Vector Problem Research Articles

Completeness of coherent state subsystems for nilpotent Lie groups

Let $G$ be a nilpotent Lie group and let $\pi$ be a coherent state representation of $G$. The interplay between the cyclicity of the restriction $\pi|_{\Gamma}$ to a lattice $\Gamma \leq G$ and the completeness of subsystems of coherent states based on a homogeneous $G$-space is considered. In particular, it is shown that necessary density conditions for Perelomov's completeness problem for the highest weight vector can be obtained via criteria for the cyclicity of $\pi|_{\Gamma}$.

A Perturbation Approach to Vector Optimization Problems: Lagrange and Fenchel–Lagrange Duality

In this paper we study the general minimization vector problem (P), concerning a perturbation mapping, defined in locally convex Hausdorff topological vector spaces where the "WInf" stands for the weak infimum with respect to an ordering generated by a convex cone $K$. Several representations of the epigraph of the conjugate mapping of the perturbation mapping are established. From these, variants vector Farkas lemmas are then proved. Armed with these basic tools, the {\it dual} and the so-called {\it loose dual problem} of (P) are defined, and then stable strong duality results between these pairs of primal-dual problems are established. The results just obtained are then applied to a general class (CCCV) of composed vector optimization problems with cone-constrained. For this classes of problems, four perturbation mappings are suggested. Each of these mappings yields several forms of vector Farkas lemmas and two forms of dual problems for (CCVP). Concretely, one of the suggested perturbation mapping give rises to well-known {\it Lagrange} and {\it loose Lagrange dual problems} for (CCVP) while each of the three others, yields two kinds of Fenchel-Lagrange dual problems for (CCVP). Stable strong duality for these pairs of primal-dual problems are proved. Several special cases of (CCVP) are also considered at the end of the paper, including: vector composite problems (without constraints), cone-constrained vector problems, and scalar composed problems. The results obtained in this papers when specified to the two concrete mentioned vector problems go some Lagrange duality results appeared recently, and also lead to new results on stable strong Fenchel-Lagrange duality results, which, to the best knowledge of the authors, appear for the first time in the literature.

Lamb waves in discrete homogeneous and heterogeneous systems: Dispersion properties, asymptotics and non-symmetric wave propagation

In this paper, we study Lamb waves propagating in a discrete strip, whose microstructure is represented by either a monatomic or a diatomic triangular lattice. In considering the in-plane vector problem, we derive an analytical solution for the dispersion relation of Lamb waves. Additionally, we investigate the main features of the eigenmodes of the system, which describe how the lattice strip vibrates at different frequencies. Further, we discuss how the dispersion properties depend on the number of the lattice’s rows and on the chosen boundary conditions. For heterogeneous systems, we focus the attention on the internal stop-band and on the flat bands appearing in the dispersion diagram. Different asymptotic models are employed to approximate the low-frequency behaviour of the lattice strip, starting from the classical Euler–Bernoulli beam. The effective behaviour of a lattice strip with dense microstructure is also investigated, and we present a comparative numerical analysis with the analogous continuum for which the classical Lamb wave problem is posed. The theory developed is exploited here to design a structured medium capable of manipulating wavemodes, and, through conversion and selection, generating uni-directional wave phenomena. We envisage that the present work can fill a gap in the research field related to the analytical study of dispersive waves in microstructured media, whose dynamic performance is influenced by the presence of multiple external boundaries.

Isomeric trees and the order of Runge–Kutta methods

The conditions for a Runge–Kutta method to be of order p with p≥5 for a scalar non-autonomous problem are a proper subset of the order conditions for a vector problem. Nevertheless, Runge–Kutta methods that were derived historically only for scalar problems happened to be of the same order for vector problems. We relate the order conditions for scalar problems to factorisations of the Runge–Kutta trees into “atomic stumps” and enumerate those conditions up to p=20. Using a special search procedure over unsatisfied order conditions, new Runge–Kutta methods of “ambiguous orders” five and six are constructed. These are used to verify the validity of the results.

MMV-Based Sequential AoA and AoD Estimation for Millimeter Wave MIMO Channels

The fact that the millimeter-wave (mmWave) multiple-input multiple-output (MIMO) channel has sparse support in the spatial domain has motivated recent compressed sensing (CS)-based mmWave channel estimation methods, where the angles of arrivals (AoAs) and angles of departures (AoDs) are quantized using angle dictionary matrices. However, the existing CS-based methods usually obtain the estimation result through one-stage channel sounding that have two limitations: (i) the requirement of large-dimensional dictionary and (ii) unresolvable quantization error. These two drawbacks are irreconcilable; improvement of the one implies deterioration of the other. To address these challenges, we propose, in this paper, a two-stage method to estimate the AoAs and AoDs of mmWave channels. In the proposed method, the channel estimation task is divided into two stages, Stage I and Stage II. Specifically, in Stage I, the AoAs are estimated by solving a multiple measurement vectors (MMV) problem. In Stage II, based on the estimated AoAs, the receive sounders are designed to estimate AoDs. The dimension of the angle dictionary in each stage can be reduced, which in turn reduces the computational complexity substantially. We then analyze the successful recovery probability (SRP) of the proposed method, revealing the superiority of the proposed framework over the existing one-stage CS-based methods. We further enhance the reconstruction performance by performing resource allocation between the two stages. We also overcome the unresolvable quantization error issue present in the prior techniques by applying the atomic norm minimization method to each stage of the proposed two-stage approach. The simulation results illustrate the substantially improved performance with low complexity of the proposed two-stage method.

Quantum mean-value approximator for hard integer-value problems

Evaluating the expectation of a quantum circuit is a classically difficult problem known as the quantum mean value problem (QMV). It is used to optimize the quantum approximate optimization algorithm and other variational quantum eigensolvers. We show that such an optimization can be improved substantially by using an approximation rather than the exact expectation. Together with efficient classical sampling algorithms, a quantum algorithm with minimal gate count can thus improve the efficiency of general integer-value problems, such as the shortest vector problem (SVP) investigated in this work.

Covering Convex Bodies and the Closest Vector Problem

We present algorithms for the (1+epsilon )-approximate version of the closest vector problem for certain norms. The currently fastest algorithm (Dadush and Kun 2016) for general norms in dimension n has running time of 2^{O(n)}(1/epsilon )^n. We improve this substantially in the following two cases. First, for ell _p-norms with p>2 (resp. p in [1,2]) fixed, we present an algorithm with a running time of 2^{O(n)}(1+1/epsilon )^{n/2} (resp. 2^{O(n)} (1+1/epsilon )^{n/p}). This result is based on a geometric covering problem, that was introduced in the context of CVP by Eisenbrand et al.: How many convex bodies are needed to cover the ball of the norm such that, if scaled by factor 2 around their centroids, each one is contained in the (1+epsilon )-scaled homothet of the norm ball? We provide upper bounds for this (2,epsilon )-covering number by exploiting the modulus of smoothness of the ell _p-balls. Applying a covering scheme, we can boost any 2-approximation algorithm for CVP to a (1+epsilon )-approximation algorithm with the improved run time, either using a straightforward sampling routine or using the deterministic algorithm of Dadush for the construction of an epsilon net. Second, we consider polyhedral and zonotopal norms. For centrally symmetric polytopes (resp. zonotopes) in {mathbb R}^n with O(n) facets (resp. generated by O(n) line segments), we provide a deterministic O(log _2(2+1/epsilon ))^{O(n)} time algorithm. This generalizes the result of Eisenbrand et al. which applies to the ell _infty -norm. Finally, we establish a connection between the modulus of smoothness and lattice sparsification. As a consequence, using the enumeration and sparsification tools developped by Dadush, Kun, Peikert, and Vempala, we present a simple alternative to the boosting procedure with the same time and space requirement for ell _p norms. This connection might be of independent interest.

Towards an efficient LWE‐based fully homomorphic encryption scheme

The security of most early fully homomorphic encryption schemes was based on the hardness of the Learning with Errors (LWE) problem. These schemes were inefficient in terms of per gate computations and public-key size. More efficient schemes were later developed based on the hardness of the Ring-LWE (RLWE) problem. While the hardness of the LWE problem is based on the hardness of the approximate shortest vector problem (GapSVPγ) over regular lattices, the hardness of the RLWE problem is based on the hardness of the approximate shortest vector problem over ideal lattices. As of now, it has not been proved that the (GapSVPγ) problem over ideal lattices is as difficult as the corresponding problem over regular lattices. In this work, the authors propose a multi-bit levelled fully homomorphic encryption scheme using multivariate polynomial evaluations whose security depends on the hardness of the LWE problem. In terms of per gate computation cost, this scheme is more efficient than existing LWE-based schemes. Further, for an appropriate choice of parameters, the per computation cost for homomorphic multiplication can be made asymptotically comparable to RLWE-based schemes in a parallel computing environment. For homomorphic multiplication, the scheme uses a polynomial-based technique that does not require relinearization (and key switching).

Some Regularity Properties on Bolza problems in the Calculus of Variations

The paper summarizes the main core of the last results that we obtained in [8, 4, 17] on the regularity of the value function for a Bolza problem of a one-dimensional, vectorial problem of the calculus of variations. We are concerned with a nonautonomous Lagrangian that is possibly highly discontinuous in the state and velocity variables, nonconvex in the velocity variable and non coercive. The main results are achieved under the assumption that the Lagrangian is convex on the one-dimensional lines of the velocity variable and satisfies a local Lipschitz continuity condition w.r.t. the time variable, known in the literature as Property (S), and strictly related to the validity of the Erdmann–Du-Bois Reymond equation.

Chiral waves in structured elastic systems: dynamics of a meta-waveguide

Summary The notion of meta-surfaces is well established for waves interacting with arrays of periodic resonators. Here, an infinite elastic rod, attached to a system of gyroscopic spinners, is considered. This forms a waveguide with unusual dispersion properties and will be referred to as a chiral meta-waveguide. A class of transient and time-harmonic formulations describing waves in elastic chiral meta-waveguides is addressed. Consideration of a chiral elastic chain leads to the governing equations of the homogenised continuous system. Both analytical and numerical solutions are presented. Features observed for chiral waveforms show resemblance with formulations of quantum physics rather than linear elasticity. For an elementary example related to an elastic chiral rod, with the absence of pre-tension, it is noted that the longitudinal velocity satisfies the Klein–Gordon equation. For full vector problems, corresponding to an elastic chiral rod with pre-tension, time-harmonic Green’s matrix functions are derived for the case of large values of the gyricity parameter. This case is especially interesting because of the presence of both exponentially localised and oscillatory terms in the representation of Green’s matrix components.

New representations of epigraphs of conjugate mappings and Lagrange, Fenchel–Lagrange duality for vector optimization problems

In this paper, we concern the vector problem: where X, Y, Z are locally convex Hausdorff topological vector spaces, and are proper mappings, C is a nonempty convex subset of X, and S is a non-empty closed, convex cone in Z. Several new presentations of epigraphs of composite conjugate mappings associated to (VP) are established under variant qualifying conditions. The significance of these representations is twofold: Firstly, they play a key role in establishing new kinds of vector Farkas lemmas which serve as tools in the study of vector optimization problems; secondly, they pay the way to define Lagrange and two new kinds of Fenchel–Lagrange dual problems for (VP). Strong and stable strong duality results corresponding to these mentioned dual problems of (VP) are established using the new Farkas-type results just obtained. It is shown that in the special case where , the Lagrange and Fenchel–Lagrange dual problems for (VP), go back to Lagrange, and Fenchel–Lagrange dual problems for scalar problems, and the resulting duality results cover, and in some cases, extend the corresponding ones for scalar problems in the literature.

LaTLS: A Lattice‐Based TLS Proxy Protocol

The function of the Internet proxy is to check and convert the data exchanged between client and server. In fact, the two-party secure communication protocol with good security is turned into an unsafe multiparty protocol. At present, there are relatively few proxy protocols that can be applied in practice. This paper analyzes the classic agent protocol mcTLS and pointed out the security issues. We focus on the security of TLS 1.3 and proposed a lattice-based multi-party proxy protocol: LaTLS. LaTLS can be proved secure in the eCK model, it can resist key-sharing attacks, counterfeiting attacks, replay attacks, and achieve forward security. Compared with traditional DH and ECDH schemes, LaTLS is more effcient. Its security is based on the shortest vector problem, therefor it has anti-quantum attack properties.

Scalarization and robustness in uncertain vector optimization problems: a non componentwise approach

The robust optimization approach can be used to tackle uncertain vector problems by considering worst case scenarios. In this context, notions of robust efficient solutions which are coherent with a set-valued minimization process have been introduced in literature in order to avoid unduly pessimistic attitudes (see e.g. Ehrgott et al. in Eur. J. Oper. Res. 239(1), 17–31, 2014). We address the question whether scalarization and robustification can be commuted in a non componentwise framework. We prove that the commutation of the two approaches is ensured under appropriate assumptions. To this purpose, we identify a class of scalarization processes that ensure necessary and sufficient robust optimality conditions through the direct scalarization of the uncertain vector optimization problem, without explicitly passing through the set-valued formulation of the problem.

Some results on OMP algorithm for MMV problem

In this paper, the robustness of the orthogonal matching pursuit (OMP) algorithm for multiple measurement vector (MMV) problem under the restricted isometry property (RIP) is presented. Moreover, it is shown that the OMP algorithm can exactly recover K‐row sparse matrices in K iterations under the RIP condition we have presented. Furthermore, the condition is sharp.

Regularity for local minima of a special class of vectorial problems with fully anisotropic growth

Regularity for local minima of a special class of vectorial problems with fully anisotropic growth

A Heuristic Method to Approximate the Closest Vector Problem

The closest vector problem, or CVP for short, is a fundamental lattice problem. The purpose of this challenge is to identify a lattice point in its ambient space that is closest to a given point. This is a provably hard problem to solve, as it is an NP-hard problem. It is considered to be more difficult than the shortest vector problem (SVP), in which the shortest nonzero lattice point is required. There are three types of algorithms that can be used to solve CVP: Enumeration algorithms, Voronoi cell computation and seiving algorithms. Many algorithms for solving the relaxed variant, APPROX-CVP, have been proposed: The Babai nearest algorithm or the embedding technique. In this work we will give a heuristic method to approximate the closest vector problem to a given vector using the embeding technique and the reduced centered law.

Topological entropy of composition and impulsive differential equations satisfying a uniqueness condition

The impulsive differential equations are examined via the associated Poincaré translation operators in terms of topological entropy. The crucial role is played by the entropy analysis of the compositions of Poincaré’s operators with the impulsive maps. For the scalar (one-dimensional) problems, the lower entropy estimations can be effectively obtained by means of horseshoes. For the vector (higher-dimensional) problems, the situation becomes more delicate and requires rather sophisticated techniques. Five main theorems are presented about a positive topological entropy (i.e. topological chaos) for given impulsive problems. For vector linear homogeneous differential equations with constant coefficients and isometric impulses, the zero entropy is deduced under commutativity restrictions imposed on the components of a mentioned composition. Several illustrative examples and numerical simulations are supplied.

Egr1/HSP70 Promoter-Driven Activation of Gene Expression for Synergistic Anti-Hepatoma Using PEI-MZF Nanoparticles and Radiation.

IntroductionSpatially restricted gene expression circumvents the gene expression and gene vector problem by enabling localized amplification. The objective of this study is to construct a spatially restricted gene expression for liver cancer therapy based upon the MFH-absorbing properties of PEI- Mn0.5zn0.5Fe2o4, gene therapy and radiation.MethodsMn0.5zn0.5Fe2O4 (MZF) magnetic nanoparticles were prepared by an improved chemical co-precipitation method, modified by polyethylene imine (PEI), and then the structure, modification characters, biocompatibility, temperature rise and control ability and binding efficiency of the plasmid were characterized. Then, the dual-promoter plasmid PCDNA3.1-EGR1-HSP70-HSVTK was constructed. The recombinant vectors were identified by enzyme digestion analysis and DNA sequencing. The TK gene expression level was detected by realtime-PCR assay in HEK293 cells. Also, the HSV-TK gene expression was detected in SMMC7721 cells with the help of PEI-Mn0.5Zn0.5Fe2O4. In vitro anti-tumor experiment, MTT assay and flow cytometry were used to evaluate the therapeutic effects of the cultured SMMC7721 cells treated by different ways. In vivo anti-tumor experiment, the xenografted mice were treated by different ways for three times to detect the antitumor effect.ResultsThe Mn0.5Zn0.5Fe2O4 magnetic nanoparticles could be successfully prepared through improved co-precipitation process and showed good biocompatibility. And PEI had been coated on MZF complex. The modified PEI-MZF presented favorable dispensability, responsibility to magnetism, good loading capability and transfect capability. Also, pCDNA3.1-Egr1-Hsp70-HSVTK plasmid had been constructed successfully and could be induced by heat and irradiation. It would be used for further target gene therapy research. The antitumor results in vitro showed: The therapeutic effects of nanosized PEI-MZF-HSV-TK complex could significantly inhibit the proliferation of cultured liver cancer cells (SMC7721), induce cell apoptosis and had a prominent cell cycle disturbance in the S phase in vitro. The results in vivo showed: The combined therapy induced by PEI-MZF-HSV-TK could inhabit the growth of hepatocellular carcinoma xenografts by killing and inhabiting the proliferation of the tumor cells.ConclusionThe novel site-directed heat/radiation-inducible expression system based upon the hyperthermia (by MFH) and radiation possessed superior antitumor effect in vivo and in vitro.

The Classical Continuous Mixed Optimal Control of Couple Nonlinear Parabolic Partial Differential Equations with State Constraints

In this work, the classical continuous mixed optimal control vector (CCMOPCV) problem of couple nonlinear partial differential equations of parabolic (CNLPPDEs) type with state constraints (STCO) is studied. The existence and uniqueness theorem (EXUNTh) of the state vector solution (SVES) of the CNLPPDEs for a given CCMCV is demonstrated via the method of Galerkin (MGA). The EXUNTh of the CCMOPCV ruled with the CNLPPDEs is proved. The Frechet derivative (FÉDE) is obtained. Finally, both the necessary and the sufficient theorem conditions for optimality (NOPC and SOPC) of the CCMOPCV with state constraints (STCOs) are proved through using the Kuhn-Tucker-Lagrange (KUTULA) multipliers theorem (KUTULATH).

Faster Provable Sieving Algorithms for the Shortest Vector Problem and the Closest Vector Problem on Lattices in ℓp Norm

In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in ℓp norm (1≤p≤∞). The running time we obtain is better than existing provable sieving algorithms. We give a new linear sieving procedure that works for all ℓp norm (1≤p≤∞). The main idea is to divide the space into hypercubes such that each vector can be mapped efficiently to a sub-region. We achieve a time complexity of 22.751n+o(n), which is much less than the 23.849n+o(n) complexity of the previous best algorithm. We also introduce a mixed sieving procedure, where a point is mapped to a hypercube within a ball and then a quadratic sieve is performed within each hypercube. This improves the running time, especially in the ℓ2 norm, where we achieve a time complexity of 22.25n+o(n), while the List Sieve Birthday algorithm has a running time of 22.465n+o(n). We adopt our sieving techniques to approximation algorithms for SVP and CVP in ℓp norm (1≤p≤∞) and show that our algorithm has a running time of 22.001n+o(n), while previous algorithms have a time complexity of 23.169n+o(n).