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  • The existence of a set of d^2 pairwise equiangular complex lines (a SIC-POVM) in d-dimensional Hilbert space is currently known only for a finite set of dimensions d. We prove that, if there exists a set of real units in a certain ray class field (depending on d) satisfying certain congruence conditions and algebraic properties, a SIC-POVM may be constructed when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact solution to the SIC-POVM problem in dimension 23.

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  • We aim to find a solution $\bm{x}\in\mathbb{R}^n/\mathbb{C}^n$ to a system of quadratic equations of the form $b_i=\lvert\langle\bm{a}_i,\bm{x}\rangle\rvert^2$, $i=1,2,\ldots,m$, e.g., the well-known phase retrieval problem, which is generally NP-hard. It has been proved that the number $m = 2n-1$ of generic random measurement vectors $\bm{a}_i\in\mathbb{R}^n$ is sufficient and necessary for uniquely determining the $n$-length real vector $\bm{x}$ up to a global sign. The uniqueness theory, however, does not provide a construction or characterization of this unique solution. As opposed to the recent nonconvex state-of-the-art solvers, we revert to the convex relaxation semidefinite programming (SDP) approach and propose to indirectly minimize the convex objective by successive and incremental nonconvex optimization, termed as \texttt{IncrePR}, to overcome the excessive computation cost of typical SDP solvers. \texttt{IncrePR} avoids sensitive dependence of initialization of nonconvex a

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  • We show that the notion of $3$-hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct polyhedral fundamental domains for the domain of discontinuity that these groups admit in the projective space or the sphere, depending on the dimension.

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  • In this paper, we consider the following weakly coupled nonlinear Schr\"{o}dinger system in $\mathbb{R}^N$ $$ \left\{ \begin{array}{ll} -\varepsilon^{2}\Delta u_1 + V_1(x)u = |u|^{2p - 2}u_1 + b|u|^{p - 2}|v|^pu, & x\in \mathbb{R}^N,\vspace{0.12cm} -\varepsilon^{2}\Delta u_2 + V_2(x)v = |v|^{2p - 2}u_2 + b|v|^{p - 2}|u|^pv, & x\in \mathbb{R}^N, \end{array} \right. $$ where $\varepsilon>0$, $b\in\mathbb{R}$ is a coupling constant, $2p\in (2 + \frac{2\sigma}{N - 2}, 2^*)$ with $\sigma \in[0,1]$, $2^* = 2N/(N - 2),\ N\geq 3$, $V_1$ and $V_2$ belong to $C(\mathbb{R}^N,[0,\infty))$. This type of systems arise in models of nonlinear optics. Let $\min_{i = 1,2}\liminf_{|x|\to\infty}V_i(x)|x|^{2\sigma} > 0$. We prove the problem has a family of nontrivial solutions $\{w_{\varepsilon} = (u_{\varepsilon},v_{\varepsilon}):0<\varepsilon<\varepsilon_{0}\}$ that concentrate at the prescribed common local minimum of $\Lambda_1 = \Lambda_2$ provided that $b>b_{\omega} > 0$ an

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  • Dictionary learning is a popular class of methods for modeling complex data by learning sparse representations directly from the data. For some large-scale applications, exploiting a known structure of the signal is often essential for reducing the complexity of algorithms and representations. One such method is tensor factorization by which a large multi-dimensional dataset can be explicitly factored or separated along each dimension of the data in order to break the representation up into smaller components. Learning dictionaries for tensor structured data is called tensor or separable dictionary learning. While there have been many recent works on separable dictionary learning, typical formulations involve solving a non-convex optimization problem and guaranteeing global optimality remains a challenge. In this work, we propose a framework that uses recent developments in matrix/tensor factorization to provide theoretical and numerical guarantees of the global optimality for the sepa

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  • We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets are bases of finite order for the positive integers. We show that generically, $A$ is a basis of order $2$ when $\operatorname{deg} p \geq 3$, but not when $\operatorname{deg} p = 2$, although then $A + A$ still has asymptotic density $1$.

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  • In this paper, motivated by the works of Bakry et. al in finding sharp Li-Yau type gradient estimate for positive solutions of the heat equation on complete Riemannian manifolds with nonzero Ricci curvature lower bound, we first introduce a general form of Li-Yau type gradient estimate and show that the validity of such an estimate for any positive solutions of the heat equation reduces to the validity of the estimate for the heat kernel of the Riemannian manifold. Then, a sharp Li-Yau type gradient estimate on the three dimensional hyperbolic space is obtained by using the explicit expression of the heat kernel and some optimal Li-Yau type gradient estimates on general hyperbolic spaces are obtained.

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  • We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. This generalizes and simplifies an earlier result of Van den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase transition on $\mathbb{Z}^d$. Our proof relies on the OSSS inequality for Boolean functions and is inspired by a series of papers by Duminil-Copin, Raoufi and Tassion in which they prove similar sharpness results for a variety of models.

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  • With the current trend of transforming a centralized power system into a decentralized one for efficiency, reliability, and environment reasons, the concept of microgrid that integrates a variety of distributed energy resources (DERs) on the distribution network is gaining popularity. In this paper, we investigate the smart charging and parking of plug-in hybrid electric vehicles (PHEVs) in microgrids with renewable energy sources (RES), such as solar panels, in grid-connected mode. To address the uncertainties associated with RES power output and PHEVs charging condition in the microgrid, we propose a two-stage scenario-based stochastic optimization approach with the objective of providing a proper scheduling for parking and charging of PHEVs that minimizes the average total operating cost while maintaining the reliability of the microgrid. A case study is conducted to show the effectiveness of the proposed approach. Extensive simulation results show that the microgrid can minimize th

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  • In this paper the 3-valued paraconsistent first-order logic QCiore is studied from the point of view of Model Theory. The semantics for QCiore is given by partial structures, which are first-order structures in which each n-ary predicate R is interpreted as a triple of paiwise disjoint sets of n-uples representing, respectively, the set of tuples which actually belong to R, the set of tuples which actually do not belong to R, and the set of tuples whose status is dubious or contradictory. Partial structures were proposed in 1986 by I. Mikenberg, N. da Costa and R. Chuaqui for the theory of quasi-truth (or pragmatic truth). In 2014, partial structures were studied by M. Coniglio and L. Silvestrini for a 3-valued paraconsistent first-order logic called LPT1, whose 3-valued propositional fragment is equivalent to da Costa-D'Otaviano's logic J3. This approach is adapted in this paper to QCiore, and some important results of classical Model Theory such as Robinson's joint consistency theore

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  • We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible.

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  • In this paper, we define a special class of elements in the algebras obtained by the Cayley Dickson process, called l elements. We find conditions such that these elements to be invertible. These conditions can be very useful for finding new identities, identities which can help us in the study of the properties of these algebras.

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  • Regularized regression problems are ubiquitous in statistical modeling, signal processing, and machine learning. Sparse regression in particular has been instrumental in scientific model discovery, including compressed sensing applications, variable selection, and high-dimensional analysis. We propose a new and highly effective approach for regularized regression, called SR3. The key idea is to solve a relaxation of the regularized problem, which has three advantages over the state-of-the-art: (1) solutions of the relaxed problem are superior with respect to errors, false positives, and conditioning, (2) relaxation allows extremely fast algorithms for both convex and nonconvex formulations, and (3) the methods apply to composite regularizers such as total variation (TV) and its nonconvex variants. We demonstrate the improved performance of SR3 across a range of regularized regression problems with synthetic and real data, including compressed sensing, LASSO, matrix completion and TV re

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  • Optimization over non-negative polynomials is fundamental for nonlinear systems analysis and control. We investigate the relation between three tractable relaxations for optimizing over sparse non-negative polynomials: sparse sum-of-squares (SSOS) optimization, diagonally dominant sum-of-squares (DSOS) optimization, and scaled diagonally dominant sum-of-squares (SDSOS) optimization. We prove that the set of SSOS polynomials, an inner approximation of the cone of SOS polynomials, strictly contains the spaces of sparse DSOS/SDSOS polynomials. When applicable, therefore, SSOS optimization is less conservative than its DSOS/SDSOS counterparts. Numerical results for large-scale sparse polynomial optimization problems demonstrate this fact, and also that SSOS optimization can be faster than DSOS/SDSOS methods despite requiring the solution of semidefinite programs instead of less expensive linear/second-order cone programs.

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  • We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm-Liouville problems. Via \Gamma-convergence theory, we study the asymptotic distribution of the minimizers as the number of intervals of the partition tends to infinity. Then we discuss several examples that fit in our framework, such as the sum of (positive and negative) powers of the eigenvalues and an approximation of the trace of the heat Sturm-Liouville operator.

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  • The main purpose of this paper is to show the global stabilization and exact controllability properties for a fourth order nonlinear fourth order nonlinear Schr\"odinger system: $$i\partial_tu +\partial_x^2u-\partial_x^4u=\lambda |u|^2u,$$ on a periodic domain $\mathbb{T}$ with internal control supported on an arbitrary sub-domain of $\mathbb{T}$. More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exact controllability ensures that fourth order nonlinear Schr\"odinger is globally exactly controllable.

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  • A noncommutative polynomial is stable if it is nonsingular on all tuples of matrices whose imaginary parts are positive definite. In this paper a characterization of stable polynomials is given in terms of strongly stable linear matrix pencils, i.e., pencils of the form $H+iP_0+P_1x_1+\cdots+P_dx_d$, where $H$ is hermitian and $P_j$ are positive semidefinite matrices. Namely, a noncommutative polynomial is stable if and only if it admits a determinantal representation with a strongly stable pencil. More generally, structure certificates for noncommutative stability are given for linear matrix pencils and noncommutative rational functions.

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  • Following Zong (arXiv:1604.07270), we define an algebraic GKM orbifold $\mathcal{X}$ to be a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. The 1-skeleton of $\mathcal{X}$ is the union of its zero-dimensional and one-dimensional $T$-orbits; its formal neighborhood $\hat{\mathcal{X}}$ in $\mathcal{X}$ determines a decorated graph, called the stacky GKM graph of $\mathcal{X}$. The $T$-equivariant orbifold Gromov-Witten (GW) invariants of $\mathcal{X}$ can be computed by localization and depend only on the stacky GKM graph of $\mathcal{X}$ with the $T$-action. We also introduce abstract stacky GKM graphs and define their formal equivariant orbifold GW invariants. Formal equivariant orbifold GW invariants of the stacky GKM graph of an algebraic GKM orbifold $\mathcal{X}$ are refinements of $T$-equivariant orbifold GW invariants of $\mathcal{X}$.

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  • We investigate the dependence of steady-state properties of Schelling's segregation model on the agents' activation order. Our basic formalism is the Pollicott-Weiss version of Schelling's segregation model. Our main result modifies this baseline scenario by incorporating contagion in the decision to move: (pairs of) agents are connected by a second, agent influence network. Pair activation is specified by a random walk on this network. The considered schedulers choose the next pair nonadaptively. We can complement this result by an example of adaptive scheduler (even one that is quite fair) that is able to preclude maximal segregation. Thus scheduler nonadaptiveness seems to be required for the validity of the original result under arbitrary asynchronous scheduling. The analysis (and our result) are part of an adversarial scheduling approach we are advocating to evolutionary games and social simulations.

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  • We consider the strictly hyperbolic Cauchy problem \begin{align*} &D_t^m u - \sum\limits_{j = 0}^{m-1} \sum\limits_{|\gamma|+j = m} a_{m-j,\,\gamma}(t,\,x) D_x^\gamma D_t^j u = 0, \newline &D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots,\,m, \end{align*} for $(t,\,x) \in [0,\,T]\times \mathbb{R}^n$ with coefficients belonging to the Zygmund class $C^s_\ast$ in $x$ and having a modulus of continuity below Lipschitz in $t$. Imposing additional conditions to control oscillations, we obtain a global (on $[0,\,T]$) $L^2$ energy estimate without loss of derivatives for $s \geq \{1+\varepsilon,\,\frac{2m_0}{2-m_0}\}$, where $m_0$ is linked to the modulus of continuity of the coefficients in time.

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  • We establish strong factorization for pairs of smooth fans which are refined by the braid arrangement fan. This includes normal fans of generalized permutahedra. We show that the realization of the toric variety of the permutahedron as an iterated blow-up of projective space can be achieved by considering a sequence of polytopes known as hyper-permutahedra. To any poset we associate a cone which is the union of some Weyl chambers of type $A$. We give conditions for when a toric variety defined by such a cone is Gorenstein and for the existence of a crepant resolution.

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  • The paper studies generalized differentiability properties of the marginal function of parametric optimal control problems of semilinear elliptic partial differential equations. We establish upper estimates for the regular and the limiting subgradients of the marginal function. With some additional assumptions, we show that the solution map of the perturbed optimal control problems has local upper H\"{o}lderian selections. This leads to a lower estimate for the regular subdifferential of the marginal function.

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  • In this paper, we improve the sup-norm bound and the lower bound of the number of nodal domains for CM Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let $\phi$ be a CM Maass form with spectral parameter $t_\phi$, then we prove that $\|\phi\|_\infty \ll t_\phi^{3/8+\varepsilon} \|\phi\|_2$, which is an improvement over the bound $t_\phi^{5/12+\varepsilon} \|\phi\|_2$ given by Iwaniec and Sarnak. As a consequence, we get a better lower bound for the number of nodal domains intersecting a fixed geodesic segment under the Lindel\"{o}f Hypothesis. Unconditionally, we prove that the number of nodal domains grows faster than $t_\phi^{1/8-\varepsilon}$ for any $\varepsilon>0$ for almost all CM Maass forms.

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  • Edge-connectivity is a classic measure for reliability of a network in the presence of edge failures. $k$-restricted edge-connectivity is one of the refined indicators for fault tolerance of large networks. Matching preclusion and conditional matching preclusion are two important measures for the robustness of networks in edge fault scenario. In this paper, we show that the DCell network $D_{k,n}$ is super-$\lambda$ for $k\geq2$ and $n\geq2$, super-$\lambda_2$ for $k\geq3$ and $n\geq2$, or $k=2$ and $n=2$, and super-$\lambda_3$ for $k\geq4$ and $n\geq3$. Moreover, as an application of $k$-restricted edge-connectivity, we study the matching preclusion number and conditional matching preclusion number, and characterize the corresponding optimal solutions of $D_{k,n}$. In particular, we have shown that $D_{1,n}$ is isomorphic to the $(n,k)$-star graph $S_{n+1,2}$ for $n\geq2$.

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  • In the line of classical work by Hardy, Littlewood and Wilton, we study a class of functional equations involving the Gauss transformation from the theory of continued fractions. This allows us to reprove, among others, a convergence criterion for a diophantine series considered by Chowla, and to give additional information about the sum of this series.

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  • We establish an invariance principle for a one-dimensional random walk in a dynamical random environment given by a speed-change exclusion process. The jump probabilities of the walk depend on the configuration of the exclusion in a finite box around the walker. The environment starts from equilibrium. After a suitable space-time rescaling, the random walk converges to a sum of two independent processes, a Brownian motion and a Gaussian process with stationary increments.

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  • We construct and study a nested sequence of finite symmetric tensor categories ${\rm Vec}=\mathcal{C}_0\subset \mathcal{C}_1\subset\cdots\subset \mathcal{C}_n\subset\cdots$ over a field of characteristic $2$ such that $\mathcal{C}_{2n}$ are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius--Perron dimension. This generalizes the category $\mathcal{C}_1$ described by Venkatesh and the category $\mathcal{C}_2$ defined by Ostrik. The Grothendieck rings of the categories $\mathcal{C}_{2n}$ and $\mathcal{C}_{2n+1}$ are both isomorphic to the ring of real cyclotomic integers defined by a primitive $2^{n+2}$-th root of unity, $\mathcal{O}_n=\mathbb Z[2\cos(\pi/2^{n+1})]$.

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  • In vehicular edge computing (VEC) system, some vehicles with surplus computing resources can provide computation task offloading opportunities for other vehicles or pedestrians. However, vehicular network is highly dynamic, with fast varying channel states and computation loads. These dynamics are difficult to model or to predict, but they have major impact on the quality of service (QoS) of task offloading, including delay performance and service reliability. Meanwhile, the computing resources in VEC are often redundant due to the high density of vehicles. To improve the QoS of VEC and exploit the abundant computing resources on vehicles, we propose a learning-based task replication algorithm (LTRA) based on combinatorial multi-armed bandit (CMAB) theory, in order to minimize the average offloading delay. LTRA enables multiple vehicles to process the replicas of the same task simultaneously, and vehicles that require computing services can learn the delay performance of other vehicles

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  • We consider the problem of testing, on the basis of a $p$-variate Gaussian random sample, the null hypothesis ${\cal H}_0:{\pmb \theta}_1= {\pmb \theta}_1^0$ against the alternative ${\cal H}_1: {\pmb \theta}_1 \neq {\pmb \theta}_1^0$, where ${\pmb \theta}_1$ is the "first" eigenvector of the underlying covariance matrix and $\thetab_1^0$ is a fixed unit $p$-vector. In the classical setup where eigenvalues $\lambda_1>\lambda_2\geq \ldots\geq \lambda_p$ are fixed, the Anderson (1963) likelihood ratio test (LRT) and the Hallin, Paindaveine, and Verdebout (2010) Le Cam optimal test for this problem are asymptotically equivalent under the null, hence also under sequences of contiguous alternatives. We show that this equivalence does not survive asymptotic scenarios where $\lambda_{n1}-\lambda_{n2}=o(r_n)$ with $r_n=O(1/\sqrt{n})$. For such scenarios, the Le Cam optimal test still asymptotically meets the nominal level constraint, whereas the LRT severely overrejects the null hypothesis.

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  • We describe the analytic calculation of the master integrals required to compute the two-mass three-loop corrections to the $\rho$ parameter. In particular, we present the calculation of the master integrals for which the corresponding differential equations do not factorize to first order. The homogeneous solutions to these differential equations are obtained in terms of hypergeometric functions at rational argument. These hypergeometric functions can further be mapped to complete elliptic integrals, and the inhomogeneous solutions are expressed in terms of a new class of integrals of combined iterative non-iterative nature.

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  • We present two families of Wilf-equivalences for consecutive and quasi-consecutive vincular patterns. These give new proofs of the classification of consecutive patterns of length $4$ and $5$. We then prove additional equivalences to explicitly classify all quasi-consecutive patterns of length $5$ into 26 Wilf-equivalence classes.

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  • For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in $S^4$ and $\pi_0(\Diff_0(S^2\times D^2)/\Diff_0(B^4))=1$. In manifolds with $\BZ_2$-torsion, one surface can be put into a normal form relative to the other.

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  • We construct a Baum--Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu_\tau$, is defined in $KK$-theory with coefficients in $\mathbb{R}$ by means of the action of the projection $[\tau]\in KK_{\mathbb{R}}^\Gamma(\mathbb{C},\mathbb{C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture.

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  • The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We prove that in dimension one the cohomology annihilator ideal and the conductor ideal coincide under mild assumptions. We present a condition on a ring homomorphism between Gorenstein rings which allows us to carry the cohomology annihilator of the domain to that of the codomain. As an application, we generalize the Milnor-Jung formula for algebraic curves to their double branched covers. We also show that the cohomology annihilator of a Gorenstein local ring is contained in the cohomology annihilator of its Henselization and in dimension one the cohomology annihilator of its completion. Finally, we investigate a relation between the cohomology annihilator of a Gorenstein ring and stable annihilators of its noncommutative resolutions.

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  • In 2010, Vladimir Voevodsky gave a lecture on "What If Current Foundations of Mathematics Are Inconsistent?" Among other things he said that he was seriously suspicious that an inconsistency in PA (first-order Peano arithmetic) might someday be found. About a year later, Edward Nelson announced that he had discovered an inconsistency not just in PA, but in a small fragment of primitive recursive arithmetic. Soon, Daniel Tausk and Terence Tao independently found a fatal error, and Nelson withdrew his claim, stating that consistency of PA was an "open problem." Many mathematicians may find such claims bewildering. Is the consistency of PA really an open problem? If so, would the discovery of an inconsistency in PA cause all of mathematics to come crashing down like a house of cards? This expository article attempts to address these questions, by sketching and discussing existing proofs of the consistency of PA (including Gentzen's proof and Friedman's relative consistency proof that appe

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  • We continue the development of the computability of the second real Johnson-Wilson theory. As ER(2) is not complex orientable, this gives some difficulty even with basic spaces. In this paper we compute the second real Johnson-Wilson theory for products of infinite complex projective spaces and for the classifying spaces for the unitary groups.

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  • The generalized second Painlev\'e equation $\Delta y -x_1 y - 2 y^3=0$ in $(x_1, x_2)\in \mathbb{R}^2$, plays an important role in the theory of light-matter interactions in nematic liquid crystals. Hastings-McLeod showed the existence of a unique, positive, entire solution of the ODE $y"-x y-2y^3=0$. In this paper we show the existence of the corresponding solution of the PDE on the plane. It has a form of a quadruple connection between the Airy function, two one dimensional Hastings-McLeod solutions $\pm h(x)$ and the heteroclinic solution $\tanh(x/\sqrt{2})$ of the one dimensional Allen-Cahn equation.

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  • In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element methods (conforming and non-conforming) as well as finite volume methods. We also use the HDM to design a novel method, based on conforming $\mathbb{P}_1$ finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.

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  • The classical inverse first passage time problem asks whether, for a Brownian motion $(B_t)_{t\geq 0}$ and a positive random variable $\xi$, there exists a barrier $b:\mathbb{R}_+\to\mathbb{R}$ such that $\mathbb{P}\{B_s>b(s), 0\leq s \leq t\}=\mathbb{P}\{\xi>t\}$, for all $t\geq 0$. We study a variant of the inverse first passage time problem for killed Brownian motion. We show that if $\lambda>0$ is a killing rate parameter and $\mathbb{1}_{(-\infty,0]}$ is the indicator of the set $(-\infty,0]$ then, under certain compatibility assumptions, there exists a unique continuous function $b:\mathbb{R}_+\to\mathbb{R}$ such that $\mathbb{E}\left[-\lambda \int_0^t \mathbb{1}_{(-\infty,0]}(B_s-b(s))\,ds\right] = \mathbb{P}\{\zeta>t\}$ holds for all $t\geq 0$. This is a significant improvement of a result of the first two authors (Annals of Applied Probability 24(1):1--33, 2014). The main difficulty arises because $\mathbb{1}_{(-\infty,0]}$ is discontinuous. We associate a semi-lin

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  • Morava $E$-theory $E$ is an $E_\infty$-ring with an action of the Morava stabilizer group $\Gamma$. We study the derived stack $\operatorname{Spf} E/\Gamma$. Descent-theoretic techniques allow us to deduce a theorem of Hopkins-Mahowald-Sadofsky on the $K(n)$-local Picard group, as well as a recent result of Barthel-Beaudry-Stojanoska on the Anderson duals of higher real $K$-theories.

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  • We prove the Schr\"oder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This result extends the Schr\"oder case of the Delta conjecture proved in (D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the $q,t$-Schr\"oder of Haglund (Haglund 2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another combinatorial interpretation of the same polynomial in terms of a new bounce statistic. Moreover, we give two more interpretations of the same polynomial in terms of doubly decorated parallelogram polyominoes, extending some of the results in (D'Adderio, Iraci 2017), which in turn extended results in (Aval et al. 2014). Also, we provide combinatorial bijections explaining some of t

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  • We generalise a definition of the special linear group due to Baez to arbitrary rings. At the infinitesimal level we get a Lie ring. We give a description of these special linear rings over some large classes of rings, including all associative rings and all algebras over a field.

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  •  

    The Theory of Interstellar Trade

    07-15 Hacker News 2

    The Theory of Interstellar Trade

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  • This paper deals with design of the secure blockchain network framework to prevent damages from an attacker. The design is based on the hybrid theoretical approaches which is named as the Blockchain Governance Game. The framework of this game finds the best strategy towards preparation for preventing attacker a network malfunction. Analytically tractable results are obtained by using hybrid of the fluctuation theory and the mixed strategy game theory which enables to predict the moment for operations and deliver the optimal portion of backup nodes to protect the blockchain network.

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  • We study a random permutation of a lattice box in which each permutation is given a Boltzmann weight with energy equal to the total Euclidean displacement. Our main result establishes the band structure of the model as the box-size $N$ tends to infinity and the inverse temperature $\beta$ tends to zero; in particular, we show that the mean displacement is of order $\min \{ 1/\beta, N\}$. In one dimension our results are more precise, specifying leading-order constants and giving bounds on the rates of convergence. Our proofs exploit a connection, via matrix permanents, between random permutations and Gaussian fields; although this connection is well-known in other settings, to the best of our knowledge its application to the study of random permutations is novel. As a byproduct of our analysis, we also provide asymptotics for the permanents of Kac-Murdock-Szego (KMS) matrices.

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  • We propose a general new method, the \emph{conditional permutation test}, for testing the conditional independence of variables $X$ and $Y$ given a potentially high-dimensional random vector $Z$ that may contain confounding factors. The proposed test permutes entries of $X$ non-uniformly, so as to respect the existing dependence between $X$ and $Z$ and thus account for the presence of these confounders. Like the conditional randomization test of \citet{candes2018panning}, our test relies on the availability of an approximation to the distribution of $X \mid Z$---while \citet{candes2018panning}'s test uses this estimate to draw new $X$ values, for our test we use this approximation to design an appropriate non-uniform distribution on permutations of the $X$ values already seen in the true data. We provide an efficient Markov Chain Monte Carlo sampler for the implementation of our method, and establish bounds on the Type~I error in terms of the error in the approximation of the condition

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  • A conjecture of Ulam states that the standard probability measure $\pi$ on the Hilbert cube $I^\omega$ is invariant under the induced metric $d_a$ provided the sequence $a = \{ a_i \}$ of positive numbers satisfies the condition $\sum\limits_{i=1}^\infty a_i^2 < \infty$. In this paper, we prove this conjecture in the affirmative. More precisely, we prove that if there exists a surjective $d_a$-isometry $f : E_1 \to E_2$, where $E_1$ and $E_2$ are Borel subsets of $I^\omega$, then $\pi(E_1) = \pi(E_2)$.

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  • Let $H_{1}(m, d, k)$ be the $k$-uniform supertree obtained from a loose path $P:v_{1}, e_{1}, v_{2}, \ldots,v_{d}, e_{d}, v_{d+1}$ with length $d$ by attaching $m-d$ edges at vertex $v_{\lfloor\frac{d}{2}\rfloor+1}.$ Let $\mathbb{S}(m,d,k)$ be the set of $k$-uniform supertrees with $m$ edges and diameter $d$ and $q_1(G)$ be the signless Laplacian spectral radius of a $k$-uniform hypergraph $G$. In this paper, we mainly determine $H_{1}(m,d,k)$ with the largest signless Laplacian spectral radius among all supertrees in $\mathbb{S}(m,d,k)$ for $3\leq d\leq m-1$. Furthermore, we determine the unique uniform supertree with the maximum signless Laplacian spectral radius among all the uniform supertrees with $n$ vertices and pendant edges (vertices).

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  • The class of nonsmooth codifferentiable functions was introduced by professor V.F.~Demyanov in the late 1980s. He also proposed a method for minimizing these functions called the method of codifferential descent (MCD) that can be applied to various nonsmooth optimization problems. However, until now almost no theoretical results on the performance of this method on particular classes of nonsmooth optimization problems were known. The main goal of this article is to improve our understanding of the MCD and provide a theoretical foundation for the comparison of the MCD with other methods of nonsmooth optimization. In the first part of the paper we study the performance of the method of codifferential descent on a class of nonsmooth convex functions satisfying some regularity assumptions, which in the smooth case are reduced to the Lipschitz continuity of the gradient. We prove that in this case the MCD has the iteration complexity bound $\mathcal{O}(1 / \varepsilon)$. In the second part

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  • Let $f=h+\overline{g}$ be a harmonic univalent map in the unit disk $\mathbb{D}$, where $h $ and $g$ are analytic. We obtain an improved estimate for the second coefficient of $h$. This indeed is the first qualitative improvement after the appearance of the papers by Clunie and Sheil-Small in 1984, and by Sheil-Small in 1990. Also, when the sup-norm of the dilatation is less than $1$, it is shown that the spherical area of the covering surface of $h$ is dominated by the spherical area of the covering surface of $f.$

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  • The amplituhedron provides a beautiful description of perturbative superamplitude integrands in N=4 SYM in terms of purely geometric objects, generalisations of polytopes. On the other hand the Wilson loop in supertwistor space also gives an explicit description of these superamplitudes as a sum of planar Feynman diagrams. Each Feynman diagram can be naturally associated with a geometrical object in the same space as the amplituhedron (although not uniquely). This suggests that these geometric images of the Feynman diagrams give a tessellation of the amplituhedron. This turns out to be the case for NMHV amplitudes. We prove however that beyond NMHV this is not true. Specifically, each Feynman diagram leads to an image with a physical boundary and spurious boundaries. The spurious ones should be "internal", matching with neighbouring diagrams. We however show that there is no choice of geometric image of the Wilson loop Feynman diagrams which yields a geometric object without leaving un

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  • In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincar\'e equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincar\'e equations. Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for opt

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  • We show that if a first-order structure $\mathcal{M}$, with universe $\mathbb{Z}$, is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,<,0)$, then $\mathcal{M}$ must be interdefinable with $(\mathbb{Z},+,0)$ or $(\mathbb{Z},+,<,0)$.

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  • Birman--Murakami--Wenzl (BMW) algebra was introduced in connection with knot theory. We treat here interaction round the face solvable (IRF) lattice models. We assume that the face transfer matrix obeys a cubic polynomial equation, which is called the three block case. We prove that the three block theories all obey the BMW algebra. We exemplify this result by treating in detail the $SU(2)$ $2\times 2$ fused models, and showing explicitly the BMW structure. We use the connection between the construction of solvable lattice models and conformal field theory. This result is important to the solution of IRF lattice models and the development of new models, as well as to knot theory.

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  • We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, for various models of random (multi)graphs. For our proofs we introduce the notion of \emph{patchworks} to describe the possible overlappings of copies of subgraphs. Furthermore, the proofs are based on analytic combinatorics to carry out asymptotic computations. The flexibility of our approach allows us to tackle a wide range of problems. We obtain the asymptotic number and the limiting distribution of the number of subgraphs which are isomorphic to a graph from a given set of graphs. The results apply to multigraphs as well as to (multi)graphs with degree constraints. One application is to scale-free multigraphs, where the degree distribution follows a power law, for which we show how to obtain the asymptotic number of copies of a given subgraph and give as an illustration the expected number of small cycles.

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  • Conditional stability estimates allow us to characterize the degree of ill-posedness of many inverse problems, but without further assumptions they are not sufficient for the stable solution in the presence of data perturbations. We here consider the stable solution of nonlinear inverse problems satisfying a conditional stability estimate by Tikhonov regularization in Hilbert scales. Order optimal convergence rates are established for a-priori and a-posteriori parameter choice strategies. The role of a hidden source condition is investigated and the relation to previous results for regularization in Hilbert scales is elaborated. The applicability of the results is discussed for some model problems, and the theoretical results are illustrated by numerical tests.

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  • A graph is k-total colourable if there is an assignment of k different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. In this paper, we determine the total chromatic number of some Cartesian and direct product graphs by giving explicit total colourings. In particular, we establish the total chromatic number of the crown graph, which we use to determine the total chromatic number of other various graphs.

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  • Boundaries, GNH, and parametrized theories. It takes three to tango. This is the motto of my doctoral thesis and the common thread along it. The thesis is structured as follows: after some acknowledgments and a brief introduction, chapter 3 is devoted to establishing the mathematical background necessary for the rest of the thesis (with special emphasis in the space of embeddings and the Fock construction). Chapter 4 is based on our papers arXiv:1701.00735, arXiv:1611.09603, and arXiv:1501.05114. We study carefully a system consisting of a string with two masses attached to the ends and try to establish if we can identify degrees of freedom at the boundary both classically and quantically (spoiler alert: it is not possible). The next chapter is a brief introduction to the parametrized theories with the simple example of the parametrized classical mechanics. The 6th chapter deals with the parametrized electromagnetism with boundaries, a generalization of our paper arXiv:1511.00826. The

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  • We study queue-based activation protocols in random-access networks. The network is modeled as an interference graph. Each node of the graph represents a server with a queue. Packets arrive at the nodes as independent Poisson processes and have independent exponentially distributed sizes. Each node can be either active or inactive. When a node is active, it deactivates at unit rate. When a node is inactive, it activates at a rate that depends on its current queue length, provided none of its neighboring nodes is active. Thus, two nodes that are connected by a bond cannot be active simultaneously. This situation arises in random-access wireless networks where, due to interference, servers that are close to each other cannot use the same frequency band. In the limit as the queue lengths at the nodes become very large, we compute the transition time between the two states where one half of the network is active and the other half is inactive. We compare the transition time with that of a

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  • Similarly as in (Blancas et al. 2018) where nested coalescent processes are studied, we generalize the definition of partition-valued homogeneous Markov fragmentation processes to the setting of nested partitions, i.e. pairs of partitions $(\zeta,\xi)$ where $\zeta$ is finer than $\xi$. As in the classical univariate setting, under exchangeability and branching assumptions, we characterize the jump measure of nested fragmentation processes, in terms of erosion coefficients and dislocation measures. Among the possible jumps of a nested fragmentation, three forms of erosion and two forms of dislocation are identified - one of which being specific to the nested setting and relating to a bivariate paintbox process.

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  • In this paper, we give two family of explicit blowup axisymmetric solution for $3$D incompressible Navier-Stokes equations in $\mathbb{R}^3$. Here one family of solutions admit the smooth initial data, and the initial data of another family of solutions are not smooth.

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  • Let $p$ be a prime and let $q$ be a power of $p$. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance- separable (MDS) codes with parameters \[ [[tq, tq-2d+2, d]]_{q} \] for any $1 \leq t \leq q, 2 \leq d \leq \lfloor \frac{tq+q-1}{q+1}\rfloor+1$, and \[ [[t(q+1)+2, t(q+1)-2d+4, d]]_{q} \] for any $1 \leq t \leq q-1, 2 \leq d \leq t+2$ with $(p,t,d) \neq (2, q-1, q)$. Our quantum codes have flexible parameters, and have minimum distances larger than $\frac{q}{2}+1$ when $t > \frac{q}{2}$. Furthermore, it turns out that our constructions generalize and improve some previous results.

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  • We use Madan-Yor's argument to construct associated submartingales to a class of two-parameter processes that are ordered by the increasing convex dominance. This class includes processes which have MTP$_2$ integrated survival functions. We prove that the integrated survival function of an integrable two-parameter process is MTP$_2$ if and only if it is TP$_2$ in each pair of arguments when the remaining argument is fixed. This result can not be deduced from known results since there are several two-parameter processes whose integrated survival functions do not have interval support. The MTP$_2$ property of certain MRL processes is useful to exhibit numerous other processes having the same property. We mention that a two-parameter process ordered by the increasing convex dominance is not necessarily associated to a two-parameter submartingale. A counterexample has been provided recently by Juillet.

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  • Energy and momentum conservation in the context of a type II, purely transmitting, defect, within a single scalar relativistic two-dimensional field theory, places a severe constraint not only on the nature of the defect but also on the potentials for the scalar fields to either side of it. The constraint is of an unfamiliar type since it requires the Poisson Bracket of the defect contributions to energy and momentum with respect to the defect discontinuity and its conjugate to be balanced by the potential difference across the defect. It is shown that the only solutions to the constraint correspond to the known integrable field theories.

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  • We prove that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with positive log cotangent sheaf are uniformly bounded. This generalizes work of Abramovich and Abramovich-Matsuki. In addition, our methods give a description of the geometry of surfaces with positive log cotangent bundle showing that all subvarieties are of log general type.

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  • In this paper we introduce and study a class of structured set-valued operators which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions where the first is convex and the second can be expressed as the minimum of finitely many convex functions.

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  • In this work, we consider a mathematical model for flow in a unsaturated porous medium containing a fracture. In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards' equation. The submodels are coupled by physical transmission conditions expressing the continuity of the normal fluxes and of the pressures. We start by analyzing the case of a fracture having a fixed width-length ratio, called $\varepsilon > 0$. Then we take the limit $\varepsilon \to 0$ and give a rigorous proof for the convergence towards effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively matrix scale with respect to $\varepsilon$, and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.

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  • We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden-Jackson-Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair.

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  • We obtain a general bound for the Wasserstein-2 distance in normal approximation for sums of locally dependent random variables. The proof is based on an asymptotic expansion for expectations of second-order differentiable functions of the sum. Applied to subgraph counts in the Erd\H{o}s-R\'enyi random graph, our result shows that the Wasserstein-1 bound of Barbour, Karo\'nski and Ruci\'nski (1989) holds for the stronger Wasserstein-2 distance.

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  • We consider a mixed moving average (MMA) process X driven by a L\'evy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the L\'evy basis. Using this property, we show conditions ensuring that sample mean and autocovariances of X have a limiting normal distribution. We extend these results to stochastic volatility models and then investigate a Generalized Method of Moments estimator for the supOU process and the supOU stochastic volatility model after choosing a suitable distribution for the mean reversion parameter. For these estimators, we analyze the asymptotic behavior in detail.

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  • Let $A$ be a finite set, $G$ a discrete countable infinite group and $(\mu_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}\mu_g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes_{g\in G}(A,\mu_g)$ is nonsingular and conservative then it is weakly mixing. This answers in positive a question by Z.~Kosloff who proved recently a weaker version of this result for $G=\Bbb Z$.

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  • We investigate polynomial patterns which can be guaranteed to appear in \emph{weakly mixing} sets introduced by introduced by Furstenberg and studied by Fish. In particular, we prove that if $A \subset \mathbb N$ is a weakly mixing set and $p(x) \in \mathbb Z[x]$ a polynomial of odd degree with positive leading coefficient, then all sufficiently large integers $N$ can be represented as $N = n_1 + n_2$, where $p(n_1) + m,\ p(n_2) + m \in A$ for some $m \in A$.

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  • We classify webs of minimal degree rational curves on surfaces and give a criterion for webs being hexagonal. In addition, we classify Neron-Severi lattices of real weak del Pezzo surfaces. These two classifications are related to root subsystems of E8.

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  • This is a survey on weight enumerators, zeta functions and Riemann hypothesis for linear and algebraic-geometry codes.

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  • We consider an arbitrary square integrable function $F$ on the phase space and look for the Wigner function closest to it with respect to the $L^2$ norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert-Schmidt operator with Weyl symbol $F$. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a {\it bona fide} Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results we are able to estimate the

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  • We exhibit a Hamel basis for the concrete $*$-algebra $\mathfrak{M}_o$ associated to monotone commutation relations realised on the monotone Fock space, mainly composed by Wick ordered words of annihilators and creators. We apply such a result to investigate spreadability and exchangeability of the stochastic processes arising from such commutation relations. In particular, we show that spreadability comes from a monoidal action implementing a dissipative dynamics on the norm closure $C^*$-algebra $\mathfrak{M} = \overline{\mathfrak{M}_o}$. Moreover, we determine the structure of spreadable and exchangeable monotone stochastic processes using their correspondence with sp\-reading invariant and symmetric monotone states, respectively.

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  • Let $H$ be a complex Hilbert space whose dimension is not less than $3$ and let ${\mathcal F}_{s}(H)$ be the real vector space formed by all self-adjoint operators of finite rank on $H$. For every non-zero natural $k<\dim H$ we denote by ${\mathcal P}_{k}(H)$ the set of all rank $k$ projections. Let $H'$ be other complex Hilbert space of dimension not less than $3$ and let $L:{\mathcal F}_{s}(H)\to {\mathcal F}_{s}(H')$ be a linear operator such that $L({\mathcal P}_{k}(H))\subset {\mathcal P}_{m}(H')$ for some natural $k,m$ and the restriction of $L$ to ${\mathcal P}_{k}(H)$ is injective. If $H=H'$ and $k=m$, then $L$ is induced by a linear or conjugate-linear isometry of $H$ to itself, except the case $\dim H=2k$ when there is another one possibility (we get a classical Wigner's theorem if $k=m=1$). If $\dim H\ge 2k$, then $k\le m$. The main result describes all linear operators $L$ satisfying the above conditions under the assumptions that $H$ is infinite-dimensional and for any

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  • In this paper we explore the geometric space parametrized by (tree level) Wilson loops in SYM $N=4$. We show that, this space can be seen as a vector bundle over a totally non-negative subspace of the Grassmannian, $\mathcal{W}_{k,cn}$. Furthermore, we explicitly show that this bundle is non-orientable in the majority of the cases, and conjecture that it is non-orientable in the remaining situation. Using the combinatorics of the Deodhar decomposition of the Grassmannian, we identify subspaces $\Sigma(W) \subset \mathcal{W}_{k,n}$ for which the restricted bundle lies outside the positive Grassmannian. Finally, while probing the combinatorics of the Deodhar decomposition, we give a diagrammatic algorithm for reading equations determining each Deodhar component as a semialgebraic set.

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  • In this paper, we compare two common modes of duplexing in wireless powered communication networks (WPCN); namely TDD and FDD. So far, TDD has been the most widely used duplexing technique due to its simplicity. Yet, TDD does not allow the energy transmitter to function continuously, which means to deliver the same amount of energy as that in FDD, the transmitter has to have a higher maximum transmit power. On the other hand, when regulations for power spectral density limits are not restrictive, using FDD may lead to higher throughput than that of TDD by allocating less bandwidth to energy and therefore leaving more bandwidth for data. Hence, the best duplexing technique to choose for a specific problem needs careful examination and evaluation.

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  • Leslie Nielsen’s career got serious when he got the chance to lighten up (2016)

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