Vector Problem Research Articles

Solution of the Vector Three-Dimensional Inverse Problem on an Inhomogeneous Dielectric Hemisphere Using a Two-Step Method

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for Maxwell’s equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography.

Design and Implementation of Hardware-Software Architecture Based on Hashes for SPHINCS+

Advances in quantum computing have posed a future threat to today’s cryptography. With the advent of these quantum computers, security could be compromised. Therefore, the National Institute of Standards and Technology (NIST) has issued a request for proposals to standardize algorithms for post-quantum cryptography (PQC), which is considered difficult to solve for both classical and quantum computers. Among the proposed technologies, the most popular choices are lattice-based (shortest vector problem) and hash-based approaches. Other important categories are public key cryptography (PKE) and digital signatures. Within the realm of digital signatures lies SPHINCS+. However, there are few implementations of this scheme in hardware architectures. In this article, we present a hardware-software architecture for the SPHINCS+ scheme. We utilized a free RISC-V (Reduced Instruction Set Computer) processor synthesized on a Field Programmable Gate Array (FPGA), primarily integrating two accelerator modules for Keccak-1600 and the Haraka hash function. Additionally, modifications were made to the processor to accommodate the execution of these added modules. Our implementation yielded a 15-fold increase in performance with the SHAKE-256 function and nearly 90-fold improvement when using Haraka, compared to the reference software. Moreover, it is more compact compared to related works. This implementation was realized on a Xilinx FPGA Arty S7: Spartan-7.

Parameterized Inapproximability of the Minimum Distance Problem over All Fields and the Shortest Vector Problem in All \({\ell_{{p}}}\) Norms

Parameterized Inapproximability of the Minimum Distance Problem over All Fields and the Shortest Vector Problem in All \({\ell_{{p}}}\) Norms

Discovering artificial viscosity models for discontinuous Galerkin approximation of conservation laws using physics-informed machine learning

Finite element-based high-order solvers of conservation laws offer large accuracy but face challenges near discontinuities due to the Gibbs phenomenon. Artificial viscosity is a popular and effective solution to this problem based on physical insight. In this work, we present a physics-informed machine learning algorithm to automate the discovery of artificial viscosity models. We refer to the proposed approach as an “hybrid” approach which stands at the edge between supervised and unsupervised learning. More precisely, the proposed “hybrid” paradigm is not supervised in the classical sense as it does not utilize labeled data in the traditional way but relies on the intrinsic properties of the reference solution. The algorithm is inspired by reinforcement learning and trains a neural network acting cell-by-cell (the viscosity model) by minimizing a loss defined as the difference with respect to a reference solution thanks to automatic differentiation. This enables a dataset-free training procedure. We prove that the algorithm is effective by integrating it into a state-of-the-art Runge-Kutta discontinuous Galerkin solver. We showcase several numerical tests on scalar and vectorial problems, such as Burgers' and Euler's equations in one and two dimensions. Results demonstrate that the proposed approach trains a model that is able to outperform classical viscosity models. Moreover, we show that the learnt artificial viscosity model is able to generalize across different problems and parameters.

An MLWE-Based Cut-and-Choose Oblivious Transfer Protocol.

The existing lattice-based cut-and-choose oblivious transfer protocol is constructed based on the learning-with-errors (LWE) problem, which generally has the problem of inefficiency. An efficient cut-and-choose oblivious transfer protocol is proposed based on the difficult module-learning-with-errors (MLWE) problem. Compression and decompression techniques are introduced in the LWE-based dual-mode encryption system to improve it to an MLWE-based dual-mode encryption framework, which is applied to the protocol as an intermediate scheme. Subsequently, the security and efficiency of the protocol are analysed, and the security of the protocol can be reduced to the shortest independent vector problem (SIVP) on the lattice, which is resistant to quantum attacks. Since the whole protocol relies on the polynomial ring of elements to perform operations, the efficiency of polynomial modulo multiplication can be improved by using fast Fourier transform (FFT). Finally, this paper compares the protocol with an LWE-based protocol in terms of computational and communication complexities. The analysis results show that the protocol reduces the computation and communication overheads by at least a factor of n while maintaining the optimal number of communication rounds under malicious adversary attacks.

SQL injection attack: Detection, prioritization & prevention

Web applications have become central in the digital landscape, providing users instant access to information and allowing businesses to expand their reach. Injection attacks, such as SQL injection (SQLi), are prominent attacks on web applications, given that most web applications integrate a database system. While there have been solutions proposed in the literature for SQLi attack detection using learning-based frameworks, the problem is often formulated as a binary, single-attack vector problem without considering the prioritization and prevention component of the attack. In this work, we propose a holistic solution, SQLR34P3R, that formulates the SQLi attack as a multi-class, multi-attack vector, prioritization, and prevention problem. For attack detection and classification, we gathered 457,233 samples of benign and malicious network traffic, as well as 70,023 samples that had SQLi and benign payloads. After evaluating several machine-learning-based algorithms, the hybrid CNN-LSTM models achieve an average F1-Score of 97% in web and network traffic filtering. Furthermore, by using CVEs of SQLi vulnerabilities, SQLR34P3R incorporates a novel risk analysis approach which reduces additional effort while maintaining reasonable coverage to assist businesses in allocating resources effectively by focusing on patching vulnerabilities with high exploitability. We also present an in-the-wild evaluation of the proposed solution by integrating SQLR34P3R into the pipeline of known vulnerable web applications such as Damn Vulnerable Web Application (DVWA) and Vulnado and via network traffic captured using Wireshark from SQLi DNS exfiltration conducted with SQLMap for real-time detection. Finally, we provide a comparative analysis with state-of-the-art SQLi attack detection and risk ratings solutions.

Smart grid stability prediction model using two-way attention based hybrid deep learning and MPSO

In smart grid management, precise stability prediction is a complicated task that adds to the effective allocation of resources with grid stability. Specifically, demand-side management is considered an essential element of the overall Smart Grids system. Hence, predicting future energy demands is crucial to regulating consumption by aligning utility offerings with consumer demand. This research presents a hybrid deep learning model (Convolutional Neural Network [CNN] with Bi-LSTM) with a two-way attention method and a multi-objective particle swarm optimization method (MPSO) for short-term load prediction from a smart grid. The proposed hybrid model utilizes a two-way attention method at its encoding and decoding stages, in which an encoding attention layer helps to recognize all the essential features from an input vector, and a decoding attention layer helps to resolve the fixed context vector problem by offering better memory capacity. A CNN and Bi-LSTM are used to capture the essential features from the dataset. We also utilize a t-Nearest Neighbours algorithm to pre-process the initial dataset. An MPSO method combines the features of CNN and Bi-LSTM methods, resulting in better prediction accuracy. As far as we know, it is the first work to suggest a dynamic short-term load prediction model that considers different significant features and enables precise predicting outcomes. The performance of the proposed model and existing well-known deep learning models such as Recurrent Neural Network, Gated Recurrent Unit, Long Short-Term Memory (LSTM), Time Series Transformer, CNN-LSTM and various performance measuring parameters MAE, MSE, MAPE and RMSE are calculated on online UCI dataset (Electrical Grid Stability Simulated Dataset). The proposed hybrid model achieved a better prediction result, which proves the efficiency of the proposed model.

Asymptotics of solutions to systems of (pseudo)differential equations with localized right-hand sides

Abstract The article describes a method for constructing semiclassical asymptotic solutions of multidimensional (pseudo)differential partial differential equations with localized right-hand sides. The method is based on the asymptotic reduction of vector problems (under certain additional conditions) to scalar (pseudo)differential problems, for which efficient asymptotic formulas have recently been constructed. The method is based on the Feynman–Maslov operator calculus. Even if the original problems are stated for differential equations, the operators in reduced problems, as a rule, turn out to be pseudodifferential. Further, nontrivial problems of calculating operator symbols in the reduced scalar problems arise in the reduction process. We discuss approaches to the transformation of these symbols so as to enable their effective use when solving real problems. The method is illustrated by several physical examples.

A High-Order Numerical Scheme for Efficiently Solving Nonlinear Vectorial Problems in Engineering Applications

In scientific and engineering disciplines, vectorial problems involving systems of equations or functions with multiple variables frequently arise, often defying analytical solutions and necessitating numerical techniques. This research introduces an efficient numerical scheme capable of simultaneously approximating all roots of nonlinear equations with a convergence order of ten, specifically designed for vectorial problems. Random initial vectors are employed to assess the global convergence behavior of the proposed scheme. The newly developed method surpasses methods in the existing literature in terms of accuracy, consistency, computational CPU time, residual error, and stability. This superiority is demonstrated through numerical experiments tackling engineering problems and solving heat equations under various diffusibility parameters and boundary conditions. The findings underscore the efficacy of the proposed approach in addressing complex nonlinear systems encountered in diverse applied scenarios.

On a Weighting Technique for Multiple Cost Optimization Problems with Interval Values

This paper deals with a weighting technique for a class of multiple cost optimization problems with interval values. More specifically, we introduce a multiobjective interval-valued controlled model and investigate it by applying the weighting method. In this regard, several characterization theorems are derived. Moreover, a numerical example is formulated. Based on the provided illustrative example and performing a comparative analysis of the results obtained using the weighting technique versus traditional optimization methods, we can easily conclude the effectiveness of the weighting technique in solving multiple cost optimization problems, that is, the conversion of a vector problem to a scalar one.

On the Selection of Weights for Difference Schemes to Approximate Systems of Differential Equations

We consider the problem of determining the weights of difference schemes whose form is specified by a particular symbolic expression. The order of approximation of the differential equation is equal to a given number. To solve it, it was propose to proceed from considering systems of differential equations of a general form to one scalar equation. This method provides us with some values for the weights, which we propose to test using Richardson’s method. The method was shown to work in the case of low-order schemes. However, when transitioning from the scalar problem to the vector and nonlinear problems, the reduction of the order of the scheme, whose weights are selected for the scalar problem, occurs in different families of schemes. This was first discovered when studying the Shanks scheme, which belongs to the family of explicit Runge–Kutta schemes. This does not deteriorate the proposed strategy itself concerning the simplification of the weight-determination problem, which should include a clause on mandatory testing of the order using the Richardson method.

Solving nonlinear vectorial problems with a stable class of Jacobian-free iterative processes

In this manuscript, a general class of Jacobian-free iterative schemes for solving systems of nonlinear equations is presented. Once its fourth-order convergence is proven, the most efficient sub-family is selected in order to make a qualitative study. It is proven that the most of elements of this family are very stable, and this is checked by means on numerical tests on several problems of different sizes. Their performance is compared with other known Jacobian-free iterative procedure, being better in the most of results.

A Study on the Average Number of LLL-based Using Statistical Learning

Lattice-based Cryptography is known as one of the key technologies in modern cryptography. This encryption scheme has the basis vectors from the lattice as the public key and a short-length vector in the lattice consisting of an integer combination of the basis vectors as the secret key. To break this encryption, we need to solve the Shortest Vector Problem (SVP), known as NP-hard. Therefore, instead of finding the shortest vector, LLL algorithm is often used to find a vector of sufficiently short length to break the encryption. The LLL algorithm is a well-known method for breaking this encryption, but there is still no clear answer to the question of how many times the LLL algorithm needs to be used to obtain the desired level of secret key, the average number of the (, )-LLL bases in dimension n is a tool to measure the probability that the LLL algorithm solves the SVP. We can expect that this number indicates how many times the appropriate algorithm should run. There is a formula for this, but it contains some functions that take a long time to compute. We apply linear regression to the formula of the average number of the (, )-LLL bases in dimension n, and therefore we obtain some formulas to approximate the average number of the (, )-LLL is based on dimension n, which contains simple functions. When the dimensions are high, our model is much better regarding the computation time.

A nonconvex sparse recovery method for DOA estimation based on the trimmed lasso

Sparse direction-of-arrival (DOA) estimation methods can be formulated as a group-sparse optimization problem. Meanwhile, sparse recovery methods based on nonconvex penalty terms have been a hot topic in recent years due to their several appealing properties. Herein, this paper studies a new nonconvex regularized approach called the trimmed lasso for DOA estimation. We define the penalty term of the trimmed lasso in the multiple measurement vector model by ℓ2,1-norm. First, we use the smooth approximation function to change the nonconvex objective function to the convex weighted problem. Next, we derive sparse recovery guarantees based on the extended Restricted Isometry Property and regularization parameter for the trimmed lasso in the multiple measurement vector problem. Our proposed method can control the desired level of sparsity of estimators exactly and give a more precise solution to the DOA estimation problem. Numerical simulations show that our proposed method overperforms traditional approaches, which is more close to the Cramer-Rao bound.

ARCHIMEDEAN NEWFORM THEORY FOR

Abstract We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$ , where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

Attack on lattice shortest vector problem using K-Nearest Neighbour

Attack on lattice shortest vector problem using K-Nearest Neighbour

Onto-semiotic analysis of Colombian engineering students’ mathematical connections to problems-solving on vectors: A contribution to the natural and exact sciences

The mathematical connections Colombian engineering students activated when they solved vector problems were explored. The study was based on the extended theory of connections and the onto-semiotic approach. We followed a qualitative methodology that consisted of three stages: (1) selection of engineering students as participants; (2) application of a questionnaire with 15 tasks on vectors to the participating students; and (3) analysis of these data based on the theoretical articulation. The results show that students perform arithmetic operations with vectors, find the scalar and vector product, the norm of a vector, the angle between vectors, and unit vector based on mathematical connections (<i>procedural, meaning, different representations, </i>and<i> implication</i>), detail from an onto-semiotic point of view. However, some students have difficulty finding the angle between vectors because they misuse the norm. Furthermore, the new metaphorical connection based on mnemonics activated by the “<i>law of the ear</i>” is reported. The connections activated by engineering students to solve problems about vectors may have been influenced by the explanations provided by their calculus teacher, who promotes connections for the teaching and learning mathematical concepts.

Quaternary Continuous Classical Boundary Optimal Control Problem Dominating by Parabolic System

In this paper, our purpose is to study the quaternary continuous classical boundary optimal control vector problem dominated by a quaternary linear parabolic boundary value problem. Under suitable assumptions and with a given quaternary continuous classical boundary control vector, the existence theorem for a unique quaternary state vector solution to the weak form is stated and demonstrated via the method of Galerkin. Furthermore, the continuity of the Lipschitz operator between the state vector solution to the weak form for the dominating equations and the corresponding are proved. The existence of a quaternary continuous classical boundary optimal control vector is stated and demonstrated under suitable Assumptions. The mathematical formulation for the quaternary adjoint boundary value problem associated with each considered boundary value problem is obtained and the Frѐchet derivative for the objective function is derived. Finally, the necessary conditions for the optimality theorem of the problem are stated and demonstrated.

A Quasilinearization Approach for Identification Control Vectors in Fractional-Order Nonlinear Systems

This paper is concerned with solving the problem of identifying the control vector problem for a fractional multi-order system of nonlinear ordinary differential equations (ODEs). We describe a quasilinearization approach, based on minimization of a quadratic functional, to compute the values of the unknown parameter vector. Numerical algorithm combining the method with appropriate fractional derivative approximation on graded mesh is applied to SIS and SEIR problems to illustrate the efficiency and accuracy. Tikhonov regularization is implemented to improve the convergence. Results from computations, both with noisy-free and noisy data, are provided and discussed. Simulations with real data are also performed.

Hierarchical categories in colored searching

In colored range counting (CRC), the input is a set of points where each point is assigned a “color” (or a “category”) and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data.However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exist or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through a reduction from the orthogonal vectors problem.