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• We prove a new classification result for (CR) rational maps from the unit sphere in some ${\mathbb C}^n$ to the unit sphere in ${\mathbb C}^N$. To so so, we work at the level of Hermitian forms, and we introduce ancestors and descendants.

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• A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any $n$-vertex path-pairable planar graph must contain a vertex of degree linear in $n$.

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• We survey the problem of separation under conjugacy and malnormality of the abelian peripheral subgroups of an orientable, irreducible $3$-manifold $X$. We shall focus on the relation between this problem and the existence of acylindrical splittings of $\pi_1(X)$ as an amalgamated product or HNN-extension along the abelian subgroups corresponding to the JSJ-tori.

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• In \cite{hien}, M. Hien introduced rapid decay homology group $\Homo^{rd}_{*}(U, (\nabla, E))$ associated to an irregular connection $(\nabla, E)$ on a smooth complex affine variety $U$, and showed that it is the dual group of the algebraic de Rham cohomology group $\Homo^*_{dR}(U,(\nabla^{\vee}, E^{\vee}))$. On the other hand, F. Pham has already introduced his version of rapid decay homology when $(\nabla, E)$ is the so-called elementary irregular connection (\cite{Sab}) in \cite{Pham}. In this report, we will state a comparison theorem of these homology groups and give an outline of its proof. This can be regarded as a homological counterpart of the result \cite{Sab} of C. Sabbah. As an application, we construct a basis of some rapid decay homologies associated to a hyperplane arrangement and hypersphere arrangement of Schl\"ofli type.

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• In the $R$-spread out, $d$-dimensional voter model, each site $x$ of $\mathbb{Z}^d$ has state (or 'opinion') 0 or 1 and, with rate 1, updates its opinion by copying that of some site $y$ chosen uniformly at random among all sites within distance $R$ from $x$. If $d \geq 3$, the set of (extremal) stationary measures of this model is given by a family $\mu_{\alpha, R}$, where $\alpha \in [0,1]$. Configurations sampled from this measure are polynomially correlated fields of 0's and 1's in which the density of 1's is $\alpha$ and the correlation weakens as $R$ becomes larger. We study these configurations from the point of view of nearest neighbor site percolation on $\mathbb{Z}^d$, focusing on asymptotics as $R \to \infty$. In \cite{RV15}, we have shown that, if $R$ is large, there is a critical value $\alpha_c(R)$ such that there is percolation if $\alpha > \alpha_c(R)$ and no percolation if $\alpha < \alpha_c(R)$. Here we prove that, as $R \to \infty$, $\alpha_c(R)$ converges to t

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• We investigate the optimal mass transport problem associated to the following "ballistic" cost functional on phase space $M\times M^*$, b_T(v, x):=\inf\{\langle v, \gamma (0)\rangle +\int_0^TL(\gamma (t), {\dot \gamma}(t))\, dt; \gamma \in C^1([0, T), M); \gamma(T)=x\}, where $M=\mathbb{R}^d$, $T>0$, and $L:M\times M^* \to \mathbb{R}$ is a Lagrangian that is jointly convex in both variables. Under suitable conditions on the initial and final probability measures, we use convex duality \a la Bolza and Monge-Kantorovich theory to lift classical Hopf-Lax formulae from state space to Wasserstein space. This allows us to relate optimal transport maps for the ballistic cost to those associated with the fixed-end cost defined on $M\times M$ by c_T(x,y):=\inf\{\int_0^TL(\gamma(t), {\dot \gamma}(t))\, dt; \gamma\in C^1([0, T), M); \gamma(0)=x, \gamma(T)=y\}. We also point to links with the theory of mean field games.

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• In 1996, Jackson and Martin proved that a strong ideal ramp scheme is equivalent to an orthogonal array. However, there was no good characterization of ideal ramp schemes that are not strong. Here we show the equivalence of ideal ramp schemes to a new variant of orthogonal arrays that we term augmented orthogonal arrays. We give some constructions for these new kinds of arrays, and, as a consequence, we also provide parameter situations where ideal ramp schemes exist but strong ideal ramp schemes do not exist.

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• In this paper, we study power-efficient resource allocation for multicarrier non-orthogonal multiple access (MC-NOMA) systems. The resource allocation algorithm design is formulated as a non-convex optimization problem which jointly designs the power allocation, rate allocation, user scheduling, and successive interference cancellation (SIC) decoding policy for minimizing the total transmit power. The proposed framework takes into account the imperfection of channel state information at transmitter (CSIT) and quality of service (QoS) requirements of users. To facilitate the design of optimal SIC decoding policy on each subcarrier, we define a channel-to-noise ratio outage threshold. Subsequently, the considered non-convex optimization problem is recast as a generalized linear multiplicative programming problem, for which a globally optimal solution is obtained via employing the branch-and-bound approach. The optimal resource allocation policy serves as a system performance benchmark du

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• We present an explicit construction of the moduli spaces of rank 2 stable parabolic bundles of parabolic degree 0 over the Riemann sphere, corresponding to an "optimum" open weight chamber of parabolic weights in the weight polytope. The complexity of the different moduli space' weight chambers is understood in terms of the complexity of the actions of the corresponding groups of bundle automorphisms on stable parabolic structures. For the given choices of parabolic weights, $\mathscr{N}$ consists entirely of isomorphism classes of strictly stable parabolic bundles whose underlying Birkhoff-Grothendieck splitting coefficients are constant and minimal, is constructed as a quotient of a set of stable parabolic structures by a group of bundle automorphisms, and is a smooth, compact complex manifold biholomorphic to$\left(\mathbb{C}\mathbb{P}^{1}\right)^{n-3}$ for even degree, and $\mathbb{C}\mathbb{P}^{n-3}$ for odd degree.

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• Three problems about recovery of a high-frequency free term in the one-dimension wave equation with homogeneous initial-boundary conditions by some information about partial asymptotics of its solution have been solved. It is shoun, that the free term can be completely recovered from a specific data about incomplete (three-terms) asymptotics of the solution. Before formulation of the each problem about recovery of free term, construction and justification of the asymptotics of the solution of original initial-boundary problem are given.

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• The fundamental group of a smooth projective variety is fibered if it maps onto the fundamental group of smooth curve of genus 2 or more. The goal of this paper is to establish some strong restrictions on these groups, and in particular on the fundamental groups of Kodaira surfaces. In the specific case of a Kodaira surface, these results are in the form of restrictions on the monodromy representation into the mapping class group. When the monodromy is composed with certain standard representations, the images are Zariski dense in a semisimple group of Hermitian type.

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• For a matrix A_2 weight W on R^p, we introduce a new notion of W-Calder\'on-Zygmund matrix kernels, following earlier work in by Isralowitz. We state and prove a T1 theorem for such operators and give a representation theorem in terms of dyadic W-Haar shifts and paraproducts, in the spirit of Hyt\"onen's Representation Theorem. Finally, by means of a Bellman function argument, we give sharp bounds for such operators in terms of bounds for weighted matrix martingale transforms and paraproducts.

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• The Randi{\' c} index of a graph $G$, written $R(G)$, is the sum of $\frac 1{\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\sum_{uv \in E(G)} \frac 1{\sqrt{d(u)d(v)}}$, which is called the Randi{\' c} index of it. Let $d$ and $D$ be positive integers $d < D$. In this paper, we prove that if $G$ is a graph with minimum degree $d$ and maximum degree $D$, then $R(G) \ge \frac{\sqrt{dD}}{d+D}n$; equality holds only when $G$ is an $n$-vertex $(d,D)$-biregular. Furthermore, we show that if $G$ is an $n$-vertex connected graph with minimum degree $d$ and maximum degree $D$, then $R(G) \le \frac n2- \sum_{i=d}^{D-1}\frac 12 \left( \frac 1{\sqrt{i}} - \frac 1{\sqrt{i+1}}\right)^2$; it is sharp for infinitely many $n$, and we characterize when equality holds in the bound.

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• We propose a continuous-time model of trading among risk-neutral agents with heterogeneous beliefs. Agents face quadratic costs-of-carry on their positions and as a consequence, their marginal valuation of the asset decreases when the magnitude of their position increases, as it would be the case for risk-averse agents. In the equilibrium models of investors with heterogeneous beliefs that followed the original work by Harrison and Kreps, investors are risk-neutral, short-selling is prohibited and agents face a constant marginal cost of carrying positions. The resulting resale option guarantees that the equilibrium price exceeds the price of the asset in a static buy-and-hold model where speculation is ruled out. Our model features three main novelties. First, increasing marginal costs entail that the price depends on the exogenous supply. Second, in addition to the resale option, agents may also value an option to delay, and this may cause the market to equilibrate \emph{below} the st

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• We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible two-sided topological Markov shifts with countable alphabets. We describe then the thermodynamic formalism for Smale skew products over countable-to-1 endomorphisms, and give several applications to measures on natural extensions of endomorphisms. We show that the exact dimensionality of conditional measures on fibers, implies the global exact dimensionality of the measure, in certain cases. We then study equilibrium states for skew products over endomor

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• A number of recent works have proposed to solve the line spectral estimation problem by applying an off-the-grid ex- tension of sparse estimation techniques. These methods are more advantageous than classical line spectral estimation algorithms because they inherently estimate the model order. However, they all have computation times which grow at least cubically in the problem size, which limits their practical applicability for large problem sizes. To alleviate this issue, we propose a low-complexity method for line spectral estimation, which also draws on ideas from sparse estimation. Our method is based on a probabilistic view of the problem. The signal covariance matrix is shown to have Toeplitz structure, allowing superfast Toeplitz inversion to be used. We demonstrate that our method achieves estimation accuracy at least as good as current methods and that it does so while being orders of magnitudes faster.

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• In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\ID$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)-1$. First we prove some sufficient conditions for univalence of such functions in $\ID$. One of these conditions enable us to consider the class $\mathcal{V}_{p}(\lambda)$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that $\mathcal{U}_{p}(\lambda)\subsetneq \mathcal{V}_{p}(\lambda)$, where $\mathcal{U}_{p}(\lambda)$ was introduced and studied in \cite{BF-1}. Finally, we discuss some coefficient problems for $\mathcal{V}_{p}(\lambda)$ and end the article with a coefficient conjecture.

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• We consider the minimization of non-convex functions that typically arise in machine learning. Specifically, we focus our attention on a variant of trust region methods known as cubic regularization. This approach is particularly attractive because it escapes strict saddle points and it provides stronger convergence guarantees than first- and second-order as well as classical trust region methods. However, it suffers from a high computational complexity that makes it impractical for large-scale learning. Here, we propose a novel method that uses sub-sampling to lower this computational cost. By the use of concentration inequalities we provide a sampling scheme that gives sufficiently accurate gradient and Hessian approximations to retain the strong global and local convergence guarantees of cubically regularized methods. To the best of our knowledge this is the first work that gives global convergence guarantees for a sub-sampled variant of cubic regularization on non-convex functions.

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• In this article we establish the validity of Prandtl layer expansions around Euler flows which are not shear. The presence of non-shear flows at the leading order creates a singularity of $o(\frac{1}{\sqrt{\epsilon}})$. A new $y$-weighted positivity estimate is developed to control this leading-order growth at the far field.

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• A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite equivalence relation of type II_1 under direct product. We show that for a countable group G and its central subgroup C, if G/C has a stable action, then so does G. Combining a previous result of the author, we obtain a characterization of a central extension having a stable action.

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• In this paper we study the structure of $\theta$-cyclic codes over the ring $B_k$ including its connection to quasi-$\tilde{\theta}$-cyclic codes over finite field $\mathbb{F}_{p^r}$ and skew polynomial rings over $B_k.$ We also characterize Euclidean self-dual $\theta$-cyclic codes over the rings. Finally, we give the generator polynomial for such codes and some examples of optimal Euclidean $\theta$-cyclic codes.

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• This paper addresses the stability analysis of adaptive control systems with the try-once-discard protocol. At every sampling time, an event trigger evaluates errors between the current value and the last released value of each measurement and determines whether to transmit the measurements and which measurements to transmit, based on the try-once-discard protocol and given lower and upper thresholds. For both gain-scheduling controllers and switching controllers that are adaptive to the maximum error of the measurements, we obtain sufficient conditions for the closed-loop stability in terms of linear matrix inequalities.

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• The paper studies functions defined on continuous branching lines connected into a system. A notion of spectrum degeneracy for these systems of functions is introduced. This degeneracy is based on the properties of the Fourier transforms for processes representing functions on the branches that are deemed to be extended onto the real axis. This spectrum degeneracy ensures some opportunities for extrapolation and sampling. The topology of the system is taken into account via a restriction that these processes coincides on certain parts of real axis. It is shown that processes with this feature are everywhere dense in the set of processes equivalent to functions on the branching lines. Some applications to extrapolation and sampling are considered.

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• We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves, under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary we show that if $X$ is a log del Pezzo surface such that for every closed point $p\in X$, there is a smooth curve (locally analytically) passing through $p$, then $X$ contains at least one smooth rational curve.

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• We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on possibly infinite dimensional complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous. Uniform asymptotics of generalized eigenvectors and conditions implying complete indeterminacy are also provided.

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• In this note, for any two orthogonal projection $P,Q$ on a Hilbert space, the characterization of spectrum of anticommutator $PQ+QP$ has been obtained. As a corollary, the norm formula $$\parallel PQ+QP\parallel=\parallel PQ\parallel+\parallel PQ\parallel^2$$ has been got an alternative proof (see Sam Waltrs, Anticommutator norm formula for projection operators, arXiv:1604.00699vl [math.FA] 3 Apr 2016)

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• Apollonian gaskets are formed by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We experimentally study the pair correlation, electrostatic energy, and nearest neighbor spacing of centers of circles from Apollonian gaskets. Even though the centers of these circles are not uniformly distributed in any ambient' space, after proper normalization, all these statistics seem to exhibit some interesting limiting behaviors.

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• P. Flajolet and B. Salvy [10] prove the famous theorem that a nonlinear Euler sum $S_{i_1i_2\cdots i_r,q}$ reduces to a combination of sums of lower orders whenever the weight $i_1+i_2+\cdots+i_r+q$ and the order $r$ are of the same parity. In this article, we develop an approach to evaluate the cubic sums $S_{1^2m,p}$ and $S_{1l_1l_2,l_3}$. By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums $S_{1^2m,m}$ and $S_{1(2l+1)^2,2l+1}$ are reducible to zeta values, quadratic and linear sums. Finally, we evaluate the alternating cubic Euler sums ${S_{{{\bar 1}^3},2r + 1}}$ and show that it are reducible to alternating quadratic and linear Euler sums. The approach is based on Tornheim type series computations.

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• Let $S$ be a finite semigroup and let $A$ be a finite dimensional $S$-graded algebra. We investigate the exponential rate of growth of the sequence of graded codimensions $c_n^S(A)$ of $A$, i.e $\lim\limits_{n \rightarrow \infty} \sqrt[n]{c_n^S(A)}$. For group gradings this is always an integer. Recently in [20] the first example of an algebra with a non-integer growth rate was found. We present a large class of algebras for which we prove that their growth rate can be equal to arbitrarily large non-integers. An explicit formula is given. Surprisingly, this class consists of an infinite family of algebras simple as an $S$-graded algebra. This is in strong contrast to the group graded case for which the growth rate of such algebras always equals $\dim (A)$. In light of the previous, we also handle the problem of classification of all $S$-graded simple algebras, which is of independent interest. We achieve this goal for an important class of semigroups that is crucial for a solution of t

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• We consider a method of construction of self-similar dendrites on a plane and establish main topological and metric properties of resulting class of dendrites.

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•   05-17 Hacker News 12

Tips for Building High-Quality Django Apps at Scale

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• In this paper fundamental Wigner coefficients are determined algebraically by considering the eigenvalues of certain generalized Casimir invariants. Here this method is applied in the context of both type 1 and type 2 unitary representations of the Lie superalgebra gl(mjn). Extensions to the non-unitary case are investigated. A symmetry relation between two classes of Wigner coefficients is given in terms of a ratio of dimensions.

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• We study a class of (focusing) nonlinear Schroedinger-type equations derived recently by Dumas, Lannes and Szeftel within the mathematical description of high intensity laser beams [5]. These equations incorporate the possibility of a (partial) off-axis variation of the group velocity of such laser beams through a second order partial differential operator acting in some, but not necessarily all, spatial directions. We study the well-posedness theory for such models and obtain a regularizing effect, even in the case of only partial off-axis dependence. This provides an answer to an open problem posed in [5].

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• In this paper, we study three asymptotic regimes that can be applied to ranking and selection (R&S) problems with general sample distributions. These asymptotic regimes are constructed by sending particular problem parameters (probability of incorrect selection, smallest difference in system performance that we deem worth detecting) to zero. We establish asymptotic validity and efficiency of the corresponding R&S procedures in each regime. We also analyze the connection among different regimes and compare the pre-limit performances of corresponding algorithms.

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• Moreau's seminal paper, introducing what is now called the Moreau envelope and the proximity operator (a.k.a. proximal mapping), appeared in 1965. The Moreau envelope of a given convex function provides a regularized version which has additional desirable properties such as differentiability and full domain. Fifty years ago, Attouch proposed to use the Moreau envelope for regularization. Since then, this branch of convex analysis has developed in many fruitful directions. In 1967, Bregman introduced what is nowadays the Bregman distance as a measure of discrepancy between two points generalizing the square of the Euclidean distance. Proximity operators based on the Bregman distance have become a topic of significant research as they are useful in algorithmic solution of optimization problems. More recently, in 2012, Kan and Song studied regularization aspects of the left Bregman-Moreau envelope even for nonconvex functions. In this paper, we complement previous works by analyzing the l

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• The {\it forward-backward algorithm} is a powerful tool for solving optimization problems with a {\it additively separable} and {\it smooth} + {\it nonsmooth} structure. In the convex setting, a simple but ingenious acceleration scheme developed by Nesterov has been proved useful to improve the theoretical rate of convergence for the function values from the standard $\mathcal O(k^{-1})$ down to $\mathcal O(k^{-2})$. In this short paper, we prove that the rate of convergence of a slight variant of Nesterov's accelerated forward-backward method, which produces {\it convergent} sequences, is actually $o(k^{-2})$, rather than $\mathcal O(k^{-2})$. Our arguments rely on the connection between this algorithm and a second-order differential inclusion with vanishing damping.

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• We consider the Dirichlet eigenvalue problem on a simple polytope. We use the Rellich identity to obtain an explicit formula expressing the Dirichlet eigenvalue in terms of the Neumann data on the faces of the polytope of the corresponding eigenfunction. The formula is particular simple for polytopes admitting an inscribed ball tangent to all the faces. Our result could be viewed as a generalization of similar identities for simplices recently found by Christianson [1][2].

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• We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible Navier-Stokes equations supplemented with H^1 initial velocity and only bounded nonnegative density. In contrast with all the previous works on that topics, we do not require regularity or positive lower bound for the initial density, or compatibility conditions for the initial velocity, and still obtain unique solutions. Those solutions are global in the two-dimensional case for general data, and in the three-dimensional case if the velocity satisfies a suitable scaling invariant smallness condition. As a straightforward application, we provide a complete answer to Lions' question in [25], page 34, concerning the evolution of a drop of incompressible viscous fluid in the vacuum.

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• This study introduces a procedure to obtain general expressions, $y = f(x)$, subject to linear constraints on the function and its derivatives defined at specified values. These constrained expressions can be used describe functions with embedded specific constraints. The paper first shows how to express the most general explicit function passing through a single point in three distinct ways: linear, additive, and rational. Then, functions with constraints on single, two, or multiple points are introduced as well as those satisfying relative constraints. This capability allows to obtain general expressions to solve linear differential equations with no need to satisfy constraints (the "subject to:" conditions) as the constraints are already embedded in the constrained expression. In particular, for expressions passing through a set of points, a generalization of the Waring's interpolation form, is introduced. The general form of additive constrained expressions is introduced as well as

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• In this note, we prove some new entropy formulae for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are simpler versions of Cao and Hamilton's entropies for Ricci flow coupled with heat-type equations.

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•   05-17 Hacker News 13

The Quest to Make a Super Tomato

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• This paper considers multiple-input multiple-output (MIMO) full-duplex (FD) two-way secrecy systems. Specifically, both multi-antenna FD legitimate nodes exchange their own confidential message in the presence of an eavesdropper. Taking into account the imperfect channel state information (CSI) of the eavesdropper, we formulate a robust sum secrecy rate maximization (RSSRM) problem subject to the outage probability constraint of the achievable sum secrecy rate and the transmit power constraint. Unlike other existing channel uncertainty models, e.g., norm-bounded and Gaussian-distribution, we exploit a moment-based random distributed CSI channel uncertainty model to recast our formulate RSSRM problem into the convex optimization frameworks based on a Markov's inequality and robust conic reformulation, i.e., semidefinite programming (SDP). In addition, difference-of-concave (DC) approximation is employed to iteratively tackle the transmit covariance matrices of these legitimate nodes. Si

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• Integer cuboids are rectangular Diophantine parallelepipeds. It has been discovered that these cuboids come in 3 varieties: Euler or body type, edge type, and face type. In all three cases, one edge or diagonal is irrational, all six others are rational. We discuss an exhaustive computer search procedure which uses the Pythagorean group Py(n) to locate all possible cuboids with a given edge n. Over the range of 44 to 155,000,000,000, for the smallest edge, 154,571 cuboids were discovered. They are listed in the Integer Cuboid Table.

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• We relate some terms on the boundary of the Newton polygon of the Alexander polynomial $\Delta(x,y)$ of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize $\Delta(x,y)$ so that no $x^{-1}$ or $y^{-1}$ terms appear, but $x^{-1}\Delta(x,y)$ and $y^{-1}\Delta(x,y)$ have negative exponents, and so that terms of even total degree are positive and terms with odd total degree are negative. If the rational link has a reduced alternating diagram with no self crossings, then $\Delta(-1, 0) = 1$. If the standard form of the rational link has $m$ monochromatic twist sites, and the $j^{\textrm{th}}$ monochromatic twist site has $\hat{q}_j$ crossings, then $\Delta(-1, 0) = \prod_{j=1}^{m}(\hat{q}_j+1)$. Our proof employs Kauffman's clock moves and a lattice for the terms of $\Delta(x,y)$ in which the $y$-power cannot decrease.

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• These are the notes of rather informal lectures given by the first co-author in UPMC, Paris, in January 2017. Practical goal is to explain how to compute or estimate the Morse index of the second variation. Symplectic geometry allows to effectively do it even for very degenerate problems with complicated constraints. Main geometric and analytic tool is the appropriately rearranged Maslov index. In these lectures, we try to emphasize geometric structure and omit analytic routine. Proofs are often substituted by informal explanations but a well-trained mathematician will easily re-write them in a conventional way.

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• A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous space $(G\times H)/{\rm diag}(H)$ is real spherical. This geometric condition is equivalent to the representation theoretic property that ${\rm dim\,Hom}_H(\pi|_H,\tau)<\infty$ for all smooth admissible representations $\pi$ of $G$ and $\tau$ of $H$. In this paper we explicitly construct for all strongly spherical pairs $(G,H)$ intertwining operators in ${\rm Hom}_H(\pi|_H,\tau)$ for $\pi$ and $\tau$ spherical principal series representations of $G$ and $H$. These so-called \textit{symmetry breaking operators} depend holomorphically on the induction parameters and we further show that they generically span the space ${\rm Hom}_H(\pi|_H,\tau)$. In the special case of multiplicity one pairs we extend our construction to vector-valued principal series representations and obtain generic formulas for the multiplicities between arbitrary principal series. As an application, we prove the Gross-Prasad conjecture f

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• We prove the existence of scaling limits for the projection on the backbone of the random walks on the Incipient Infinite Cluster and the Invasion Percolation Cluster on a regular tree. We treat these projected random walks as randomly trapped random walks (as defined in [BC\v{C}R15]) and thus describe these scaling limits as spatially subordinated Brownian motions

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• In this article, we discuss the rationality of the Betti Series of the universal module of nth order derivations of R_{m} where m is a maximal ideal of R. We proved that if R is a coordinate ring of an affine irreducible curve and if it has at most one singularity point, then the Betti series is rational.

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• A linear automorphism of Euclidean space is called bi-circular its eigenvalues lie in the disjoint union of two circles $C_1$ and $C_2$ in the complex plane where the radius of $C_1$ is $r_1$, the radius of $C_2$ is $r_2$, and $0 < r_1 < 1 < r_2$. A well-known theorem of Philip Hartman states that a local $C^{1,1}$ diffeomorphism $T$ of Euclidean space with a fixed point $p$ whose derivative $DT_p$ is bi-circular is $C^{1,\beta}$ linearizable near $p$. We generalize this result to $C^{1,\alpha}$ diffeomorphisms $T$ where $0 < \alpha < 1$. We also extend the result to local diffeomorphisms in Banach spaces with $C^{1,\alpha}$ bump functions. The results apply to give simpler proofs under weaker regularity conditions of classical results of L. P. Shilnikov on the existence of horseshoe dynamics near so-called saddle-focus critical points of vector fields in $R^3$.

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• In this paper, based on the block operator technique and operator spectral theory, the general explicit expressions for intertwining operators and direct rotations of two orthogonal projections have been established. As a consequence, it is an improvement of Kato's result (Perturbation Theory of Linear operators, Springer-Verlag, Berlin/Heidelberg, 1996); J. Avron, R. Seiler and B. Simon's Theorem 2.3 (The index of a pair of projections, J. Funct. Anal. 120(1994) 220-237) and C. Davis, W.M. Kahan, (The rotation of eigenvectors by a perturbation, III. SIAM J. Numer. Anal. 7(1970) 1-46).

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• We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that conservatively approximates the set of reachable states of a nonlinear dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF), which is derived from an over-approximation of the gradient of the solution flows. The SF measures the discrepancy between two states propagated by the system solution from two initial states lying in a well-defined region, thereby allowing LRT to compute a reachtube with a ball-overestimate in a metric where the computed enclosure is as tight as possible. To evaluate its performance, we implemented a prototype of LRT in C++/Matlab, and ran it on a set of well-established benchmarks. Our results show that LRT compares very favorably with respect to the CAPD and Flow* tools.

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• We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, right-angled Coxeter groups, most 3-manifold groups, right-angled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a "best" one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindri

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• In \cite{JS} Jensen and Su constructed 0-Schur algebras on double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver approach naturally gives rise to a new class of algebras. That is, the path algebras defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters $\ut$. In particular, when all the entries of $\ut$ are 1, we have 0-Schur algerbas. When all the entries of $\ut$ are zero, we obtain a class of degenerate 0-Schur algebras. We prove that the degenerate algebras are associated graded algebras and quotients of 0-Schur algebras. Moreover, we give a geometric interpretation of the degenerate algebras using double flag varieties, in the same spirit as \cite{JS}, and show how the centralizer algebras are related to nil-Hecke algebras and nil-Temperly-Lieb algebras

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• We show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. This completes the result of Jordan on the simplicity of the derivation algebra \cite{Jo}. Given proof is self-contained and does not depend on the results of Jordan. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.

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• We consider the Dirichlet Laplacian $H_\gamma$ on a 3D twisted waveguide with random Anderson-type twisting $\gamma$. We introduce the integrated density of states $N_\gamma$ for the operator $H_\gamma$, and investigate the Lifshits tails of $N_\gamma$, i.e. the asymptotic behavior of $N_\gamma(E)$ as $E \downarrow \inf {\rm supp}\, dN_\gamma$. In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.

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• We study the 3-D compressible barotropic radiation fluid dynamics system describing the motion of the compressible rotating viscous fluid with gravitation and radiation confined to a straight layer. We show that weak solutions in the 3-D domain converge to the strong solution of the rotating 2-D Navier-Stokes-Poisson system with radiation for all times less than the maximal life time of the strong solution of the 2-D system when the Froude number is small or to the strong solution of the rotating pure 2-D Navier- Stokes system with radiation.

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• We investigate a mixed $0-1$ conic quadratic optimization problem with indicator variables arising in mean-risk optimization. The indicator variables are often used to model non-convexities such as fixed charges or cardinality constraints. Observing that the problem reduces to a submodular function minimization for its binary restriction, we derive three classes of strong convex valid inequalities by lifting the polymatroid inequalities on the binary variables. Computational experiments demonstrate the effectiveness of the inequalities in strengthening the convex relaxations and, thereby, improving the solution times for mean-risk problems with fixed charges and cardinality constraints significantly.

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• Let $W$ be the ring of the Witt vectors of a perfect field of characteristic $p$, $\mathfrak{X}$ a smooth formal scheme over $W$, $\mathfrak{X}'$ the base change of $\mathfrak{X}$ by the Frobenius morphism of $W$, $\mathfrak{X}_{2}'$ the reduction modulo $p^{2}$ of $\mathfrak{X}'$ and $X$ the special fiber of $\mathfrak{X}$. We lift the Cartier transform of Ogus-Vologodsky defined by $\mathfrak{X}_{2}'$ modulo $p^{n}$. More precisely, we construct a functor from the category of $p^{n}$-torsion $\mathscr{O}_{\mathfrak{X}'}$-modules with integrable $p$-connection to the category of $p^{n}$-torsion $\mathscr{O}_{\mathfrak{X}}$-modules with integrable connection, each subject to suitable nilpotence conditions. Our construction is based on Oyama's reformulation of the Cartier transform of Ogus-Vologodsky in characteristic $p$. If there exists a lifting $F:\mathfrak{X}\to \mathfrak{X}'$ of the relative Frobenius morphism of $X$, our functor is compatible with a functor constructed by Shiho f

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• We show that points in specific degree 2 hypersurfaces in the Grassmannian $Gr(3, n)$ correspond to generic arrangements of $n$ hyperplanes in $\mathbb{C}^3$ with associated discriminantal arrangement having intersections of multiplicity three in codimension two.

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• In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills {\alpha}-energy. More specifically, we show that for the SU(2) Hopf fibration over the four sphere, for sufficiently small {\alpha} values the SO(4) invariant ADHM instanton is the unique {\alpha}-critical point which has Yang-Mills {\alpha}-energy lower than a specific threshold.

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• We study Riemannian metrics on compact, torsionless, non-geometric $3$-manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "\`a la Margulis" for the systole and a volume estimate for these manifolds, only in terms of an upper bound of entropy and diameter. We then deduce corresponding local topological rigidy results in the class $\mathscr{M}_{ngt}^\partial (E,D)$ of compact non-geometric 3-manifolds with torsionless fundamental group (with possibly empty, non-spherical boundary) whose entropy and diameter are bounded respectively by $E, D$. For instance, this class locally contains only finitely many topological types; and closed, irreducible manifolds in this class which are close enough (with respect to $E,D$) are diffeomorphic. Several examples and counter-examples are produced to stress the differences with the geometric case.

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• A survey of real differential geometry and loop theory is given in order to introduce the construction of an analytic loop associated to p-adic differential manifold.

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• Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen and the intracellular trafficking of vesicles into dendritic spines, involve the near-contact of elastic structures separated by thin layers of fluid. Motivated by such problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We demonstrate 2nd-order accurate convergence for simple two-dimensional flows with known exact solutions to showcase the increased accuracy of this method compared to the standard immersed boundary method. Motivated by the phenomenon of wall-induced migration, we apply the lubricated immersed boundary method to simulate an elastic capsule near a wall in shear flow. We also simulate the dynamics of a deformable vesicle traveling through a narrow channel and observe that the ability of the lubricated method to capture the vesicle motion is

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• The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly. This article confirms this conjecture. As a special case, this gives a new method of uniformization of the Riemann sphere.

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• We unveil the existence of a precise mapping between the ground state of non-interacting free fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture b

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• Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order convergence. We define the unbounded order dual (or uo-dual) $X_{uo}^\sim$ of a Banach lattice $X$ and identify it as the order continuous part of the order continuous dual $X_n^\sim$. The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists. Applications to the Fenchel-Moreau duality theory of convex functionals are given. The applications are of interest in the theory of risk measures in Mathematical Finance.

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• This paper concerns the continuous time mean-variance portfolio selection problem with a special nonlinear wealth equation. This nonlinear wealth equation has nonsmooth random coefficients and the dual method developed in [7] does not work. To apply the completion of squares technique, we introduce two Riccati equations to cope with the positive and negative part of the wealth process separately. We obtain the efficient portfolio strategy and efficient frontier for this problem. Finally, we find the appropriate sub-derivative claimed in [7] using convex duality method.

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• This paper analyzes the iteration-complexity of a generalized alternating direction method of multipliers (G-ADMM) for solving linearly constrained convex problems. This ADMM variant, which was first proposed by Bertsekas and Eckstein, introduces a relaxation parameter $\alpha \in (0,2)$ into the second ADMM subproblem. Our approach is to show that the G-ADMM is an instance of a hybrid proximal extragradient framework with some special properties, and, as a by product, we obtain ergodic iteration-complexity for the G-ADMM with $\alpha\in (0,2]$, improving and complementing related results in the literature. Additionally, we also present pointwise iteration-complexity for the G-ADMM.

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• Efficient beam alignment is a crucial component in millimeter wave systems with analog beamforming, especially in fast-changing vehicular settings. This paper uses the vehicle's position (e.g., available via GPS) to query the multipath fingerprint database, which provides prior knowledge of potential pointing directions for reliable beam alignment. The approach is the inverse of fingerprinting localization, where the measured multipath signature is compared to the fingerprint database to retrieve the most likely position. The power loss probability is introduced as a metric to quantify misalignment accuracy and is used for optimizing candidate beam selection. Two candidate beam selection methods are derived, where one is a heuristic while the other minimizes the misalignment probability. The proposed beam alignment is evaluated using realistic channels generated from a commercial ray-tracing simulator. Using the generated channels, an extensive investigation is provided, which includes

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• We consider the estimation of hidden Markovian process by using information geometry with respect to transition matrices. We consider the case when we use only the histogram of $k$-memory data. Firstly, we focus on a partial observation model with Markovian process and we show that the asymptotic estimation error of this model is given as the inverse of projective Fisher information of transition matrices. Next, we apply this result to the estimation of hidden Markovian process. For this purpose, we define an exponential family of ${\cal Y}$-valued transition matrices. We carefully discuss the equivalence problem for hidden Markovian process on the tangent space. Then, we propose a novel method to estimate hidden Markovian process.

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• We propose new iterative methods for computing nontrivial extremal generalized singular values and vectors. The first method is a generalized Davidson-type algorithm and the second method employs a multidirectional subspace expansion technique. Essential to the latter is a fast truncation step designed to remove a low quality search direction and to ensure moderate growth of the search space. Both methods rely on thick restarts and may be combined with two different deflation approaches. We argue that the methods have monotonic and (asymptotic) linear convergence, derive and discuss locally optimal expansion vectors, and explain why the fast truncation step ideally removes search directions orthogonal to the desired generalized singular vector. Furthermore, we identify the relation between our generalized Davidson-type algorithm and the Jacobi--Davidson algorithm for the generalized singular value decomposition. Finally, we generalize several known convergence results for the Hermitian

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• This paper is devoted to strict $K$- monotonicity and $K$-order continuity in symmetric spaces. Using the local approach to the geometric structure in a symmetric space $E$ we investigate a connection between strict $K$-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of $K$-order continuity in a symmetric space $E$ on $[0,\infty)$ implies that the embedding $E\hookrightarrow{L^1}[0,\infty)$ does not hold. We finish this article with a complete characterization of $K$-order continuity in a symmetric space $E$ that is written using a notion of order continuity under some assumptions on the fundamental function of $E$.

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• We study ${\mathbb Z}$-graded thread $W^+$-modules $$V=\oplus_i V_i, \; \dim{V_i}=1, -\infty \le k< i < N\le +\infty, \; \dim{V_i}=0, \; {\rm \; otherwise},$$ over the positive part $W^+$ of the Witt (Virasoro) algebra $W$. There is well-known example of infinite-dimensional ($k=-\infty, N=\infty$) two-parametric family $V_{\lambda, \mu}$ of $W^+$-modules induced by the twisted $W$-action on tensor densities $P(x)x^{\mu}(dx)^{-\lambda}, \mu, \lambda \in {\mathbb K}, P(x) \in {\mathbb K}[t]$. Another family $C_{\alpha, \beta}$ of $W^+$-modules is defined by the action of two multiplicative generators $e_1, e_2$ of $W^+$ as $e_1f_i=\alpha f_{i+1}$ and $e_2f_j=\beta f_{j+2}$ for $i,j \in {\mathbb Z}$ and $\alpha, \beta$ are two arbitrary constants ($e_if_j=0, i \ge 3$). We classify $(n+1)$-dimensional graded thread $W^+$-modules for $n$ sufficiently large $n$ of three important types. New examples of graded thread $W^+$-modules different from finite-dimensional quotients of $V_{\lam 收起 • Let$X = \{1,-1\}^\mathbb{N}$be the symbolic space endowed with the product order. A Borel probability measure$\mu$over$X$is said to satisfy the FKG inequality if for any pair of continuous increasing functions$f$and$g$we have$\mu(fg)-\mu(f)\mu(g)\geq 0$. In the first part of the paper we prove the validity of the FKG inequality on Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the Ruelle operator, including several examples of interest in Statistical Mechanics. In addition to deducing this inequality in cases not covered by classical results about attractive specifications our proof has advantage of to be easily adapted for suitable subshifts. We review (and provide proofs in our setting) some classical results about the long-range Ising model on the lattice$\mathbb{N}$and use them to deduce some monotonicity properties of the associated Ruelle operator and their relations with phase transitions. As is widely known, for some continuous potentia 收起 • We study the class of groups having the property that every non-nilpotent subgroup is equal to its normalizer. These groups are either soluble or perfect. We completely describe the structure of soluble groups and finite perfect groups with the above property. Furthermore, we give some structural information in the infinite perfect case. 收起 • In this paper, we define a general class of abstract aerial robotic systems named Laterally Bounded Force (LBF) vehicles, in which most of the control authority is expressed along a principal thrust direction, while in the lateral directions a (smaller and possibly null) force may be exploited to achieve full-pose tracking. This class approximates well platforms endowed with non-coplanar/non-collinear rotors that can use the tilted propellers to slightly change the orientation of the total thrust w.r.t. the body frame. For this broad class of systems, we introduce a new geometric control strategy in SE(3) to achieve, whenever made possible by the force constraints, the independent tracking of position-plus-orientation trajectories. The exponential tracking of a feasible full-pose reference trajectory is proven using a Lyapunov technique in SE(3). The method can deal seamlessly with both under- and fully-actuated LBF platforms. The controller guarantees the tracking of at least the posi 收起 • We give sharp remainder terms of$L^{p}$and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy-Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types. 收起 • These notes present a first graduate course in harmonic analysis. The first part emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle, for example, where it is used to prove L^p convergence of Fourier series. Maximal functions and Calderon--Zygmund decompositions are treated in R^d, so that they can be applied again in the second part of the course, where the Fourier transform is studied. The final part of the course treats band limited functions, Poisson summation and uncertainty principles. Distribution functions and interpolation are covered in the Appendices. The references at the beginning of each chapter provide guidance to students who wish to delve more deeply, or roam more widely, in the subject. 收起 • Notions and results from quantum harmonic analysis, such as the convolution between functions and operators or between two operators, is identified as the appropriate setting for Berezin quantization and Berezin-Lieb inequalities. Based on this insight we provide a rigorous approach to generalized phase-space representation introduced by Klauder-Skagerstam and their variants of Berezin-Lieb inequalities in this setting. Hence our presentation of the results of Klauder-Skagerstam gives a more conceptual framework, which yields as a byproduct an interesting perspective on the connection between Berezin quantization and Weyl quantization. 收起 • In the paper, we present a high order fast algorithm with almost optimum memory for the Caputo fractional derivative, which can be expressed as a convolution of$u'(t)$with the kernel$(t_n-t)^{-\alpha}$. In the fast algorithm, the interval$[0,t_{n-1}]$is split into nonuniform subintervals. The number of the subintervals is in the order of$\log n$at the$n$-th time step. The fractional kernel function is approximated by a polynomial function of$K$-th degree with a uniform absolute error on each subinterval. We save$K+1$integrals on each subinterval, which can be written as a convolution of$u'(t)$with a polynomial base function. As compared with the direct method, the proposed fast algorithm reduces the storage requirement and computational cost from$O(n)$to$O((K+1)\log n)$at the$n\$-th time step. We prove that the convergence rate of the fast algorithm is the same as the direct method even a high order direct method is considered. The convergence rate and efficiency of th

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