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We give normal forms for generic kdimensional parametric families $(Z_\varepsilon)_\varepsilon$ of germs of holomorphic vector fields near $0\in\mathbb{C}^2$ unfolding a saddlenode singularity $Z_0$, under the condition that there exists a family of invariant analytic curves unfolding the weak separatrix of $Z_0$. These normal forms provide a moduli space for these parametric families. In our former 2008 paper, a modulus of a family was given as the unfolding of the MartinetRamis modulus, but the realization part was missing. We solve the realization problem in that partial case and show the equivalence between the two presentations of the moduli space. Finally, we completely characterize the families which have a modulus depending analytically on the parameter. We provide an application of the result in the field of nonlinear, parameterized differential Galois theory.
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In this paper, an Artificial Neural Network (ANN) technique is developed to find solution of celebrated Fractional order Differential Equations (FDE). Compared to integer order differential equation, FDE has the advantage that it can better describe sometimes various real world application problems of physical systems. Here we have employed multilayer feed forward neural architecture and error back propagation algorithm with unsupervised learning for minimizing the error function and modification of the parameters (weights and biases). Combining the initial conditions with the ANN output gives us a suitable approximate solution of FDE. To prove the applicability of the concept, some illustrative examples are provided to demonstrate the precision and effectiveness of this method. Comparison of the present results with other available results by traditional methods shows a close match which establishes its correctness and accuracy of this method.
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We give an introduction into some aspects of the emerging mathematical theory of manybody localization (MBL) for disordered quantum spin chains. In particular, we discuss manifestations of MBL such as zerovelocity LiebRobinson bounds, quasilocality of the time evolution of local observables, as well as exponential clustering and low entanglement of eigenstates. Explicit models where such properties have recently been verified are the XY and XXZ spin chain, in each case with disorder introduced in the form of a random exterior field. We introduce these models, state many of the available results and try to provide some general context. We discuss methods and ideas which enter the proofs and, in a few illustrative examples, include more detailed arguments. Finally, we also mention some directions for future mathematical work on MBL.
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We prove upper bounds for the average size of the $\ell$torsion $\Cl_K[\ell]$ of the class group of $K$, as $K$ runs through certain natural families of number fields and $\ell$ is a positive integer. We refine a key argument, used in almost all results of this type, which links upper bounds for $\Cl_K[\ell]$ to the existence of many primes splitting completely in $K$ that are small compared to the discriminant of $K$. Our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument, in conjunction with new counting results for these invariants. This leads to significantly improved upper bounds for the average and sometimes even higher moments of $\Cl_K[\ell]$ for many families of number fields $K$ considered in the literature, for example, for the families of all degree$d$fields for $d\in\{2,3,4,5\}$ (and non$D_4$ if $d=4$). As an application of the case $d=2$ we obtain the best upper bou
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In this paper, bicomplex kFibonacci quaternions are defined. Also, some algebraic properties of bicomplex kFibonacci quaternions which are connected with bicomplex numbers and kFibonacci numbers are investigated. Furthermore, the Honsberger identity, the d'Ocagne's identity, Binet's formula, Cassini's identity, Catalan's identity for these quaternions are given.
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We prove that nearthreshold negative energy solutions to the 2D cubic ($L^2$critical) focusing ZakharovKuznetsov (ZK) equation blowup in finite or infinite time. The proof consists of several steps. First, we show that if the blowup conclusion is false, there are negative energy solutions arbitrarily close to the threshold that are globally bounded in $H^1$ and are spatially localized, uniformly in time. In the second step, we show that such solutions must in fact be exact remodulations of the ground state, and hence, have zero energy, which is a contradiction. This second step, a nonlinear Liouville theorem, is proved by contradiction, with a limiting argument producing a nontrivial solution to a (linear) linearized ZK equation obeying uniformintime spatial localization. Such nontrivial linear solutions are excluded by a localviral spacetime estimate. The general framework of the argument is modeled on Merle [29] and Martel & Merle [24], who treated the 1D problem of the
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We study strong types and Galois groups in model theory from a topological and descriptivesettheoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems related to cardinality and Borel cardinality of strong types, quotients of definable groups and related objecets, generalising (and often improving) essentially all hitherto known results in this area. In particular, we show that under reasonable assumptions, strong type spaces are "locally" quotients of compact Polish groups. It follows that they are smooth if and only if they are typedefinable, and that a quotient of a typedefinable group by an analytic subgroup is either finite or of cardinality at least continuum.
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In this article we prove a new central limit theorem (CLT) for coupled particle filters (CPFs). CPFs are used for the sequential estimation of the difference of expectations w.r.t. filters which are in some sense close. Examples include the estimation of the filtering distribution associated to different parameters (finite difference estimation) and filters associated to partially observed discretized diffusion processes (PODDP) and the implementation of the multilevel Monte Carlo (MLMC) identity. We develop new theory for CPFs and based upon several results, we propose a new CPF which approximates the maximal coupling (MCPF) of a pair of predictor distributions. In the context of ML estimation associated to PODDP with discretization $\Delta_l$ we show that the MCPF and the approach in Jasra et al. (2018) have, under assumptions, an asymptotic variance that is upperbounded by an expression that is (almost) $\mathcal{O}(\Delta_l)$, uniformly in time. The $\mathcal{O}(\Delta_l)$ rate pr
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We propose the following definition of topological quantum phases valid for mixed states: two states are in the same phase if there exists a time independent, fast and local Lindbladian evolution driving one state into the other. The underlying idea, motivated by Koenig and Pastawski in 2013, is that it takes time to create new topological correlations, even with the use of dissipation. We show that it is a good definition in the following sense: (1) It divides the set of states into equivalent classes and it establishes a partial order between those according to their level of "topological complexity". (2) It provides a path between any two states belonging to the same phase where observables behave smoothly. We then focus on pure states to relate the new definition in this particular case with the usual definition for quantum phases of closed systems in terms of the existence of a gapped path of Hamiltonians connecting both states in the corresponding ground state path. We show first
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Despite the fact that many important problems (including clustering) can be described using hypergraphs, theoretical foundations as well as practical algorithms using hypergraphs are not well developed yet. In this paper, we propose a hypergraph modularity function that generalizes its well established and widely used graph counterpart measure of how clustered a network is. In order to define it properly, we generalize the ChungLu model for graphs to hypergraphs. We then provide the theoretical foundations to search for an optimal solution with respect to our hypergraph modularity function. Two simple heuristic algorithms are described and applied to a few small illustrative examples. We show that using a strict version of our proposed modularity function often leads to a solution where a smaller number of hyperedges get cut as compared to optimizing modularity of 2section graph of a hypergraph.
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In this paper the authors obtain a new equivalent norms of the Besov spaces of variable smoothness and integrability. Our main tools are the continuous version of Calderon reproducing formula, maximal inequalities and variable exponent technique, but allowing the parameters to vary from point to point will raise extra difficulties which, in general, are overcome by imposing regularity assumptions on these exponents.
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We address the question of convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Samesign particles repel each other, and oppositesign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blowup in finite time. We remedy this situation by regularising the interaction potential at a lengthscale $\delta_n>0$, which converges to zero as the number of particles $n$ tends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of
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Under $\mathfrak{p} = \mathfrak{c}$, we answer Question 24 of \cite{dikranjan&shakhmatov3} for cardinality ${\mathfrak c}$ , by showing that if a nontorsion Abelian group of size continuum admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with nontrivial convergent sequences.
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We study wavespackets in nonlinear periodic media in arbitrary ($d$) spatial dimension, modeled by the cubic GrossPitaevskii equation. In the asymptotic setting of small and broad wavespackets with $N\in \mathbb{N}$ carrier Bloch waves the effective equations for the envelopes are first order coupled mode equations (CMEs). We provide a rigorous justification of the effective equations. The estimate of the asymptotic error is carried out in an $L^1$norm in the Bloch variables. This translates to a supremum norm estimate in the physical variables. In order to investigate the existence of gap solitons of the $d$dimensional CMEs, we discuss spectral gaps of the CMEs. For $N=4$ and $d=2$ a family of time harmonic gap solitons is constructed formally asymptotically and numerically. Moving gap solitons have not been found for $d>1$ and for the considered values of $N$ due to the absence of a spectral gap in the standard moving frame variables.
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Throughout history, recreational mathematics has always played a prominent role in advancing research. Following in this tradition, in this paper we extend some recent work with crazy sequential representations of numbers equations made of sequences of one through nine (or nine through one) that evaluate to a number. All previous work on this type of puzzle has focused only on base ten numbers and whether a solution existed. We generalize this concept and examine how this extends to arbitrary bases, the ranges of possible numbers, the combinatorial challenge of finding the numbers, efficient algorithms, and some interesting patterns across any base. For the analysis, we focus on bases three through ten. Further, we outline several interesting mathematical and algorithmic complexity problems related to this area that have yet to be considered.
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We give a generalization of Dorronsoro's Theorem on critical $\mathrm{L}^p$Taylor expansions for $\mathrm{BV}^k$maps on $\mathrm{R}^n$, i.e., we characterize homogeneous linear differential operators $\mathbb{A}$ of $k$th order such that $D^{kj}u$ has $j$th order $\mathrm{L}^{n/(nj)}$Taylor expansion a.e. for all $u\in\mathrm{BV}^\mathbb{A}_{\text{loc}}$ (here $j=1,\ldots, k$, with an appropriate convention if $j\geq n$). The space $\mathrm{BV}^\mathbb{A}_{\text{loc}}$ consists of those locally integrable maps $u$ such that $\mathbb{A} u$ is a Radon measure on $\mathbb{R}^n$. A new $\mathrm{L}^\infty$Sobolev inequality is established to cover higher order expansions. Lorentz refinements are also considered. The main results can be seen as pointwise regularity statements for linear elliptic systems with measuredata.
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We consider random multimodal $C^3$ maps with negative Schwarzian derivative, defined on a finite union of closed intervals in $[0,1]$, onto the interval $[0,1]$ with the base space $\Omega$ and a base invertible ergodic map $\theta:\Omega\to\Omega$ preserving a probability measure $m$ on $\Omega$. We denote the corresponding skew product map by $T$ and call it a critically finite random map of an interval. We prove that there exists a subset $AA(T)$ of $[0,1]$ with the following properties: (1) For each $t\in AA(T)$ a $t$conformal random measure $\nu_t$ exists. We denote by $\lambda_{t,\nu_t,\omega}$ the corresponding generalized eigenvalues of the corresponding dual operators $\mathcal{L}_{t,\omega}^*$, $\omega\in\Omega$. (2) Given $t\ge 0$ any two $t$conformal random measures are equivalent. (3) The expected topological pressure of the parameter $t$: $$\mathcal{E}P(t):=\int_{\Omega}\log\lambda_{t,\nu,\omega}dm(\omega) $$ is independent of the choice of a $t$conformal random measu
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Consider the random Cayley graph of a finite, Abelian group $G = \oplus_{j=1}^d \mathbb{Z}_{m_j}$ with respect to $k$ generators chosen uniformly at random. We prove that the simple random walk on this graph exhibits abrupt convergence to equilibrium, known as cutoff, subject to $k \gg 1$, $\log k \ll \log G$ and mild conditions on $d$ and $\min_j m_j$ in terms of $G$ and $k$. In accordance with spirit of a conjecture of Aldous and Diaconis, the cutoff time is shown to be independent of the algebraic structure of the group; it occurs around the time that the entropy of the simple random walk on $\mathbb{Z}^k$ is $\logG$, independent of $d$ and $\{m_j\}_{j=1}^d$. Moreover, we prove a Gaussian profile of convergence to equilibrium inside the cutoff window. We also prove that the order of the spectral gap is $G^{2/k}$ with high probability (as $G$ and $k$ diverge); this extends a celebrated result of Alon and Roichman.
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There is a canonical derived Poisson structure on the universal enveloping algebra $\mathcal{U}\mathfrak{a}$ of a (DG) Lie algebra $\mathfrak{a}$ that is Koszul dual to a cyclic cocommutative (DG) coalgebra. Interesting special cases of this derived Poisson structure include (an analog of) the ChasSullivan bracket on string topology. We study how certain derived character of $\mathfrak{a}$ intertwine this derived Poisson structure with the induced Poisson structure on the representation homology of $\mathfrak{a}$. In addition, we obtain an analog of one of our main results for associative algebras.
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We study the decentralized caching scheme in a twolayer network, which includes a sever, multiple helpers, and multiple users. Basically, the proposed caching scheme consists of two phases, i.e, placement phase and delivery phase. In the placement phase, each helper/user randomly and independently selects contents from the server and stores them into its memory. In the delivery phase, the users request contents from the server, and the server satisfies each user through a helper. Different from the existing caching scheme, the proposed caching scheme takes into account the prestored contents at both helpers and users in the placement phase to design the delivery phase. Meanwhile, the proposed caching scheme exploits index coding in the delivery phase and leverages multicast opportunities, even when different users request distinct contents. Besides, we analytically characterize the performance limit of the proposed caching scheme, and show that the achievable rate region of the propo
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Resonant power converters offer improved levels of efficiency and power density. In order to implement such systems, advanced control techniques are required to take the most of the power converter. In this context, model predictive control arises as a powerful tool that is able to consider nonlinearities and constraints, but it requires the solution of complex optimization problems or strong simplifying assumptions that hinder its application in real situations. Motivated by recent theoretical advances in the field of deep learning, this paper proposes to learn, offline, the optimal control policy defined by a complex model predictive formulation using deep neural networks so that the online use of the learned controller requires only the evaluation of a neural network. The obtained learned controller can be executed very rapidly on embedded hardware. We show the potential of the presented approach on a HardwareintheLoop setup of an FPGAcontrolled resonant power converter.
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A space has $\sigma$compact tightness if the closures of $\sigma$compact subsets determines the topology. We consider a dense set variant that we call densely kseparable. We consider the question of whether every densely kseparable space is separable. The somewhat surprising answer is that this property, for compact spaces, implies that every dense set is separable. The path to this result relies on the known connections established between $\pi$weight and the density of all dense subsets, or more precisely, the cardinal invariant $\delta(X)$.
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The derived category $D[C,V]$ of the Grothendieck category of enriched functors $[C,V]$, where $V$ is a closed symmetric monoidal Grothendieck category and $C$ is a small $V$category, is studied. We prove that if the derived category $D(V)$ of $V$ is a compactly generated triangulated category with certain reasonable assumptions on compact generators or $K$injective resolutions, then the derived category $D[C,V]$ is also compactly generated triangulated. Moreover, an explicit description of these generators is given.
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Let X be a compact Kahler manifold of dimension n. Let G be a group of zero entropy automorphisms of X. Let G0 be the set of elements in G which are isotopic to the identity. We prove that after replacing G by a suitable finiteindex subgroup, G/G0 is a unipotent group of derived length at most n1. This is a corollary of an optimal upper bound of length involving the Kodaira dimension of X. We also study the algebrogeometric structure of X when it admits a group action with maximal derived length n1.
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We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R^n. We prove wellposedness in SobolevKato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wavefront sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of randomfield solutions for the associated stochastic Cauchy problems. To this aim, we first discuss algebraic properties for iterated integrals of suitable parameterdependent families of Fourier integral operators, associated with the characteristic roots, which are involved in the construction of the fundamental solution. In particular, we show that, also for this operator class, the involutiveness of the characteristics implies commutative properties for such e
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A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The representation theory of quantum groups is remarkably similar to the representation theory of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for finite quantum groups. The Upper Bound Lemma is used to study the convergence of ergodic random walks on the dual group $\widehat{S_n}$ as well as on the truly quantum groups of Sekine.
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This paper gives convex conditions for synthesis of a distributed control system for largescale networked nonlinear dynamic systems. It is shown that the technique of control contraction metrics (CCMs) can be extended to this problem by utilizing separable metric structures, resulting in controllers that only depend on information from local sensors and communications from immediate neighbours. The conditions given are pointwise linear matrix inequalities, and are necessary and sufficient for linear positive systems and certain monotone nonlinear systems. Distributed synthesis methods for systems on chordal graphs are also proposed based on SDP decompositions. The results are illustrated on a problem of vehicle platooning with heterogeneous vehicles, and a network of nonlinear dynamic systems with over 1000 states that is not feedback linearizable and has an uncontrollable linearization
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Strong stability preserving (SSP) RungeKutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear noninnerproduct stability properties to be satisfied. Unlike the case for L2 linear stability, implicit methods do not significantly alleviate the timestep restriction when the SSP property is needed. For this reason, when handling problems with a linear component that is stiff and a nonlinear component that is not, SSP integrating factor RungeKutta methods may offer an attractive alternative to traditional timestepping methods. The strong stability properties of integrating factor RungeKutta methods where the transformed problem is evolved with an explicit SSP RungeKutta method with nondecreasing abscissas was recently established. In this work, we consider the use of downwinded spatial operators to preserve the strong stability properties of integrating factor RungeKutta methods where the RungeKutta method
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In this paper, dual complex Pell numbers and quaternions are defined. Also, some algebraic properties of dualcomplex Pell numbers and quaternions which are connected with dual complex numbers and Pell numbers are investigated. Furthermore, the Honsberger identity, Binet's formula, Cassini's identity, Catalan's identity for these quaternions are given.
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In this paper, dualcomplex kPell numbers and dualcomplex kPell quaternions are defined. Also, some algebraic properties of dualcomplex kPell numbers and quaternions which are connected with dualcomplex numbers and kPell numbers are investigated. Furthermore, the Honsberger identity, the d'Ocagne's identity, Binet's formula, Cassini's identity, Catalan's identity for these quaternions are given.
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Oneparameter interpolations between any two unitary matrices (e.g., quantum gates) $U_1$ and $U_2$ along efficient paths contained in the unitary group are constructed. Motivated by applications, we propose the continuous unitary path $U(\theta)$ obtained from the QRfactorization \[ U(\theta)R(\theta)=(1\theta)A+\theta B, \] where $U_1 R_1=A$ and $U_2 R_2=B$ are the QRfactorizations of $A$ and $B$, and $U(\theta)$ is a unitary for all $\theta$ with $U(0)=U_1$ and $U(1)=U_2$. The QRalgorithm is modified to, instead of $U(\theta)$, output a matrix whose columns are proportional to the corresponding columns of $U(\theta)$ and whose entries are polynomial or rational functions of $\theta$. By an extension of the BerlekampWelch algorithm we show that rational functions can be efficiently and exactly interpolated with respect to $\theta$. We then construct probability distributions over unitaries that are arbitrarily close to the Haar measure. Demonstration of computational advantages
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We give an introduction to the theory and to some applications of eigenvectors of tensors (in other words, invariant onedimensional subspaces of homogeneous polynomial maps), including a review of some concepts that are useful for their discussion. The intent is to give practitioners an overview of fundamental notions, results and techniques.
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We review Aomoto's generalized hypergeometric functions on Grassmannian spaces Gr(k +1, n+1). Particularly, we clarify integral representations of the generalized hypergeometric functions in terms of twisted homology and cohomology. With an example of the Gr(2, 4) case, we consider in detail Gauss' original hypergeometric functions in Aomoto's framework. This leads us to present a new systematic description of Gauss' hypergeometric differential equation in a form of a first order Fuchsian differential equation.
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It is proved that the projective model structure of the category of topologically enriched diagrams of topological spaces over a topologically enriched locally contractible small category is Quillen equivalent to the standard Quillen model structure of topological spaces. We give a geometric interpretation of this fact in directed homotopy.
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Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic objects over the model; two generics are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relations for various partial orders, with particular focus on Cohen and random forcing. We prove, amongst other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, while the latter is neither amenable nor treeable.
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We estimate size of recurrence of an action of a nilpotent group by homeomorphisms of a compact space for polynomial mappings into a nilpotent group form the partial semigroup $(\mathcal{P}_{f}(\mathbb{N}),\uplus)$. To do this we have used algebraic structure of the Stone\v{C}ech copactification partial semigroup and that of the given nilpotent group.
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Onedimensional potentials defined by $V^{(S)}(x) =S(S+1) \hbar^2 \pi^2 /[2ma^2\sin^2(\pi x/a)]$ (for integer $S$) arise in the repeated supersymmetrization of the infinite square well, here defined over the region $(0,a)$. We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closedform expressions for the normalized solutions, $\psi_n^{(S)}(x)$, for all $V^{(S)}(x)$ in terms of wellknown special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shapeinvariant potentials, the trigonometric P\"oschlTeller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behavior of the corresponding momentumspace wave functions $\phi_n^{(S)}(p)$ for large $
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The purpose of this work is to rigorously prove the existence of traveling waves in neural field models with lateral inhibition synaptic coupling types and sigmoidal firing rate functions. In the case of traveling fronts, we utilize theory of linear operators and the implicit function theorem on Banach spaces, providing a variation of the homotopy approach originally proposed by Ermentrout and McLeod (1992) in their seminal study of monotone fronts in neural field models. After establishing the existence of traveling fronts, we move to a wellstudied singularly perturbed system with linear feedback. For the special case where the synaptic coupling kernel is a difference of exponential functions, we are able to combine our results for the front with theory of invariant manifolds in autonomous dynamical systems to prove the existence of fast traveling pulses that are comparable to singular homoclinical orbits. Finally, using a numerical approximation scheme, we derive the ubiquitous Evan
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We investigate the algebraic structure of complex Lie groups equipped with leftinvariant metrics which are expanding semialgebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semialgebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding algebraic solitons on their nirladicals.
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We study the Lie bialgebra structures that can be built on the onedimensional central extension of the Poincar\'e and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, $U(1)$ or $SO(2)$ on the (1+1) dimensional Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the ColemanMandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensorproduct states. We obtain all Lie bialgebra structures on centrallyextended Poincar\'e and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore all of them admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous M
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Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over arbitrary base fields, and give examples illustrating how behaviour can differ from the algebraically closed case.
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In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the ChekanovEliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.
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Error bound condition has recently gained revived interest in optimization. It has been leveraged to derive faster convergence for many popular algorithms, including subgradient methods, proximal gradient method and accelerated proximal gradient method. However, it is still unclear whether the FrankWolfe (FW) method can enjoy faster convergence under error bound condition. In this short note, we give an affirmative answer to this question. We show that the FW method (with a line search for the step size) for optimization over a strongly convex set is automatically adaptive to the error bound condition of the problem. In particular, the iteration complexity of FW can be characterized by $O(\max(1/\epsilon^{1\theta}, \log(1/\epsilon)))$ where $\theta\in[0,1]$ is a constant that characterizes the error bound condition. Our results imply that if the constrained set is characterized by a strongly convex function and the objective function can achieve a smaller value outside the considered
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We consider stochastic (partial) differential equations appearing as Markovian lifts of affine Volterra processes with jumps from the point of view of the generalized Feller property which was introduced in, e.g., D\"orsekTeichmann (2010). In particular we provide new existence, uniqueness and approximation results for Markovian lifts of affine rough volatility models of general jump diffusion type. We demonstrate that in this Markovian light the theory of stochastic Volterra processes becomes almost classical.
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We study the oneparameter family of twisted Kahler TaubNUT metrics (discovered by Donaldson), along with two exceptional TaubNUTlike instantons, and understand them to the extend that should be sufficient for blowup and gluing arguments. In particular we parametrize their geodesics from the origin, determine curvature falloff rates, volume growth rates for metric balls, and find blowdown limits.
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3Dprinted plastic objects can track their own use without any electronics
1011 MIT Technology 193We propose a formula for the enumeration of closed lattice random walks of length $n$ enclosing a given algebraic area. The information is contained in the Kreft coefficients which encode, in the commensurate case, the Hofstadter secular equation for a quantum particle hopping on a lattice coupled to a perpendicular magnetic field. The algebraic area enumeration is possible because it is split in $2^{n/21}$ pieces, each tractable in terms of explicit combinatorial expressions.
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An anonymous reader writes: The Kaiser Family Foundation's annual review of employerbased insurance shows that 21% of large employers collect health information from employees' mobile apps or wearable devices, as part of their wellness programs  up from 14% last year. Wellness programs are voluntary, and so is contributing your health information to them. But among companies that offer a wellness program, just 9% of employers (including 35% of large employers) offer workers an incentive to participate.
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At least two U.S. states are investigating a breach at Alphabet's Google that may have exposed private profile data of at least 500,000 users to hundreds of external developers. From a report: The investigation follows Google's announcement on Monday that it would shut down the consumer version of its social network Google+ and tighten its datasharing policies after a "bug" potentially exposed user data that included names, email addresses, occupations, genders and ages. "We are aware of public reporting on this matter and are currently undertaking efforts to gain an understanding of the nature and cause of the intrusion, whether sensitive information was exposed, and what steps are being taken or called for to prevent similar intrusions in the future," Jaclyn Severance, a spokeswoman for Connecticut Attorney General George Jepsen, told Reuters in an email. The New York Attorney General's office also said it was looking into the breach.
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Fully selfdriving cars may be on the fast lane to U.S. roads under a pilot program the Trump administration said on Tuesday it was considering, which would allow realworld road testing for a limited number of the vehicles. Reuters: Selfdriving cars used in the program would potentially need to have technology disabling the vehicle if a sensor fails or barring vehicles from traveling above safe speeds, the National Highway Traffic Safety Administration (NHTSA) said in a document made public Tuesday. NHTSA said it was considering whether it would have to be notified of any accident within 24 hours and was seeking public input on what other data should be disclosed including near misses. The U.S. House of Representatives passed legislation in 2017 to speed the adoption of selfdriving cars, but the Senate has not approved it. Several safety groups oppose the bill, which is backed by carmakers. It has only a slender chance of being approved in 2018, congressional aides said.
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An anonymous reader shares a report: Last month, an activist from the German art collective Peng! walked into her local government office in Berlin and applied for a new passport. "I probably have broken the law," the woman, a chemist living in the Western Saxony region, told Motherboard, "but our lawyers don't know which one." The woman applied for a passport using a photo of two separate people. Using specialized software created by Peng!, the collective merged the facial vectors from two different faces from two different images into one. Billie Hoffman (a pseudonym used by everyone in the Peng! Collective when talking to journalists), she told me how easy the whole process was: "Officials didn't mention fraud at any point." Hoffman's passport application was approved, and now she has an official German passport using the digitally altered photo. The photo is half her, half Federica Mogherini, an Italian politician who is the High Representative of the European Union for Foreign Aff
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At its Pixel 3 launch event, Google announced a smart speaker called the Google Home Hub, featuring a 7inch display to give you visual information, making it easier to control smart home devices and view photos and the weather. Interestingly, Google decided not to include a camera in this device for privacy reasons, as they want you to feel comfortable placing it in an intimate location, such as a bedroom. PhoneDog reports: Google explains that Home Hub will be able to recognize who is speaking to it using Voice Match to provide info for that specific person, which should help to make the device more useful in homes with multiple people. And when you're not using Home Hub, a feature called Live Albums will let you select certain people and have Google Photos create albums with images of these people. Another feature of Google's Home Hub is the Home View. With it, you can easily see and control your smart home devices. And then there's Ambient EQ, which uses a sensor that'll adjust the
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Not everything got leaked before Google's event today. One surprise announcement that wowed was Call Screen, a new feature that lets the Google Assistant answer your incoming calls and politely ask what the caller wants. A realtime transcript will appear on your screen, allowing you to decide whether or not you want to pick up. When your Pixel rings, a "Screen call" button shows up alongside the usual controls. Tapping it will prompt the Google Assistant to tell your caller that the call is being screened and ask what it's about. Their explanation is transcribed on your screen, and you have options to mark the call as spam or tell the caller you'll get back to them, among others. This is an amazing feature that will save a lot of people a lot of frustration. I want this feature on my phone now. On a related note, Google Duplex, the feature whereby the Google Assistant will call restaurants and such on your behalf, will be rolled out to Pixel phones next month.
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Alongside the new Pixel smartphones, and the Pixel Slate laptoptablet hybrid, Google on Tuesday also announced a new version of its Chromecast streaming adapter, the third generation of the company's streaming device, which supports playback video at higher frame rates and can also stream multiroom audio. From a report: The new device goes on sale Tuesday in the U.S., Australia, Canada, Denmark, Finland, Great Britain, Japan, Netherlands, New Zealand, Norway, Singapore and Sweden. Stateside, the new Chromecast once again costs $35  the same as its predecessor. [...] The bigger changes are on the inside: The new Chromecast is 15% faster than the previous model, which allows it to stream 1080p HD video with a rate of up to 60 frames per second (fps). "Everything becomes much smoother," said Google Home product manager Chris Chan during a recent interview with Variety. He specifically cited the growth of 60fps content on YouTube as one of the reasons Google added the new feature.
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Google on Tuesday unveiled the Pixel 3 and Pixel 3 XL, its latest flagship Android smartphones. "For life on the go, we designed the world's best camera and put it in the world's most helpful phone," said Google's hardware chief Rick Osterloh. From a report: The Pixel 3 starts at $799 for 64GB, with the 3 XL costing $899. Add $100 to either for the 128GB storage option. Core specs for both include a Snapdragon 845, 4GB RAM (there's no option for more), Bluetooth 5.0, and frontfacing stereo speakers. Also inside is a new Titan M security chip, which Google says provides "ondevice protection for login credentials, disk encryption, app data, and the integrity of the operating system." Preorders for both phones begin today, and buyers will get six months of free YouTube Music service. The Pixel 3 and 3 XL both feature larger screens than last year's models thanks to slimmed down bezels  and the controversial notch in the case of the bigger phone. The 3 XL has a 6.3inch display (up fro
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Google intends to launch a censored version of its Search app for China sometime in the next six to nine months, according to a leaked transcript from a private employee meeting held last month. The Intercept's Ryan Gallagher today reported the company's Search engine chief, Ben Gomes, held a meeting to congratulate a room full of employees working on the platform, dubbed Project Dragonfly. From a report: According to The Intercept, Gomes talked about the launch timeline: "While we are saying it's going to be six and nine months [to launch], the world is a very dynamic place." He goes on to point out that the current political climate makes it difficult to pinpoint a definite timeline, but indicates employees should be ready to launch whenever a "window opens." These comments come in stark contrast to public statements given recently by both Gomes and Google's chief privacy officer, Kieth Enright. Speaking to members of Congress last month, Enright tried to skirt the issue of the Drago
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In addition to announcing new flagship phones today, Google took the wraps off a new premium tablet called the Pixel Slate. It's a Chrome OSpowered slate with a 12.3inch display that's supposed to be the sharpest in its class. Google claims this isn't just a laptop pretending to be a tablet or a phone pretending to be a computer. From a report: It has a resolution of 3,000 x 2,000  i.e., a pixel density of 293 ppi, which Google says is the highest for a premium 12inch tablet. For reference, the Surface Pro 6 and iPad Pro (12.9 inch) come in at 267 ppi and 264 ppi, respectively. Google was able to make the screen so sharp because of an energyefficient LCD technology called Low Temperature PolySilicon (LTPS), which let the company pack in more pixels without sacrificing size or battery. In fact, the Pixel Slate is supposed to last up to 12 hours on a charge, which is impressive for its skinny 7mm profile. [...] What stands out about the Pixel Slate is the version of Chrome OS it ru
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Google unveiled its new Pixel phones today, as well as the Pixel Slate, a ChromeOS tablet/laptop device that's basically a cross between an iPad Pro and a Surface Pro. Virtually everything from the event was leaked over the past few weeks, so there were few  if any  surprises. The new devices are certainly interesting, but Google continues its policy of not making these products available in most of the world, so there's little for me to say about them  I have never seen them, let alone used them. One thing that stood out to me about the Pixel Slate are its specifications  it runs on Intel processors, and in order to get a processor that isn't a slow Celeron or m3, you need to shell out some big bucks. I don't have particularly good experiences with Celeron or m3 processors, and even Intel's mobile i5 chips have never really managed to impress me  hence why I opted for the i7 version of the latest Dell XPS 13 when I bought a new laptop a few weeks ago. In The Verge's video, yo
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Google's humansounding AI software that makes calls for you is coming to Pixel smartphones next month in select markets, like New York, Atlanta, Phoenix, and the San Francisco Bay Area. Google Duplex, as it is called, will be a feature of Google Assistant and, for now, will only be able to call restaurants without online booking systems, which are already supported by the assistant. Wired reports: A Google spokesperson told WIRED that the company now has a policy to always have the bot disclose its true nature when making calls. Duplex still retains the humanlike voice and "ums," "ahs," and "ummhmms" that struck some as spooky, though. Nick Fox, the executive who leads product and design for Google search and the company's assistant, says those interjections are necessary to make Duplex calls shorter and smoother. "The person on the other end shouldn't be thinking about how do I adjust my behavior, I should be able to do what I normally do and the system adapts to that," he says. Fo
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A relation between the precanonical quantization of pure YangMills fields and the functional Schr\"odinger representation in the temporal gauge is discussed. It is shown that the latter can be obtained from the former when the ultraviolet parameter $\varkappa$ introduced in precanonical quantization goes to infinity. In this limiting case, the Schr\"odinger wave functional can be expressed as the trace of the Volterra product integral of Cliffordalgebravalued precanonical wave functions restricted to a certain field configuration, and the canonical functional derivative Schr\"odinger equation together with the quantum Gau\ss\ constraint are derived from the Diraclike precanonical Schr\"odinger equation.
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