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Japanese retailer Uniqlo in Tokyo's Ariake district has managed to cut 90% of its staff and replace them with robots that are capable of inspecting and sorting the clothing housed there. The automation also allows them to operate 24 hours a day. Quartz reports: The company recently remodeled the existing warehouse with an automated system created in partnership with Daifuku, a provider of material handling systems. Now that the system is running, the company revealed during a walkthrough of the new facility, Uniqlo has been able to cut staff at the warehouse by 90%. The Japan News described how the automation works: "The robotic system is designed to transfer products delivered to the warehouse by truck, read electronic tags attached to the products and confirm their stock numbers and other information. When shipping, the system wraps products placed on a conveyor belt in cardboard and attaches labels to them. Only a small portion of work at the warehouse needs to be done by employees,
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The BBC had to replace live broadcasts with recorded material on its TV news channels for about an hour on Wednesday following a technical glitch. BBC News reports: The News at Six was also presented from the BBC's Millbank studio instead of its usual home of New Broadcasting House. The issue affected OpenMedia, a new computer system rolled out across BBC News outlets over the past six months. OpenMedia supplier Annova has been helping to investigate the fault. Engineers believe they have now addressed the problem. BBC News Home Editor Mark Easton shared on social media that he was rushing across London to the Millbank studio.
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The Android Developers Blog describes the control-flow integrity work that is shipping on the Pixel 3 handset. "LLVM's CFI implementation adds a check before each indirect branch to confirm that the target address points to a valid function with a correct signature. This prevents an indirect branch from jumping to an arbitrary code location and even limits the functions that can be called. As C compilers do not enforce similar restrictions on indirect branches, there were several CFI violations due to function type declaration mismatches even in the core kernel that we have addressed in our CFI patch sets for kernels 4.9 and 4.14."
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During a hearing in front of the Senate Homeland Security Committee on Wednesday, FBI Director Christopher Wray told senators to "be careful what you read," when asked about a recent story involving spy chips from China being secretly embedded into servers owned by Apple, Amazon and other big companies. From a report: Senator Ron Johnson, R-Wis., chairman of the committee, asked Wray when his agency found out about the chips that server manufacturer Super Micro implanted into server hardware, as reported last week by Bloomberg Businessweek. "I would say to the newspaper article or, I mean, the magazine article, I would say be careful what you read," Wray replied. "Especially in this context." Johnson called on Wray to speak to the accuracy of the story, telling the FBI director that, "We don't want false information out there." Wray said he couldn't offer much detail because the agency has a policy of not confirming or denying that an investigation is underway. "I do want to be careful
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We discuss the constant $\sigma_{2}$ problem for conic 4-spheres. Based on earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a necessary condition for the existence problem. In particular, when the condition is sharp, we have the uniqueness result similar to that of Troyanov in dimension 2. It indicates that the boundary of the moduli of all conic 4-spheres with constant $\sigma_{2}$ metrics consists of conic spheres with 2 conic points and rotational symmetry.
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Let $S$ be a smooth, totally real, compact immersion in $\mathbb{C}^n$ of real dimension $m \leq n$, which is locally polynomially convex and it has finitely many points where it self-intersects finitely many times, transversely or non-transversely. We prove that $S$ is rationally convex if and only if it is isotropic with respect to a "degenerate" K\"ahler form in $\mathbb{C}^n$.
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This paper presents a theoretical analysis of numerical integration based on interpolation with a Stein kernel. In particular, the case of integrals with respect to a posterior distribution supported on a general Riemannian manifold is considered and the asymptotic convergence of the estimator in this context is established. Our results are considerably stronger than those previously reported, in that the optimal rate of convergence is established under a basic Sobolev-type assumption on the integrand. The theoretical results are empirically verified on $\mathbb{S}^2$.
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We introduce a direct, linear sampling approach to imaging in an acoustic waveguide with sound hard walls. The waveguide terminates at one end and has unknown geometry due to compactly supported wall deformations. The goal of imaging is to determine these deformations and to identify localized scatterers in the waveguide, using a remote array of sensors that emits time harmonic probing waves and records the echoes. We present a theoretical analysis of the imaging approach and illustrate its performance with numerical simulations.
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In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants.
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This paper concerns a Fokker-Planck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system. The first main result concerns the well-posedness and regularity of the Cauchy problem. The well-posedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield H\"older regularity of the non-local order parameter, which is improved by an iteration argument. The asymptotic behavior of solutions depends on some order parameter $\rho$ depending on the initial data. The system shows different behavior depending on a value $\rho_s>0$, determined from the potentials and diffusion coefficient. For $\rho \leq \rho_s$, there exists an equilibrium solution $c^{\text{eq}}_{(\rho)}$. If $\rho\le\rho_s$ the solution converges strongly to $c^{\text{eq}}
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The purpose of this note is to prove the $G$-equivariant Sarkisov program for a connected algebraic group $G$ following the proof of the Sarkisov program by Hacon and McKernan. As a consequence, we obtain a characterisation of connected subgroups of $Bir(Z)$ acting rationally on $Z$.
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For any rational prime $p$, we define a certain $p$-stabilization of holomorphic Siegel Eisenstein series for the symplectic group ${\rm Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for the Fourier coefficients and conclude their $p$-adic interpolation problems. Consequently, for any odd prime $p$, we deduce the existence of a $\Lambda$-adic form (in the sense of A. Wiles, H. Hida and R.L. Taylor) such that after taking a suitable constant multiple, it interpolates $p$-adic analytic families of the above-mentioned $p$-stabilized Siegel Eisenstein series with nebentypus characters locally trivial at $p$ and Siegel Eisenstein series with nebentypus characters locally non-trivial at $p$ simultaneously. This can be viewed as a quite natural generalization of the ordinary $\Lambda$-adic Eisenstein series for ${\rm GL}(2)$.
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We study a basic linear elliptic equation on a lower dimensional rectifiable set $S$ in $\mathbb{R}^N$ with the Neumann boundary data. Set $S$ is a support of a finite Borel measure $\mu$. We will use the measure theoretic tools to interpret the equation and the Neumann boundary condition. For this purpose we recall the Sobolev-type space dependent on the measure $\mu$. We establish existence and uniqueness of weak solutions provided that an appropriate source term is given.
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We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian.
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The analyses of cellular network performance based on stochastic geometry generally ignore the traffic dynamics in the network. This restricts the proper evaluation and dimensioning of the network from the perspective of a mobile operator. To address the effect of dynamic traffic, recently, the mean cell approach has been introduced, which approximates the average network load by the zero cell load. However, this is not a realistic characterization of the network load, since a zero cell is statistically larger than a random cell drawn from the population of cells, i.e., a typical cell. In this paper, we analyze the load of a noise-limited network characterized by high signal to noise ratio (SNR). The noise-limited assumption can be applied to a variety of scenarios, e.g., millimeter wave networks with efficient interference management mechanisms. First, we provide an analytical framework to obtain the cumulative density function of the load of the typical cell. Then, we obtain two appr
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Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer's velocity. These restrictions define certain `manoeuvres' of the saucer, which we call `attacking,' `landing,' or `G2 mode' manoeuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manoeuvre corresponds to the flat Legendrean contact structure, the landing manoeuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1,1), and the most complicated G2 manoeuvre corresponds to the contact Engel structure with split real form of the exceptional Lie group G2 as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic G2 geometries is used to construct a `G2 joystic
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We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of `parabolic geometry') and we provide examples when this geometry is `flat,' meaning that its symmetries comprise the split form of the exceptional Lie algebra G2.
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Generation expansion planning (GEP) models have been useful aids for long-term planning. Recent growth in intermittent renewable generation has increased the need to represent the capability for non-renewables to respond to rapid changes in daily loads, leading research to bring unit commitment (UC) features into GEPs. Such GEP+UC models usually contain discrete variables which, along with many details, make computation times impractically long for analysts who need to develop, debug, modify and use the GEP for many alternative runs. We propose a GEP with generation aggregated by technology type, and with the minimal UC content necessary to represent the limitations on generation to respond to rapid changes in demand, i.e., ramp-up and ramp-down constraints, with ramp limits estimated from historical data on maximum rates of change of each generation type. We illustrate with data for the province of Ontario in Canada; the GEP is a large scale linear program that solves in less than one
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Given a rational map $f:\overline{\mathbb C}\to \overline{\mathbb C}$ and a finite graph $G\subset \overline{\mathbb C}$ such that $f(G)\subset G$ and $f$ is expanding on some neighbourhood of $G$, we show that there is another finite graph $G'\subset \bigcup _{n\ge 0}f^{-n}(G)$ in an arbitrarily small neighbourhood of $G$ such that $f^N(G')\subset G'$ for some integer $N$ but $\bigcup _{i=0}^{N-1}f^{i}(G')$ contains accumulating {\em{plaits}} and {\em{nests}}
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In this article, we analyse a stabilised equal-order finite element approximation for the Stokes equations on anisotropic meshes. In particular, we allow arbitrary anisotropies in a sub-domain, for example along the boundary of the domain, with the only condition that a maximum angle is fulfilled in each element.This discretisation is motivated by applications on moving domains as arising e.g. in fluid-structure interaction or multiphase-flow problems. To deal with the anisotropies, we define a modification of the original Continuous Interior Penalty stabilisation approach. We show analytically the discrete stability of the method and convergence of order ${\cal O}(h^{3/2})$ in the energy norm and ${\cal O}(h^{5/2})$ in the $L^2$-norm of the velocities. We present numerical examples for a linear Stokes problem and for a non-linear fluid-structure interaction problem, that substantiate the analytical results and show the capabilities of the approach.
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In noisy evolutionary optimization, sampling is a common strategy to deal with noise. By the sampling strategy, the fitness of a solution is evaluated multiple times (called \emph{sample size}) independently, and its true fitness is then approximated by the average of these evaluations. Previous studies on sampling are mainly empirical. In this paper, we first investigate the effect of sample size from a theoretical perspective. By analyzing the (1+1)-EA on the noisy LeadingOnes problem, we show that as the sample size increases, the running time can reduce from exponential to polynomial, but then return to exponential. This suggests that a proper sample size is crucial in practice. Then, we investigate what strategies can work when sampling with any fixed sample size fails. By two illustrative examples, we prove that using parent or offspring populations can be better. Finally, we construct an artificial noisy example to show that when using neither sampling nor populations is effecti
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We give normal forms for generic k-dimensional parametric families $(Z_\varepsilon)_\varepsilon$ of germs of holomorphic vector fields near $0\in\mathbb{C}^2$ unfolding a saddle-node 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 Martinet-Ramis 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 non-linear, 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 multi-layer 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 many-body localization (MBL) for disordered quantum spin chains. In particular, we discuss manifestations of MBL such as zero-velocity Lieb-Robinson bounds, quasi-locality 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 k-Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex k-Fibonacci quaternions which are connected with bicomplex numbers and k-Fibonacci 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 near-threshold negative energy solutions to the 2D cubic ($L^2$-critical) focusing Zakharov-Kuznetsov (ZK) equation blow-up in finite or infinite time. The proof consists of several steps. First, we show that if the blow-up 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 uniform-in-time spatial localization. Such nontrivial linear solutions are excluded by a local-viral space-time 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 descriptive-set-theoretical 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 type-definable, and that a quotient of a type-definable 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 upper-bounded 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 Chung-Lu 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 2-section 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. Same-sign particles repel each other, and opposite-sign 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 blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale $\delta_n>0$, 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 non-torsion Abelian group of size continuum admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.
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We study waves-packets in nonlinear periodic media in arbitrary ($d$) spatial dimension, modeled by the cubic Gross-Pitaevskii equation. In the asymptotic setting of small and broad waves-packets 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 $D^{k-j}u$ has $j$-th order $\mathrm{L}^{n/(n-j)}$-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 measure-data.
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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 $\log|G|$, 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 Chas-Sullivan 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 two-layer 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 pre-stored 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 Hardware-in-the-Loop setup of an FPGA-controlled 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 k-separable. We consider the question of whether every densely k-separable 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 finite-index subgroup, G/G0 is a unipotent group of derived length at most n-1. This is a corollary of an optimal upper bound of length involving the Kodaira dimension of X. We also study the algebro-geometric structure of X when it admits a group action with maximal derived length n-1.
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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 well-posedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of random-field solutions for the associated stochastic Cauchy problems. To this aim, we first discuss algebraic properties for iterated integrals of suitable parameter-dependent 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 large-scale 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) Runge-Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-inner-product stability properties to be satisfied. Unlike the case for L2 linear stability, implicit methods do not significantly alleviate the time-step 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 Runge--Kutta methods may offer an attractive alternative to traditional time-stepping methods. The strong stability properties of integrating factor Runge--Kutta methods where the transformed problem is evolved with an explicit SSP Runge--Kutta method with non-decreasing abscissas was recently established. In this work, we consider the use of downwinded spatial operators to preserve the strong stability properties of integrating factor Runge--Kutta methods where the Runge--Kutta method
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In this paper, dual complex Pell numbers and quaternions are defined. Also, some algebraic properties of dual-complex 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, dual-complex k-Pell numbers and dual-complex k-Pell quaternions are defined. Also, some algebraic properties of dual-complex k-Pell numbers and quaternions which are connected with dual-complex numbers and k-Pell 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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One-parameter 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 QR-factorization \[ U(\theta)R(\theta)=(1-\theta)A+\theta B, \] where $U_1 R_1=A$ and $U_2 R_2=B$ are the QR-factorizations of $A$ and $B$, and $U(\theta)$ is a unitary for all $\theta$ with $U(0)=U_1$ and $U(1)=U_2$. The QR-algorithm 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 Berlekamp-Welch 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 one-dimensional 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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