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This summary of the doctoral thesis is created to emphasize the close connection of the proposed spectral analysis method with the Discrete Fourier Transform (DFT), the most extensively studied and frequently used approach in the history of signal processing. It is shown that in a typical application case, where uniform data readings are transformed to the same number of uniformly spaced frequencies, the results of the classical DFT and proposed approach coincide. The difference in performance appears when the length of the DFT is selected to be greater than the length of the data. The DFT solves the unknown data problem by padding readings with zeros up to the length of the DFT, while the proposed Extended DFT (EDFT) deals with this situation in a different way, it uses the Fourier integral transform as a target and optimizes the transform basis in the extended frequency range without putting such restrictions on the time domain. Consequently, the Inverse DFT (IDFT) applied to the res
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We introduce a category of dual pairs of finite locally free algebras over a ring. This gives an efficient way to represent finite locally free commutative group schemes. We give a number of algorithms to compute with dual pairs of algebras, and we apply our results to Galois representations on finite Abelian groups.
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We introduce a class of distributed control policies for networks of discretetime linear systems with polytopic additive disturbances. The objective is to restrict the networklevel state and controls to userspecified polyhedral sets for all times. This problem arises in many safetycritical applications. We consider two problems. First, given a communication graph characterizing the structure of the information flow in the network, we find the optimal distributed control policy by solving a single linear program. Second, we find the sparsest communication graph required for the existence of a distributed invarianceinducing control policy. Illustrative examples, including one on platooning, are presented.
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We construct a fixed parameter algorithm parameterized by d and k that takes as an input a graph G' obtained from a ddegenerate graph G by complementing on at most k arbitrary subsets of the vertex set of G and outputs a graph H such that G and H agree on all but f(d,k) vertices. Our work is motivated by the first order model checking in graph classes that are first order interpretable in classes of sparse graphs. We derive as a corollary that if G_0 is a graph class with bounded expansion, then the first order model checking is fixed parameter tractable in the class of all graphs that can obtained from a graph G from G_0 by complementing on at most k arbitrary subsets of the vertex set of G; this implies an earlier result that the first order model checking is fixed parameter tractable in graph classes interpretable in classes of graphs with bounded maximum degree.
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We say that an exact equivalence between the derived categories of two algebraic varieties is tiltingtype if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tiltingtype equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.
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We prove a decomposition formula of logarithmic GromovWitten invariants in a degeneration setting. A oneparameter log smooth family X>B with singular fibre over b_0 \in B yields a family M(X/B,\beta) > B of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over b_0 in terms of rigid tropical curves. This generalizes one aspect of known results in the case that the fibre X_{b_0} is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.
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Motivated by the recent interest in formal methodsbased control of multiagent systems, we adopt to a bottomup approach. Each agent is subject to a local signal temporal logic task that may depend on other agents behavior. These dependencies pose control challenges since some of the tasks may be opposed to each other. We first develop a local continuous feedback control law and identify conditions under which this control law guarantees satisfaction of the local tasks. If these conditions do not hold, we propose to use the developed control law in combination with an online detection & repair scheme, expressed as a local hybrid system. After detection of a critical event, a twostage procedure is initiated to resolve the problem. The theoretical results are illustrated in simulations.
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Let A be the jacobian variety of a hyperelliptic curve defined over a number field k. We provide a decomposition formula for the Faltings height of A and for the N\'eronTate height of krational points on A. We formulate a question of Bogomolov type on the space of principally polarized abelian varieties of dimension g.
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Lubiw showed that several variants of Graph Isomorphism are NPcomplete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs $X_1=(V,E_1)$ and $X_2=(V,E_2)$ and a subset $R\subseteq V\times V$ of forbidden pairs whether there is an isomorphism $\pi$ from $X_1$ to $X_2$ such that $\pi(i)\neq j$ for all $(i,j)\in R$. We prove that this problem and several of its generalizations are in fact in FPT:  The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiwstyle constraints is in FPT, with k and the size of $R$ as parameters. The problem remains in FPT if a CNF of such constraints is allowed. It follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017].  When the w
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We give explicit numerical estimates for the generalized Chebyshev functions. Explicit results of this kind are useful for estimating of computational complexity of algorithms which generates special primes. Such primes are needed to construct an elliptic curve over prime field using complex multiplication method.
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We show that nonflatness of a morphism f of complexanalytic spaces with a locally irreducible target Y of dimension n manifests in the existence of vertical components in the nfold fibred power of the pullback of f to the desingularization of Y. An algebraic analogue follows: Let R be a locally (analytically) irreducible finite type complexalgebra and an integral domain of Krull dimension n, and let S be a regular ndimensional algebra of finite type over R (but not necessarily a finite Rmodule), such that the induced morphism of spectra is dominant. Then a finite type Ralgebra A is Rflat if and only if the tensor product of S with the nfold tensor power of A over R is a torsionfree Rmodule.
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We establish exact recovery for the Least Unsquared Deviations (LUD) algorithm of \"{O}zyesil and Singer. More precisely, we show that for sufficiently many cameras with given corrupted pairwise directions, where both camera locations and pairwise directions are generated by a special probabilistic model, the LUD algorithm exactly recovers the camera locations with high probability. A similar exact recovery guarantee was established for the ShapeFit algorithm by Hand, Lee and Voroninski. Comparing the two results, we conclude that in theory LUD can tolerate more corruption than ShapeFit.
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Let $G$ be a simple graph of $n$ vertices. We consider the problem $\mathrm{IS}^i_\ell$ of deciding whether there exists an induced subtree with exactly $i \leq n$ vertices and $\ell$ leaves in $G$. We also study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by $L_G(i)$, realized by an induced subtree with $i$ vertices, for $2\le i \le n$. We compute the values of the map $L_G$ for some classical families of graphs and in particular for the $d$dimensional hypercubic graphs $Q_d, d\leq 6$. Then we prove that the $\mathrm{IS}^i_\ell$ problem is in general NPcomplete. We also describe a nontrivial branch and bound algorithm that computes the function $L_G$ for any simple graph $G$. In the special case where $G$ is a tree, we provide a $\mathcal{O}(n^3\delta)$ time and $\mathcal{0}(n^2)$ space algorithm, where $\delta$ is the maximum degree of $G$. Finally, we exhibit a bijection between the set of discrete derivative of the sequenc
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In this work, we study a family of wireless channel simulation models called geometrybased stochastic channel models (GBSCMs). Compared to more complex raytracing simulation models, GBSCMs do not require an extensive characterization of the propagation environment to provide wireless channel realizations with adequate spatial and temporal statistics. The tradeoff they achieve between the quality of the simulated channels and the computational complexity makes them popular in standardization bodies. Using the generic formulation of the GBSCMs, we identify a matrix structure that can be used to improve the performance of their implementations. Furthermore, this matrix structure allows us to analyze the spatial covariance of the channel realizations. We provide a way to efficiently compute the spatial covariance matrix in most implementations of GBSCMs. In accordance to widesense stationary and uncorrelated scattering hypotheses, this covariance is static in frequency and does not evo
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We extend earlier numerical and analytical considerations of the conformally invariant wave equation on a Schwarzschild background from the case of spherically symmetric solutions, discussed in Class. Quantum Grav. 34, 045005 (2017), to the case of general, nonsymmetric solutions. A key element of our approach is the modern standard representation of spacelike infinity as a cylinder. With a decomposition into spherical harmonics, we reduce the fourdimensional wave equation to a family of twodimensional equations. These equations can be used to study the behaviour at the cylinder, where the solutions turn out to have logarithmic singularities at infinitely many orders. We derive regularity conditions that may be imposed on the initial data, in order to avoid the first singular terms. We then demonstrate that the fully pseudospectral time evolution scheme can be applied to this problem leading to a highly accurate numerical reconstruction of the nonsymmetric solutions. We are particula
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When sales of a product are affected by randomness in demand, etailers use dynamic pricing strategies to maximize their profits. In this article the pricing problem is formulated as a continuoustime stochastic optimal control problem, where the optimal policy can be found by solving the associated HamiltonJacobiBellman (HJB) equation. We propose a new approach to modelling the randomness in the dynamics of sales based on diffusion processes. The model assumes a continuum approximation to the stock levels of the retailer, which should scale much better to largeinventory problems than the existing models in the revenue management literature, which are based on Poisson processes. We present closedform solutions to the HJB equation when there is no randomness in the system. It turns out that the deterministic pricing policy is nearoptimal for systems with demand uncertainty. Numerical errors in calculating the optimal pricing policy may in fact result in lower profit on average than
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We introduce a notion of cocycleinduction for strong uniform approximate lattices in locally compact second countable groups and use it to relate (relative) Kazhdan and Haageruptype of approximate lattices to the corresponding properties of the ambient locally compact groups. Our approach applies to large classes of uniform approximate lattices (though not all of them) and is flexible enough to cover the $L^p$versions of Property (FH) and a(FH)menability as well as quasified versions thereof a la BurgerMonod and Ozawa.
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We prove weighted uniform estimates for the resolvent of the Laplace operator in Schatten spaces, on nontrapping asymptotically conic manifolds of dimension $n\ge 3$, generalizing a result of Frank and Sabin, obtained in the Euclidean setting. As an application of these estimates we establish LiebThirring type bounds for eigenvalues of Schr\"odinger operators with complex potentials on nontrapping asymptotically conic manifolds, extending those of Frank, Frank and Sabin, and Frank and Simon proven in the Euclidean setting. In particular, our results are valid for the metric Schr\"odinger operator in the Euclidean space, with a metric being a sufficiently small compactly supported perturbation of the Euclidean one. To the best of our knowledge, these are the first LiebThirring type bounds for nonselfadjoint elliptic operators, with principal part having variable coefficients.
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The polynomial Fre\u{\i}manRuzsa conjecture is a fundamental open question in additive combinatorics. However, over the integers (or more generally $\mathbb{R}^d$ or $\mathbb{Z}^d$) the optimal formulation has not been fully pinned down. The conjecture states that a set of small doubling is controlled by a very structured set, with polynomial dependence of parameters. The ambiguity concerns the class of structured sets needed. A natural formulation in terms of generalized arithmetic progressions was recently disproved by Lovett and Regev. A more permissive alternative is in terms of \emph{convex progressions}; this avoids the obstruction, but uses is a significantly larger class of objects, yielding a weaker statement. Here we give another formulation of PFR in terms of Euclidean ellipsiods (and some variations). We show it is in fact equivalent to the convex progression version; i.e. that the full range of convex progressions is not needed. The key ingredient is a strong result from
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We investigate the universal cover of a Lie group that is not necessarily connected. Its existence as a Lie group is governed by a Taylor cocycle, an obstruction in 3cohomology. Alternatively, a Lie group can be thought of as a Lie 2group, and there is a natural notion of universal cover in this context. The splitness of this universal cover is also governed by an obstruction in 3cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are inverse of each other.
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We show that, for each integer n, there exist infinitely many pairs of nframed knots representing homeomorphic but nondiffeomorphic (Stein) 4manifolds, which are the simplest possible exotic 4manifolds regarding handlebody structures. To produce these examples, we introduce a new description of cork twists and utilize satellite maps. As an application, we produce knots with the same 0surgery which are not concordant for any orientations, disproving the AkbulutKirby conjecture given in 1978.
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In this paper we present a Boltzmanntype kinetic approach to the modelling of road traffic, which includes control strategies at the level of microscopic binary interactions aimed at the mitigation of speeddependent road risk factors. Such a description is meant to mimic a system of driverassist vehicles, which by responding locally to the actions of their drivers can impact on the largescale traffic dynamics, including those related to the collective road risk and safety.
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For all positive integers $\ell$, we prove nontrivial bounds for the $\ell$torsion in the class group of $K$, which hold for almost all number fields $K$ in certain families of cyclic extensions of arbitrarily large degree. In particular, such bounds hold for almost all cyclic degree$p$extensions of $F$, where $F$ is an arbitrary number field and $p$ is any prime for which $F$ and the $p$th cyclotomic field are linearly disjoint. Along the way, we prove precise asymptotic counting results for the fields of bounded discriminant in our families with prescribed splitting behavior at finitely many primes.
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An optimizationbased approach for the Tucker tensor approximation of parameterdependent data tensors and solutions of tensor differential equations with low Tucker rank is presented. The problem of updating the tensor decomposition is reformulated as fitting problem subject to the tangent space without relying on an orthogonality gauge condition. A discrete Euler scheme is established in an alternating least squares framework, where the quadratic subproblems reduce to trace optimization problems, that are shown to be explicitly solvable and accessible using SVD of small size. In the presence of small singular values, instability for larger ranks is reduced, since the method does not need the (pseudo) inverse of matricizations of the core tensor. Regularization of Tikhonov type can be used to compensate for the lack of uniqueness in the tangent space. The method is validated numerically and shown to be stable also for larger ranks in the case of small singular values of the core unfol
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We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layeradapted piecewise equidistant meshes proposed by Sun and Stynes. We also study the streamlinediffusion finite element method (SDFEM) for such problems. For these methods error estimates uniform with respect to $\varepsilon$ are proven in the energy norm and in the stronger SDFEMnorm, respectively. Numerical experiments confirm the theoretical findings.
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The largest order $n(d,k)$ of a graph of maximum degree $d$ and diameter $k$ cannot exceed the Moore bound, which has the form $M(d,k)=d^k  O(d^{k1})$ for $d\to\infty$ and any fixed $k$. Known results in finite geometries on generalised $(k+1)$gons imply, for $k=2,3,5$, the existence of an infinite sequence of values of $d$ such that $n(d,k)=d^k  o(d^k)$. This shows that for $k=2,3,5$ the Moore bound can be asymptotically approached in the sense that $n(d,k)/M(d,k)\to 1$ as $d\to\infty$; moreover, no such result is known for any other value of $k\ge 2$. The corresponding graphs are, however, far from vertextransitive, and there appears to be no obvious way to extend them to vertextransitive graphs giving the same type of asymptotic result. The second and the third author (2012) proved by a direct construction that the Moore bound for diameter $k=2$ can be asymptotically approached by Cayley graphs. Subsequently, the first and the third author (2015) showed that the same construct
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We use Fuchsian Reduction to study the behavior near the singularity of a class of solutions of Einstein's vacuum equations. These solutions admit two commuting spacelike Killing fields like the Gowdy spacetimes, but their twist does not vanish. The spacetimes are also polarized in the sense that one of the `gravitational degrees of freedom' is turned off. Examining an analytic family of solutions with the maximum number of arbitrary functions, we find that they are all asymptotically velocityterm dominated as one approaches the singularity.
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A polynomial assignment for a continuous action of a compact torus $T$ on a topological space $X$ assigns to each $p\in X$ a polynomial function on the Lie algebra of the isotropy group at $p$ in such a way that a certain compatibility condition is satisfied. The space ${\mathcal{A}}_T(X)$ of all polynomial assignments has a natural structure of an algebra over the polynomial ring of ${\rm Lie}(T)$. It is an equivariant homotopy invariant, canonically related to the equivariant cohomology algebra. In this paper we prove various properties of ${\mathcal{A}}_T(X)$ such as Borel localization, a ChangSkjelbred lemma, and a GoreskyKottwitzMacPherson presentation. In the special case of Hamiltonian torus actions on symplectic manifolds we prove a surjectivity criterion for the assignment equivariant Kirwan map corresponding to a circle in $T$. We then obtain a TolmanWeitsman type presentation of the kernel of this map.
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We investigate the risk of overestimating the performance gain when applying neural network based receivers in systems with pseudo random bit sequences or with limited memory depths, resulting in repeated short patterns. We show that with such sequences, a large artificial gain can be obtained which comes from pattern prediction rather than predicting or compensating the studied channel/phenomena.
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Determining entropy rates of stochastic processes is a fundamental and difficult problem, with closedform solutions known only for specific cases. This paper pushes the stateoftheart by solving the problem for Hidden Markov Models (HMMs) and Renyi entropies. While the problem for Markov chains reduces to studying the growth of a matrix product, computations for HMMs involve \emph{products of random matrices}. As a result, this case is much harder and no explicit formulas have been known so far. We show how to circumvent this issue for Renyi entropy of integer orders, reducing the problem again to a \emph{single matrix products} where the matrix is formed from transition and emission probabilities by means of tensor product. To obtain results in the asymptotic setting, we use a novel technique for determining the growth of nonnegative matrix powers. The classical approach is the FrobeniusPerron theory, but it requires positivity assumptions; we instead work directly with the spect
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We use Fuchsian Reduction to construct singular solutions of Einstein's equations which belong to the class of Gowdy spacetimes. The solutions have the maximum number of arbitrary functions. Special cases correspond to polarized, or other known solutions. The method provides precise asymptotics at the singularity, which is Kasnerlike. All of these solutions are asymptotically velocitydominated. The results account for the fact that solutions with velocity parameter uniformly greater than one are not observed numerically. They also provide a justification of formal expansions proposed by Grubi\v si\'c and Moncrief.
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We consider weighted Radon transforms RW along hyperplanes in R 3 with strictly positive weights W. We construct an example of such a transform with nontrivial kernel KerRW in the space of infinitely smooth compactly supported functions and with continuous weight. Moreover, in this example the weight W is rotation invariant. In particular, by this result we continue studies of Quinto (1983), Markoe, Quinto (1985), Boman (1993) and Goncharov, Novikov (2017). We also extend our example to the case of weighted Radon transforms along twodimensional planes in R d , d $\ge$ 3.
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For a global field, local field, or finite field $k$ with infinite Galois group, we show that there can not exist a functor from the MorelVoevodsky $\mathbb{A}^1$homotopy category of schemes over $k$ to a genuine Galois equivariant homotopy category satisfying a list of hypotheses one might expect from a genuine equivariant category and an \'etale realization functor. For example, these hypotheses are satisfied by genuine $\mathbb{Z}/2$spaces and the $\mathbb{R}$realization functor constructed by MorelVoevodsky. This result does not contradict the existence of \'etale realization functors to (pro)spaces, (pro)spectra or complexes of modules with actions of the absolute Galois group when the endomorphisms of the unit is not enriched in a certain sense. It does restrict enrichments to representation rings of Galois groups.
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We consider a Boussinesq system of KdVKdV type introduced by J. Bona, M. Chen and J.C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the PaleyWiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained
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We investigate the computabilitytheoretic properties of valued fields, and in particular algebraically closed valued fields and $p$adically closed valued fields. We give an effectiveness condition, related to Hensel's lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally $p$adic field which does not embed into any computable $p$adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is sufficient to find such an embedding. By checking that algebraically closed valued fields and $p$adically closed valued fields of infinite transcendence degree have the Mal'cev property, we show that they have computable dimension $\omega$.
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Securing the physical components of a city's interdependent critical infrastructure (ICI) such as power, natural gas, and water systems is a challenging task due to their interdependence and large number of involved sensors. Using a novel integrated statespace model that captures the interdependence, a twostage cyber attack on ICI is studied in which the attacker first compromises the ICI's sensors by decoding their messages, and, subsequently, it alters the compromised sensors' data to cause state estimation errors. To thwart such attacks, the administrator of the CIs must assign protection levels to the sensors based on their importance in the state estimation process. To capture the interdependence between the attacker and the ICI administrator's actions and analyze their interactions, a Colonel Blotto game framework is proposed. The mixedstrategy Nash equilibrium of this game is derived analytically. At this equilibrium, it is shown that the administrator can strategically rando
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We use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinitedimensional separable Hilbert spaces: these were longmissing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetryobservable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Furthermore, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theoryindependent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature provid
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J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of latticebased algebras. After reviewing this evolution we make two contributions. First it is shown that the question of whether or not a variety of algebras is closed under canonical extensions reduces to the question of whether or not it contains the canonical extension of a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operator
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Given a set E in a complex space and a point p in E, there is a unique smallest complexanalytic germ containing the germ of E at p, called the holomorphic closure of E at p. We study the holomorphic closure of semialgebraic arcsymmetric sets. Our main application concerns CRcontinuation of semialgebraic arcanalytic mappings: A mapping f on a realanalytic CR manifold M which is semialgebraic arcanalytic and CR on a nonempty open subset of M is CR on the whole M.
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We investigate the longtime behavior of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finitetime stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition.
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Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a standalone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than $100$ the asymptotic methods are enough for a double precision accuracy computation ($15$$16$ digits) of the nodes and weights of the GaussHermite and GaussLaguerre quadratures.
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This paper was inspired by four articles: surface cluster algebras studied by FominShapiroThurston \cite{fst}, the mutation theory of quivers with potentials initiated by DerksenWeymanZelevinsky \cite{dwz}, string modules associated to arcs on unpunctured surfaces by AssemBr$\ddot{u}$stleCharbonneauPlamondon \cite{acbp} and Quivers with potentials associated to triangulated surfaces, part II: Arc representations by LabardiniFragoso. \cite{lf2}. For a surface with marked points ($\Sigma,M$) LabardiniFragoso associated a quiver with potential $(Q(\tau),S(\tau))$ then for an ideal triangulation of ($\Sigma,M$) and an ideal arc LabardiniFragoso defined an arc representation of $(Q(\tau),S(\tau))$. This paper focuses on extent the definition of arc representation to a more general context by considering a tagged triangulation and a tagged arc. We associate in an explicit way a representation of the quiver with potential constructed LabardiniFragoso and prove that the Jacobian rel
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We survey the dynamics of functions in the EremenkoLyubich class, $\mathcal{B}$. Among transcendental entire functions, those in this class have properties that make their dynamics markedly accessible to study. Many authors have worked in this field, and the dynamics of class $\mathcal{B}$ functions is now particularly wellunderstood and welldeveloped. There are many striking and unexpected results. Several powerful tools and techniques have been developed to help progress this work. We consider the fundamentals of this field, review some of the most important results, techniques and ideas, and give steppingstones to deeper inquiry.
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We give several versions of Siu's $\partial\bar{\partial}$formula for maps from a strictly pseudoconvex pseudoHermitian manifold $(M^{2m+1}, \theta)$ into a K\"ahler manifold $(N^n, g)$. We also define and study the notion of pseudoHermitian harmonicity for maps from $M$ into $N$. In particular, we prove a CR version of Siu Rigidity Theorem for pseudoHermitian harmonic maps from a pseudoHermitian manifold with vanishing Webster torsion into a K\"ahler manifold having strongly negative curvature.
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In dimension $d\geq 3$, a variational principle for the size of the pure point spectrum of (discrete) Schr\"odinger operators $H(\mathfrak{e},V)$ on the hypercubic lattice $\mathbb{Z}^{d}$, with dispersion relation $\mathfrak{e}$ and potential $V$, is established. The dispersion relation $\mathfrak{e}$ is assumed to be a Morse function and the potential $V(x)$ to decay faster than $x^{2(d+3)}$, but not necessarily to be of definite sign. Our estimate on the size of the purepoint spectrum yields the absence of embedded and threshold eigenvalues of $H(\mathfrak{e},V)$ for a class ot potentials of this kind. The proof of the variational principle is based on a limiting absorption principle combined with a positive commutator (Mourre) estimate, and a Virial theorem. A further observation of crucial importance for our argument is that, for any selfadjoint operator $B$ and positive number $\lambda >0$, the number of negative eigenvalues of $\lambda B$ is independent of $\lambda$.
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We study the boundary regularity for the normalised $\infty$heat equation $u_t = \Delta_{\infty}^Nu$ in arbitrary domains. Perron's Method is used for constructing solutions. We characterize regular boundary points with barrier functions, and prove an Exterior Ball result. A Petrovskylike criterion is established.
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The main result of this article is a BerryEsseenlike bound, which states the convergence to the normal distribution of sums of independent, identically distributed random variables in chisquare distance, defined as the variance of the density with respect to the normal distribution. Our main assumption is that the random variables involved in the sum are independent and have polynomial density; the identical distribution hypothesis can in fact be relaxed. The method consists of taking advantage of the underlying time nonhomogeneous Markovian structure and providing a Poincar{\'e}like inequality for the nonreversible transition operator, which allows to find the optimal rate in the convergence above under matching moments assumptions.
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In 2007, Dubouloz introduced Danielewski varieties. Such varieties generalize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to MakarLimanov.
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In this paper, we analyze the toggle group on the set of antichains of a poset. Toggle groups, generated by simple involutions, were first introduced by Cameron and FonDerFlaass for order ideals of posets. Recently Striker has motivated the study of toggle groups on general families of subsets, including antichains. This paper expands on this work by examining the relationship between the toggle groups of antichains and order ideals, constructing an explicit isomorphism between the two groups. We also focus on the rowmotion action that has been wellstudied in dynamical algebraic combinatorics, describing it as the composition of antichain toggles. We also describe a piecewiselinear analogue of toggling to the Stanley's chain polytope. We examine the connections with the piecewiselinear toggling Einstein and Propp introduced for order polytopes and prove that almost all of our results for antichain toggles extend to the piecewiselinear setting.
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A lattice Maxwell system is developed with gaugesymmetry, symplectic structure and discrete spacetime symmetry. Noether's theorem for Lie group symmetries is generalized to discrete symmetries for the lattice Maxwell system. As a result, the lattice Maxwell system is shown to admit a discrete local energymomentum conservation law corresponding to the discrete spacetime symmetry. These conservative properties make the discrete system an effective algorithm for numerically solving the governing differential equations on continuous spacetime. Moreover, the lattice model, respecting all conservation laws and geometric structures, is as good as and probably more preferable than the continuous Maxwell model. Under the simulation hypothesis by Bostrom and in consistent with the discussion on lattice QCD by Beane et al., the two interpretations of physics laws on spacetime lattice could be essentially the same.
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Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and optimization, inference, and several statistical techniques. The solution for even a single case may be quite expensive; whereas parallel computing may be applied, this reduces the total elapsed time but not the total computational effort. In the case of flows governed by the NavierStokes equations, a method has been devised for computing an ensemble of solutions. Recently, a reducedorder model derived from a proper orthogonal decomposition (POD) approach was incorporated into a firstorder accurate in time version of the ensemble algorithm. In this work, we expand on that work by incorporating the POD reduced order model into a secondorder accurate ensemble algorithm. Stability and convergence results for this method are updated to account for the POD/ROM ap
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We show a general framework of parallelizing Dykstra splitting that includes the classical Dykstra's algorithm and the product space formulation as special cases, and prove their convergence. The key idea is to split up the function whose conjugate takes in the sum of all dual variables in the dual formulation.
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We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let $G$ be a quasirandom $n$vertex graph and suppose $H_1,\dots,H_s$ are bounded degree $n$vertex graphs with $\sum_{i=1}^{s} e(H_i) \leq (1o(1)) e(G)$. Then $H_1,\dots,H_s$ can be packed edgedisjointly into $G$. The case when $G$ is the complete graph $K_n$ implies an approximate version of the tree packing conjecture of Gy\'arf\'as and Lehel for bounded degree trees, and of the Oberwolfach problem. We provide a more general version of the above approximate decomposition result which can be applied to superregular graphs and thus can be combined with Szemer\'edi's regularity lemma. In particular our result can be viewed as an extension of the classical blowup lemma of Koml\'os, S\'ark\H{o}zy and Szemer\'edi to the setting of approximate decompositions.
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As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial differential equations since initial data given by sums of Dirac masses would be smoothed instantaneously: particles do not remain particles. Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocity field that ensures particles do remain particles and apply this to develop a numerical blob method for a range of diffusive partial differential equations of Wasserstein gradient flow type, including the heat equation, the porous medium equation, the FokkerPlanck equation, the KellerSegel equation, and its variants. Our choice of regularization is guided by the Wasserstein gradient flow structure, and the corresponding energy has a novel form, combining aspects of the wellknown interaction and potential energie
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In this paper we prove that for a complete manifold without conjugate points with sectional curvatures bounded below by $c^2$ and whose geodesic flow is of Anosov type, then constant of contraction of the flow is $\geq e^{c}$. Moreover, if M has finite volume the equality is hold if and only if the sectional curvature is constant. We also show some results similar to Oseledet's theorem for Anosov geodesic flows on a complete surface with finite volume.
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This paper proposes a novel consensusbased distributed control algorithm for solving the economic dispatch problem of distributed generators. A legacy central controller can be eliminated in order to avoid a single point of failure, relieve computational burden, maintain data privacy, and support plugandplay functionalities. The optimal economic dispatch is achieved by allowing the iterative coordination of local agents (consumers and distributed generators). As coordination information, the local estimation of power mismatch is shared among distributed generators through communication networks and does not contain any private information, ultimately contributing to a fair electricity market. Additionally, the proposed distributed algorithm is particularly designed for easy implementation and configuration of a large number of agents in which the distributed decision making can be implemented in a simple proportionalintegral (PI) or integral (I) controller. In MATLAB/Simulink simul
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Distributed Generation (DG) is an effective way of integrating renewable energy sources to conventional power grid, which improves the reliability and efficiency of power systems. Photovoltaic (PV) systems are ideal DGs thanks to their attractive benefits, such as availability of solar energy and low installation costs. Battery groups are used in PV systems to balance the power flows and eliminate power fluctuations due to change of operating condition, e.g., irradiance and temperature variation. In an attempt to effectively manage the power flows, this paper presents a novel power control and management system for gridconnected PVBattery systems. The proposed system realizes the maximum power point tracking (MPPT) of the PV panels, stabilization of the DC bus voltage for load plugandplay access, balance among the power flows, and quick response of both active and reactive power demands.
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We consider the global existence and asymptotic behavior of classical solutions for the polyatomic BGK model when the initial data starts sufficiently close to a global polyatomic Maxwellian. We observe that the linearized relaxation operator is decomposed into a polyatomic part and a monatomiclike part, leading to a dichotomy in the dissipative estimate in the sense that the degeneracy of the dissipation changes abruptly as the relaxation parameter $\theta$ reaches zero. Accordingly, we employ two different sets of micromacro system to derive the full coercivity and close the energy estimate.
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The grid Ramsey number $ G(r) $ is the smallest number $ n $ such that every edgecolouring of the grid graph $\Gamma_{n,n} := K_n \times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G(r) \leq r^{\binom{r+1}{2}}  \left( 1/4  o(1) \right) r^{\binom{r}{2}+1} $, slightly improving the currently best known upper bound due to Gy\'arf\'as.
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Subspace based techniques for direction of arrival (DOA) estimation need large amount of snapshots to detect source directions accurately. This poses a problem in the form of computational burden on practical applications. The introduction of compressive sensing (CS) to solve this issue has become a norm in the last decade. In this paper, a novel CS beamformer rootMUSIC algorithm is presented with a revised optimal measurement matrix bound. With regards to this algorithm, the effect of signal subspace deviation under low snapshot scenario (e.g. target tracking) is analysed. The CS beamformer greatly reduces computational complexity without affecting resolution of the algorithm, works on par with rootMUSIC under low snapshot scenario and also, gives an option of nonuniform linear array sensors unlike the case of rootMUSIC algorithm. The effectiveness of the algorithm is demonstrated with simulations under various scenarios.
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We prove a sparse bound for the $m$sublinear form associated to vectorvalued maximal functions of FeffermanStein type. As a consequence, we show that the sparse bounds of multisublinear operators are preserved via $\ell^r$valued extension. This observation is in turn used to deduce vectorvalued, multilinear weighted norm inequalities for multisublinear operators obeying sparse bounds, which are out of reach for the extrapolation theory recently developed by CruzUribe and Martell. As an example, vectorvalued multilinear weighted inequalities for bilinear Hilbert transforms are deduced from the scalar sparse domination theorem of the authors.
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An effective Chebotarev density theorem for a fixed normal extension $L/\mathbb{Q}$ provides an asymptotic, with an explicit error term, for the number of primes of bounded size with a prescribed splitting type in $L$. In many applications one is most interested in the case where the primes are small (with respect to the absolute discriminant of $L$); this is wellknown to be closely related to the Generalized Riemann Hypothesis for the Dedekind zeta function of $L$. In this work we prove a new effective Chebotarev density theorem, independent of GRH, that improves the previously known unconditional error term and allows primes to be taken quite small (certainly as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Such a family has fixed degree, fixed Galois group of the Galois closure, and in certain cases a ramification restriction on all tame
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This paper gives an introduction to certain classical physical theories described in the context of locally Minkowskian causal structures (LMCSs). For simplicity of exposition we consider LMCSs which have locally Euclidean topology (i.e. are manifolds) and hence are M\"{o}bius structures. We describe natural principal bundle structures associated with M\"{o}bius structures. Fermion fields are associated with sections of vector bundles associated to the principal bundles while interaction fields (bosons) are associated with endomorphisms of the space of fermion fields. Classical quantum field theory (the Dirac equation and Maxwell's equations) is obtained by considering representations of the structure group $K \subset SU(2;2)$ of a principal bundle associated with a given M\"{o}bius structure where $K$, while being a subset of $SU(2;2)$ is also locally isomorphic to $O(1;3)$. The analysis requires the use of an intertwining operator between the action of $K$ on $R^4$ and the adjoint ac
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The notion of a separable extension is an important concept in Galois theory. Traditionally, this concept is introduced using the minimal polynomial and the formal derivative. In this work, we present an alternative approach to this classical concept.Based on our approach, we will give new proofs of some basic results about separable extensions (such as the existence of the separable closure, Theorem of the primitive element and the transitivity of separability).
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An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worstcase, takes $\mathcal{O}(\epsilon^{3/2})$ iterations to drive the norm of the gradient of the objective function below a prescribed positive real number $\epsilon$ and can take $\mathcal{O}(\varepsilon^{3})$ iterations to drive the leftmost eigenvalue of the Hessian of the objective above $\varepsilon$. The proposed algorithm is a general framework that covers a wide range of techniques including quadratically and cubically regularized Newton methods, such as the Adaptive Regularisation using Cubics (ARC) method, and the recently proposed TrustRegion Algorithm with Contractions and Expansions (TRACE). The generality of our method is achieved through the introduction of generic conditions that each trial step is required to satisfy, which in particular allow for inexact regularized Newton steps to be used. These conditions center around a new subproblem that can be approximately solved to
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The spectrum of an admissible subalgebra $\mathscr{A}(G)$ of $\mathscr{LUC}(G)$, the algebra of right uniformly continuous functions on a locally compact group $G$, constitutes a semigroup compactification $G^\mathscr{A}$ of $G$. In this paper we analyze the algebraic behaviour of those points of $G^\mathscr{A}$ that lie in the closure of $\mathscr{A}(G)$sets, sets whose characteristic function can be approximated by functions in $\mathscr{A}(G)$. This analysis provides a common ground for far reaching generalizations of Veech's property (the action of $G$ on $G^\mathscr{LUC}$ is free) and Pym's Local Structure Theorem. This approach is linked to the concept of translationcompact set, recently developed by the authors, and leads to characterizations of strongly prime points in $G^\mathscr{A}$, points that do not belong to the closure of $G^\ast G^\ast$, where $G^\ast=G^\mathscr{A}\setminus G.$ All these results will be applied to show that, in many of the most important algebras, lef
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It is experimentally known that achiral hyperbolic 3manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3sphere. In this paper, we show that there exist infinitely many achiral 1cusped hyperbolic 3manifolds not homeomorphic to any amphicheiral nullhomologous knot complement in any closed achiral 3manifold.
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We obtain a two weight local Tb theorem for any gradient elliptic fractional singular integral operator T on the real line, and any pair of locally finite positive Borel measures on the line. This includes the Hilbert transform and improves on the T1 theorem by the authors and M. Lacey.
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