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Let $K$ be a Henselian, nontrivially valued field with separated analytic structure. We prove the existence of definable retractions onto an arbitrary closed definable subset of $K^{n}$. Hence directly follow definable nonArchimedean versions of the extension theorems by TietzeUrysohn and Dugundji. This generalizes our previous paper dealing with complete nonArchimedean fields with separated power series and remains true for Henselian valued fields with strictly convergent analytic structure, because every such a structure can be extended in a definitional way to a separated analytic structure. Our proof uses a variant of the one from that paper, based on canonical resolution of singularities, and a modeltheoretic compactness argument.
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Calabi observed that there is a natural correspondence between the solutions of the minimal surface equation in $\mathbb{R}^3$ with those of the maximal spacelike surface equation in $\mathbb{L}^3$. We are going to show how this correspondence can be extended to the family of $\varphi $minimal graphs in $\mathbb{R}^3 $ when the function $\varphi$ is invariant under a twoparametric group of translations. We give also applications in the study and description of new examples.
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Detecting the components common or correlated across multiple data sets is challenging due to a large number of possible correlation structures among the components. Even more challenging is to determine the precise structure of these correlations. Traditional work has focused on determining only the model order, i.e., the dimension of the correlated subspace, a number that depends on how the modelorder problem is defined. Moreover, identifying the model order is often not enough to understand the relationship among the components in different data sets. We aim at solving the complete modelselection problem, i.e., determining which components are correlated across which data sets. We prove that the eigenvalues and eigenvectors of the normalized covariance matrix of the composite data vector, under certain conditions, completely characterize the underlying correlation structure. We use these results to solve the modelselection problem by employing bootstrapbased hypothesis testing.
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Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index one or two in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.
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Thomassen formulated the following conjecture: Every $3$connected cubic graph has a redblue vertex coloring such that the blue subgraph has maximum degree $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$edge path. We prove the conjecture for Generalized Petersen graphs. We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.
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Singular boundary value problems (SBVPs) arise in various fields of Mathematics, Engineering and Physics such as boundary layer theory, gas dynamics, nuclear physics, nonlinear optics, etc. The present monograph is devoted to systems of SBVPs for ordinary differential equations (ODEs). It presents existence theory for a variety of problems having unbounded nonlinearities in regions where their solutions are searched for. The main focus is to establish the existence of positive solutions. The results are based on regularization and sequential procedure. First chapter of this monograph describe the motivation for the study of SBVPs. It also include some available results from functional analysis and fixed point theory. The following chapters contain results from author's PhD thesis, National University of Sciences and Technology, Islamabad, Pakistan. These results provide the existence of positive solutions for a variety of systems of SBVPs having singularity with respect to independent
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Let $W$ be a manifold with boundary $M$ given together with a conformal class $\bar C$ which restricts to a conformal class $C$ on $M$. Then the relative Yamabe constant $Y_{\bar C}(W,M;C)$ is welldefined. We study the shorttime behavior of the relative Yamabe constant $Y_{[\bar g_t]}(W,M;C)$ under the Ricci flow $\bar g_t$ on $W$ with boundary conditions that mean curvature $H_{\bar g_t}\equiv 0$ and $\bar{g}_t_M\in C = [\bar{g}_0]$. In particular, we show that if the initial metric $\bar{g}_0$ is a Yamabe metric, then, under some natural assumptions, $\left.\frac{d}{dt}\right_{t=0}Y_{[\bar g_t]}(W,M;C)\geq 0$ and is equal to zero if and only the metric $\bar{g}_0$ is Einstein.
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Given two $\left( n+1\right) \times\left( n+1\right)$matrices $A$ and $B$ over a commutative ring, and some $k\in\left\{ 0,1,\ldots,n\right\}$, we consider the $\dbinom{n}{k}\times\dbinom{n}{k}$matrix $W$ whose entries are $\left( k+1\right) \times\left( k+1\right)$minors of $A$ multiplied by corresponding $\left( k+1\right) \times\left( k+1\right)$minors of $B$. Here we require the minors to use the last row and the last column (which is why we obtain an $\dbinom{n}{k}\times\dbinom{n}{k}$matrix, not a $\dbinom{n+1}{k+1}\times\dbinom{n+1}{k+1}$matrix). We prove that the determinant $\det W$ is a multiple of $\det A$ if the $\left( n+1,n+1\right)$th entry of $B$ is $0$. Furthermore, if the $\left( n+1,n+1\right)$th entries of both $A$ and $B$ are $0$, then $\det W$ is a multiple of $\left( \det A\right) \left( \det B\right)$. This extends a previous result of Olver and the author ( arXiv:1802.02900 ).
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We present a framework for characterizing injectivity of classes of maps (on cosets of a linear subspace) by injectivity of classes of matrices. Using our formalism, we characterize injectivity of several classes of maps, including generalized monomial and monotonic (not necessarily continuous) maps. In fact, monotonic maps are special cases of {\em componentwise affine} maps. Further, we study compositions of maps with a matrix and other composed maps, in particular, rational functions. Our framework covers classical injectivity criteria based on mean value theorems for vectorvalued maps and recent results obtained in the study of chemical reaction networks.
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We consider a onedimensional analogue of the threedimensional FokkerPlanck equation for bosons. The latter is still only partially understood, and, in particular, the physically relevant question of whether this equation has solutions which form a BoseEinstein condensate has remained unanswered. After a change of variables, we establish globalintime existence and uniqueness for our 1D model (and generalisations thereof) using the concept of viscosity solutions. We show that such solutions enjoy good regularity properties, which guarantee that in the original variables blowup can only occur at the origin and with a fixed spatial profile, up to leading order, following a power law linked to the steady states of the equation. This enables us to extend entropy methods beyond the first blowup time. As a consequence, in the masssupercritical case, solutions will blow up in $L^\infty$ in finite time and  understood in an extended, measurevalued sense  they will eventually have a c
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For a lowmobile Poisson bipolar network and under lineofsight/nonlineofsight (LOS/NLOS) pathloss model, we study repetitive retransmissions (RR) and blocked incremental redundancy (BIR). We consider spatiallycoded multipleinput multipleoutput (MIMO) zeroforcing beamforming (ZFBF) multiplexing system, whereby the packet success reception is determined based on the aggregate data rate across spatial dimensions of the MIMO system. Characterization of retransmission performance in this lowmobile configuration is practically important, but inherently complex due to a substantial rate correlation across retransmissions and intractability of evaluating the probability density function (pdf) of aggregate data rate. Adopting tools of stochastic geometry, we firstly characterize the rate correlation coefficient (RCC) for both schemes. Our results show that, compared to RR scheme, BIR scheme has higher RCC while its coverage probability is substantially larger. We demonstrate that t
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Let $(\bf U, \bf U^\imath)$ be a quasisplit quantum symmetric pair of arbitrary KacMoody type, where "quasisplit" means the corresponding Satake diagram contains no black node. We give a presentation of the $\imath$quantum group $\bf U^\imath$ with explicit $\imath$Serre relations. The verification of new $\imath$Serre relations is reduced to some new qbinomial identities. Consequently, $\bf U^\imath$ is shown to admit a bar involution under suitable conditions on the parameters.
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This paper introduces a hierarchical interpolative decomposition butterflyLU factorization (HIDBFLU) preconditioner for solving twodimensional electricfield integral equations (EFIEs) in electromagnetic scattering problems of perfect electrically conducting objects with open surfaces. HIDBFLU leverages the interpolative decomposition butterfly factorization (IDBF) to compress dense blocks of the discretized EFIE operator to expedite its application; this compressed operator also serves as an approximate LU factorization of the EFIE operator leading to an efficient preconditioner in iterative solvers. Both the memory requirement and computational cost of the HIDBFLU solver scale as $O(N\log^2 N)$ in one iteration; the total number of iterations required for a reasonably good accuracy scales as $O(1)$ to $O(\log^2N)$ in all of our numerical tests. The efficacy and accuracy of the proposed preconditioned iterative solver are demonstrated via its application to a broad range of sc
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We initiate the study of Boolean function analysis on highdimensional expanders. We give a randomwalk based definition of high dimensional expansion, which coincides with the earlier definition in terms of twosided link expanders. Using this definition, we describe an analogue of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analogue is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. We then use this decomposition to extend the FriedgutKalaiNaor theorem to highdimensional expanders. Our results demonstrate that a highdimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only $X(k1)=O(n)$ points in contrast to $\binom{n}{k}$ points
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In this note, we extend work of Farkas and Rim\'anyi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of possible values of $g\in \{1,\ldots,N\}$ has density $\Omega(\log(N)^{0.087})$ for $N>>0$. Furthermore, no divisorial condition defined using hypersurfaces of degree greater than 2 give counterexamples to the Slope Conjecture, and every divisor in our family has slope at least $6+\frac{8}{g+1}$.
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We prove that the KLR algebra associated with the cyclic quiver of length $e$ is a subquotient of the KLR algebra associated with the cyclic quiver of length $e+1$. We also give a geometric interpretation of this fact. This result has an important application in the theory of categorical representations. We prove that a category with an action of $\widetilde{\mathfrak{sl}}_{e+1}$ contains a subcategory with an action of $\widetilde{\mathfrak{sl}}_{e}$. We also give generalizations of these results to more general quivers and Lie types.
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Kontsevich and Manin gave a formula for the number $N_e$ of rational plane curves of degree $e$ through $3e1$ points in general position in the plane. When these $3e1$ points have coordinates in the rational numbers, the corresponding set of $N_e$ rational curves has a natural Galoismodule structure. We make some extremely preliminary investigations into this Galois module structure, and relate this to the deck transformations of the generic fibre of the product of the evaluation maps on the moduli space of maps. We then study the asymptotics of the number of rational points on hypersurfaces of low degree, and use this to generalise our results by replacing the projective plane by such a hypersurface.
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It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$ary trees (levelgreedy trees) with $n$~leaves, the number of `proper' words, the number of bounded degree sequences, and the number of ways of writing $1= \frac{1}{t^{x_1}}+ \dots + \frac{1}{t^{x_n}}$ with integers $0 \leq x_1 \leq x_2 \leq \dots \leq x_n$. In this work, we show that one can compute this sequence for \textbf{all} $n<N$ with essentially one power series division. In total we need at most $N^{1+\varepsilon}$ additions and multiplications of integers of $cN$ bits, $c<1$, or $N^{2+\varepsilon}$ bit operations, respectively. This improves an earlier bound by Even and Lempel who needed $O(N^3)$ operations in the integer ring or $O(N^4)$ bit operations, respectively.
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Dimension profiles were introduced in [8,11] to give a formula for the boxcounting and packing dimensions of the orthogonal projections of a set $R^n$ onto almost all $m$dimensional subspaces. However, these definitions of dimension profiles are indirect and are hard to work with. Here we firstly give alternative definitions of dimension profiles in terms of capacities of $E$ with respect to certain kernels, which lead to the boxcounting and packing dimensions of projections fairly easily, including estimates on the size of the exceptional sets of subspaces where the dimension of projection is smaller the typical value. Secondly, we argue that with this approach projection results for different types of dimension may be thought of in a unified way. Thirdly, we use a Fourier transform method to obtain further inequalities on the size of the exceptional subspaces.
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Using dual perturbation theory in a nonsunreflexive context, we establish a correspondence between 1. a class of nonlinear abstract delay differential equations (DDEs) with unbounded linear part and an unknown taking values in an arbitrary Banach space and 2. a class of abstract weak* integral equations of convolution type involving the sunstar adjoint of a translationlike strongly continuous semigroup. For this purpose we also characterize the sun dual of the underlying state space. More generally we consider bounded linear perturbations of an arbitrary strongly continuous semigroup and we comment on some implications for the particular case of abstract DDEs.
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Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for analyzing nonnegative data. A key aspect of NMF is the choice of the objective function that depends on the noise model (or statistics of the noise) assumed on the data. In many applications, the noise model is unknown and difficult to estimate. In this paper, we define a multiobjective NMF (MONMF) problem, where several objectives are combined within the same NMF model. We propose to use Lagrange duality to judiciously optimize for a set of weights to be used within the framework of the weightedsum approach, that is, we minimize a single objective function which is a weighted sum of the all objective functions. We design a simple algorithm using multiplicative updates to minimize this weighted sum. We show how this can be used to find distributionally robust NMF (DRNMF) solutions, that is, solutions that minimize the largest error among all objectives. We illustrate the effectiveness of this
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Extending earlier work by Sommers and Tymoczko, in 2016 Abe, Barakat, Cuntz, Hoge, and Terao established that each arrangement of ideal type $\mathcal{A}_\mathcal{I}$ stemming from an ideal $\mathcal{I}$ in the set of positive roots of a reduced root system is free. Recently, R\"ohrle showed that a large class of the $\mathcal{A}_\mathcal{I}$ satisfy the stronger property of inductive freeness and conjectured that this property holds for all $\mathcal{A}_\mathcal{I}$. In this article, we confirm this conjecture.
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We show that the geometry of $4n$dimensional quaternionic K\"ahler spaces with a locally free $\mathbb{R}^{n+1}$action admits a GibbonsHawkinglike description based on the GalickiLawson notion of quaternionic K\"ahler moment map. This generalizes to higher dimensions a fourdimensional construction of Calderbank and Pedersen of selfdual manifolds with two linearly independent commuting Killing vector fields. As an application, we use this new Ansatz to give an explicit equivariant completion of the twistor space construction of the local cmap proposed by Ro\v{c}ek, Vafa and Vandoren.
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We consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two nonisogenous elliptic curves. For this twoparameter family we prove Manin's unity, using the presentation of the Kummer surfaces as isotrivial elliptic fibration and as double cover of the modular elliptic surface of level two. By carrying out the rational pointcount with respect to either of the two elliptic fibrations explicitly, we obtain an interesting new identity between twoparameter counting functions.
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Inspired by a PDEODE system of aggregation developed in the biomathematical literature, an interacting particle system representing aggregation at the level of individuals is investigated. It is proved that the empirical density of the individual converges to solution of the PDEODE system.
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We present an information theoretic proof of the nonsignalling multiprover parallel repetition theorem, a recent extension of its twoprover variant that underlies many hardness of approximation results. The original proofs used de Finetti type decomposition for strategies. We present a new proof that is based on a technique we introduced recently for proving strong converse results in multiuser information theory and entails a change of measure after replacing hard information constraints with soft ones.
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Caching the content closer to the user equipments (UEs) in heterogenous cellular networks (HetNets) improves userperceived QualityofService (QoS) while lowering the operators backhaul usage/costs. Nevertheless, under the current networking strategy that promotes aggressive densification, it is unclear whether cacheenabled HetNets preserve the claimed costeffectiveness and the potential benefits. This is due to 1) the collective cost of caching which may inevitably exceed the expensive cost of backhaul in a dense HetNet, and 2) the excessive interference which affects the signal reception irrespective of content placement. We analyze these significant, yet overlooked, issues, showing that while densification reduces backhaul load and increases spectral efficiency in cacheenabled dense networks, it simultaneously reduces cachehit probability and increases the network cost. We then introduce a caching efficiency metric, area spectral efficiency per unit spent cost, and find it enou
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We recall some of the fundamental achievements of formal deformation quantization to argue that one of the most important remaining problems is the question of convergence. Here we discuss different approaches found in the literature so far. The recent developments of finding convergence conditions are then outlined in three basic examples: the Weyl star product for constant Poisson structures, the Gutt star product for linear Poisson structures, and the Wick type star product on the Poincar\'e disc.
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Our aim is to study the existence and uniqueness of the $L^{p}$  variational solution, with $p>1,$ of the following multivalued backward stochastic differential equation with $p$integrable data: \[ \left\{ \begin{align*} &dY_{t}+\partial_{y}\Psi\left( t,Y_{t}\right) dQ_{t} \ni H\left( t,Y_{t},Z_{t}\right) dQ_{t}Z_{t}dB_{t},\;t\in\left[ 0,T\right] ,\\ &Y_{T} =\eta, \end{align*} \right. \] where $Q$ is a progresivelly measurable increasing continuous stochastic process and $\partial_{y}\Psi$ is the subdifferential of the convex lower semicontinuous function $y\mapsto\Psi(t,y)$. In the framework $p\geq2$ of Maticiuc, R\u{a}\c{s}canu from [Bernoulli, 2015], the strong solution found it there is the unique variational solution, via the uniqueness property proved in the present article.
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We consider the number of critical points of a stationary planar Gaussian field, restricted to a large domain, whose heights lie in a certain interval. Asymptotics for the mean of this quantity are simple to establish via the KacRice formula, and recently Estrade and Fournier proved a second moment bound that is optimal in the case that the height interval does not depend on the size of the domain. Here we establish a bound that remains optimal in the more delicate case of height windows that are shrinking with the size of the domain.
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We show that if $M$ is a subRiemannian manifold and $N$ is a Carnot group such that the nilpotentization of $M$ at almost every point is isomorphic to $N$, then there are subsets of $N$ of positive measure that embed into $M$ by bilipschitz maps. Furthermore, $M$ is countably $N$rectifiable, i.e., all of $M$ except for a null set can be covered by countably many such maps.
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We investigate the effect of surface tension on the linear RayleighTaylor (RT) instability in stratified incompressible viscous fluids with or without (interface) surface tension. The existence of linear RT instability solutions with largest growth rate $\Lambda$ is proved under the instability condition (i.e., the surface tension coefficient $\vartheta$ is less than a threshold $\vartheta_{\mm{c}}$) by modified variational method of PDEs. Moreover we find a new upper bound for $\Lambda$. In particular, we observe from the upper bound that $\Lambda$ decreasingly converges to zero, as $\vartheta$ goes from zero to the threshold $\vartheta_{\mm{c}}$. The convergence behavior of $\Lambda$ mathematically verifies the classical RT instability experiment that the instability growth is limited by surface tension during the linear stage.
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Apple has now shut down Google’s ability to distribute its internal iOS apps, following a similar shutdown that was issued to Facebook earlier this week. A person familiar with the situation tells The Verge that early versions of Google Maps, Hangouts, Gmail, and other prerelease beta apps have stopped working today, alongside employeeonly apps like a Gbus app for transportation and Google’s internal cafe app. “We’re working with Apple to fix a temporary disruption to some of our corporate iOS apps, which we expect will be resolved soon,” says a Google spokesperson in a statement to The Verge. Apple has not yet commented on the situation. There are two sides to this story. One the one hand, I’m glad Apple is taking measures and revoking some of these companies’ developer rights. These kinds of privacyinvading apps are a terrible idea, even if people get paid for them, and no platform should allow them. On the other hand, though, I would much rather have such tactics be w
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An obsession with computer vision shows the lopsided nature of the AI boom
0201 MIT Technology 1174 
Consider the minimization of a nonconvex differentiable function over a polyhedron. A popular primaldual firstorder method for this problem is to perform a gradient projection iteration for the augmented Lagrangian function and then update the dual multiplier vector using the constraint residual. However, numerical examples show that this approach can exhibit "oscillation" and may not converge. In this paper, we propose a proximal alternating direction method of multipliers for the multiblock version of this problem. A distinctive feature of this method is the introduction of a "smoothed" (i.e., exponentially weighted) sequence of primal iterates, and the inclusion, at each iteration, to the augmented Lagrangian function a quadratic proximal term centered at the current smoothed primal iterate. The resulting proximal augmented Lagrangian function is inexactly minimized (via a gradient projection step) at each iteration while the dual multiplier vector is updated using the residual o
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In this paper we study axially symmetric solutions of AllenCahn equation with finite Morse index. It is shown that there does not exist such a solution in dimensions between $4$ and $10$. In dimension $3$, we prove that these solutions have finitely many ends. Furthermore, the solution has exactly two ends if its Morse index equals $1$.
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A nested Schur complement solver is proposed for iterative solution of linear systems arising in exponential and implicit time integration of the Maxwell equations with perfectly matched layer (PML) nonreflecting boundary conditions. These linear systems are the socalled double saddle point systems whose structure is handled by the Schur complement solver in a nested, twolevel fashion. The solver is demonstrated to have a meshindependent convergence at the outer level, whereas the inner level system is of elliptic type and thus can be treated efficiently by a variety of solvers.
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We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all firstorder systems and all secondorder scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergencefree tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.
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The presented splitting lemma extends the techniques of Gromov and Forstneri\v{c} to glue local sections of a given analytic sheaf, a key step in the proof of all Oka principles. The novelty on which the proof depends is a lifting lemma for transition maps of coherent sheaves, which yields a reduction of the proof to the work of Forstneri\v{c}. As applications we get shortcuts in the proofs of Forster and Ramspott's Oka principle for admissible pairs and of the interpolation property of sections of elliptic submersions, an extension of Gromov's results obtained by Forstneri\v{c} and Prezelj.
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We study stochastic differential equations (SDEs) of McKeanVlasov type with distribution dependent drifts and driven by pure jump L\'{e}vy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with L\'{e}vy noise to the corresponding McKeanVlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results that are novel even for a broad class of L\'{e}vydriven SDEs with distribution independent coefficients.
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Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the $d$dimensional upper halfsphere. We show that, as $n\to\infty$, the $f$vector of the $(d+1)$dimensional convex cone $C_n$ generated by $U_1,\ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of B\'ar\'any, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the halfsphere at its north pole is the convex hull of th
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This paper studies curves on quartic K3 surfaces, or more generally K3 surfaces which are complete intersection in weighted projective spaces. A folklore conjecture concerning rational curves on K3 surfaces states that all K3 surfaces contain infinite number of irreducible rational curves. It is known that all K3 surfaces, except those contained in the countable union of hypersurfaces in the moduli space of K3 surfaces satisfy this property. In this paper we present a new approach for constructing curves on varieties which admit nice degenerations. We apply this technique to the above problem and prove that there is a Zariski open dense subset in the moduli space of quartic K3 surfaces whose members satisfy the conjecture. Various other curves of positive genus can be also constructed.
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We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for DushnikMiller dimension, we have that if $\dim(C)\le d$ for every component $C$ of a poset $P$, then $\dim(P)\le \max\{2,d\}$; also if $\dim(B)\le d$ for every block $B$ of a poset $P$, then $\dim(P)\le d+2$. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if $\text{ldim}(C)\le d$ for every component $C$ of a poset $P$, then $\text{ldim}(P)\le d+2$; however, for every $d\ge 4$, there exists a poset $P$ with $\text{ldim}(P)=d$ and $\dim(B)\le 3$ for every block $B$ of $P$. In this paper we show that Boolean dimension behaves like DushnikMiller dimension with respect to both components and blocks: if $\text{bdim}(C)\le d$ for every component $C$ of $P$, then $\text{bdim}(P)\le 2+d+4\cdot2^d$; also if $\text{bdim}(B)\le d$ for every block of $P$, then $\text{bdim}(P)\le 19+d+18\cdot 2^d$.
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In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (\Delta)_{p}^{s}u+ (\Delta)_{q}^{s}u + V(\varepsilon x) (u^{p2}u + u^{q2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p<q<\frac{N}{s}$, $(\Delta)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $\mathcal{C}^{1}$function with subcritical growth. Applying minimax theorems and the LjusternikSchnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that $\varepsilon$ is sufficiently small.
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In this paper, a modified van der Pol oscillator equation is considered which appears in several heart action models. We study its global dynamics and verify many interesting bifurcations such as a Hopf bifurcation, a heteroclinic saddle connection, and a homoclinic saddle connection. Some of these bifurcations are detected by using Conley index methods. We demonstrate how the study of connection matrices and transition matrices shows how to select interesting parameter values for simulations.
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We introduce the $2$nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$manifolds homeomorphic but not diffeomorphic to ${\mathbb{CP}}^{2}\#6({\overline{\mathbb{CP}}^{2}})$, ${\mathbb{CP}}^{2}\#7({\overline{\mathbb{CP}}^{2}})$, and $3{\mathbb{CP}}^{2}\#k({\overline{\mathbb{CP}}^{2}})$ for $k=16, 17, 18, 19$ using combinations of such deformations, symplectic blowups, and (generalized) rational blowdown surgery. We also discuss generalizing our constructions to higher genus fibrations using $g$nodal spherical deformations of certain singular fibers of genus $g \geq 3$ fibrations.
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Recently, in the paper \cite{CJKM1} we suggested the two conjectures about the diameter of iodecomposable Riordan graphs of the Bell type. In this paper, we give a counterexample for the first conjecture. Then we prove that the first conjecture is true for the graphs of some particular size and propose a new conjecture. Finally, we show that the second conjecture is true for some special iodecomposable Riordan graphs.
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We present, to the best of the authors' knowledge, all known results for the crossing numbers of specific graphs and graph families. The results are separated into various categories; specifically, results for general graph families, results for graphs arising from various graph products, and results for recursive graph constructions.
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Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions for two finiteorder mapping classes to have commuting conjugates in $\text{Mod}(S_g)$. As an application of this result, we show that any finiteorder mapping class, whose corresponding orbifold is not a sphere, has a conjugate that lifts under any finitesheeted cover of $S_g$. Furthermore, we show that any torsion element in the centralizer of an irreducible finite order mapping class is of order at most $2$. We also obtain conditions for the primitivity of a finiteorder mapping class. Finally, we describe a procedure for determining the explicit hyperbolic structures that realize twogenerator finite abelian groups of $\text{Mod}(S_g)$ as isometry groups.
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Presented is a method to compute certain classes of HamiltonJacobi equations that result from optimal control and trajectory generation problems with time delays. Many robotic control and trajectory problems have limited information of the operating environment a priori and must continually perform online trajectory optimization in real time after collecting measurements. The sensing and optimization can induce a significant time delay, and must be accounted for when computing the trajectory. This paper utilizes the generalized Hopf formula, which avoids the use of grids and numerical gradients that is typical of other methods for computing solutions to the HamiltonJacobi equation, which suffer exponential dimensional scaling. We present as an example a robot that incrementally predicts a communication channel from measurements as it travels. As part of this example, we introduce a seemingly new generalization of a nonparametric formulation of robotic communication channel estimatio
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Unmanned aerial vehicles (UAVs) have recently found abundant applications in the public and civil domains. To ensure reliable control and navigation, connecting UAVs to controllers via existing cellular network infrastructure, i.e., ground base stations (GBSs), has been proposed as a promising solution. Nevertheless, it is highly challenging to characterize the communication performance of cellularconnected UAVs, due to their unique propagation conditions. This paper proposes a tractable framework for the coverage analysis of cellularconnected UAV networks, which consists of a new blockage model and an effective approach to handle general fading channels. In particular, a lineofsight (LoS) ball model is proposed to capture the probabilistic propagation in UAV communication systems, and a tractable expression is derived for the Laplace transform of the aggregate interference with general Nakagami fading. This framework leads to a tractable expression for the coverage probability, wh
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In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology. We introduce several measures related to fixed points for continuous selfmaps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum $F(X)$ of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous selfmap. We give a complete computation of $F(C_n)$ where $C_n$ is the digital cycle of $n$ points. For other digital images, we show that, if $X$ has at least 4 points, then $F(X)$ always contains the numbers 0, 1, 2, 3, and the cardinality of $X$. We give several examples, including $C_n$, in which $F(X)$ does not equal $\{0,1,\dots,\#X\}$. We examine how fixed point
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China and Russia could disrupt critical national infrastructure in the US,
0131 MIT Technology 1008 
Firefox 65.0 is out. The release notes list a few new features, including: "Enhanced tracking protection: Simplified content blocking settings give users standard, strict, and custom options to control online trackers. A redesigned content blocking section in the site information panel (viewed by expanding the small “i” icon in the address bar) shows what Firefox detects and blocks on each website you visit."
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Brain implants, AI, and a speech synthesizer have turned brain activity int
0130 MIT Technology 1177 
Yesterday, a worrying and invasive bug that allowed callers to secretly listen in on unknowing recipients through Apple’s FaceTime app quickly made news headlines. It was discovered that people could initiate a FaceTime call and, with a couple short steps, tap into the microphone on the other end as the call rang — without the other person accepting the FaceTime request. Apple said last night that an iOS update to eliminate the privacy bug is coming this week; in the meantime, the company took the step of disabling group FaceTime at the server level as an immediate emergency fix. However, new information suggests that Apple has already had several days to respond; the company was tipped off about it last week. Back on January 20th, a Twitter user tweeted at Apple’s support account clearly outlining the gist of the FaceTime bug: “My teen found a major security flaw in Apple’s new iOS. He can listen in to your iPhone/iPad without your approval.” The parent’s teenager had discovered the p
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Great reporting by TechCrunch’s Josh Constine: Desperate for data on its competitors, Facebook has been secretly paying people to install a “Facebook Research” VPN that lets the company suck in all of a user’s phone and web activity, similar to Facebook’s Onavo Protect app that Apple banned in June and that was removed in August. Facebook sidesteps the App Store and rewards teenagers and adults to download the Research app and give it root access in what may be a violation of Apple policy so the social network can decrypt and analyze their phone activity, a TechCrunch investigation confirms. Facebook admitted to TechCrunch it was running the Research program to gather data on usage habits, and it has no plans to stop. Since 2016, Facebook has been paying users ages 13 to 35 up to $20 per month plus referral fees to sell their privacy by installing the iOS or Android “Facebook Research” app. Facebook even asked users to screenshot their Amazon order history page. The program is ad
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Astronomers have spotted a small firstofitskind object in the Kuiper Bel
0129 MIT Technology 1319 
An anonymous reader shares a report: It seems the stuff of fantasy. Giant ships sail the seas burning fuel that has been extracted from water using energy provided by the winds, waves and tides. A dramatic but implausible notion, surely. Yet this grand green vision could soon be realised thanks to a remarkable technological transformation that is now under way in Orkney. Perched 10 miles beyond the northern edge of the British mainland, this archipelago of around 20 populated islands  as well as a smattering of uninhabited reefs and islets  has become the centre of a revolution in the way electricity is generated. Orkney was once utterly dependent on power that was produced by burning coal and gas on the Scottish mainland and then transmitted through an undersea cable. Today the islands are so festooned with wind turbines, they cannot find enough uses for the emissionfree power they create on their own. Communityowned wind turbines generate power for local villages; islanders d
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