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Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions such as polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [7], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state-of-the-art for this problem, Umans [25] and Kedlaya & Umans [16] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields, respectively, provided that the number of variables n is at most do(1) where the degree of the input polynomial in every variable is less than d. They also stated the question of designing fast algorithms for the large variable case (i.e., n x do(1)) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field Fq of characteristic p, which evaluates an n-variate polynomial of degree less than d in each variable on N inputs in time provided that p is at most do(1), and q is at most (exp(exp(exp(· · · (exp(d))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g., n x do(1)), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas, and this algebraic structure naturally leads to the following two independently interesting applications: --We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [21] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant's program [26]. More precisely, for every fixed prime p, we show that for every constant e > 0, and large enough n, the rank of any n × n Vandermonde matrix V over the field Fpa can be reduced to (n/exp(O(poly(e)log0.53 n))) by changing at most nT(e) entries in every row of V, provided a = poly(logn). Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression [9]. [ABSTRACT FROM AUTHOR]

The aim of this note is to give a simplified proof of the induced version of the Ramsey theorem for vector spaces first proved by H. J. Prömel in 1986.Mathematics Subject Classifications: 05D10, 15A03Keywords: Ramsey theory, vector spaces

The objective of knowledge embedding (KE) is to represent entities and relations in a knowledge graph (KG) in a continuous low-dimensional vector space, thereby facilitating the integration of the KG into various downstream applications. Existing KE methods can be classified into two categories: closed-world knowledge embedding methods (CWKEs) and open-world knowledge embedding methods (OWKEs). CWKEs necessitate the re-training of the entire model upon the addition of new entities to the KG. In contrast, OWKEs offer greater practical application value as they do not require re-training due to their capacity to generate vector representations for new entities. The current major OWKEs employ the description text of new entities to generate vector representations for new entities. However, these methods are highly dependent on the richness of entity relationships and attribute information present in the text, and they are ineffective when there are only a few textual resources, such as entity names. In reality, most KGs have only a limited number of textual resources. For this reason, we propose the first open-world Knowledge Embedding method (LTKE) in a low textual resource environment. LTKE is designed to efficiently generate vector representations for new entities when KGs have only a few textual resources. By efficiently serializing the local structure of entities, LTKE can obtain information-rich text when an entity has only the entity name. Additionally, we develop a new scoring function to enhance model efficiency. Extensive experiments on real-world KG datasets (Freebase, WordNet) have demonstrated the effectiveness of the model. [ABSTRACT FROM AUTHOR]

Space vector pulse width modulation (SVPWM) is widely used in three-phase inverters. As the performance requirements of inverters increase, there is a demand to suppress common-mode voltages (CMVs) generated by SVPWM. In order to suppress the CMVs, it is necessary to mathematically analyze the CMVs. By using a mathematical analysis method based on double Fourier series, general expressions of CMV harmonic amplitudes and spectra are obtained for seven-segment SVPWM and five-segment SVPWM. Comparative analyses on the CMV general expressions are performed for the two SVPWMs, and the CMV harmonics characteristics for the two SVPWMs are summarized. Simulations are carried out in an inverter-driven permanent magnet motor system, and simulation results are in good agreement with calculation ones, which verifies the correctness and validity of the mathematical analysis. Based on these analyses, a more in-depth research can be conducted on the CMV suppression. [ABSTRACT FROM AUTHOR]

An affine spread is a set of subspaces of AG (n , q) of the same dimension that partitions the points of AG (n , q) . Equivalently, an affine spread is a set of projective subspaces of PG (n , q) of the same dimension which partitions the points of PG (n , q) \ H ∞ ; here H ∞ denotes the hyperplane at infinity of the projective closure of AG (n , q) . Let Q be a non-degenerate quadric of H ∞ and let Π be a generator of Q , where Π is a t-dimensional projective subspace. An affine spread P consisting of (t + 1) -dimensional projective subspaces of PG (n , q) is called hyperbolic, parabolic or elliptic (according as Q is hyperbolic, parabolic or elliptic) if the following hold: Each member of P meets H ∞ in a distinct generator of Q disjoint from Π ; Elements of P have at most one point in common; If S , T ∈ P , | S ∩ T | = 1 , then ⟨ S , T ⟩ ∩ Q is a hyperbolic quadric of Q . In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of PG (n , q) is equivalent to a spread of Q + (n + 1 , q) , Q (n + 1 , q) or Q - (n + 1 , q) , respectively. [ABSTRACT FROM AUTHOR]

Most of the modern natural language processing (NLP) techniques are based on the vector space models of language, in which each word is represented by a vector in a high dimensional space. One of the earliest successes was demonstrated by the four-term analogical reasoning task: what is to C as B is to A? The trained word vectors form "parallelograms" representing the quadruple of words in analogy. This discovery in NLP offers us insight into our understanding of human semantic representation of words via analogical reasoning. Despite successful applications of the large-scale language models, it has not been fully understood why such parallelograms emerge by learning through natural language data. As the vector space model is not optimized to form parallelograms, the key structure related to geometric shapes of word vectors is expected to be in the data, rather than the models. In the present article, we test our hypothesis that such parallelogram arrangement of word vectors readily exists in the co-occurrence statistics of language. Our approach focuses more on the data itself, and it is different from the existing theoretical approach trying to find the mechanism of parallelogram formation in the algorithms and/or vector arithmetic operations on word vectors. First, our analysis suggested that analogical reasoning is possible by decomposition of the bigram co-occurrence matrix. Second, we demonstrated the formation of a parallelepiped, a more structured geometric object than a parallelogram, by creating a small artificial corpus and its word vectors. With these results, we propose a refined form of distributional hypothesis pointing out an isomorphism between a sort of symmetry or exchangeability and word co-occurrence statistics. [ABSTRACT FROM AUTHOR]

Heart sound auscultation plays a crucial role in the early diagnosis of cardiovascular diseases. In recent years, great achievements have been made in the automatic classification of heart sounds, but most methods are based on segmentation features and traditional classifiers and do not fully exploit existing deep networks. This paper proposes a cardiac audio classification method based on image expression of multidimensional features (CACIEMDF). First, a 102-dimensional feature vector is designed by combining the characteristics of heart sound data in the time domain, frequency domain and statistical domain. Based on the feature vector, a two-dimensional feature projection space is constructed by PCA dimensionality reduction and the convex hull algorithm, and 102 pairs of coordinate representations of the feature vector in the two-dimensional space are calculated. Each one-dimensional component of the feature vector corresponds to a pair of 2D coordinate representations. Finally, the one-dimensional feature component value and its divergence into categories are used to fill the three channels of a color image, and a Gaussian model is used to dye the image to enrich its content. The color image is sent to a deep network such as ResNet50 for classification. In this paper, three public heart sound datasets are fused, and experiments are conducted using the above methods. The results show that for the two-classification/five-classification task of heart sounds, the method in this paper can achieve a classification accuracy of 95.68%/94.53% when combined with the current deep network. [ABSTRACT FROM AUTHOR]

Harmonic synthesis describes translation invariant linear spaces of continuous complex valued functions on locally compact abelian groups. The basic result due to L. Schwartz states that such spaces on the reals are topologically generated by the exponential monomials in the space; in other words, the locally compact abelian group of the reals is synthesizable. This result does not hold for continuous functions in several real variables, as was shown by D.I. Gurevich's counterexamples. On the other hand, if two discrete abelian groups have this synthesizability property, then so does their direct sum, as well. In this paper, we show that if two locally compact abelian groups have this synthesizability property and at least one of them is discrete, then their direct sum is synthesizable. In fact, more generally, we show that any extension of a synthesizable locally compact abelian group by a synthesizable discrete abelian group is synthesizable. This is an important step toward the complete characterization of synthesizable locally compact abelian groups. [ABSTRACT FROM AUTHOR]

In the context of energy conversion from renewable sources to distribution grids (insulated or not), a converter is often required to transfer energy from a low voltage source towards three-phase grids. This paper presents the HW design, the simulation results, and the conversion performance of a CSI converter intended to interface low-voltage renewable sources to three-phase grids. The main focus of this paper is to obtain the best performance in terms of voltage increase towards the output stage while maximizing the conversion efficiency. In comparison with the currently used energy conversion systems for small photovoltaic systems, hereafter some solutions were adopted to level and maximize the energy flow from the source to the DC-link and improve the quality of current supplied in terms of harmonic distortion. The proposed system is composed of two conversion stages: the first, voltage-to-current, the second current-to-current via a three-phase CSI bridge modulated with the SVM technique. The stages are not completely decoupled from an electrical point of view; therefore, in order to mitigate the effects of these interactions, synchronization strategies have been adopted. [ABSTRACT FROM AUTHOR]

In this paper, we establish several proper separation theorems for an element and a convex set and for two convex sets in terms of their quasi-relative interiors. Then, we prove that the separation theorem given by [Cammaroto, F and B Di Bella (2007). On a separation theorem involving the quasi-relative. Proceedings of the Edinburgh Mathematical Society, 50(3), 605–610] in Theorem 2.5, is in fact a proper separation theorem for two convex sets in which the classical interior is replaced by the quasi-relative interior. Besides, we extend some known results in the literature, such as [Adán, M and V Novo (2004). Proper efficiency in vector optimization on real linear spaces. Journal of Optimization Theory and Applications, 121, 515–540] in Theorem 2.1 and [Edwards, R (1965). Functional Analysis: Theory and Applications. New York: Reinhart and Winston] in Corollary 2.2.2, through the quasi-relative interior and the quasi-interior, respectively. As an application, we provide Karush–Kuhn–Tucker multipliers for quasi-relative solutions of vector optimization problems. Several examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Following the definition of interval-valued fuzzy n-normed linear space given by S. Vijayabalaji et al., in this paper, notion of interval-valued generalized fuzzy n-normed linear space is introduced. The notion of convergent sequence, Cauchy sequence and their relation are studied. Some basic results are established on finite dimensional interval-valued generalized fuzzy n-normed linear space. [ABSTRACT FROM AUTHOR]

In long‐range missions based on angle‐of‐arrival positioning, the angle measurement error of unmanned aerial vehicles is a major source of error. Therefore, reducing the unmanned aerial vehicle angle measurement error is crucial to achieve accurate remote positioning. In this paper, we propose a space vector–based method to correct the space vector of the target for the unmanned aerial vehicles when there are fewer than three available reference points, which in turn corrects the angular value of the target relative to the unmanned aerial vehicles. Simulation results show that when the distance between the reference point and the unmanned aerial vehicles is smaller than the distance between the target and the unmanned aerial vehicles, the azimuth measurement error can be reduced to 55% of the original error for the case of a single reference point, while the pitch angle measurement error remains almost unchanged. In the case of more than two reference points, the azimuth measurement error can be reduced to 1e5 and the pitch angle measurement error can be reduced to 30% of the original error. This method can be adapted to the rapid positioning task for high‐speed and high‐mobility targets without iteration, low computation, good correction effect, and the need of prior known data set reference. [ABSTRACT FROM AUTHOR]

Inverter-based systems encounter significant challenges in mitigating common-mode voltage (CMV) and minimizing inverter losses. Despite various space vector pulse-width modulation (SVPWM) techniques proposed to address these issues, a comprehensive comparative analysis has been lacking. This paper addresses this gap through an experimental and simulation-based evaluation of nine SVPWM techniques. A new discontinuous SVPWM technique, DSVPWM-K4, is introduced, which involves reversing the use of the two zero vectors in DSVPWM-K3. DSVPWM-K3 delivers superior performance in terms of CMV reduction, total harmonic distortion (THD), and inverter losses across all modulation indices (MI = 1, 0.75, 0.5, and 0.25), making it the most effective overall. Although DSVPWM-K4 is a novel approach, it ranks second in effectiveness. The RSPWM technique achieves the lowest CMV with a zero peak-to-peak value but is most effective at lower modulation indices (0.25 and 0.5) due to higher harmonic distortion at higher modulation indices. AZSPWM performs optimally at higher modulation indices, providing a 66.66% reduction in CMV compared to continuous SVPWM and significantly lower THD compared to RSPWM. In contrast, NSPWM exhibits nearly double the THD compared to continuous SVPWM. [ABSTRACT FROM AUTHOR]

With the development of power conversion systems or bidirectional grid-connected inverters characterized by high DC voltage, high efficiency, and high-power density, high-switching-frequency SiC power switches are being widely used, and these require a short computational time of control algorithm. Based on the sector judgment of a space voltage vector and the midpoint potential control balancing of a DC bus, an improved finite control set method prediction control (FCS-MPC) method was proposed for a T-type three-level PCS. This improved FCS-MPC first judges the sector of the space voltage vector and locates the position of the corresponding large sector according to phase lock information; then, it analyzes the sampled voltage of the upper and lower bus capacitors to obtain the midpoint potential situation and selects appropriate small vectors based on the midpoint potential situation. This simple improved strategy can reduce the computational complexity of traditional MPC for rolling optimization, resulting in a reduction in the vectors from 27 to 8. A SiC-based 10 kW bidirectional PCS prototype verified the improved FCS-MPC, and the computational time was reduced about by 56% in comparison to traditional FCS-MPC. [ABSTRACT FROM AUTHOR]

A novel and unified design approach on higher-dimensional switching chaos generators is derived in this paper. The whole n-dimensional linear space is divided into two parts by a closed hyper-polyhedron. Two higher-dimensional linear systems with the simplest structures as switching chaos generators are designed successfully to generate chaos. State matrix of the first linear system is Hurwitz stable. State matrix of the second linear system is not Hurwitz stable. Chaotic dynamical behaviors take place due to switching two systems. The switching trajectories go through the boundary of the closed hyper-polyhedron endlessly. Moreover, the size of the hyper-polyhedron can determine and control the amplitude of the chaotic signals. Specific numerical examples on four-dimensional, five-dimensional and six-dimensional switching chaos generators are employed, respectively, to illustrate the effectiveness of the novel and advanced approach presented in this paper. The proposed approach can also be applied to designing other switching chaos generators with the higher dimension beyond six. [ABSTRACT FROM AUTHOR]

Spin textures, i.e., the distribution of spin polarization vectors in reciprocal space, exhibit diverse patterns determined by symmetry constraints, resulting in a variety of spintronic phenomena. Here, we propose a universal theory to comprehensively describe the nature of spin textures by incorporating three symmetry flavors of reciprocal wavevector, atomic orbital, and atomic site. Such an approach enables us to establish a complete classification of spin textures constrained by the little co-group and predict some exotic spin texture types, such as Zeeman-type spin splitting in antiferromagnets and quadratic spin texture. To illustrate the influence of atomic orbitals and sites on spin textures, we predict orbital-dependent spin texture and anisotropic spin-momentum-site locking effects, and corresponding material candidates validated through first-principles calculations. The comprehensive classification and the predicted new spin textures in realistic materials are expected to trigger future spin-based functionalities in electronics. [ABSTRACT FROM AUTHOR]

In this article, we study the properties of 4-by-4 metric matrices and characterize their dependence and independence by M 4 × 4 = (M 4 × 4 − D M 4 × 4) ∪ D M 4 × 4 , where D M 4 × 4 is the set of all dependent metric matrices. D M 4 × 4 is further characterized by D M 4 × 4 = D M 1 4 × 4 ∪ D M 2 4 × 4 , where D M 2 4 × 4 is characterized by D M 2 4 × 4 = D M 21 4 × 4 ∪ D M 22 4 × 4 . These characterizations provide some insightful findings that go beyond the Euclidean distance or Euclidean distance matrix and link the distance functions to vector spaces, which offers some theoretical and application-related advantages. In the application parts, we show that the metric matrices associated with all Minkowski distance functions over four different points are linearly independent, and that the metric matrices associated with any four concyclic points are also linearly independent. [ABSTRACT FROM AUTHOR]

The new solution concepts of interval-valued multiobjective optimization problems using ordering cone are proposed in this paper. An equivalence relation is introduced to divide the collection of all bounded closed intervals into the equivalence classes. The family of all equivalence classes is also called a quotient set. In this case, this quotient set can turn into a vector space under some suitable settings for vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. It means that an ordering in the quotient set can be defined to study the Pareto optimal solution in multiobjective optimization problems. In this paper, we consider the multiobjective optimization problem such that its coefficients are taken to be the bounded closed intervals. With the help of the convex cone, we can study the Pareto optimal solutions of the multiobjective optimization problem with interval-valued coefficients. [ABSTRACT FROM AUTHOR]

In recent years, research on attribute-based encryption (ABE) has expanded into the quantum domain. Because a traditional single authority can cause the potential single point of failure, an improved lattice-based quantum-resistant identity authentication and policy attribute encryption scheme is proposed, in which the generation of random values is optimized by adjusting parameters in the Gaussian sampling algorithm to improve overall performance. Additionally, in the key generation phase, attributes are processed according to their shared nature, which reduces the computational overhead of the authorization authority. In the decryption phase, the basis transformation of the Lenstra–Lenstra–Lovász (LLL) lattice reduction algorithm is utilized to rapidly convert shared matrices into the shortest vector form, which can reduce the computational cost of linear space checks. The experimental results demonstrate that the proposed method not only improves efficiency but also enhances security compared with related schemes. [ABSTRACT FROM AUTHOR]

We consider fixed boundary flows with canonical interpretability as principal components extended on nonlinear Riemannian manifolds. We aim to find a flow with fixed start and end points for noisy multivariate datasets lying near an embedded nonlinear Riemannian manifold. In geometric terms, the fixed boundary flow is defined as an optimal curve that moves in the data cloud with two fixed end points. At any point on the flow, we maximize the inner product of the vector field, which is calculated locally, and the tangent vector of the flow. The rigorous definition is derived from an optimization problem using the intrinsic metric on the manifolds. For random datasets, we name the fixed boundary flow the random fixed boundary flow and analyze its limiting behavior under noisy observed samples. We construct a high-level algorithm to compute the random fixed boundary flow, and provide the convergence of the algorithm. We show that the fixed boundary flow yields a concatenate of three segments, one of which coincides with the usual principal flow when the manifold is reduced to the Euclidean space. We further prove that the random fixed boundary flow converges largely to the population fixed boundary flow with high probability. Finally, we illustrate how the random fixed boundary flow can be used and interpreted, and demonstrate its application in real datasets. for this article are available online. [ABSTRACT FROM AUTHOR]

To address the harmonic issues associated with the pulse width modulation (PWM) method, this article introduces variable delayed time (VDT) space vector pulse width modulation (SVPWM) with double central symmetry for dual three-phase permanent magnet synchronous machines (DTP-PMSMs). Firstly, the switching sequence of four traditional vectors undergoes double central symmetry, resulting in the doubling of the frequency of the phase voltage. This alteration eliminates harmonics occurring at odd multiples of carrier frequency. Additionally, the adopting of a single null vector minimizes the additional switching times introduced by the double central symmetry. Subsequently, VDT-SVPWM is employed to further suppress the harmonics at the even multiples of carrier frequency. The implementation of the proposed double central symmetry method involves directly utilizing the calculated duty cycle of the traditional four vectors. Moreover, integrating VDT-SVPWM with the double central symmetry method is straightforward. Simulation and experimental results validate the efficacy of the proposed method in suppressing harmonics and mitigating vibrations in the DTP-PMSM. [ABSTRACT FROM AUTHOR]

Incorporating AC-type flying capacitors (FC) between series-connected devices is an effective way to enhance the rated voltage for high-power applications based on current source converters (CSCs). Through appropriate modulation and FC voltage control, it is possible to achieve improved DC bus voltage quality with reduced common-mode voltage (CMV) and low dv/dt. On the other hand, the parallel CSC is a popular choice for increasing the system's rated current to accommodate higher power applications. The use of interleaved modulation techniques can improve the harmonic performance of parallel converters while reducing the need for passive filters. The modular flying capacitor clamped (FCC)-CSC structure can combine these advantages, achieving higher rated power along with improved power quality on both the DC and AC sides. Moreover, the enhanced AC quality contributes to the regulation of FC voltage and further improves the DC-side voltage quality. This paper analyzes the operation principle of the parallel FCC-CSC structure and proposes an interleaved space vector modulation (SVM) method to enhance the harmonic performance of the AC output. Additionally, an optimized zero-state replacement (ZSR) based FC voltage control and a DC-link current balance strategy built on this control are introduced. Simulation and experimental results validate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In computational homogenization approaches, data-driven methods entail advantages due to their ability to capture complex behavior without assuming a specific material model. Within this domain, constitutive model-based and model-free data-driven methods are distinguished. The former employ artificial neural networks as models to approximate a constitutive relation, whereas the latter directly incorporate stress–strain data in the analysis. Neural network-based constitutive descriptions are one of the most widely used data-driven approaches in computational mechanics. In contrast, distance-minimizing data-driven computational mechanics enables substituting the material modeling step entirely by iteratively obtaining a physically consistent solution close to the material behavior represented by the data. The maximum entropy data-driven solver is a generalization of this method, providing increased robustness concerning outliers in the underlying data set. Additionally, a tensor voting enhancement based on incorporating locally linear tangent spaces enables interpolating in regions of sparse sampling. In this contribution, a comparison of neural network-based constitutive models and data-driven computational mechanics is made. General differences between machine learning, distance minimizing, and entropy maximizing data-driven methods are explored. These include the pre-processing of data and the required computational effort for optimization as well as evaluation. Numerical examples with synthetically generated datasets obtained by numerical material tests are employed to demonstrate the capabilities of the investigated methods. An anisotropic nonlinear elastic constitutive law is chosen for the investigation. The resulting constitutive representations are then applied in structural simulations. Thereby, differences in the solution procedure as well as use-case accuracy of the methods are investigated. [ABSTRACT FROM AUTHOR]

This paper is dedicated to defining and studying for the first time the concept of fuzzy metric spaces based on two-fold fuzzy algebras, where the elementary properties of this new concept will be studied and presented by many theorems and related examples that explain the validity of our work. Also, many different types of open and closed balls will be discussed, as well as the relationships between these metric substructures. Keywords: fuzzy metric space, two-fold algebra, open ball, closed ball, torus Introduction and basic concepts The applications of neutrosophic sets and fuzzy sets are very wide and open research areas. In the literature, we can find many neutrosophic and fuzzy algebraic structures with deep connection with applied mathematics and number theory [5-11]. The concept of two-fold algebra was presented by Smarandache in [4], where many suggestions for the algebraic structure related to this algebra were defined and presented. This new idea has been used in [1] to study the two-fold algebra based on the standard fuzzy number theoretical system [3]. In [2], Hatip et.al. proposed the two-fold vector space and two-fold algebraic module based on fuzzy mappings, where they have studied the elementary properties of these new generalizations with many interesting examples. This work is motivated by the modern idea of two-fold algebra, and metric spaces, where we can combine those to different structures in one algebraic structure called two-fold. [ABSTRACT FROM AUTHOR]

We consider bounded linear operators from the view of functional reproducing property. For some bounded linear operators associated with orthogonal polynomials we define an inner product space associated with a kernel constructed with orthogonal polynomials, show that it is a functional reproducing kernel Hilbert space (FRKHS) associated with these bounded linear operators and give decay rate for the best FRKHS approximation with a K -functional associated with the FRKHS. On this basis, we provide a learning rate for kernel regularized regression whose hypothesis space is the defined FRKHS. As applications, we define some concrete FRKHSs associated with polynomial operators such as the Bernstein–Durrmeyer operators, the de la Vallée Poussin operators on both the unit sphere S d − 1 and the unit ball B d . We show that these polynomial operators have reproducing property with respect to the corresponding concrete FRKHSs and show the learning rate for the kernel regularized regression. In short, we provide a way of constructing FRKHS operators with Fourier multipliers and show a learning framework from the view of operator approximation. [ABSTRACT FROM AUTHOR]

A new concept known as the orthogonal component graph associated with a finite-dimensional vector space over a finite field has been recently added as another class of algebraic graphs. In these types of graphs, the vertices will be all the possible non-zero linear combinations of orthogonal basis vectors, and any two vertices will be adjacent if the corresponding vectors are orthogonal. In this paper, we discuss the various colorings and structural properties of orthogonal component graphs. [ABSTRACT FROM AUTHOR]

In a famous article published in 1975, Kramosil and Michálek introduced a notion of fuzzy metric that was the origin of numerous researches and publications in several frameworks and fields. In 2010, Chistyakov introduced and discussed in detail the concept of modular metric. Since then, some authors have investigated the problem of establishing connections between the notions of fuzzy metric and modular metric, obtaining positive partial solutions. In this paper, we are interested in determining the precise relationship between these two concepts. To achieve this goal, we examine a proof, based on the use of uniformities, of the important result that the topology induced by a fuzzy metric is metrizable. As a consequence of that analysis, we introduce the notion of a weak fuzzy metric and show that every weak fuzzy metric, with continuous t-norm the minimum t-norm, generates a modular metric and, conversely, we show that every modular metric generates a weak fuzzy metric, with continuous t-norm the product t-norm. It follows that every modular metric can be generated from a suitable weak fuzzy metric, and that several examples and properties of modular metrics can be directly deduced from those previously obtained in the field of fuzzy metrics. [ABSTRACT FROM AUTHOR]

Feature fusion for multimodal sentiment analysis is a challenging but worthwhile research topic. With the extension of the time dimension, there are interactions between multimodal signals and the lack of control over the target modal representations during the fusion process leads to erroneous shifts of vectors in the feature space. Moreover, ignoring the representation of target modal features under different fusion orders may lead to insufficient fusion of complementary information. To address the above issues, this paper proposes a transformer-encoder-based multimodal multi-attention fusion network model. The model constructs a multi-attention fusion transformer-encoder to learn inter-modal consistent features and enhance the representation of target modal features. Meanwhile, for each target modality, we construct multi-attention fusion transformer-encoder with different fusion orders in the model to capture the complementary features among the sequences with different fusion orders. Then, the three target modal representations containing consistent features and complementary features are fused with initial features through residual connections to guide the final sentiment analysis. We conduct extensive experiments on three public multimodal datasets. The results show that our approach outperforms the compared multimodal sentiment analysis methods on most metrics and can explain the contributions of inter- and intra-modal interactions in multiple modalities. [ABSTRACT FROM AUTHOR]

In this paper, the fuzzy counterparts of the collineations defined in classical projective spaces are defined in a 3-dimensional fuzzy projective space derived from a 4-dimensional fuzzy vector space. The properties of fuzzy projective space (λ,S) left invariant under the fuzzy collineations are characterized depending on the membership degrees of the given fuzzy projective space and also depending on the pointwise invariant of the lines. Moreover, some relations between membership degrees of the fuzzy projective space are presented according to which are of the base point, base line, and base plane invariant under a fuzzy collineation. Specifically, when all membership degrees of (λ,S) are distinct, the base point, base line, and base plane of (λ,S) are invariant under the fuzzy collineation ¯f . Conversely, if none of the base point, base line, or base plane remain invariant, then the system becomes crisp in (λ,S). Additionally, some relations between the membership degrees of the fuzzy projective space, concerning the invariance of the base point, base line, and base plane, are presented. [ABSTRACT FROM AUTHOR]

For a finite group acting linearly on a vector space, a separating set is a subset of the invariant ring that separates the orbits. In this paper, we determined explicit separating sets in the corresponding rings of invariants for four families of finite dimensional representations of the elementary abelian p-groups (Z/p)² of rank two over an algebraically closed field of characteristic p, where p is an odd prime. Our construction was recursive. The separating sets consisted only of transfers and norms, and the size of every separating set depended only on the dimension of the representation. [ABSTRACT FROM AUTHOR]

We introduce the general notion of a rank on a vector space, which includes both tensor rank and conventional matrix rank, but incorporates other examples as well. Extending this concept, we investigate vector spaces consisting of vectors with a lower bound on their rank. Our main result shows that bases for such spaces of maximum dimension can be chosen to consist exclusively of vectors of minimal rank. This generalization extends the results of [15,36], with potential applications in different areas. [ABSTRACT FROM AUTHOR]

A new concept known as the orthogonal component graph associated with a finite-dimensional vector space over a finite field has been recently added as another class of algebraic graphs. In these types of graphs, the vertices will be all the possible non-zero linear combinations of orthogonal basis vectors, and any two vertices will be adjacent if the corresponding vectors are orthogonal. In this paper, we discuss the various colorings and structural properties of orthogonal component graphs. [ABSTRACT FROM AUTHOR]

The revitalization of historic districts is crucial for the sustainable development of cities, with street corner spaces being a vital component of the public space in these districts. However, street corner spaces have been largely overlooked in previous research on crowd dynamics within historic districts. This study investigates the key factors influencing crowd dynamics in street corner spaces within historic districts. First, a hierarchical model of vitality-influencing factors was developed based on prior research. Potential factors influencing the vitality of street corners were quantified using multi-source data collection methods, including deep learning algorithms, and crowd vitality within these spaces was assessed through multidimensional measurements. The impact of each element on crowd vitality was then analyzed through a multivariate linear regression model. The findings revealed that eight factors—corner building historicity, first-floor functional communality, transparency, openness, density of functional facilities, greenness, functional variety of buildings, and walkability—significantly influence the vitality of corner spaces, collectively explaining 77.5% of the vitality of these spaces. These conclusions offer new perspectives and scientific evidence for the revitalization and conservation of historic districts. [ABSTRACT FROM AUTHOR]

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class A p with p ∈ (1 , ∞). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces. [ABSTRACT FROM AUTHOR]

As there is a great need for high-quality electricity on the distribution side, distribution side generation (DSG) has become increasingly important. The increased weight of EVs on the distribution side is the cause of this. There are numerous power quality mitigation techniques employed to address this type of issue, but many of the solutions suggest the usage of a separate device, such as an active power filter. But while construction the DSG the solution to this problem may be addressed using the proposed solution in this paper. Power quality (PQ) problems are being caused by the grid's integration of Photo-Voltaic (PV) and its application to all connected loads. With the aid of Direct Quardrature (DQ) controller and Multicarrier Space Vector Pulse Width Modulation (SVPWM) technology, the overall power quality disturbance is decreased. A Simulink model for the PV-EV-Grid system was built to measure voltage and current Total Harmonic Distortion (THD) percentages under linear, non-linear, and Plug in Hybrid Vehicle (PHEV) load situations. The model shows that the THD values are well within the IEEE 519. [ABSTRACT FROM AUTHOR]

In the semialgebraic range searching problem, we are given a set of n points in ℝd, and we want to preprocess the points such that for any query range belonging to a family of constant complexity semialgebraic sets (Tarski cells), all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, the problem is well-understood: It can be solved using S(n) space and with Q(n) query time with S(n)Q(n)d=Õ(nd), where the Õ(⋅) notation hides polylogarithmic factors and this trade-off is tight (up to no(1) factors). In particular, there exist “low space” structures that use O(n) space with O(n1-1/d) query time [8, 25] and “fast query” structures that use O(nd) space with O(log n) query time [9]. However, for general semialgebraic ranges, only “low space” solutions are known, but the best solutions [7] match the same trade-off curve as simplex queries, with O(n) space and Õ(n1−1/d) query time. It has been conjectured that the same could be done for the “fast query” case, but this open problem has stayed unresolved. Here, we disprove this conjecture. We give the first nontrivial lower bounds for semialgebraic range searching and other related problems. More precisely, we show that any data structure for reporting the points between two concentric circles, a problem that we call 2D annulus reporting, with Q(n) query time must use S(n)=...(n³/Q(n)5) space, where the ...(⋅) notation hides no(1) factors, meaning, for Q(n)=logO(1)n, ...(n³) space must be used. In addition, we study the problem of reporting the subset of input points in a polynomial slab defined by {(x,y)∈R²:P(x)≤y≤P(x)+w}, where P(x)=∑i=0Δaixi is a univariate polynomial of degree Δ and a0,…,aΔ,w are given at the query time, a problem that we call polynomial slab reporting. For this, we show a space lower bound of ...(nΔ+1/Q(n)(Δ+3)Δ/2), which implies that for Q(n)=logO(1)n, we must use ...(nΔ+1) space. We also consider the dual semialgebraic stabbing problems of semialgebraic range searching and present lower bounds for them. In particular, we show that in linear space, any data structure that solves 2D annulus stabbing problems must use ...(n2/3) query time. Note that this almost matches the upper bound obtained by lifting 2D annuli to 3D. Like semialgebraic range searching, we also present lower bounds for general polynomial slab stabbing problems. Again, our lower bounds are almost tight for linear size data structures in this case. [ABSTRACT FROM AUTHOR]

The identification of cancer driver genes is crucial for early detection, effective therapy, and precision medicine of cancer. Cancer is caused by the dysregulation of several genes at various levels of regulation. However, current techniques only capture a limited amount of regulatory information, which may hinder their efficacy. In this study, we present IMI-driver, a model that integrates multi-omics data into eight biological networks and applies Multi-view Collaborative Network Embedding to embed the gene regulation information from the biological networks into a low-dimensional vector space to identify cancer drivers. We apply IMI-driver to 29 cancer types from The Cancer Genome Atlas (TCGA) and compare its performance with nine other methods on nine benchmark datasets. IMI-driver outperforms the other methods, demonstrating that multi-level network integration enhances prediction accuracy. We also perform a pan-cancer analysis using the genes identified by IMI-driver, which confirms almost all our selected candidate genes as known or potential drivers. Case studies of the new positive genes suggest their roles in cancer development and progression. Author summary: Cancer driver identification is of great value in cancer research. Recent advancements have demonstrated that incorporating the intricate regulatory relationships among genes can facilitate driver genes identification. However, current methods do not fully capture the regulatory relationships at different layers. In this study, we propose a new method for predicting cancer driver genes (IMI-driver), which integrates multi-omics data into eight biological networks (including common networks such as protein-protein interaction networks and various cancer-specific networks) and embeds multi-level network information as features to predict driver gene sets. Experiment demonstrates that our method outperforms mainstream tools, and can discover new drivers. [ABSTRACT FROM AUTHOR]

The identification and classification of railway turnout faults are essential for guaranteeing train safety. Traditional diagnostic methods for these faults face challenges due to limited accuracy, stemming from the scarcity of fault samples, and often fail to provide detailed fault classification. In response to these issues, we introduce an advanced two‐stage model for the classification of railway turnout faults, utilizing the FastDTW algorithm, known for its efficient approximation of DTW (dynamic time warping) with linear time and space complexity. In the first stage, we employ a Shapelets feature extraction algorithm, based on a greedy strategy, to efficiently identify the most representative segments from long sequence action curves. Progressing to the second stage, the model tackles the inherent singularities in the FastDTW algorithm by incorporating a novel curve segmentation technique, also rooted in a greedy strategy. This technique fine‐tunes the fault classification process, leading to more accurate outcomes. The effectiveness and precision of our proposed model were validated empirically using a dataset of 540 faulty curves from a specific high‐speed railway station, achieving an impressive classification accuracy of 97%. This substantial accuracy in fault curve classification underscores the potential of our model to significantly enhance the safety and efficiency of railway operations, marking a notable advancement in the field of railway turnout fault classification. [ABSTRACT FROM AUTHOR]

A variational heteroencoder based on recurrent neural networks, trained with SMILES linear notations of molecular structures, was used to derive the following atomic descriptors: delta latent space vectors (DLSVs) obtained from the original SMILES of the whole molecule and the SMILES of the same molecule with the target atom replaced. Different replacements were explored, namely, changing the atomic element, replacement with a character of the model vocabulary not used in the training set, or the removal of the target atom from the SMILES. Unsupervised mapping of the DLSV descriptors with t-distributed stochastic neighbor embedding (t-SNE) revealed a remarkable clustering according to the atomic element, hybridization, atomic type, and aromaticity. Atomic DLSV descriptors were used to train machine learning (ML) models to predict 19F NMR chemical shifts. An R2 of up to 0.89 and mean absolute errors of up to 5.5 ppm were obtained for an independent test set of 1046 molecules with random forests or a gradient-boosting regressor. Intermediate representations from a Transformer model yielded comparable results. Furthermore, DLSVs were applied as molecular operators in the latent space: the DLSV of a halogenation (H→F substitution) was summed to the LSVs of 4135 new molecules with no fluorine atom and decoded into SMILES, yielding 99% of valid SMILES, with 75% of the SMILES incorporating fluorine and 56% of the structures incorporating fluorine with no other structural change. [ABSTRACT FROM AUTHOR]

It is of paramount importance to accurately predict the maintenance demands of vehicles in order to guarantee their sustainable use. Nevertheless, the current methodologies merely predict a partial aspect of a vehicle's maintenance demands, rather than the comprehensive maintenance demands. Moreover, the process of predicting vehicle maintenance demands must give due consideration to the influence of mileage on such demands. In light of the aforementioned considerations, we put forth a vehicle overall maintenance demand prediction method that incorporates vehicle mileage awareness. In order to address the discrepancy between the vector space of mileage and that of the project, we put forth a mileage representation method for the maintenance demand prediction task. To capture the significant impact of key mileage and projects on future demand, we propose a learning module for key temporal information using a fusion of Long Short-Term Memory (LSTM) networks and attention mechanism. Moreover, to integrate maintenance mileage and projects, we propose a fusion method based on a gated unit. The experimental results obtained from real datasets demonstrate that the proposed model exhibits a superior performance compared to existing methods. [ABSTRACT FROM AUTHOR]

The Euclidean hull of a linear code C is defined as C ∩ C⊥, where C⊥ denotes the dual of C under the Euclidean inner product. A linear code with the trivial hull is called a linear complementary dual (LCD) code. A pair (C, D) of linear codes of length n over the finite field Fq is called a linear complementary pair (LCP) of codes if C ⊕ D = Fnq . More generally, a pair (C, D) of linear codes of the same length over Fq is called a linear ℓ -intersection pair of codes if C ∩ D has dimension ℓ as a vector space over Fq . In this paper, we give characterizations of LCD, LCP of cyclic codes and one-dimensional hull cyclic codes of length qm− 1, m ≥ 1, over Fq in terms of their basic dual zero sets and their trace representations. We also formulate the hull dimension of a cyclic code of arbitrary length over Fq with respect to its basic dual zero set. Moreover, we provide a general formula for the dimension ℓ of the intersection of two cyclic codes of arbitrary length over Fq based on their basic dual zero sets. [ABSTRACT FROM AUTHOR]

The non-zero component graph of a finite dimensional vector space V over a finite field F is the graph G(Va) = (V, E), where vertices of G(Va) are the non-zero vectors in V, two of which are adjacent if they share at least one basis vector with non-zero coefficient in their basic representation. In this paper, we study the various types of colorings of non-zero component graph. [ABSTRACT FROM AUTHOR]

The concept of summability is crucial in deriving formal solutions to partial differential equations. This paper explores the connection between the methods of statistical convergence of sequences and statistical Cesàro summability in intuitionistic fuzzy n-normed linear space (IFnNLS). While the existing literature covers Cesàro summability and its statistical variant in fuzzy, intuitionistic fuzzy, and classical normed spaces, this study stands out not only for its methodology but also for its comprehensive approach, encompassing a broader range of spaces and detailing the pathway from the statistical Cesàro summability method to statistical convergence. These results will lead us to Tauberian theorems in IFnNLS. [ABSTRACT FROM AUTHOR]

In this paper, three discrete transforms related to vector spaces over finite fields are studied. For our purposes, and according to the properties of the finite fields, the most suitable transforms are as follows: for binary fields, this is the Walsh–Hadamard transform; for odd prime fields, the Vilenkin–Chrestenson transform; and for composite fields, the trace transform. A factorization of the transform matrices using Kronecker power is given so that the considered discrete transforms are reduced to the fast discrete transforms. Examples and applications are also presented of the considered transforms in coding theory for calculating the weight distribution of a linear code. [ABSTRACT FROM AUTHOR]

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R , C , H , O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element, we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley–Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, which is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure using vectorization. Endowed with this introduced product, the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the "imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not the subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm, we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary "half-octonions" is unitless and totally associative. [ABSTRACT FROM AUTHOR]

The complete characterization of the almost-entropic region yields rate regions for network coding problems. However, this characterization is difficult and open. In this paper, we propose a novel algorithm to determine whether an arbitrary vector in the entropy space is entropic or not, by parameterizing and generating probability mass functions by neural networks. Given a target vector, the algorithm minimizes the normalized distance between the target vector and the generated entropic vector by training the neural network. The algorithm reveals the entropic nature of the target vector, and obtains the underlying distribution, accordingly. The proposed algorithm was further implemented with convolutional neural networks, which naturally fit the structure of joint probability mass functions, and accelerate the algorithm with GPUs. Empirical results demonstrate improved normalized distances and convergence performances compared with prior works. We also conducted optimizations of the Ingleton score and Ingleton violation index, where a new lower bound of the Ingleton violation index was obtained. An inner bound of the almost-entropic region with four random variables was constructed with the proposed method, presenting the current best inner bound measured by the volume ratio. The potential of a computer-aided approach to construct achievable schemes for network coding problems using the proposed method is discussed. [ABSTRACT FROM AUTHOR]

The paralleled configuration of three-phase two-level (3P2L) inverters has been put forward to increase the output power rating, operating efficiency, and system reliability. Nevertheless, this architecture brings about the serious circulating current problem, which distorts the quality of output currents, results in additional power losses, and reduces the system efficiency. Another problem is the common-mode voltage (CMV), which causes electromagnetic interference and threatens the safe operation of the system. There exists interconnection between these two issues in the paralleled 3P2L inverters. To suppress the CMV and circulating current simultaneously, an improved control method is presented. At first, the discrete model of paralleled 3P2L inverters is established, based on which the improved control method is designed to restrain the circulating current, while the parameter tuning is avoided. In addition, the zero-sequence component injection associated with the optimized configuration of carrier phase is conducted, and the CMV magnitude of each inverter is limited within one-sixth of dc-side voltage. When comparing with the traditional space vector modulation (SVM) approach, the CMV magnitude is restrained by two-thirds by the presented method. The hardware-based evaluation results have been provided to validate the presented approach. • This article investigates a control method for paralleled three-phase inverters. • The control parameters are directly obtained by the discrete model. • The configuration of carrier phase is optimized to reduce common-mode voltage. [ABSTRACT FROM AUTHOR]

Network representation learning aims to map the relationship between network nodes and context nodes to a low-dimensional representation vector space. Directed network representation learning considers mapping directional of node vector. Currently, only sporadic work on direct network representation has been reported. In this work, we propose a novel algorithm that takes into account the direction of the edge with text's attribute of the node in directed network representation learning. We then define the matrix based on in-degree of Laplacian and signless Laplacian for digraph, and it utilizes web page datasets from universities in the USA to evaluate the performance of vertex classification. We compare our algorithm with other directed representation learning algorithms. The experimental results show that our algorithm outperforms the baseline by over 20% when the training ratio ranges from 10 to 90%. We apply the in-degree-Laplacian and In-degree-signless-Laplacian to directed representation learning, which is one of the main contributions of this algorithm. Additionally, we incorporate text information through matrix completion in directed network representation learning and the experimental results show an increase in performance of up to 20% compared to the baseline, especially when the training ratio is 10%. [ABSTRACT FROM AUTHOR]

This research work extends the least squares criterion. The regression models which have been treated so far in the literature do not study multilinear relationships between variables. Such relationships are of a nonlinear nature. They take place whenever two or more than two univariate variables are the components of a multiple variable of order 2 or an order greater than 2. A multiple variable of order 2 is not a bivariate variable, and a multiple variable of an order greater than 2 is not a multivariate variable. A multiple variable allows for the construction of a tensor. The α -norm of this tensor gives rise to an aggregate measure of a multilinear nature. In particular, given a multiple variable of order 2, four regression lines can be estimated in the same subset of a two-dimensional linear space over R. How these four regression lines give rise to an aggregate measure of a multilinear nature is shown by this paper. In this research work, such a measure is an estimate concerning the expected return on a portfolio of financial assets. The metric notion of α -product is used to summarize the sampling units which are observed. [ABSTRACT FROM AUTHOR]

In this study, first the pedal curves as the geometric locus of perpendicular projections to the Frenet vectors of a space curve were defined and the Frenet vectors, curvature, and torsion of these pedal curves were calculated. Second, for each pedal curve, Smarandache curves were defined by taking the Frenet vectors as position vectors. Finally, the expressions of Frenet vectors, curvature, and torsion related to the main curves were obtained for each Smarandache curve. Thus, new curves were added to the curve family. [ABSTRACT FROM AUTHOR]