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Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.

Historical biogeography is increasingly studied from an explicitly statistical perspective, using stochastic models to describe the evolution of species range as a continuous-time Markov process of dispersal between and extinction within a set of discrete geographic areas. The main constraint of these methods is the computational limit on the number of areas that can be specified. We propose a Bayesian approach for inferring biogeographic history that extends the application of biogeographic models to the analysis of more realistic problems that involve a large number of areas. Our solution is based on a \"data-augmentation\" approach, in which we first populate the tree with a history of biogeographic events that is consistent with the observed species ranges at the tips of the tree. We then calculate the likelihood of a given history by adopting a mechanistic interpretation of the instantaneous-rate matrix, which specifies both the exponential waiting times between biogeographic events and the relative probabilities of each biogeographic change. We develop this approach in a Bayesian framework, marginalizing over all possible biogeographic histories using Markov chain Monte Carlo (MCMC). Besides dramatically increasing the number of areas that can be accommodated in a biogeographic analysis, our method allows the parameters of a given biogeographic model to be estimated and different biogeographic models to be objectively compared. Our approach is implemented in the program, BayArea.

In 1967, I. J. Maddox generalized the spaces [c.sub.0], c, [l.sub.[infinity]] by adding the powers [p.sub.k] (k [member of] N) in the definitions of the spaces to the terms of elements of sequences ([x.sub.k]). Gokhan and Colak in 2004-2006 defined the corresponding double sequence spaces for the Pringsheim and the bounded Pringsheim convergence. We will additionally define the corresponding double sequence spaces for the regular convergence. In 2009, Gokhan, Colak and Mursaleen characterized some classes of matrix transformations involving these double sequence spaces with powers. However, many of their results appeared to be wrong. In this paper, we give corresponding counterexamples and prove the correct results. Moreover, we present the conditions for a wider class of matrices.

Starting from a representation formula for 2x2 non-singular complex matrices in terms of 2nd kind Chebyshev polynomials, a link is observed between the 1st kind Chebyshev polinomials and traces of matrix powers. Then, the standard composition of matrix powers is used in order to derive composition identities of 2nd and 1st kind Chebyshev polynomials. Before concluding the paper, the possibility to extend this procedure to the multivariate Chebyshev and Lucas polynomials is touched on. Keywords: Chebyshev polynomials; matrix powers; composition identities MSC: 33C45; 15A16

We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of randomized iterative algorithms based on similar principles to address a variety of common tasks in numerical linear algebra. Prom the point of view of numerical linear algebra, the main novelty of the fast randomized iteration schemes described in this article is that they have dramatically reduced operations and storage cost per iteration (as low as constant under appropriate conditions) and are rather versatile: we will show how they apply to the solution of linear systems, eigenvalue problems, and matrix exponentiation, in dimensions far beyond the present limits of numerical linear algebra. While traditional iterative methods in numerical linear algebra were created in part to deal with instances where a matrix (of size O(n²)) is too big to store, the algorithms that we propose are effective even in instances where the solution vector itself (of size O(n)) may be too big to store or manipulate. In fact, our work is motivated by recent diffusion Monte Carlo based quantum Monte Carlo schemes that have been applied to matrices as large as 10¹⁰⁸ × 10¹⁰⁸. We provide basic convergence results, discuss the dependence of these results on the dimension of the system, and demonstrate dramatic cost savings on a range of test problems.

A complete theory of dimensional and scaling analysis is presented and its power is demonstrated through a series of examples. A vector-matrix exponentiation is introduced to simplify notation and calculus.

With the development of the Internet of Things, smart grids have become indispensable in our daily life and can provide people with reliable electricity generation, transmission, distribution and control. Therefore, how to design a privacy-preserving data aggregation protocol has been a research hot-spot in smart grid technology. However, these proposed protocols often contain some complex cryptographic operations, which are not suitable for resource-constrained smart meter devices. In this paper, we combine data aggregation and the outsourcing of computations to design two privacy-preserving outsourcing algorithms for the modular exponentiation operations involved in the multi-dimensional data aggregation, which can allow these smart meter devices to delegate complex computation tasks to nearby servers for computing. By utilizing our proposed outsourcing algorithms, the computational overhead of resource-constrained smart meter devices can be greatly reduced in the process of data encryption and aggregation. In addition, the proposed algorithms can protect the input’s privacy of smart meter devices and ensure that the smart meter devices can verify the correctness of results from the server with a very small computational cost. From three aspects, including security, verifiability and efficiency, we give a detailed analysis about our proposed algorithms. Finally, through carrying out some experiments, we prove that our algorithms can improve the efficiency of performing the data encryption and aggregation on the smart meter device side.

With the fast development of cloud computing, clients without enough computational power can widely outsource their heavy computations to cloud service providers. One of the most widely used and costly operations in cryptographic protocols is modular exponentiation, which can be computed at a lower cost by enjoying advantages of cloud computing, however, at the same time we need to address new challenges such as data privacy and verification of results. In this paper, first, we propose a secure outsourcing of single modular exponentiation protocol with verifiability one. Although the proposed single exponentiation scheme has the same verifiability as Ren’2018, but our scheme requires one less modular multiplication. However, the main contribution of this paper is proposing a scheme for outsourcing of multiplications of several modular exponentiations, hereafter called as composite exponentiation, which to the best of our knowledge, and is the first outsourcing scheme with full verification for composite exponentiation. As the evaluation results show, the advantages of this scheme, in comparison with state of the art schemes, are evident in terms of performance and verifiability criteria.

Terahertz time domain spectroscopy imaging systems suffer from the problems of long image acquisition time and massive data processing. Reducing the sampling rate will lead to the degradation of the imaging reconstruction quality. To solve this issue, a novel terahertz imaging model, named the dual sparsity constraints terahertz image reconstruction model (DSC-THz), is proposed in this paper. DSC-THz fuses the sparsity constraints of the terahertz image in wavelet and gradient domains into the terahertz image reconstruction model. Differing from the conventional wavelet transform, we introduce a non-linear exponentiation transform into the shift invariant wavelet coefficients, which can amplify the significant coefficients and suppress the small ones. Simultaneously, the sparsity of the terahertz image in gradient domain is used to enhance the sparsity of the image, which has the advantage of edge preserving property. The split Bregman iteration scheme is utilized to tackle the optimization problem. By using the idea of separation of variables, the optimization problem is decomposed into subproblems to solve. Compared with the conventional single sparsity constraint terahertz image reconstruction model, the experiments verified that the proposed approach can achieve higher terahertz image reconstruction quality at low sampling rates.

In this paper, we introduce spherical fuzzy matrices (SFMs) which is an advanced tool of the fuzzy matrices, intuitionistic fuzzy matrices and picture fuzzy matrices. We investigate the basic properties of SFMs and compare the idea SFMs with picture fuzzy matrices. Then some algebraic operations, such as max-min, min-max, complement, algebraic sum, algebraic product are defined and investigated their algebraic properties. Further, scalar multiplication (nA) and exponentiation ([A.sup.n]) operations of a SFM A using algebraic operations are constructed, and their desirable properties are studied. Finally, we define a new operation(@) on spherical fuzzy matrices and discuss distributive laws in the case where the operations of [[direct sum] or [symmetry].sub.s], [[cross product].sub.s], [[disjunction].sub.s] and [[disjunction].sub.s] are combined each other. Keywords: Intuitionistic fuzzy matrix, Pythagorean fuzzy matrix, Picture fuzzy matrix. Spherical fuzzy matrix, Algebraic sum, Algebraic product, Scalar multiplication, Exponentiation operations. AMS Subject Classification: 03E72, 08A72, 15B15.