with the company from a single online portal. U.S. users will be able to download data such as all of their address book contacts, calendar appointments, music streaming preferences and details about past Apple product repairs. Previously, customers could get their data by contacting Apple directly. In May, when Apple first launched the online privacy portal, it only allowed U.S. users to either correct their data or delete their Apple accounts.
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People's Facebook posts might predict whether they are suffering from depression, researchers reported this week. From a report: The researchers found that the words people used seemed to indicate whether they would later be diagnosed with depression. The findings offer a way to flag people who may be in need of help, but they also raise important questions about people's health privacy, the team reported in the Proceedings of the National Academy of Sciences. People who were later clinically diagnosed with depression used more "I" language, according to Johannes Eichstaedt of the University of Pennsylvania and his colleagues. They also used more words reflecting loneliness, sadness and hostility. "We observed that users who ultimately had a diagnosis of depression used more firstperson singular pronouns, suggesting a preoccupation with the self," they wrote. That is an indicator of depression in some people. The team recruited 683 people who visited an emergency room for their study
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Three state treasurers and a top official from New York have joined a shareholders' motion to install an independent chairman at Facebook, claiming the move would improve governance and accountability. [...] The move comes as Facebook was presented with a new legal challenge. The technology company has been accused of misleading advertisers by inflating the viewing figures for videos on its site. A group of US advertisers launched a fraud claim against the social media giant on Tuesday, stating that it had overstated the average viewing time of advertising videos on the site by between 100 and 900pc before reporting them in 2016. All tech companies are pretty terrible as far as companies go, but Facebook really seems to be going out of its way to lead the pack. As far as I'm concerned, we shut it down. Would anyone really miss it?
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An anonymous reader shares a report: Amazon cracked down on fake reviews two years ago by prohibiting shoppers from getting free products directly from merchants in exchange for writing reviews. It was a major turning point for the world's largest online retailer, which had previously seen "incentivized reviews" as a key way for consumers to discover new products. Amazon changed course because it realized some merchants were using such reviews to game its search algorithm, undermining faith in the customer feedback that helps drive ecommerce. Amazon instead used its "Vine" program, in which Amazon serves as a middleman between prolific Amazon reviewers and vendors eager for exposure. Amazon would still allow freebies in exchange for feedback so long as there was no direct contact between its retail partners and reviewers, theoretically lessening the chance of quidproquo. Amazon would select shoppers eligible for the program, and Amazon vendors would pay a fee and provide free produc
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Zorro shares a report from The Mercury News: Not only did Facebook inflate adwatching metrics by up to 900 percent (Warning: source may be paywalled, alternative source), it knew for more than a year that its averageviewership estimates were wrong and kept quiet about it, a new legal filing claims. A group of small advertisers suing the Menlo Park social media titan alleged in the filing that Facebook "induced" advertisers to buy video ads on its platform because advertisers believed Facebook users were watching video ads for longer than they actually were. That "unethical, unscrupulous" behavior by Facebook constituted fraud because it was "likely to deceive" advertisers, the filing alleged. The latest allegations arose out of a lawsuit that the advertisers filed against Mark Zuckerbergled Facebook in federal court in 2016 over alleged inflation of adwatching metrics. "Suggestions that we in any way tried to hide this issue from our partners are false," the company told The Wall S
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OpenSourceAllTheWay writes: There are many fantastic opensource tools out there for everything from scanning documents to making interactive music to creating 3D assets for games. Many of these tools have an Achilles heel though  while the code quality is great and the tool is fully functional, the user interface (UI) and user experience (UX) are typically significantly inferior to what you get in competing commercial tools. In an nutshell, with open source, the code is great, the tool is free, there is no DRM/activation/telemetry bullshit involved in using the tool, but you very often get a weak UI/UX with the tool that  unfortunately  ultimately makes the tool far less of a joy to use daily than should be the case. A prime example would be the FOSS 3D tool Blender, which is great technically, but ultimately flops on its face because of a poorly designed UI that is a decade behind commercial 3D software. So here is the question: should opensource developer teams for larger FOS
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Essential Products, a startup founded in 2015 by Android creator Andy Rubin, was started to create a smartphone with highend design features that wasn't associated with a particular operatingsystem maker. Unfortunately, reaching that goal has been harder than anticipated as the company has laid off about 30 percent of its staff. Fortune reports: Cuts were particularly deep in hardware and marketing. The company's website indicates it has about 120 employees. A company spokesperson didn't confirm the extent of layoffs, but said that the decision was difficult for the firm to make and, "We are confident that our sharpened product focus will help us deliver a truly game changing consumer product." The firm was Rubin's first startup after leaving Google in 2014, which had acquired his cofounded firm, Android, in 2005. Essential's first phone came out in August 2017, a few weeks later than initially promised. It received mixed reviews, with most critics citing its lower quality and missi
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An anonymous reader shares a report: Sellers have been modifying lower end NVIDIA graphics cards and selling them more powerful cards online. In a recent version of the GPUZ graphics card information utility, TechPowerUp has added the ability to now detect these fake NVIDIA cards. This new feature allows buyers of cards to detect if the card is actually a relabled NVIDIA G84, G86, G92, G94, G96, GT215, GT216, GT218, GF108, GF106, GF114, GF116, GF119, or GK106 GPU by displaying an exclamation point where the NVIDIA logo would normally appear and also prepends the string "[FAKE]" before the card's name.
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Millions of dollars in funding and billions of dollars in valuations have made scooters the next big thing since the last big thing. From a report: When Michael Ramsey, an analyst for technology research firm Gartner, started in February to put together his 2018 "hype cycle" report for the future of transportation, he had plenty of topics to choose from: electric vehicles, flying cars, 5G, blockchain, and, of course, autonomous vehicles. But one type of transportation is conspicuously absent from the results of the report: electric scooters. "At the time, outside of California, these scooters were really not that common," Ramsey said. "That's how much has happened." As for autonomous vehicles, which have enjoyed years of hype as the next big thing, Ramsey labeled them sliding into "the trough of disillusionment," which Ramsey described as "when expectations don't meet the truth." In a matter of months, electric scooter startups have gone from tech oddity to global phenomenon. In some c
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At its annual Adobe Max conference, Adobe announced plans to bring a complete version of Photoshop to the iPad in 2019. Photoshop CC for iPad will feature a revamped interface designed specifically for a touch experience, but it will bring the power and functionality people are accustomed to on the desktop. This is the real, full photoshop  the same codebase as the regular Photoshop, but running on the iPad with a touch UI. The Verge's Dami Lee and artist colleagues at The Verge got to test this new version of Photoshop, and they are very clear to stress that the biggest news here isn't even having the "real" Photoshop on the iPad, but the plans Adobe has for the PSD file format. But the biggest change of all is a total rethinking of the classic .psd file for the cloud, which will turn using Photoshop into something much more like Google Docs. Photoshop for the iPad is a big deal, but Cloud PSD is the change that will let Adobe bring Photoshop everywhere. This does se
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In this work, we show that along a particular choice of Hermitian curvature flow, the nonpositivity of ChernRicci curvature will be preserved if the initial metric has nonpositive bisectional curvature. As a corollary, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisectional curvature and quasinegative ChernRicci curvature is ample.
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A tetravalent $2$arctransitive graph of order $728$ is either the known $7$arctransitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$transitive graph of order $182$ denoted $A[182,1]$ or $A[182,2]$ in the $2009$ list of Poto\v{c}nik.
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Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: $(i)$ ${\rm height}(I)=n1$; $(ii)$ $I$ contains at least $n3$ pure powers of the variables $x_1^d,...,x_{n3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables.
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A sequence is called $r$sparse if every contiguous subsequence of length $r$ has no repeated letters. A $DS(n, s)$sequence is a $2$sparse sequence with $n$ distinct letters that avoids alternations of length $s+2$. Pettie and Wellman (2018) asked whether there exist $r$sparse $DS(n, s)$sequences of length $\Omega(s n^{2})$ for $s \geq n$ and $r > 2$, which would generalize a result of Roselle and Stanton (1971) for the case $r = 2$. We construct $r$sparse $DS(n, s)$sequences of length $\Omega(s n^{2})$ for $s \geq n$ and $r > 2$. Our construction uses linear hypergraph edgecoloring bounds. We also use the construction to generalize a result of Pettie and Wellman by proving that if $s = \Omega(n^{1/t} (t1)!)$, then there are $r$sparse $DS(n, s)$sequences of length $\Omega(n^{2} s / (t1)!)$ for all $r \geq 2$. In addition, we find related results about the lengths of sequences avoiding $(r, s)$formations.
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We study the continuoustime evolution of the recombination equation of population genetics. This evolution is given by a differential equation that acts on a product probability space, and its solution can be described by a Markov chain on a set of partitions that converges to the finest partition. We study an explicit form of the law of this process by using a family of trees. We also describe the geometric decay rate to the finest partition and the quasistationary behavior of the Markov chain when conditioned on the event that the chain does not hit the limit.
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We propose an index for gapped quantum lattice systems that conserve a $\mathrm{U}(1)$charge. This index takes integer values and it is therefore stable under perturbations. Our formulation is general, but we show that the index reduces to (i) an index of projections in the noninteracting case, (ii) the filling factor for translational invariant systems, (iii) the quantum Hall conductance in the twodimensional setting without any additional symmetry. Example (ii) recovers the LiebSchultzMattis theorem, (iii) provides a new and short proof of quantization of Hall conductance in interacting manybody systems. Additionally, we provide a new proof of Bloch's theorem on the vanishing of ground state currents.
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In this work we present an algorithm to construct sparsepaving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any SparsePaving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparsepaving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the $r$subsets of $S$, $\binom{S}{r}=\sqcup_{i=1}^{\gamma }\mathcal{U}_{i}$ such that each $\mathcal{U}_{i}$ defines a sparsepaving matroid of rank $r$.
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In this paper we study the contribution of monopole bubbling to the expectation value of supersymmetric 't Hooft defects in Lagrangian theories of class $\mathcal{S}$ on $\mathbb{R}^3\times S^1$. This can be understood as the Witten index of an SQM living on the world volume of the 't Hooft defect that couples to the bulk 4D theory. The computation of this Witten index has many subtleties originating from a continuous spectrum of scattering states along the noncompact vacuum branches. We find that even after properly dealing with the spectral asymmetry, the standard localization result for the 't Hooft defect does not agree with the result obtained from the AGT correspondence. In this paper we will explicitly show that one must correct the localization result by adding an extra term to the standard JeffreyKirwan residue formula. This extra term accounts for the contribution of ground states localized along the noncompact branches. This extra term restores both the expected symmetry
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For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. As a consequence, we prove a conjecture of Charles and KerrPearlstein, that zero loci of normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, this gives a short proof of a special, algebraically motivated case of a result of Saito, BrosnanPearlstein, and Schnell, conjectured by GreenGriffiths, on zero loci of admissible normal functions.
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Recently, Misanantenaina and Wagner characterized the set of induced $N$free and bowtiefree posets as a certain class of recursively defined subposets which they term "$\mathcal{V}$posets". Here we offer a new characterization of $\mathcal{V}$posets by introducing a property we refer to as \emph{autonomy}. A poset $\mathcal{P}$ is said to be \emph{autonomous} if there exists a directed acyclic graph $D$ (with adjacency matrix $U$) whose transitive closure is $\mathcal{P}$, with the property that any total ordering of the vertices of $D$ so that Gaussian elimination of $U^TU$ proceeds without row swaps is a linear extension of $\mathcal{P}$. Autonomous posets arise from the theory of pressing sequences in graphs, a problem with origins in phylogenetics. The pressing sequences of a graph can be partitioned into families corresponding to posets; because of the interest in enumerating pressing sequences, we investigate when this partition has only one block, that is, when the pressing
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In this paper, we prove a new cohomology theory that is an invariant of a planar trivalent graph with a given perfect matching. This bigraded cohomology theory appears to be very powerful: the graded Euler characteristic of the cohomology is a one variable polynomial (called the 2factor polynomial) that, if nonzero when evaluated at one, implies that the perfect matching is even. This polynomial can be used to construct a polynomial invariant of the graph called the even matching polynomial. We conjecture that the even matching polynomial is positive when evaluated at one for all bridgeless planar trivalent graphs. This conjecture, if true, implies the existence of an even perfect matching for the graph, and thus the trivalent planar graph is 3edgecolorable. This is equivalent to the four color theorema famous conjecture in mathematics that was proven using a computer program in 1970s. While these polynomial invariants may not have enough strength as invariants to prove such a co
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Under the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable GromovHausdorff topology, we establish asymptotic bounds on the distribution of the $\varepsilon$blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the $\varepsilon$blanket times of the random walks if the $\varepsilon$blanket time of the limiting diffusion is continuous with probability one at $\varepsilon$. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical GaltonWatson trees, the Erd\H{o}sR\'enyi random graph in the critical window and the configuration model in the scaling critical window. We highlight that proving continuity of the $\varepsilon$blanket time of the limiting diffusion relies on the scale invariance of a finite measure that gives rise to realizations of the limiting compact random metric space, and t
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Let $\{T(t)\}_{t\ge 0}$ be a $C_0$semigroup on a separable Hilbert space $H$. We characterize that $T(t)$ is an $m$isometry for every $t$ in terms that the mapping $t\in \Bbb R^+ \rightarrow \T(t)x\^2$ is a polynomial of degree less than $m$ for each $x\in H$. This fact is used to study $m$isometric right translation semigroup on weighted $L^p$spaces. We characterize the above property in terms of conditions on the infinitesimal generator operator or in terms of the cogenerator operator of $\{ T(t)\}_{t\geq 0}$. Moreover, we prove that a nonunitary $2$isometry on a Hilbert space satisfying the kernel condition, that is, $$ T^*T(KerT^*)\subset KerT^*\;, $$ then $T$ can be embedded into a $C_0$semigroup if and only if $dim (KerT^*)=\infty$.
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It is known that fuzzy set theory can be viewed as taking place within a topos. There are several equivalent ways to construct this topos, one is as the topos of \'{e}tal\'{e} spaces over the topological space $Y=[0,1)$ with lower topology. In this topos, the fuzzy subsets of a set $X$ are the subobjects of the constant \'{e}tal\'{e} $X\times Y$ where $X$ has the discrete topology. Here we show that the type2 fuzzy truth value algebra is isomorphic to the complex algebra formed from the subobjects of the constant relational \'{e}tal\'{e} given by the type1 fuzzy truth value algebra $\mathfrak{I}=([0,1],\wedge,\vee,\neg,0,1)$. More generally, we show that if $L$ is the lattice of open sets of a topological space $Y$ and $\mathfrak{X}$ is a relational structure, then the convolution algebra $L^\mathfrak{X}$ is isomorphic to the complex algebra formed from the subobjects of the constant relational \'{e}tal\'{e} given by $\mathfrak{X}$ in the topos of \'{e}tal\'{e} spaces over $Y$.
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Facebook is developing hardware for the TV, news outlet Cheddar reported Tuesday. From the report: The world's largest social network is building a cameraequipped device that sits atop a TV and allows video calling along with entertainment services like Facebook's YouTube competitor, according to people familiar with the matter. The project, internally codenamed "Ripley," uses the same core technology as Facebook's recently announced Portal video chat device for the home. Portal begins shipping next month and uses A.I. to automatically detect and follow people as they move throughout the frame during a video call. Facebook currently plans to announce project Ripley in the spring of 2019, according to a person with direct knowledge of the project. But the device is still in development and the date could be changed.
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This paper presents a modified quasireversibility method for computing the exponentially unstable solution of a nonlocal terminalboundary value parabolic problem with noisy data. Based on data measurements, we perturb the problem by the socalled filter regularized operator to design an approximate problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semilinear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there is likely a myriad of models derived from this simpler case, such as source localization problems for brain tumors and heat conduction problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the FaedoGalerkin meth
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This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of SasakiEinstein metrics.
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The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix { entries} is obtained. In the limit $M\to\infty$ this identity induces some requirements, which should satisfy the scattering data of the resulting infinitedimensional Jacobi operator in the halfline, which super and subdiagonal matrix elements are equal to 1. We obtain such requirements in the simplest case of the discrete Schr\"odinger operator acting in ${l}^2( \mathbb{N})$, which does not have bound and semibound states, and which potential has a compact support.
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The performance of alloptical dualhop relayed freespace optical communication systems is analytically studied and evaluated. We consider the case when the total received signal undergoes turbulenceinduced channel fading, modeled by the versatile mixtureGamma distribution. Also, the misalignmentinduced fading due to the presence of pointing errors is jointly considered in the enclosed analysis. The performance of both amplifyandforward and decodeandforward relaying transmission is studied, when heterodyne detection is applied. New closedform expressions are derived regarding some key performance metrics of the considered system; namely, the system outage probability and average biterror rate.
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In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints in our framework, when present, are treated either by using the extreme barrier function or through a projection approach. The introduced extension guaranties to the regarded class of evolution strategies global convergence properties for first order stationary constraints. Preliminary numerical experiments are carried out on a set of known test problems as well as on a multidisciplinary design optimization problem
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In many contemporary optimization problems, such as hyperparameter tuning for deep learning architectures, it is computationally challenging or even infeasible to evaluate an entire function or its derivatives. This necessitates the use of stochastic algorithms that sample problem data, which can jeopardize the guarantees classically obtained through globalization techniques via a trust region or a line search. Using subsampled function values is particularly challenging for the latter strategy, that relies upon multiple evaluations. On top of that all, there has been an increasing interest for nonconvex formulations of datarelated problems. For such instances, one aims at developing methods that converge to secondorder stationary points, which is particularly delicate to ensure when one only accesses subsampled approximations of the objective and its derivatives. This paper contributes to this rapidly expanding field by presenting a stochastic algorithm based on negative curvature a
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We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the righthand side has very few regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.
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We exhibit a new construction of edgeregular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edgeregular graph with parameters $(24,8,2)$. We also show that edgeregular graphs with $1$regular cliques that are not strongly regular must have at least $24$ vertices.
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We look at the number $L(n)$ of $O$sequences of length $n$. This interesting and naturallydefined sequence $L(n)$ was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partitiontheoretic argument. In particular, it turns out that, for suitable positive constants $c_1$ and $c_2$ and all $n\ge 1$, $$e^{c_1\sqrt{n}}\le L(n)\le e^{c_2\sqrt{n}\log n}.$$ It remains an open problem to determine an exact asymptotic estimate for $L(n)$.
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Emerging applications of sensor networks for detection sometimes suggest that classical problems ought be revisited under new assumptions. This is the case of binary hypothesis testing with independent  but not necessarily identically distributed  observations under the two hypotheses, a formalism so orthodox that it is used as an opening example in many detection classes. However, let us insert a new element, and address an issue perhaps with impact on strategies to deal with "big data" applications: What would happen if the structure were streamlined such that data flowed freely throughout the system without provenance? How much information (for detection) is contained in the sample values, and how much in their labels? How should decisionmaking proceed in this case? The theoretical contribution of this work is to answer these questions by establishing the fundamental limits, in terms of error exponents, of the aforementioned binary hypothesis test with unlabeled observations draw
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The set of all perfect matchings of a plane (weakly) elementary bipartite graph equipped with a partial order is a poset, moreover the poset is a finite distributive lattice and its Hasse diagram is isomorphic to $Z$transformation directed graph of the graph. A finite distributive lattice is matchable if its Hasse diagram is isomorphic to a $Z$transformation directed graph of a plane weakly elementary bipartite graph, otherwise nonmatchable. We introduce the meetirreducible cell with respect to a perfect matching of a plane (weakly) elementary bipartite graph and give its equivalent characterizations. Using these, we extend a result on nonmatchable distributive lattices, and obtain a class of new nonmatchable distributive lattices.
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An anonymous reader quotes a report from Recode: Facebook announced Portal last week, its take on the inhome, voiceactivated speaker to rival competitors from Amazon, Google and Apple. Last Monday, we wrote: "No data collected through Portal  even call log data or app usage data, like the fact that you listened to Spotify  will be used to target users with ads on Facebook." We wrote that because that's what we were told by Facebook executives. But Facebook has since reached out to change its answer: Portal doesn't have ads, but data about who you call and data about which apps you use on Portal can be used to target you with ads on other Facebookowned properties. "Portal voice calling is built on the Messenger infrastructure, so when you make a video call on Portal, we collect the same types of information (i.e. usage data such as length of calls, frequency of calls) that we collect on other Messengerenabled devices. We may use this information to inform the ads we show you acr
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The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2microlocal frontier, a monotone concave downward curve in $\mathbb {R}^2$, provides a complete and profound classification of pointwise singularity. In \cite{Meyer1998}, \cite{GuiJaffardLevy1998} and \cite{LevySeuret2004} the authors show the following: given a monotone concave downward curve in the plane it is possible to exhibit one function (or distribution) such that its 2microlocal frontier al $x_0$ is the given curve. In this work we are able to unify the previous results, by obtaining a large class of functions (or distributions), that includes the three examples mentioned above, for which the 2microlocal frontier is the given curve. The three examples above are in this class. Further, if the curve is a line, we characterize all the functions whose 2microlocal frontier at $x_0$ is the given line.
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We prove a blowup criterion in terms of an $L_2$bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve has fixed contact angle $\alpha \in (0, \pi)$ with that line and satisfies a noflux condition. The proof is led by contradiction: A compactness argument combined with the short time existence result enables us to extend the flow, which contradicts the maximality of the solution.
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