results for au:Zhang_L in:math

- In this paper, we present a simple analysis of \bf fast rates with \it high probability of \bf empirical minimization for \it stochastic composite optimization over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledge, this result is the first of its kind. As a byproduct, we can directly obtain the fast rate with \it high probability for exponential concave empirical risk minimization with and without any convex regularization, which not only extends existing results of empirical risk minimization but also provides a unified framework for analyzing exponential concave empirical risk minimization with and without \it any convex regularization. Our proof is very simple only exploiting the covering number of a finite-dimensional bounded set and a concentration inequality of random vectors.
- Sep 05 2017 math.OC arXiv:1709.00559v1We propose two basic assumptions, under which the rate of convergence of the augmented Lagrange method for a class of composite optimization problems is estimated. We analyze the rate of local convergence of the augmented Lagrangian method for a nonlinear semidefinite nuclear norm composite optimization problem by verifying these two basic assumptions. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold \bar c>0. The analysis is based on variational analysis about the proximal mapping of the nuclear norm and the projection operator onto the cone of positively semidefinite symmetric matrices.
- Sep 01 2017 math.NA arXiv:1708.09845v1We use a unified framework to summarize sixteen randomized iterative methods including Kaczmarz method, coordinate descent method, etc. Some new iterative schemes are given as well. Some relationships with \textscmg and \textscddm are also discussed. We analyze the convergence properties of the iterative schemes by using the matrix integrals associated with alternating projectors, and demonstrate the convergence behaviors of the randomized iterative methods by numerical examples.
- Investigation about various random quantum state ensembles is always focused on its eigenvalue statistics. Analytical derivation of joint probability distribution density functions of quantum state ensembles can be used to study asymptotical phenomenon and concentration of measure phenomenon in high dimensions. In this paper, however, we will derive the spectral density of equiprobable mixture of two random density matrices for qubits. Besides, we will also work out the spectral density of mixture under the so-called quantum addition rule (see details in the text), proposed in http://dx.doi.org/10.1063/1.4950785 (J.Math.Phys. 57, 052202 (2016)] of two random density matrices for qubits. We also introduce a new divergence by mimicking the quantum Jensen-Shannon divergence. Unfortunately, our numerical experiments show that both the divergence and its square root are not valid metric.
- In this paper, we focus on some properties of the spreading speeds which can be estimated by linear operators approach, such as the sign, the continuity and a limiting case which admits no spreading phenomenon. These theoretical results are well applied to study the effect of templates on propagation speeds for cellular neural networks (CNNs), which admit three kinds of propagating phenomenon.
- Jul 25 2017 math.AP arXiv:1707.07156v1In this paper, we study the degree counting formula of the rank two Toda system with simple singular source when $\rho_1\in(0,4\pi)\cup(4\pi,8\pi)$ and $\rho_2\notin 4\pi\mathbb{N}.$ The key step is to derive the degree formula of the shadow system, which arises from the bubbling solutions as $\rho_1$ tends to $4\pi$. In order to compute the topological degree of the shadow system, we need to find some suitable deformation. During this deformation, we shall deal with \textitnew difficulty arising from the new phenomena: blow up does not necessarily imply concentration of mass. This phenomena occurs due to the collapsing of singularities. This is a continuation of the previous work Lee, Lin, Wei and Yang.
- Identifying the spectrum of eigenvalues of the sum of two given Hermitian matrices with fixed eigenvalues is the famous Horn's problem. In this note, we investigate the variant of Horn's problem, i.e., we identify the probability density function (abbr. PDF) of the diagonals of the sum of two Hermitian matrices of specific spectrum.We then use it to re-derive the PDF of the eigenvalues of the sum of two Hermitian matrices with given eigenvalues via \emphderivative principle, a powerful tool used to get the exact probability distribution by reducing to the corresponding distribution of diagonal entries. We can recover Jean-Bernard Zuber's recent results on the probability distribution function (PDF) of the eigenvalues of two Hermitian matrices with given eigenvalues. The more general derivative principle relates invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra. Some potential applications in quantum information theory, such as uniform average distance and average coherence of uniform mixture of two orbits, are discussed.
- Jul 18 2017 math.DG arXiv:1707.04846v1In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with $div^4Rm^\pm=0$, we show that it is either Einstein or a finite quotient of $\mathbb{R}^4$, $\mathbb{S}^2\times\mathbb{R}^2$ or $\mathbb{S}^3\times\mathbb{R}$. The same result can be obtained under the condition of $div^4W^\pm=0$. We also present some classification results of four-dimensional complete non-compact gradient expanding Ricci soliton with non-negative Ricci curvature and gradient steady Ricci solitons under certain curvature conditions.
- Jul 03 2017 math.AP arXiv:1706.10264v2We study the Gauss curvature equation with negative singularities. For a local mean field type equation with only one negative index we prove a uniqueness property. For a global equation with one or two negative indexes we prove the non-degeneracy of the linearized equations.
- In this paper, a new generalized $5\times5$ matrix spectral problem of Ablowitz-Kaup-Newell-Segur(AKNS) type associated with the enlarged matrix Lie super algebra is proposed and its corresponding super soliton hierarchy is established. The super variational identities is used to furnish super-Hamiltonian structures for the resulting super soliton hierarchy.
- In this manuscript, we study quantile regression in partial functional linear model where response is scalar and predictors include both scalars and multiple functions. Wavelet basis are adopted to better approximate functional slopes while effectively detect local features. The sparse group lasso penalty is imposed to select important functional predictors while capture shared information among them. The estimation problem can be reformulated into a standard second-order cone program and then solved by an interior point method. We also give a novel algorithm by using alternating direction method of multipliers (ADMM) which was recently employed by many researchers in solving penalized quantile regression problems. The asymptotic properties such as the convergence rate and prediction error bound have been established. Simulations and a real data from ADHD-200 fMRI data are investigated to show the superiority of our proposed method.
- Jun 07 2017 math.OC arXiv:1706.01698v1This paper aims to study the convergence rate of a majorized alternating direction method of multiplier with indefinite proximal terms (iPADMM) for solving linearly constrained convex composite optimization problems. We establish the Q-linear rate convergence theorem for 2-block majorized iPADMM under mild conditions. Based on this result, the convergence rate analysis of symmetric Gaussian-Seidel based majorized ADMM, which is designed for solving multi-block composite convex optimization problems, are given. We apply the majorized iPADMM to solve three types of regularized logistic regression problems: constrained regression, fused lasso and overlapping group lasso. The efficiency of majorized iPADMM are demonstrated on both simulation experiments and real data sets.
- Jun 06 2017 math.AP arXiv:1706.01187v1In this paper, we are concerned with the global existence and stability of a 3-D perturbed viscous circulatory flow around an infinite long cylinder. This flow is described by 3-D compressible Navier-Stokes equations. By introducing some suitably weighted energy spaces and establishing a priori estimates, we show that the 3-D cylindrical symmetric circulatory flow is globally stable in time when the corresponding initial states are perturbed suitably small.
- We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack $X$ over a field $k$. This is done by describing the Nori fundamental gerbe of an essentially finite cover of $X$. A similar result is also obtained for the $S$-fundamental gerbe.
- May 30 2017 math.DG arXiv:1705.09754v1We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\mathbb{R}^4$, $\mathbb{R}^2\times\mathbb{S}^2$ or the round cylinder $\mathbb{R}\times\mathbb{S}^3$. Under the condition of fourth order divergence-free Weyl tensor, we have the same results.
- May 19 2017 math.NA arXiv:1705.06415v1This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.
- May 11 2017 physics.flu-dyn math.DS arXiv:1705.03486v1Microscopic artificial swimmers have recently become highly attractive due to their promising potential for biomedical applications. The pioneering work of Dreyfus et al (2005) has demonstrated the motion of a microswimmer with an undulating chain of superparamagnetic beads, which is actuated by an oscillating external magnetic field. Interestingly, it has also been theoretically predicted that the swimming direction of this swimmer will undergo a $90^\circ$-transition when the magnetic field's oscillations amplitude is increased above a critical value of $\sqrt{2}$. In this work, we further investigate this transition both theoretically and experimentally by using numerical simulations and presenting a novel flexible microswimmer with a superparamagnetic head. We realize the $90^\circ$-transition in swimming direction, prove that this effect depends on both frequency and amplitude of the oscillating magnetic field, and demonstrate the existence of an optimal amplitude, under which, maximal swimming speed can be achieved. By asymptotically analyzing the dynamic motion of microswimmer with a minimal two-link model, we reveal that the stability transitions representing the changes in the swimming direction are induced by the effect of nonlinear parametric excitation.
- May 03 2017 math.AG arXiv:1705.00847v1In this paper, we prove abundance for 3-folds with non-trivial Albanese maps, over an algebraically closed field of characteristic $p > 5$.
- Apr 19 2017 math.AP arXiv:1704.05323v1Consider a class of non-homogenous ultraparabolic differential equations with drift terms or lower order terms arising from some physical models, and we prove that weak solutions are Hölder continuous, which also generalizes the classic results of parabolic equations of second order. The main ingredients are a type of weak Poincaré inequality satisfied by non-negative weak sub-solutions and Moser iteration.
- Apr 06 2017 math.NA arXiv:1704.01268v3Stochastic nonlinear Schrödinger equation with quadratic potential models Bose-Einstein condensations under a magnetic trap when $\theta<0$, which is not only an infinite-dimensional stochastic Hamiltonian system but also a stochastic Hamiltonian partial differential equation essentially. We firstly indicate that this equation possesses stochastic symplectic structure and stochastic multi-symplectic conservation law. It is also shown that the charge and energy have the evolution laws of linear growth with respect to time in the sense of expectation. Moreover, we propose a stochastic symplectic scheme in the temporal discretization, and give the error estimate of this scheme in probability. In order to simulate the evolution laws of charge and energy numerically, we also give a stochastic multi-symplectic scheme, of which the corresponding discrete charge and energy are in accordance with the continuous case. Numerical experiments are performed to test the proposed numerical scheme.
- Apr 03 2017 math.DS arXiv:1703.10707v1We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the half space. We impose a boundary condition that the curves are perpendicular to the boundary for $t>0$, however, the initial curve intersects the boundary tangentially. In other words, the initial curve is oriented singularly. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.
- Apr 03 2017 math.DS arXiv:1703.10709v1In this paper, we consider the mean curvature flow with driving force on fixed extreme points in the plane. We give a general local existence and uniqueness result of this problem with $C^2$ initial curve. For a special family of initial curves, we classify the solutions into three categories. Moreover, in each category, the asymptotic behavior is given.
- Mar 21 2017 math.AG arXiv:1703.06631v1Let $k$ be an algebraically closed field of characteristic $p>0$. We give a birational characterization of ordinary abelian varieties over $k$: a smooth projective variety $X$ is birational to an ordinary abelian variety if and only if $\kappa_S(X)=0$ and $b_1(X)=2 \dim X$. We also give a similar characterization of abelian varieties as well: a smooth projective variety $X$ is birational to an abelian variety if and only if $\kappa(X)=0$, and the Albanese morphism $a: X \to A$ is generically finite. Along the way, we also show that if $\kappa _S (X)=0$ (or if $\kappa(X)=0$ and $a$ is generically finite) then the Albanese morphism $a:X\to A$ is surjective and in particular $\dim A\leq \dim X$.
- We investigate the local descents for special orthogonal groups over p-adic local fields of characteristic zero, and obtain explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence. The main result can be regarded as a refinement of the local Gan-Gross-Prasad conjecture.
- In this paper, we are concerned with Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$ on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of $P_n^{(\alpha,\beta)}(x)$ is derived in the variable of parametrization. This formula further allows us to show that the maximum value of $\left|P_n^{(\alpha,\beta)}(z)\right|$ over the Bernstein ellipse is attained at one of the endpoints of the major axis if $\alpha+\beta\geq -1$. For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., $\alpha=\beta$), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.
- Mar 08 2017 math.NA arXiv:1703.02351v1This paper discusses the multiscale approach and the convergence of the time-dependent Maxwell-Schrödinger system with rapidly oscillating discontinuous coefficients arising from the modeling of a heterogeneous nanostructure with a periodic microstructure. The homogenization method and the multiscale asymptotic method for the nonlinear coupled equations are presented. The efficient numerical algorithms based on the above methods are proposed. Numerical simulations are then carried out to validate the method presented in this paper.
- The speed at which two remote parties can exchange secret keys over a fixed-length fiber-optic cable in continuous-variable quantum key distribution (CV-QKD) is currently limited by the computational complexity of post-processing algorithms for key reconciliation. Multi-edge low-density parity-check (LDPC) codes with low code rates and long block lengths were proposed for CV-QKD, in order to extend the maximum reconciliation distance between the two remote parties. Key reconciliation over multiple dimensions has been shown to further improve the error-correction performance of multi-edge LDPC codes in CV-QKD, thereby increasing both the secret key rate and distance. However, the computational complexity of LDPC decoding for long block lengths on the order of 10^6 bits remains a challenge. This work introduces a quasi-cyclic (QC) code construction for multi-edge LDPC codes that is highly suitable for hardware-accelerated decoding on a modern graphics processing unit (GPU). When combined with an 8-dimensional reconciliation scheme, the LDPC decoder achieves a raw decoding throughput of 1.72Mbit/s and an information throughput of 7.16Kbit/s using an NVIDIA GeForce GTX 1080 GPU at a maximum distance of 160km with a secret key rate of 4.10x10^-7 bits/pulse for a rate 0.02 multi-edge code with block length of 10^6 bits when finite-size effects are considered. This work extends the previous maximum CV-QKD distance of 100km to 160km, while delivering between 1.07x and 8.03x higher decoded information throughput over the upper bound on the secret key rate for a lossy channel. The GPU-based QC-LDPC decoder achieves a 1.29x improvement in throughput over the best existing GPU decoder implementation for a rate 1/10 multi-edge LDPC code with block length of 2^20 bits. These results show that LDPC decoding is no longer the computational bottleneck in long-distance CV-QKD.
- Feb 14 2017 math.AG arXiv:1702.03751v1Let $X$ be a normal, connected and projective variety over an algebraically closed field $k$. It is known that a vector bundle $V$ on $X$ is essentially finite if and only if it is trivialized by a proper surjective morphism $f:Y\to X$. In this paper we introduce a different approach to this problem which allows to extend the results to normal, connected and strongly pseudo-proper algebraic stack of finite type over an arbitrary field $k$.
- Feb 10 2017 math.NA arXiv:1702.02701v1Atomistic/continuum coupling methods aim to achieve optimal balance between accuracy and efficiency. Adaptivity is the key for the efficient implementation of such methods. In this paper, we carry out a rigorous a posterior error analysis which includes the residual estimate, the stability constant estimate and the error bound, for a consistent atomistic/continuum coupling method in 2D. Adaptive mesh refinement algorithm can be designed correspondingly and the convergence rate with respect to degrees of freedom is optimal compare with a priori error estimates.
- In this paper a new hp-adaptive strategy for elliptic problems based on refinement history is proposed, which chooses h-, p- or hp-refinement on individual elements according to a posteriori error estimate, as well as smoothness estimate of the solution obtained by comparing the actual and expected error reduction rate. Numerical experiments show that exponential convergence can be achieved with this strategy.
- Jan 25 2017 math.AP arXiv:1701.06911v1This paper mainly focus on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function $J$ may not be symmetric here. The asymmetry of $J$ has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of $f'(0)=0$, we can no longer use the common method which mainly depending on Ikehara theorem and bilateral Laplace transform to study the asymptotic rates of the nondecreasing traveling wave and the nonincreasing one tending to 0, respectively, thus we adopt another method to investigate them. Afterwards, we establish a new entire solution and obtain its qualitative properties by constructing proper supersolution and subsolution and by classifying the sign and size of the wave speeds.
- Jan 24 2017 math.NA arXiv:1701.05979v1A high order wavelet integral collocation method (WICM) is developed for general nonlinear boundary value problems in physics. This method is established based on Coiflet approximation of multiple integrals of interval bounded functions combined with an accurate and adjustable boundary extension technique. The convergence order of this approximation has been proven to be N as long as the Coiflet with N-1 vanishing moment is adopted, which can be any positive even integers. Before the conventional collocation method is applied to the general problems, the original differential equation is changed into its equivalent form by denoting derivatives of the unknown function as new functions and constructing relations between the low and high order derivatives. For the linear cases, error analysis has proven that the proposed WICM is order N, and condition numbers of relevant matrices are almost independent of the number of collocation points. Numerical examples of a wide range of nonlinear differential equations in physics demonstrate that accuracy of the proposed WICM is even greater than N, and most interestingly, such accuracy is independent of the order of the differential equation to be solved. Comparison to existing numerical methods further justifies the accuracy and efficiency of the proposed method.
- Jan 24 2017 math.NA arXiv:1701.06441v1A high precision, and space time fully decoupled, wavelet formulation numerical method is developed for a class of nonlinear initial boundary value problems. This method is established based on a proposed Coiflet based approximation scheme with an adjustable high order for a square integrable function over a bounded interval, which allows expansion coefficients to be explicitly expressed by function values at a series of single points. In applying the solution method, the nonlinear initial boundary value problems are first spatially discretized into a nonlinear initial value problem by combining the proposed wavelet approximation scheme and the conventional Galerkin method. A novel high order step by step time integrating approach is then developed for the resulting nonlinear initial value problem using the same function approximation scheme based on wavelet theory. The solution method is shown to have Nth-order accuracy, as long as the Coiflet with [0, 3N-1] compact support is adopted, where N can be any positive even number. In addition, the stability property of the method is analyzed, and the stable domain is determined. Numerical examples are considered to justify both the accuracy and efficiency of the method. Results show that the proposed solution method has better accuracy and efficiency than most other methods.
- In this paper, we examine the physical layer security for cooperative wireless networks with multiple intermediate nodes, where the decode-and-forward (DF) protocol is considered. We propose a new joint relay and jammer selection (JRJS) scheme for protecting wireless communications against eavesdropping, where an intermediate node is selected as the relay for the sake of forwarding the source signal to the destination and meanwhile, the remaining intermediate nodes are employed to act as friendly jammers which broadcast the artificial noise for disturbing the eavesdropper. We further investigate the power allocation among the source, relay and friendly jammers for maximizing the secrecy rate of proposed JRJS scheme and derive a closed-form sub-optimal solution. Specificially, all the intermediate nodes which successfully decode the source signal are considered as relay candidates. For each candidate, we derive the sub-optimal closed-form power allocation solution and obtain the secrecy rate result of the corresponding JRJS scheme. Then, the candidate which is capable of achieving the highest secrecy rate is selected as the relay. Two assumptions about the channel state information (CSI), namely the full CSI (FCSI) and partial CSI (PCSI), are considered. Simulation results show that the proposed JRJS scheme outperforms the conventional pure relay selection, pure jamming and GSVD based beamforming schemes in terms of secrecy rate. Additionally, the proposed FCSI based power allocation (FCSI-PA) and PCSI based power allocation (PCSI-PA) schemes both achieve higher secrecy rates than the equal power allocation (EPA) scheme.
- In this paper, we characterize all the irreducible Darboux polynomials and polynomial first integrals of FitzHugh-Nagumo (F-N) system. The method of the weight homogeneous polynomials and the characteristic curves is widely used to give a complete classification of Darboux polynomials of a system. However, this method does not work for F-N system. Here by considering the Darboux polynomials of an assistant system associated to F-N system, we classified the invariant algebraic surfaces of F-N system. Our results show that there is no invariant algebraic surface of F-N system in the biological parameters region.
- Jan 02 2017 math.NT arXiv:1612.09362v1In the present paper, we investigate the case of more general imaginary cyclic quartic field $F=\mathbb{Q}\Big(\sqrt{-(D+B\sqrt{D})}\Big)$, in particular, with class number one. When $S$ is a set of primes of $F,$ containing the infinite ones, according to a theorem of Bass and Tate the tame kernel is contained in the natural image of $U_S\otimes U_S$ in $K_2F$ ($U_S$ is the $S$-unit group of $F$) if $S$ contains all finite primes of $F$ up to some bound. A theoretical bound for any number field has been obtained by R.Groenewegen, but unfortunately the bound is usually very large. So the problem is reduced to that of decreasing the theoretical bound to a manageable bound and the main difficulty is the large-scale data emerged in the process of computation. To overcome this difficulty, in this paper, we have done: (1) the PARI's functions are invoked in C++ codes; (2) the parallel programming approach is used in C++ codes; (3) in the design of algorithms and codes, the object-oriented viewpoint is used, so an extensible program is obtained. As a result, modifying the theoretical bound in the imaginary quartic field case, we prove that $K_2\mathcal{O}_F$ is trivial when $F=\mathbb{Q}\Big(\sqrt{-(D+B\sqrt{D})}\Big)$ with $B=1,D=2$ or $B=2, D=13.$
- Owing to their low-complexity iterations, Frank-Wolfe (FW) solvers are well suited for various large-scale learning tasks. When block-separable constraints are present, randomized block FW (RB-FW) has been shown to further reduce complexity by updating only a fraction of coordinate blocks per iteration. To circumvent the limitations of existing methods, the present work develops step sizes for RB-FW that enable a flexible selection of the number of blocks to update per iteration while ensuring convergence and feasibility of the iterates. To this end, convergence rates of RB-FW are established through computational bounds on a primal sub-optimality measure and on the duality gap. The novel bounds extend the existing convergence analysis, which only applies to a step-size sequence that does not generally lead to feasible iterates. Furthermore, two classes of step-size sequences that guarantee feasibility of the iterates are also proposed to enhance flexibility in choosing decay rates. The novel convergence results are markedly broadened to encompass also nonconvex objectives, and further assert that RB-FW with exact line-search reaches a stationary point at rate $\mathcal{O}(1/\sqrt{t})$. Performance of RB-FW with different step sizes and number of blocks is demonstrated in two applications, namely charging of electrical vehicles and structural support vector machines. Extensive simulated tests demonstrate the performance improvement of RB-FW relative to existing randomized single-block FW methods.
- We study the asymptotic of the global fluctuations for the difference between two adjacent levels in the $\beta$-Jacobi corners process (multilevel and general $\beta$ extension of the classical Jacobi ensemble of random matrices). The limit is identified with the derivative of the $2d$ Gaussian Free Field. Our main tool is integral forms for the (Macdonald-type) difference operators originating from the shuffle algebra.
- By a result of Biswas and Dos Santos, on a smooth and projective variety over an algebraically closed field, a vector bundle trivialized by a proper and surjective map is essentially finite, that is it corresponds to a representation of the Nori fundamental group scheme. In this paper we obtain similar results for non-proper non-smooth algebraic stacks over arbitrary fields of characteristic $p>0$. As by-product we have the following partial generalization of the Biswas-Dos Santos' result in positive characteristic: on a pseudo-proper and inflexible stack of finite type over $k$ a vector bundle which is trivialized by a proper and flat map is essentially finite.
- Dec 01 2016 math.CA arXiv:1611.10275v2The Strichartz estimates for Schrödinger equations can be improved when the data is spread out in either physical or frequency space. In this paper we give refinements of the 2-dimensional homogeneous Strichartz estimate on the maximum size of a single wave packet. Different approaches are used in the proofs, including arithmetic approaches, polynomial partitioning, and the $l^2$ Decoupling Theorem, for different cases. We also give examples to show that the refinements we obtain cannot be further improved when $2 \leq p \leq 4$ and $p = 6$.
- Dec 01 2016 math.OC arXiv:1611.10296v1In Part I of this paper we formulate an optimal scheduling problem for battery swapping that assigns to each electric vehicle (EV) a best station to swap its depleted battery based on its current location and state of charge. The schedule aims to minimize a weighted sum of total travel distance and generation cost over both station assignments and power flow variables, subject to EV range constraints, grid operational constraints and AC power flow equations. We propose there a centralized solution based on the second-order cone programming (SOCP) relaxation of optimal power flow (OPF) and generalized Benders decomposition that is suitable when global information is available. In this paper we propose two distributed solutions based on the alternating direction method of multipliers (ADMM) and dual decomposition respectively that are suitable for cases where the distribution grid, battery stations and EVs are managed by separate entities. Our algorithms allow these entities to make individual decisions but coordinate through privacy-preserving information exchanges to jointly solve an approximate version of the joint battery swapping scheduling and OPF problem. We evaluate our algorithms through simulations.
- The purpose of this article is to investigate the geometry of the set of locally diagonalizable bipartite quantum states. Instead of characterizing which bipartite quantum states may be locally diagonalizable except the two-qubit situation, we calculate the Hilbert-Schmidt volume of all locally diagonalizable bipartite quantum states. Besides, we partition the set of all locally diagonalizable states as local unitary orbits (or co-adjoint orbits) of diagonal forms. It is well-known that the Riemannian volume of a co-adjoint orbit for a regular point in a specified Weyl chamber can be calculated exactly by Harish-Chandra's volume formula. By modifying Harish-Chandra's volume formula, we firstly give a specific formula for the Riemannian volume of a co-adjoint local unitary orbit of a regular point in a specified Weyl chamber. Several open questions are presented as well.
- We show that any Kahler extension of a finitely generated abelian group by a surface group of genus g at least 2 is virtually a product. Conversely, we prove that any homomorphism of an even rank, finitely generated abelian group into the genus g mapping class group with finite image gives rise to a Kahler extension. The main tools come from surface topology and known restrictions on Kahler groups.
- Nov 29 2016 math.AP arXiv:1611.09133v2Let $0<\alpha,\beta<2$ be any real number. In this paper, we investigate the following semilinear system involving the fractional Laplacian \beginequation* \left{\beginarraylll (-\lap)^\alpha/2 u(x)=f(v(x)), & (-\lap)^\beta/2 v(x)=g(u(x)), & \qquad x∈\mathbbR^n_+, u,v\geq0, & \qquad x∈\mathbbR^n∖\mathbbR^n_+. \endarray\right. \endequation* Applying a direct method of moving planes for the fractional Laplacian, without any decay assumption on the solutions at infinity, we prove Liouville theorems of nonnegative solutions under some natural conditions on $f$ and $g$.
- We study the coexistence problem in a two-tier heterogeneous network (HetNet) with cognitive small cells. In particular, we consider an underlay HetNet, where the cognitive small base station (C-SBS) is allowed to use the frequency bands of the macro cell with an access probability (AP) as long as the C-SBS satisfies a preset interference probability (IP) constraint at macro users (MUs). To enhance the AP (or transmission opportunity) of the C-SBS, we propose a learning-based algorithm for the C-SBS and exploit the distance information between the macro base station (MBS) and MUs. Generally, the signal from the MBS to a specific MU contains the distance information between the MBS to the MU. We enable the C-SBS to analyze the MBS signal on a target frequency band, and learn the distance information between the MBS and the corresponding MU. With the learnt distance information, we calculate the upper bound of the probability that the C-SBS may interfere with the MU, and design an AP with a closed-form expression under the IP constraint. Numerical results indicate that the proposed algorithm outperforms the existing methods up to $60\%$ AP (or transmission opportunity).
- Nov 28 2016 math.AP arXiv:1611.08124v1We show the existence of Hölder continuous periodic solution with compact support in time of the Boussinesq equations with partial viscosity. The Hölder regularity of the solution we constructed is anisotropic which is compatible with partial viscosity of the equations.
- This paper develops a novel linear-time algorithm, termed \emphSPARse Truncated Amplitude flow (SPARTA), to reconstruct a sparse signal from a small number of magnitude-only measurements. It deals with what is also known as sparse phase retrieval (PR), which is \emphNP-hard in general and emerges in many science and engineering applications. Upon formulating sparse PR as an amplitude-based nonconvex optimization task, SPARTA works iteratively in two stages: In stage one, the support of the underlying sparse signal is recovered using an analytically well-justified rule, and subsequently a sparse orthogonality-promoting initialization is obtained via power iterations restricted on the support; and, in stage two, the initialization is successively refined by means of hard thresholding based truncated gradient iterations. SPARTA is a simple yet effective, scalable, and fast sparse PR solver. On the theoretical side, for any $n$-dimensional $k$-sparse ($k\ll n$) signal $\bm{x}$ with minimum (in modulus) nonzero entries on the order of $(1/\sqrt{k})\|\bm{x}\|_2$, SPARTA recovers the signal exactly (up to a global unimodular constant) from about $k^2\log n$ random Gaussian measurements with high probability. Furthermore, SPARTA incurs computational complexity on the order of $k^2n\log n$ with total runtime proportional to the time required to read the data, which improves upon the state-of-the-art by at least a factor of $k$. Furthermore, SPARTA is robust against additive noise of bounded support. Extensive numerical tests corroborate markedly improved recovery performance and speedups of SPARTA relative to existing alternatives.
- Nov 08 2016 math.PR arXiv:1611.01619v3The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes, especially stochastic integrals and differential equations. In this paper, general central limit theorems and functional central limit theorems are obtained for martingale like random variables under the sub-linear expectation. As applications, the Lindeberg central limit theorem and functional central limit theorem are obtained for independent but not necessarily identically distributed random variables, and a new proof of the Lévy characterization of a G-Brownian motion without using stochastic calculus is given. For proving the results, we have also established Rosenthal's inequality and the exceptional inequality for the martingale like random variables.
- Nov 04 2016 math.DG arXiv:1611.00920v1In this paper, we explore the similarity between normal homogeneity and $\delta$-homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected $\delta$-homogeneous Finsler space is $G$-$\delta$-homo-geneous, for some suitably chosen connected quasi-compact $G$. So $\delta$-homogeneous Finsler metrics can be defined by a bi-invariant singular metric on $G$ and submersion, just as normal homogeneous metrics, using a bi-invariant Finsler metric on $G$ instead. More careful analysis shows, in the space of all Finsler metrics on $G/H$, the subset of all $G$-$\delta$-homogeneous ones is in fact the closure for the subset of all $G$-normal ones, in the local $C^0$-topology (Theorem \refmain-thm-1). Using this approximation technique, the classification work for positively curved normal homogeneous Finsler spaces can be applied to classify positively curved $\delta$-homogeneous Finsler spaces, which provides the same classification list. As a by-product, this argument tells more about $\delta$-homogeneous Finsler metrics satisfying the (FP) condition (a weaker version of positively curved condition).
- Oct 13 2016 math.AG arXiv:1610.03637v3In this paper, we prove abundance for non-uniruled 3-folds with non-trivial Albanese maps, over an algebraically closed field of characteristic $p > 5$. As an application we get a characterization of abelian 3-folds.
- Oct 10 2016 math.PR arXiv:1610.02088v1A $d$-dimensional branching diffusion, $Z$, is investigated, where the linear attraction or repulsion between particles is competing with an Ornstein-Uhlenbeck drift, with parameter $b$ (we take $b>0$ for inward O-U and $b<0$ for outward O-U). This work has been motivated by [4], where a similar model was studied, but without the drift component. We show that the large time behavior of the system depends on the interaction and the drift in a nontrivial way. Our method provides, inter alia, the SLLN for the non-interactive branching (inward) O-U process. First, regardless of attraction ($\gamma >0$) or repulsion ($\gamma <0$), a.s., as time tends to infinity, the center of mass of $Z$ (i) converges to the origin, when $b>0$; (ii) escapes to infinity exponentially fast (rate $|b|$), when $b<0$. We then analyze $Z$ as viewed from the center of mass, and finally, for the system as a whole, we show a number of results/conjectures regarding the long term behavior of the system; some of these are scaling limits, while some others concern local extinction.
- General isometries of cyclic codes, including multipliers and translations, are introduced; and isometrically self-dual cyclic codes are defined. In terms of Type-I duadic splittings given by multipliers and translations, a necessary and sufficient condition for the existence of isometrically self-dual cyclic codes is obtained. A program to construct isometrically self-dual cyclic codes is provided, and illustrated by several examples. In particular, a class of isometrically self-dual MDS cyclic codes, which are alternant codes from a class of generalized Reed-Solomon codes, is presented.
- Oct 04 2016 math.AP arXiv:1610.00122v2Let $0<\alpha<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \beginequation(-\bigtriangleup)^\alpha/2 u(x)=f(u),\endequation on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Applying a direct method of moving planes for the fractional Laplacian, we prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ under mild conditions on $f$.
- Oct 03 2016 math.AG arXiv:1609.09592v2Let $f:X\to Y$ be a fibration from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p >5$. We prove that if the generic fiber $X_{\eta}$ has big canonical divisor $K_{X_{\eta}}$, then $$\kappa(X)\ge\kappa(Y) + \kappa(X_\eta).$$
- Sep 28 2016 math.NA arXiv:1609.08299v2In this papers, we couple the parareal algorithm with projection methods of the trajectory on a specific manifold, defined by the preservation of some conserved quantities of the differential equations. First, projection methods are introduced as the coarse and fine propagators. Second, we also apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm. Finally, three numerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration, and preservation in conserved quantities of model systems.
- In underlay heterogeneous networks (HetNets), the distance between a macro base station (MBS) and a macro user (MU) is crucial for a small-cell based station (SBS) to control the interference to the MU and achieve the coexistence. To obtain the distance between the MBS and the MU, the SBS needs a backhaul link from the macro system, such that the macro system is able to transmit the information of the distance to the SBS through the backhaul link. However, there may not exist any backhaul link from the macro system to the SBS in practical situations. Thus, it is challenging for the SBS to obtain the distance. To deal with this issue, we propose a median based (MB) estimator for the SBS to obtain the distance between the MBS and the MU without any backhaul link. Numerical results show that the estimation error of the MB estimator can be as small as $4\%$.
- Sep 12 2016 math.AP arXiv:1609.02772v1For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses $(\sigma_1,\sigma_2)$ at each blowup point has the expression $$\sigma_i=2(N_i1\mu_1+N_i2\mu_2+N_i3),$$ where $N_{ij}\in\mathbb{Z},~i=1,2,~j=1,2,3.$ (ii) Suppose at each vortex point $p_t$, $(\alpha_1^t,\alpha_2^t)$ are integers and $\rho_i\notin 4\pi\mathbb{N}$, then all the solutions of Toda systems are uniformly bounded. (iii) If the blow up point $q$ is not a vortex point, then $$u^k(x)+2\log|x-x^k|≤C,$$ where $x^k$ is the local maximum point of $u^k$ near $q$. (iv) If the blow up point $q$ is a vortex point $p_t$ and $\alpha_t^1,\alpha_t^2$ and $1$ are linearly independent over $Q$, then $$u^k(x)+2\log|x-p_t|≤C.$$ The Harnack type inequalities of (iii) or (iv) is important for studying the bubbling behaves near each blow up point.
- Sep 07 2016 math.NT arXiv:1609.01386v1We prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate Eisenstein Series on GL(n).
- The goal of this paper is to calculate exactly the average of uncertainty-product of two bounded observables and to establish its typicality over the whole set of finite dimensional quantum pure states. Firstly, we investigate the average uncertainty of an observable over isospectral density matrices with a fixed spectrum. By letting the isospectral density matrices be of rank-one, we get the average uncertainty of an observable restricted to pure quantum states. Physically, the ensemble of a large number of particles as a closed system is represented by mixed state $\rho$. When we measure observable $A$ at a mixed state $\rho$ with many repetitions, we suggest that each time we measure $A$ at a point within the isospectral density matrices, i.e. the unitary orbit $\cU_\rho$ of $\rho$. Thus it is suitable for taking average of uncertainty of observable $A$ over the whole unitary orbit $\cU_\rho$. Based on this result, we finally get the calculation of the average of uncertainty-product over the whole set of mixed quantum states. These results can help us check how large the gap is between the uncertainty-product and any obtained lower bounds about the uncertainty-product. Although our method in the present paper cannot give a tighter lower bound of uncertainty-product for bounded observables, it can help us drop any one that is not tighter than the known one substantially.
- This paper focuses on convex constrained optimization problems, where the solution is subject to a convex inequality constraint. In particular, we aim at challenging problems for which both projection into the constrained domain and a linear optimization under the inequality constraint are time-consuming, which render both projected gradient methods and conditional gradient methods (a.k.a. the Frank-Wolfe algorithm) expensive. In this paper, we develop projection reduced optimization algorithms for both smooth and non-smooth optimization with improved convergence rates under a certain regularity condition of the constraint function. We first present a general theory of optimization with only one projection. Its application to smooth optimization with only one projection yields $O(1/\epsilon)$ iteration complexity, which improves over the $O(1/\epsilon^2)$ iteration complexity established before for non-smooth optimization and can be further reduced under strong convexity. Then we introduce a local error bound condition and develop faster algorithms for non-strongly convex optimization at the price of a logarithmic number of projections. In particular, we achieve an iteration complexity of $\widetilde O(1/\epsilon^{2(1-\theta)})$ for non-smooth optimization and $\widetilde O(1/\epsilon^{1-\theta})$ for smooth optimization, where $\theta\in(0,1]$ appearing the local error bound condition characterizes the functional local growth rate around the optimal solutions. Novel applications in solving the constrained $\ell_1$ minimization problem and a positive semi-definite constrained distance metric learning problem demonstrate that the proposed algorithms achieve significant speed-up compared with previous algorithms.
- In this paper, we consider the Wright's generalized Bessel kernel $K^{(\alpha,\theta)}(x,y)$ defined by $$\theta x^\alpha\int_0^1J_\frac\alpha+1\theta,\frac1\theta(ux)J_\alpha+1,\theta((uy)^\theta)u^\alpha\,\mathrmd u, \qquad \alpha>-1, \qquad \theta>0,$$ where $$J_a,b(x)=\sum_j=0^∞\frac(-x)^jj!\Gamma(a+bj),\qquad a∈\mathbbC,\qquad b>-1,$$ is Wright's generalization of the Bessel function. This non-symmetric kernel, which generalizes the classical Bessel kernel (corresponding to $\theta=1$) in random matrix theory, is the hard edge scaling limit of the correlation kernel for certain Muttalib-Borodin ensembles. We show that, if $\theta$ is rational, i.e., $\theta=\frac{m}{n}$ with $m,n\in\mathbb{N}$, $gcd(m,n)=1$, and $\alpha > m-1-\frac{m}{n}$, the Wright's generalized Bessel kernel is integrable in the sense of Its-Izergin-Korepin-Slavnov. We then come to the Fredholm determinant of this kernel over the union of several scaled intervals, which can also be interpreted as the gap probability (the probability of finding no particles) on these intervals. The integrable structure allows us to obtain a system of coupled partial differential equations associated with the corresponding Fredholm determinant as well as a Hamiltonian interpretation. As a consequence, we are able to represent the gap probability over a single interval $(0,s)$ in terms of a solution of a system of nonlinear ordinary differential equations.
- Aug 03 2016 math.PR arXiv:1608.00710v1Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have raised progressive interest recently. The purpose of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng (2008b). We introduce a concept of extended negative dependence which is an extension of this kind of weak independence and the extended negative independence relative to classical probability appeared in recent literatures. Powerful tools as the moment inequality and Kolmogorov's exponential inequality are established for this kind of extended negatively independent random variables, which improve those of Chen, Chen and Ng(2010) a lot. And the strong law of large numbers and the law of iterated logarithm are obtained by applying these inequalities.
- Aug 02 2016 math.NA arXiv:1608.00419v1We propose a new method for computing the $\varphi$-functions of large sparse matrices with low rank or fast decaying singular values. The key is to reduce the computation of $\varphi_{\ell}$-functions of a large matrix to $\varphi_{\ell+1}$-functions of some $r$-by-$r$ matrices, where $r$ is the numerical rank of the large matrix in question. Some error analysis on the new method is given. Furthermore, we propose two novel strategies for estimating 2-norm condition numbers of the $\varphi$-functions. Numerical experiments illustrate the numerical behavior of the new algorithms and show the effectiveness of our theoretical results.
- Jul 27 2016 math.NA arXiv:1607.07433v1Noise in initial conditions from measurement errors can create unwanted oscillations which propagate in numerical solutions. We present a technique of prohibiting such oscillation errors when solving initial-boundary-value problems of semilinear diffusion equations. Symmetric Strang splitting is applied to the equation for solving the linear diffusion and nonlinear remainder separately. An oscillation-free scheme is developed for overcoming any oscillatory behavior when numerically solving the linear diffusion portion. To demonstrate the ills of stable oscillations, we compare our method using a weighted implicit Euler scheme to the Crank-Nicolson method. The oscillation-free feature and stability of our method are analyzed through a local linearization. The accuracy of our oscillation-free method is proved and its usefulness is further verified through solving a Fisher-type equation where oscillation-free solutions are successfully produced in spite of random errors in the initial conditions.
- Dirac delta function of matrix argument is employed frequently in the development of diverse fields such as Random Matrix Theory, Quantum Information Theory, etc. The purpose of the article is pedagogical, it begins by recalling detailed knowledge about Heaviside unit step function and Dirac delta function. Then its extensions of Dirac delta function to vector spaces and matrix spaces are discussed systematically, respectively. The detailed and elementary proofs of these results are provided. Though we have not seen these results formulated in the literature, there certainly are predecessors. Applications are also mentioned.
- Wishart ensemble is a useful and important random matrix model used in diverse fields. By realizing induced random mixed quantum states as Wishart ensemble with the fixed-trace one, using matrix integral technique we give a fast track to the average coherence for random mixed quantum states induced via partial-tracing of the Haar-distributed bipartite pure states. As a direct consequence of this result, we get a compact formula of the average subentropy of random mixed states. These obtained compact formulae extend our previous work.
- The decentralized caching is studied in two-layer networks, where users request contents through intermediate nodes (helpers) from a file server. By placing contents randomly and independently in each node and carefully designing the data delivery, the correlations of the pre-stored contents across layers can be utilized to reduce the transmission rates in the network. A hybrid caching scheme is developed by exploiting the cross-layer storage correlations, the single-layer multicast opportunities from the server (each helper) to helpers (the attached users), and the cross-layer multicast opportunities from the server to users. It is observed that, by the hybrid caching scheme, the achievable rate in the first layer is reduced without compromising the achievable rate in the second layer compared with the state of art. Furthermore, the achievable rate region is shown to be order-optimal and lies within constant margins to the information theoretic optimum. In particular, the multiplicative and additive factors are carefully sharpened to be $\frac{1}{48}$ and $4$, respectively.
- Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of gamblets introduced in \citeOwhadiMultigrid:2015 enabling the resolution of these implicit systems in near-linear complexity and provide rigorous a-priori error bounds on the resulting numerical approximations of hyperbolic and parabolic PDEs. These generalized gamblets induce a multiresolution decomposition of the solution space that is adapted to both the underlying (hyperbolic and parabolic) PDE (and the system of ODEs resulting from space discretization) and to the time-steps of the numerical scheme.
- In cognitive radio networks, the channel gain between primary transceivers, namely, primary channel gain, is crucial for a cognitive transmitter (CT) to control the transmit power and achieve spectrum sharing. Conventionally, the primary channel gain is estimated in the primary system and thus unavailable at the CT. To deal with this issue, two estimators are proposed by enabling the CT to sense primary signals. In particular, by adopting the maximum likelihood (ML) criterion to analyze the received primary signals, a ML estimator is first developed. After demonstrating the high computational complexity of the ML estimator, a median based (MB) estimator with proved low complexity is then proposed. Furthermore, the estimation accuracy of the MB estimation is theoretically characterized. By comparing the ML estimator and the MB estimator from the aspects of the computational complexity as well as the estimation accuracy, both advantages and disadvantages of two estimators are revealed. Numerical results show that the estimation errors of the ML estimator and the MB estimator can be as small as $0.6$ dB and $0.7$ dB, respectively.
- Jun 15 2016 math.FA arXiv:1606.04377v1A generalized Krein-Rutman theorem for a strongly positive bounded linear operator whose spectral radius is larger than essential spectral radius is established: the spectral radius of the operator is an algebraically simple eigenvalue with strongly positive eigenvector and other eigenvalues are less than the spectral radius.