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Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$invariant LUR or strictly convex renormings of $X$.
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Given a generic PL map or a generic smooth immersion $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\hookrightarrow M\times\mathbb R^k$ if and only if its double point locus $(f\times f)^{1}(\Delta_M)\setminus\Delta_N$ admits an equivariant map to $S^{k1}$. As a corollary we answer a 1990 question of P. Petersen on whether the universal coverings of the lens spaces $L(p,q)$, $p$ odd, lift to embeddings in $L(p,q)\times\mathbb R^3$. We also show that if a nondegenerate PL map $N\to M$ lifts to a topological embedding in $M\times\mathbb R^k$ then it lifts to a PL embedding in there.
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Highdimensional data have recently been analyzed because of data collection technology evolution. Although many methods have been developed to gain sparse recovery in the past two decades, most of these methods require selection of tuning parameters. As a consequence of this feature, results obtained with these methods heavily depend on the tuning. In this paper we study the theoretical properties of signconstrained generalized linear models with convex loss function, which is one of the sparse regression methods without tuning parameters. Recent studies on this topic have shown that, in the case of linear regression, signconstrains alone could be as efficient as the oracle method if the design matrix enjoys a suitable assumption in addition to a traditional compatibility condition. We generalize this kind of result to a much more general model which encompasses the logistic and quantile regressions. We also perform some numerical experiments to confirm theoretical findings obtained
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One of the most famous applications of Graph Theory is in the field of Channel Assignment Problems. There are varieties of graph colouring concepts that are used for different requirements of frequency assignments in communication channels. We introduce here L(t, 1)colouring of graphs. This has its foundation in Tcolouring and L(p, q)colouring. For a given finite set T including zero, an L(t, 1)colouring of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the colours of adjacent vertices must not belong to the set T and the colours of vertices that are at distance two must be distinct. The variable t in L(t, 1) denotes the elements of the set T. For a graph G, the L(t, 1)span of G is the minimum of the highest colour used to colour the vertices of a graph out of all the possible L(t, 1)colourings. It is denoted by $\lambda_{t,1} (G)$. We study some properties of L(t, 1)colouring. We also find upper bounds of $\lambda_{t,1}
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In this paper we study the KyleBack strategic insider trading equilibrium model in which the insider has an instantaneous information on an asset, assumed to follow an OrnsteinUhlenbacktype dynamics that allows possible influence by the market price. Such a model exhibits some further interplay between insider's information and the market price, and it is the first time being put into a rigorous mathematical framework of the recently developed {\it conditional meanfield} stochastic differential equation (CMFSDEs). With the help of the "reference probability measure" concept in filtering theory, we shall first prove a general wellposedness result for a class of linear CMFSDEs, which is new in the literature of both filtering theory and meanfield SDEs, and will be the foundation for the underlying strategic equilibrium model. Assuming some further Gaussian structures of the model, we find a closed form of optimal intensity of trading strategy as well as the dynamic pricing rules. W
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In this paper we propose a generalized Koopman operator framework for discrete and continuous time random dynamical systems. For the particular classes of random dynamical systems, we provide the results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operator. We discuss the relationship between the spectral properties of the generator of the evolution and the Koopman operator family. The numerical approximations of the spectral objects (eigenvalues, eigenfunctions) of the stochastic Koopman operator are computed by using the state of the art DMD RRR algorithm. We explore its behavior in the stochastic case on several test examples. Moreover, the DMD RRR algorithm is applied in combination with the Hankel matrix and a convergence theorem for Hankel DMD RRR in the stochastic case is proved.
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We describe a procedure naturally associating relativistic KleinGordon equations in static curved spacetimes to nonrelativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is given by a quadratic function of the energy level number, since for such systems the spacetimes one obtains possess evenly spaced, resonant spectra of frequencies for scalar fields of a certain mass. This construction emerges as a generalization of the previously studied correspondence between the Higgs oscillator and Antide Sitter spacetime, which has been useful for both understanding weakly nonlinear dynamics in Antide Sitter spacetime and algebras of conserved quantities of the Higgs oscillator. Our conversion procedure ("KleinGordonization") reduces to a nonlinear elliptic equation closely reminiscent of the one emerging in relation to the celebrated Yamabe problem of
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We derandomize the famous Isolation Lemma by Mulmuley, Vazirani, and Vazirani for polytopes given by totally unimodular constraints. That is, we construct a weight assignment such that one vertex in such a polytope is isolated, i.e., there is a unique minimum weight vertex. Our weights are quasipolynomially bounded and can be constructed in quasipolynomial time. In fact, our isolation technique works even under the weaker assumption that every face of the polytope lies in an affine space defined by a totally unimodular matrix. This generalizes the recent derandomization results for bipartite perfect matching and matroid intersection. We prove our result by associating a lattice to each face of the polytope and showing that if there is a totally unimodular kernel matrix for this lattice, then the number of nearshortest vectors in it is polynomially bounded. The proof of this latter geometric fact is combinatorial and follows from a polynomial bound on the number of nearshortest circ
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This paper is concerned with the sensitivity of invariant states in linear quantum stochastic systems with respect to nonlinear perturbations. The system variables are governed by a Markovian HudsonParthasarathy quantum stochastic differential equation (QSDE) driven by quantum Wiener processes of external bosonic fields in the vacuum state. The quadratic system Hamiltonian and the linear systemfield coupling operators, corresponding to a nominal open quantum harmonic oscillator, are subject to perturbations represented in a Weyl quantization form. Assuming that the nominal linear QSDE has a Hurwitz dynamics matrix and using the WignerMoyal phasespace framework, we carry out an infinitesimal perturbation analysis of the quasicharacteristic function for the invariant quantum state of the nonlinear perturbed system. The resulting correction of the invariant states in the spatial frequency domain may find applications to their approximate computation, analysis of relaxation dynamics a
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Let $K$ be a standard H\"older continuous Calder\'onZygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed degree is bounded on $L^p(\mathbb{R}^{\mathbf{d}})$ for $2\leq p<\infty$. This extends Sj\"olin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.
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Fix an integer partition lambda that has no more than n parts. Let beta be a weakly increasing ntuple with entries from {1,..,n}. The flagged Schur function indexed by lambda and beta is a polynomial generating function in x_1, .., x_n for certain semistandard tableaux of shape lambda. Let pi be an npermutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by lambda and pi is another such polynomial generating function. Reiner and Shimozono and then Postnikov and Stanley studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general beta, and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let R be the set of lengths of columns in the shape of lambda that are less than n. Ordered set partitions of
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We find that Koschorke's $\beta$invariant and the triple $\mu$invariant of link maps in the critical dimension can be computed as degrees of certain maps of configuration spaces  just like the linking number. Both formulas admit geometric interpretations in terms of Vassiliev's ornaments via new operations akin to the Jin suspension, and both were unexpected for the author, because the only known direct ways to extract $\mu$ and $\beta$ from invariants of maps between configuration spaces involved some homotopy theory (Whitehead products and the stable Hopf invariant, respectively).
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We give a characterization of the covering Schreier graphs of groups generated by bounded automata to be Galois. We also investigate the zeta and $L$ functions of Schreier graphs of few groups namely the Grigorchuk group, GuptaSidki $p$ group, GuptaFabrykowski group and BSV torsionfree group.
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A noncommutative extension of an ideal (Hamiltonian) fluid model in $3+1$dimensions is proposed. The model enjoys several interesting features: it allows a multiparameter central extension in Galilean boost algebra (which is significant being contrary to existing belief that similar feature can appear only in $2+1$dim.); noncommutativity generates vorticity in a canonically irrotational fluid; it induces a nonbarotropic pressure leading to a nonisentropic system. (Barotropic fluids are entropy preserving as pressure depends only on matter density.) Our fluid model is termed "Exotic" since it has close resemblance with the extensively studied planar (2+1dim.) Exotic models and Exotic (noncommutative) field theories.
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The correspondence between definable connected groupoids in a theory $T$ and internal generalised imaginary sorts of $T$, established by Hrushovski in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics, 2012], is here extended in two ways: First, it is shown that the correspondence is in fact an equivalence of categories, with respect to appropriate notions of morphism. Secondly, the equivalence of categories is shown to vary uniformly in definable families, with respect to an appropriate relativisation of these categories. Some elaboration on Hrushovki's original constructions are also included.
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Continuoustime branching processes (CTBPs) are powerful tools in random graph theory, but are not appropriate to describe realworld networks, since they produce trees rather than (multi)graphs. In this paper we analyze collapsed branching processes (CBPs), obtained by a collapsing procedure on CTBPs, in order to define multigraphs where vertices have fixed outdegree $m\geq 2$. A key example consists of preferential attachment models (PAMs), as well as generalized PAMs where vertices are chosen according to their degree and age. We identify the degree distribution of CBPs, showing that it is closely related to the limiting distribution of the CTBP before collapsing. In particular, this is the first time that CTBPs are used to investigate the degree distribution of PAMs beyond the tree setting.
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We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to LandauGinzburg potentials. We also treat the higher Maslov index versions of LG potentials introduced in a more restricted setting. We discover a relation between higher disk potentials and symplectic cohomology rings of anticanonical divisor complements (themselves related to closedstring GromovWitten invariants), and explore several other applications to the geometry of Liouville domains.
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A wellknown theorem of Wolpert shows that the WeilPetersson symplectic form on Teichm\"uller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichm\"uller space can be represented. We then prove a generalization of Wolpert's formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.
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In this paper we consider the Prym map for double coverings of curves of genus $g$ ramified at $r>0$ points. That is, the map associating to a double ramified covering its Prym variety. The generic Torelli theorem states that the Prym map is generically injective as soon as the dimension of the space of coverings is less or equal to the dimension of the space of polarized abelian varieties. We prove the generic injectivity of the Prym map in the cases of double coverings of curves with: (a) $g=2$, $r=6$, and (b) $g= 5$, $r=2$. In the first case the proof is constructive and can be extended to the range $r\ge \max \{6,\frac 23(g+2) \}$. For (b) we study the fibre along the locus of the intermediate Jacobians of cubic threefolds to conclude the generic injectivity. This completes the work of Marcucci and Pirola who proved this theorem for all the other cases, except for the bielliptic case $g=1$ (solved later by Marcucci and the first author), and the case $g=3, r=4$ considered previo
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We introduce new characteristic classes of manifold bundles with fiber a closed $4k$dimensional manifold $M$ with indefinite intersection form of signature $(p,q)$. These characteristic classes originate in the homology of arithmetic subgroups of SO$(p,q)$. We prove that our characteristic classes are nontrivial for $M = \#_g(S^{2k}\times S^{2k})$. In this case, the classes we produce live in degree $g$ and are independent from the algebra generated by the stable (i.e. MMM) classes.
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This paper extends the considerations of the works [1, 2] regarding curseofdimensionalityfree numerical approaches to solve certain types of HamiltonJacobi equations arising in optimal control problems, differential games and elsewhere. A rigorous formulation and justification for the extended HopfLax formula of [2] is provided together with novel theoretical and practical discussions including useful recommendations. By using the method of characteristics, the solutions of some problem classes under convexity/concavity conditions on Hamiltonians (in particular, the solutions of HamiltonJacobiBellman equations in optimal control problems) are evaluated separately at different initial positions. This allows for the avoidance of the curse of dimensionality, as well as for choosing arbitrary computational regions. The corresponding feedback control strategies are obtained at selected positions without approximating the partial derivatives of the solutions. The results of numerical
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The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the HartmannTzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports.
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In this investigation we use a simple model of the dynamics of an inviscid vortex sheet given by the BirkhoffRott equation to obtain fundamental insights about the potential for stabilization of shear layers using feedback. First, we demonstrate using analytical computations that the BirkhoffRott equation linearized around the flatsheet configuration is in fact controllable when a pair of point vortices located on both sides of the sheet is used as actuation. On the other hand, this system is not controllable when the actuation has the form of a pair of sinks/sources with zero net mass flux. Next we design a statebased LQR stabilization strategy where the key difficulty is the numerical solution of the Riccati equation in the presence of severe illconditioning resulting from the properties of the BirkhoffRott equation and the chosen form of actuation, an issue which is overcome by performing computations with a suitably increased arithmetic precision. Analysis of the linear close
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We establish a relation between the "large r" asymptotics of the TuraevViro invariants $TV_r$ and the Gromov norm of 3manifolds. We show that for any orientable, compact 3manifold $M$, with (possibly empty) toroidal boundary, $\log TV_r (M)$ is bounded above by a function linear in $r$ and whose slope is a positive universal constant times the Gromov norm of $M$. The proof combines TQFT techniques, geometric decomposition theory of 3manifolds and analytical estimates of $6j$symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $\log TV_r (M)\geqslant B \ r$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3manifolds whose $SO(3)$ TuraevViro invariants grow exponentially. These constructions are essential for the results of [DK:AMU] where the authors make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class gro
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In this paper, we introduce a large system of interacting financial agents in which each agent is faced with the decision of how to allocate his capital between a risky stock or a riskless bond. The investment decision of investors, derived through an optimization, drives the stock price. The model has been inspired by the econophysical LevyLevySolomon model (Economics Letters, 45). The goal of this work is to gain insights into the stock price and wealth distribution. We especially want to discover the causes for the appearance of powerlaws in financial data. We follow a kinetic approach similar to (D. Maldarella, L. Pareschi, Physica A, 391) and derive the mean field limit of our microscopic agent dynamics. The novelty in our approach is that the financial agents apply model predictive control (MPC) to approximate and solve the optimization of their utility function. Interestingly, the MPC approach gives a mathematical connection between the two opponent economic concepts of mode
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In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)=\sum_{k=1}^n Y_{k,1} \cos(kt)+Y_{k,2}\sin(kt)$ for a given sequence of i.i.d. random variables $\{Y_{k,1},Y_{k,2}\}_{k\ge 1}$ which are centered and standardized. We set $\mathcal{N}([0,\pi],Y)$ the number of real roots over $[0,\pi]$ and $\mathcal{N}([0,\pi],G)$ the corresponding quantity when the coefficients follow a standard Gaussian distribution. We prove under a Doeblin's condition on the distribution of the coefficients that $$ \lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,\pi],Y)\right)}{n} =\lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,\pi],G)\right)}{n} +\frac{1}{30}\left(\mathbb{E}(Y_{1,1}^4)3\right). $$ The latter establishes that the behavior of the variance is not universal and depends on the distribution of the underlying coefficients through their kurtosis. Actually, a more general result is proven in this article, which does not require that the coeffici
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We prove that the moduli space of polarized $K3$ surfaces of genus eleven with $n$ marked points is unirational when $n\leq 6$ and uniruled when $n\leq7$. As a consequence, we settle a long standing but not proved assertion about the unirationality of $\cal{M}_{11,n}$ for $n\leq6$. We also prove that the moduli space of polarized $K3$ surfaces of genus eleven with $9$ marked points has nonnegative Kodaira dimension.
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We study the images of the complex Ginibre eigenvalues under the power maps $\pi_M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M {\rm{Gin}} (N,M,k), $$ where the socalled PowerGinibre distributions ${\rm{Gin}}(N,M,k)$ form $M$ independent determinantal point processes. This result can be extended to any radially symmetric normal matrix ensemble, and generalizes Rains' superposition theorem for the CUE to a wider class of point processes. In the same spirit, we prove a generalization of Kostlan's and Rains' independence theorems for twodimensional beta ensembles with radial symmetry and even parameter $\beta$, replacing independence by conditional independence. Our proof technique also allows us to recover a result by Edelman and La Croix for the GUE, and gives new insight into some of the questions they raised.
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Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz reflection and line bundles defined on the Schottky double of a planar domain is briefly discussed in the same context.
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The leading asymptotic behaviour of the Humbert functions $\Phi_2$, $\Phi_3$, $\Xi_2$ of two variables is found, when the absolute values of the two independent variables become simultaneosly large. New integral representations of these functions are given. These are reexpressed as inverse Laplace transformations and the asymptotics is then found from a Tauberian theorem. Some integrals of the Humbert functions are also analysed.
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We use BonahonWong's trace map to study character varieties of oncepunctured torus and of 4punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structure of cluster algebra. It is also shown that cluster mutations give automorphism of character varieties. Motivated by a work of Chekhov, Mazzocco & Rubtsov, we revisit confluences of punctures on sphere from cluster algebraic viewpoint, and we obtain associated affine cubic surfaces constructed by van der Put & Saito based on the RiemannHilbert correspondence. Further studied are quantizations of character varieties by use of quantum cluster algebra.
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Information bottleneck [IB] is a technique for extracting information in some `input' random variable that is relevant for predicting some different 'output' random variable. IB works by encoding the input in a compressed 'bottleneck variable' from which the output can then be accurately decoded. IB can be difficult to compute in practice, and has been mainly developed for two limited cases: (1) discrete random variables with small state spaces, and (2) continuous random variables that are jointly Gaussian distributed (in which case the encoding and decoding maps are linear). We propose a method to perform IB in more general domains. Our approach can be applied to discrete or continuous inputs and outputs, and allows for nonlinear encoding and decoding maps. The method uses a novel upper bound on the IB objective, derived using a nonparametric estimator of mutual information and a variational approximation. We show how to implement the method using neural networks and gradientbased o
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We recall known and establish new properties of the Dieudonn\'e and Moore determinants of quaternionic matrices.Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables. Then we introduce and briefly discuss quaternionic MongeAmp\'ere equations.
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Using the thinning method, we explain the link between classical FisherTippettGnedenko classification of extreme events and their free analogue obtained by Ben Arous and Voiculescu in the context of free probability calculus. In particular, we present explicit examples of large random matrix ensembles, realizing free Weibull, free Fr\'{e}chet and free Gumbel limiting laws, respectively. We also explain, why these free laws are identical to Balkemade HaanPickands limiting distribution for exceedances, i.e. why they have the form of the generalized Pareto distributions.
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We study infinite families of quadratic and cubic twists of the elliptic curve $E = X_0(27)$. For the family of quadratic twists, we establish a lower bound for the $2$adic valuation of the algebraic part of the value of the complex $L$series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$adic valuation of the algebraic part of the same $L$value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and SwinnertonDyer.
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In this paper we consider the field of local times of a discretetime Markov chain on a general state space, and obtain uniform (in time) upper bounds on the total variation distance between this field and the one of a sequence of $n$ i.i.d. random variables with law given by the invariant measure of that Markov chain. The proof of this result uses a refinement of the soft local time method of [11].
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The use of quadratic residues to construct matrices with specific determinant values is a familiar problem with connections to many areas of mathematics and statistics. Our research has focused on using cubic residues to construct matrices with interesting and predictable determinants.
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Let $M$ be a compact Riemannian manifold, $\pi:\tilde M\rightarrow M$ be the universal covering and $\omega$ be a smooth $2$form on $M$ with $\pi^*\omega$ cohomologous to zero. Suppose the fundamental group $\pi_1(M)$ satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth $1$form $\eta$ on $\tilde M$ of linear (resp. bounded) growth such that $\pi^*\omega=d \eta$. As applications, we prove that on a compact K\"ahler manifold $(M,\omega)$ with $\pi^*\omega$ cohomologous to zero, if $\pi_1(M)$ is hyperbolic, then $M$ is K\"ahler hyperbolic and the Euler characteristic $(1)^{\frac{\dim_\mathbb{R} M}{2}} \chi(M)>0$; if $\pi_1(M)$ is $\mathrm{CAT}(0)$ or automatic, then $M$ is K\"ahler nonelliptic and $(1)^{\frac{\dim_\mathbb{R} M}{2}} \chi(M)\geq 0$.
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In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlayatype inequality $$ \mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big( x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big) $$ holds for an arbitrary $(x_n)$ ($\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_n)$ is a vector with corresponding (nonnormalized) weights $(\lambda_n)$ and $\mathscr{M}_{i=1}^n(x_i,\lambda_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$ \mathscr{A}_{i=1}^n \big( \mathscr{M}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big) \le \mathscr{M}_{i=1}^n \big( \mathscr{A}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big) $$ holds for every $(x_n)$ and $(\lambda_n)$ such that the sequence $(\frac{\lambda_k}{\lambda_1+\cdots+\lambda_k})$ is decreasing.
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It is well known that topological classification of dynamical systems with hyperbolic dynamics is significantly defined by dynamics on nonwandering set. F. Przytycki generalized axiom $A$ for smooth endomorphisms that was previously introduced by S. Smale for diffeomorphisms and proved spectral decomposition theorem which claims that nonwandering set of an $A$endomorphism is a union of a finite number basic sets. In present paper the criterion for a basic sets of an $A$endomorphism to be an attractor is given. Moreover, dynamics on basic sets of codimension one is studied. It is shown, that if an attractor is a topological submanifold of codimension one of type $(n1, 1)$, then it is smoothly embedded in ambient manifold and restriction of the endomorphism to this basic set is an expanding endomorphism. If a basic set of type $(n, 0)$ is a topological submanifold of codimension one, then it is a repeller and restriction of the endomorphism to this basic set is also an expanding endom
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We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map $f$ on a topological graph $G$ has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak's M\"obius Disjointness Conjecture is true for graph maps with zero topological entropy. We also extend several results known in interval dynamics to graph maps. We show that a graph map has zero topological entropy if and only if there is no $3$scrambled tuple if and only if the proximal relation is an equivalence relation; a graph map has no scrambled pairs if and only if it is null if and only if it is tame.
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We study an $\ell$adic Galois analogue of the distribution formulas for polylogarithms with special emphasis on path dependency and arithmetic behaviors. As a goal, we obtain a notion of certain universal KummerHeisenberg measures that enable interpolating the $\ell$adic polylogarithmic distribution relations for all degrees.
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In this paper we give an algebraic description of the category of $n$slices for an arbitrary group $G$, in the sense of HillHopkinsRavenel. Specifically, given a finite group $G$ and an integer $n$, we construct an explicit $G$spectrum $W$ (called an isotropic slice $n$sphere) with the following properties: (i) the $n$slice of a $G$spectrum $X$ is equivalent to the data of a certain quotient of the Mackey functor $\underline{[W,X]}$ as a module over the endomorphism Green functor $\underline{[W,W]}$; (ii) the category of $n$slices is equivalent to the full subcategory of right modules over $\underline{[W,W]}$ for which certain restriction maps are injective. We use this theorem to recover the known results on categories of slices to date, and exhibit the utility of our description in several new examples. We go further and show that the Green functors $\underline{[W,W]}$ for certain slice $n$spheres have a special property (they are "geometrically split") which reduces the amo
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In [3] L.Zapponi studied the arithmetic of plane bipartite trees with prime number of edges. He obtained a lower bound on the degree of tree's definition field. Here we obtain a similar lower bound in the following case. There exists a prime number $p$ such, that: a) the number of edges is divisible by $p$, but not by $p^2$; b) for any proper subset of the set of white (or black) vertices the sum of their degrees is not divisible by this $p$.
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We consider a registrationbased approach for localizing sensor networks from range measurements. This is based on the assumption that one can find overlapping cliques spanning the network. That is, for each sensor, one can identify geometric neighbors for which all intersensor ranges are known. Such cliques can be efficiently localized using multidimensional scaling. However, since each clique is localized in some local coordinate system, we are required to register them in a global coordinate system. In other words, our approach is based on transforming the localization problem into a problem of registration. In this context, the main contributions are as follows. First, we describe an efficient method for partitioning the network into overlapping cliques. Second, we study the problem of registering the localized cliques, and formulate a necessary rigidity condition for uniquely recovering the global sensor coordinates. In particular, we present a method for efficiently testing rigi
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Shannon's sampling theorem provides a link between the continuous and the discrete realms stating that bandlimited signals are uniquely determined by its values on a discrete set. This theorem is realized in practice using so called analogtodigital converters (ADCs). Unlike Shannon's sampling theorem, the ADCs are limited in dynamic range. Whenever a signal exceeds some preset threshold, the ADC saturates, resulting in aliasing due to clipping. The goal of this paper is to analyze an alternative approach that does not suffer from these problems. Our work is based on recent developments in ADC design, which allow for ADCs that reset rather than to saturate, thus producing modulo samples. An open problem that remains is: Given such modulo samples of a bandlimited function as well as the dynamic range of the ADC, how can the original signal be recovered and what are the sufficient conditions that guarantee perfect recovery? In this paper, we prove such sufficiency conditions and compl
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Let K be a field and let L/K be a finite extension. Let X/K be a scheme of finite type. A point of X(L) is said to be new if it does not belong to the union of X(F), when F runs over all proper subextensions of L. Fix now an integer g>0 and a finite separable extension L/K of degree d. We investigate in this article whether there exists a smooth proper geometrically connected curve of genus g with a new point in X(L). We show for instance that if K is infinite of characteristic different from 2 and g is bigger or equal to [d/4], then there exist infinitely many hyperelliptic curves X/K of genus g, pairwise nonisomorphic over the algebraic closure of K, and with a new point in X(L). When d is between 1 and 10, we show that there exist infinitely many elliptic curves X/K with pairwise distinct jinvariants and with a new point in X(L).
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The question we deal with here, which was presented to us by Joe Auslander and Anima Nagar, is whether there is a nontrivial cascade (X,T) whose enveloping semigroup, as a dynamical system, is topologically weakly mixing (WM). After an introductory section recalling some definitions and classic results, we establish some necessary conditions for this to happen, and in the final section we show, using Ratner's theory, that the enveloping semigroup of the `time one map' of a classical horocycle flow is weakly mixing.
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We design a stochastic algorithm to train any smooth neural network to $\varepsilon$approximate local minima, using $O(\varepsilon^{3.25})$ backpropagations. The best result was essentially $O(\varepsilon^{4})$ by SGD. More broadly, it finds $\varepsilon$approximate local minima of any smooth nonconvex function in rate $O(\varepsilon^{3.25})$, with only oracle access to stochastic gradients and Hessianvector products.
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We present several applications of mode matching methods in spectral and scattering problems. First, we consider the eigenvalue problem for the Dirichlet Laplacian in a finite cylindrical domain that is split into two subdomains by a "perforated" barrier. We prove that the first eigenfunction is localized in the larger subdomain, i.e., its $L_2$ norm in the smaller subdomain can be made arbitrarily small by setting the diameter of the "holes" in the barrier small enough. This result extends the well known localization of Laplacian eigenfunctions in dumbbell domains. We also discuss an extension to noncylindrical domains with radial symmetry. Second, we study a scattering problem in an infinite cylindrical domain with two identical perforated barriers. If the holes are small, there exists a low frequency at which an incident wave is fully transmitted through both barriers. This result is counterintuitive as a single barrier with the same holes would fully reflect incident waves with lo
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This paper presents a novel method for reformulating nondifferentiable collision avoidance constraints into smooth nonlinear constraints using strong duality of convex optimization. We focus on a controlled object whose goal is to avoid obstacles while moving in an ndimensional space. The proposed reformulation does not introduce approximations, and applies to general obstacles and controlled objects that can be represented in an ndimensional space as the finite union of convex sets. Furthermore, we connect our results with the notion of signed distance, which is widely used in traditional trajectory generation algorithms. Our method can be used in generic navigation and trajectory planning tasks, and the smoothness property allows the use of generalpurpose gradient and Hessianbased optimization algorithms. Finally, in case a collision cannot be avoided, our framework allows us to find "leastintrusive" trajectories, measured in terms of penetration. We demonstrate the efficacy o
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Based on the impressive features that network coding and compressed sensing paradigms have separately brought, the idea of bringing them together in practice will result in major improvements and influence in the upcoming 5G networks. In this context, this paper aims to evaluate the effectiveness of these key techniques in a clusterbased wireless sensor network, in the presence of temporal and spatial correlations. Our goal is to achieve better compression gains by scaling down the total payload carried by applying temporal compression as well as reducing the total number of transmissions in the network using real field network coding. In order to further reduce the number of transmissions, the clusterheads perform a low complexity spatial precoding consisting of sending the packets with a certain probability. Furthermore, we compare our approach with benchmark schemes. As expected, our numerical results run on NS3 simulator show that on overall our scheme dramatically drops the num
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The capacity of cellular networks can be improved by the unprecedented array gain and spatial multiplexing offered by Massive MIMO. Since its inception, the coherent interference caused by pilot contamination has been believed to create a finite capacity limit, as the number of antennas goes to infinity. In this paper, we prove that this is incorrect and an artifact from using simplistic channel models and suboptimal precoding/combining schemes. We show that with multicell MMSE precoding/combining and a tiny amount of spatial channel correlation or largescale fading variations over the array, the capacity increases without bound as the number of antennas increases, even under pilot contamination. More precisely, the result holds when the channel covariance matrices of the contaminating users are asymptotically linearly independent, which is generally the case. If also the diagonals of the covariance matrices are linearly independent, it is sufficient to know these diagonals (and not t
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It is shown that transient graphs for the simple random walk do not admit a nearest neighbor transient Markov chain (not necessarily a reversible one), that crosses all edges with positive probability, while there is such chain for the square grid $\mathbb{Z}^2$. In particular, the $d$dimensional grid $\mathbb{Z}^d$ admits such a Markov chain only when $d=2$. For $d=2$ we present a relevant example due to Gady Kozma, while the general statement for transient graphs is obtained by proving that for every transient irreducible Markov chain on a countable state space, its trace is a.s. recurrent for simple random walk. The case that the Markov chain is reversible is due to GurelGurevich, Lyons and the first named author. We exploit recent results in potential theory of nonreversible Markov chains in order to extend their result to the nonreversible setup.
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This article considers the inverse problem of Magnet resonance electrical impedance tomography (MREIT) in two dimensions. A rigorous mathematical framework for this inverse problem as well as the existing Harmonic $B_z$ Algorithm as a solution algorithm are presented. The convergence theory of this algorithm is extended, such that the usage an approximative forward solution of the underlying partial differential equation (PDE) in the algorithm is sufficient for convergence. Motivated by this result, a novel algorithm is developed where it is the aim to speedup the existing Harmonic $B_z$ Algorithm. This is achieved by combining it with an adaptive variant of the reduced basis method, a model order reduction technique. In a numerical experiment a highresolution image of the shepplogan phantom is reconstructed and both algorithms are compared.
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An integral domain is said to have the IDF property, when every nonzero element of it, has only a finite number of nonassociate irreducible divisors. A counterexample has already been found showing that IDF property does not necessarily ascend in polynomial extensions. In this paper, we introduce a new class of integral domains, called MCDfinite domains, and show that for any domain $D$, $D[X]$ is an IDF domain if and only if $D$ is both IDF and MCDfinite. This in particular entails all the previously known sufficient conditions for the ascent of IDF property. Our new characterization of polynomial domains with IDF property, enables us to use a different construction and build another counterexample which in particular strengthen the previously known result on this matter.
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This paper deals with the problem of finding biLipschitz behavior in nondegenerate Lipschitz maps between metric measure spaces. Specifically, we study maps from (subsets of) Ahlfors regular PI spaces into subRiemannian Carnot groups. We prove that such maps have many biLipschitz tangents, verifying a conjecture of Semmes. As a stronger conclusion, one would like to know whether such maps decompose into countably many biLipschitz pieces. We show that this is true when the Carnot group is Euclidean. For general Carnot targets, we show that the existence of a biLipschitz decomposition is equivalent to a condition on the geometry of the image set.
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This paper introduces a micronexus of water and energy which can be considered as one of the physical infrastructures of the future building/city/village systems. For the electricity side, an alternating current (AC) power flow model integrated with battery energy storage and renewable generation is adopted. The nonlinear hydraulic characteristics in pipe networks is also considered in the proposed micro waterenergy nexus (WEN) model. Integer variables are involved to represent the on/off state of pumps. Base on the proposed nexus model, a cooptimization framework of water and energy networks is developed. The overall cooptimization model is a mixedinteger nonlinear programming problem which is tested on a waterenergy nexus which consists of the IEEE 13bus distribution system and a 8node water distribution network. The simulation results demonstrate that the costefficiency of the cooptimization framework is higher than optimizing two systems independently.
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For any integers $d, n \geq 2$ and $1/({\min\{n,d\}})^{0.4999} < \varepsilon<1$, we show the existence of a set of $n$ vectors $X\subset \mathbb{R}^d$ such that any embedding $f:X\rightarrow \mathbb{R}^m$ satisfying $$ \forall x,y\in X,\ (1\varepsilon)\xy\_2^2\le \f(x)f(y)\_2^2 \le (1+\varepsilon)\xy\_2^2 $$ must have $$ m = \Omega(\varepsilon^{2} \lg n). $$ This lower bound matches the upper bound given by the JohnsonLindenstrauss lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of $\varepsilon$ of interest, since there is always an isometric embedding into dimension $\min\{d, n\}$ (either the identity map, or projection onto $\mathop{span}(X)$). Previously such a lower bound was only known to hold against linear maps $f$, and not for such a wide range of parameters $\varepsilon, n, d$ [LN16]. The best previously known lower bound for general $f$ was $m = \Omega(\varepsilon^{2}\lg n/\lg(1/\varepsilon))$ [Wel74, Lev83, Alo03], which is subop
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In this research we study a finite horizon optimal purchasing problem for items with a mean reverting price process. Under this model a fixed amount of identical items are bought under a given deadline, with the objective of minimizing the cost of their purchasing price and associated holding cost. We prove that the optimal policy for minimizing the expected cost is in the form of a timevariant threshold function that defines the price region in which a purchasing decision is optimal. We construct the threshold function with a simple algorithm that is based on a dynamic programming procedure that calculates the cost function. As part of this procedure we also introduce explicit equations for the crossing time probability and the overshoot expectation of the price process with respect to the threshold function. The characteristics and dynamics of the threshold function are analyzed with respect to time, holding cost, and different parameters of the price process, and yields meaningful
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We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges in the sense of strong distribution convergence to Lamperti's generalized arcsine distribution. Our results can be applied to interval maps and Markov chains. We adopt the double Laplace transform method, which has been utilized in the study of occupation times of diffusions on multiray. We also discuss the inverse problem.
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One of the most significant 5G technology enablers will be DevicetoDevice (D2D) communications. D2D communications constitute a promising way to improve spectral, energy and latency performance, exploiting the physical proximity of communicating devices and increasing resource utilization. Furthermore, network infrastructure densification has been considered as one of the most substantial methods to increase system performance, taking advantage of base station proximity and spatial reuse of system resources. However, could we improve system performance by leveraging both of these two 5G enabling technologies together in a multicell environment? How does spectrum sharing affect performance enhancement? This article investigates the implications of interference, densification and spectrum sharing in D2D performance gain. The inband D2D approach, where legacy users coexist with potential D2D pairs, is considered in a multicell system. Overlay and underlay spectrum sharing approaches
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A single cell downlink scenario is considered where a multipleantenna base station delivers contents to multiple cacheenabled user terminals. Using the ideas from multiserver coded caching (CC) scheme developed for wired networks, a joint design of CC and general multicast beamforming is proposed to benefit from spatial multiplexing gain, improved interference management and the global CC gain, simultaneously. Utilizing the multiantenna multicasting opportunities provided by the CC technique, the proposed method is shown to perform well over the entire SNR region, including the low SNR regime, unlike the existing schemes based on zero forcing (ZF). Instead of nulling the interference at users not requiring a specific coded message, general multicast beamforming strategies are employed, optimally balancing the detrimental impact of both noise and interstream interference from coded messages transmitted in parallel. The proposed scheme is shown to provide the same degreesoffreedom
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Given the high throughput requirement for 5G, merging millimeter wave technologies and multiuser MIMO seems a very promising strategy. As hardware limitations impede to realize a full digital architecture, hybrid MIMO architectures, using digital precoding and phased antenna arrays, are considered a feasible solution to implement multiuser MIMO at millimeter wave. However, real channel propagation and hardware nonidealities can significantly degrade the performance of such systems. Experimenting the new architecture is thus crucial to confirm and to support system design. Nevertheless, hybrid MIMO systems are not yet understood as the effects of the wide channel bandwidths at millimeter wave, the nonideal RF front end as well as the imperfections of the analog beamforming are often neglected. In this paper, we present a 60 GHz MUMIMO testbed using phased antenna arrays at both transmitter and receiver. The base station equipped with a 32 phased antenna array allocates simultaneous
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An anonymous reader quotes a report from Bloomberg: How should we think about new and future technologies? The two main stances seem to be extreme optimism and extreme pessimism. A better approach would be careful planning and management. Optimists tend to overlook the fact that the technological successes of the past required a lot of social engineering before their benefits became widely shared. Countries like Maoist China and North Korea implemented perverse economic systems that withheld the bounty of modern technology from most of their citizens. And poor countries didn't really begin to beat poverty until decades after colonialism ended. Pessimists, meanwhile, often assume that new technologies can be stopped in their tracks by act of popular will. They probably can't. Even the most impoverished, repressive regimes of the 20th century adopted new technologies, and often suffered their worst consequences. Scientific research and invention, meanwhile, can be forbidden in one countr
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We study Bernoulli bond percolation on nonunimodular quasitransitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasitransitive subgroup. We prove that percolation on any such graph has a nonempty phase in which there are infinite light clusters, which implies the existence of a nonempty phase in which there are infinitely many infinite clusters. That is, we show that $p_c<p_h \leq p_u$ for any such graph. This answers a question of Haggstrom, Peres, and Schonmann (1999), and verifies the nonunimodular case of a wellknown conjecture of Benjamini and Schramm (1996). We also prove that the triangle condition holds at criticality on any such graph, which is known to imply that the critical exponents governing the percolation probability and the cluster volume take their meanfield values. Finally, we also prove that the susceptibility exponent, the gap exponent, and the cluster radius exponent each take their meanfield values on any such graph;
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