Complex Vector Research Articles

A General Theory of Micro and Macro Mechanics in Complex Vector Space: Applications to Quantum Theory and Gravitation

A General Theory of Micro and Macro Mechanics in Complex Vector Space: Applications to Quantum Theory and Gravitation

TQFTs and quantum computing

Quantum computing is captured in the formalism of the monoidal subcategory of VectC generated by C2 — in particular, quantum circuits are diagrams in VectC — while topological quantum field theories, in the sense of Atiyah, are diagrams in VectC indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call FVectC. We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to FVectC, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category.

On-axis complex-amplitude modulation for the generation of super-stable vector modes

We propose a technique to generate complex vector beams with high quality and stability. Our approach relies on the combination of complex amplitude modulation (CAM) and on-axis modulation, two techniques that seem incompatible at first glance. The first one produces scalar structured light fields in phase and amplitude with high accuracy, while the second one is preferred for generating vector beams of great stability although of reduced quality. Specifically, the idea behind our technique is to send the shaped light produced by CAM co-axially with the zeroth order, rather than diffracted to the first order, as it is commonly done. We first describe our technique, explaining the generation of the hologram and experimental setup to isolate the desired vector mode, and then present experimental results that corroborate our approach. We first address the quality of the generated beams using Stokes polarimetry to reconstruct their transverse polarisation distribution, and then compare their stability against the same mode produced using a Sagnac interferometric method. Our vector beams are of good quality and remarkably stable, two qualities that we expect will appeal to the community working with vector modes.

Enhancing Knowledge Graph Embedding with Hierarchical Self-Attention and Graph Neural Network Techniques for Drug-Drug Interaction Prediction in Virtual Reality Environments

In biomedicine, the critical task is to decode Drug–Drug Interactions (DDIs) from complex biomedical texts. The scientific community employs Knowledge Graph Embedding (KGE) methods, enhanced with advanced neural network technologies, including capsule networks. However, existing methodologies primarily focus on the structural details of individual entities or relations within Biomedical Knowledge Graphs (BioKGs), overlooking the overall structural context of BioKGs, molecular structures, positional features of drug pairs, and their critical Relational Mapping Properties. To tackle the challenges identified, this study presents HSTrHouse an innovative hierarchical self-attention BioKGs embedding framework. This architecture integrates self-attention mechanisms with advanced neural network technologies, including Convolutional Neural Network (CNN) and Graph Neural Network (GNN), for enhanced computational modeling in biomedical contexts. The model bifurcates the BioKGs into entity and relation layers for structural analysis. It employs self-attention across these layers, utilizing PubMedBERT and CNN for position feature extraction, and a GNN for drug pair molecular structure analysis. Then, we connect the position and molecular structure features to integrate them into the self-attention calculation of entity and relation. After that, the output of the self-attention layer is combined with the connected vectors of the position feature and molecular structure feature to obtain the final representation vector, and finally, to model the Relational Mapping Properties (RMPs), the representation vector is embedded into the complex vector space using Householder projections to obtain the BioKGs model. The paper validates HSTrHouse’s efficacy by comparing it with advanced models on three standard BioKGs for DDIs research.

Invariant Grassmannians and a K3 surface with an action of order 192*2

Given a complex vector space V of finite dimension, its Grassmannian variety parametrizes all subspaces of V of a given dimension. Similarly, if a finite group G acts on V, its invariant Grassmannian parametrizes all the G−invariant subspaces of V of a given dimension. Based on this fact, we develop an algorithm for finding equations of G−invariant projective varieties arising as an intersection of hypersurfaces of the same degree.We apply the algorithm to find equations describing a polarized K3 surface with a faithful action of T192⋊μ2 and some further symmetric K3 surfaces with a degree 8 polarization.

The amoeba dimension of a linear space

Given a complex vector subspace V V of C n \mathbb {C}^n , the dimension of the amoeba of V ∩ ( C ∗ ) n V \cap (\mathbb {C}^*)^n depends only on the matroid that V V defines on the ground set { 1 , … , n } \{1,\ldots ,n\} . Here we prove that this dimension is given by the minimum of a certain function over all partitions of the ground set, as previously conjectured by Rau. We also prove that this formula can be evaluated in polynomial time.

Kähler-Yang-Mills Equations and Vortices

The Kähler-Yang-Mills equations are coupled equations for a Kähler metric on a compact complex manifold and a connection on a complex vector bundle over it. After briefly reviewing the main aspects of the geometry of the Kähler-Yang-Mills equations, we consider dimensional reductions of the equations related to vortices — solutions to certain Yang-Mills-Higgs equations.

THE RELEVANCE OF THE SYSTEM OF DIRECT CONTROL OF THE TORQUE OF AN ASYNCHRONOUS MOTOR

In the article, a mathematical model of the DTC (direct torque control) system of an alternating current electric drive is developed for the tasks of urban electric transport rolling stock, as an alternative to complex vector control of an asynchronous motor, taking into account its relevance, principles of operation and causes of occurrence. The research of the mathematical model of the direct control of the torque of the asynchronous motor in MATLAB was carried out, its main shortcomings, which limit the scope of its application, were revealed. The main task of the DTC system is to identify the stator flux linkage, which is solved by integration with error accumulation. The source of error accumulation is the inaccuracy of determining the active resistance of the stator and its changes during the operation of the asynchronous motor. The DTC system has a disadvantage associated with the appearance of torque ripples and rotor speed fluctuations at low loads.

A Synthesis Method of Open-Chain Coupled Serial Linkage Mechanism Fusing Trajectory and Posture

In order to meet the motion characteristics requirements of multi-objective fusion of execution components, a synthesis method of open-chain coupled serial linkage mechanism fusing trajectory and posture is proposed. According to periodic change characteristics of the execution component trajectory and posture of the linkage mechanism, the motion characteristics equation including trajectory and posture information is established by using Fourier series. The harmonic parameters of the link trajectory and posture are calculated according to the principle of discrete Fourier transform. The complex vector theory is applied to determine the functional relationship between the harmonic parameters of the trajectory and posture and the independent variable of the linkage mechanism, and furthermore the motion synthesis equation of the open-chain linkage mechanism fusing trajectory and posture information is established. The linkage mechanism parameters are calculated using Mathcad Prime software. The single-degree-of-freedom open-chain coupled serial linkage mechanism fusing trajectory and posture information is obtained by using a noncircular gear planetary gear train to couple the connection points of the linkage mechanism. The correctness of the method is verified through an example.

Estimates of Picard modular cusp forms

Abstract In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

Biomedical Knowledge Graph Embedding with Householder Projection (Student Abstract)

Researchers have applied knowledge graph embedding (KGE) techniques with advanced neural network techniques, such as capsule networks, for predicting drug-drug interactions (DDIs) and achieved remarkable results. However, most ignore molecular structure and position features between drug pairs. They cannot model the biomedical field's significant relational mapping properties (RMPs,1-N, N-1, N-N) relation. To solve these problems, we innovatively propose CDHse that consists of two crucial modules: 1) Entity embedding module, we obtain position feature obtained by PubMedBERT and Convolutional Neural Network (CNN), obtain molecular structure feature with Graphic Nuaral Network (GNN), obtain entity embedding feature of drug pairs, and then incorporate these features into one synthetic feature. 2) Knowledge graph embedding module, the synthetic feature is Householder projections and then embedded in the complex vector space for training. In this paper, we have selected several advanced models for the DDIs task and performed experiments on three standard BioKG to validate the effectiveness of CDHse.

Longitudinally continuous varying high-order cylindrical vector fields enabled by spin-decoupled metasurfaces

Abstract The manipulation of vector optical fields in three-dimensional (3D) space plays a vital role in both fundamental research and practical implementations of polarization optics. However, existing studies mostly focus on 3D vector optical fields with limited modes. Here, an approach of spin-decoupled spatial partitioning is proposed to generate complex 3D vector optical fields with a customizable number of modes on demand. The crosstalk among different modes is effectively suppressed by the decoupling capability of asymmetric photonic spin–orbit interactions (PSOIs) and the design of region displacement for opposite spin states. As a proof-of-concept demonstration, a metasurface is designed to generate longitudinally varying high-order cylindrical vector fields, ranging from the 2nd to the 10th order in even sequences along the propagation direction. The experimental results demonstrate the effectiveness and potential of our approach to enabling precise control of 3D vector optical fields with arbitrary mode combinations. This work holds promising applications in biophotonics, quantum optics, and communications.

Design of Complex Vector Controller for High-Power Induction Machine Drive

In high-power drives of large commercial vehicles, the switching frequency of the inverter is always strictly limited to avoid high switching losses. In this paper, a high-performance complex vector controller in discrete domain is proposed to reduce the cross-coupling effect and improve control performance under low switching frequency. The discretization error of the induction motor model under different coordinate systems is explored, and a high-precision discrete induction motor model is proposed for the design of the complex vector controller and flux observer. In order to improve the dynamic response of the rotor flux linkage, the designed controller takes the current as the inner loop and the rotor flux linkage as the outer loop. Besides, the signal reconstruction strategy is proposed to solve the contradiction between the variable-step characteristics caused by synchronous modulation and the discretization process based on constant step size. Finally, the designed control strategy and analysis of the system are validly verified by simulation and experimental results.

Comparative Study of Advanced Modulation and Control Schemes for Dual Three-Phase PMSM Drives With Low Switching Frequencies

In this paper, multiple advanced low-switching-frequency modulation and control schemes are investigated for dual three-phase permanent-magnet synchronous motor (PMSM) drives. The modulation schemes under investigation include multisampling space vector modulation (MS-SVM), selective harmonic elimination pulse width modulation (SHEPWM) and synchronous optimal pulse width modulation (SOPWM) while the control schemes include complex vector control, model predictive control (MPC) and flux trajectory control-based model predictive pulse pattern control (MP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> C). The difference between three-phase and dual three-phase PMSM drives at low switching frequencies is analyzed. Moreover, an optimal pulse width modulation (PWM)-based MP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> C scheme is proposed for the dual three-phase PMSM, where the optimal PWM modulation and MP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> C control are combined such that the optimal steady-state and dynamic control performance is achieved among the modulation and control schemes investigated. Experimental results are given to compare and validate the proposed control scheme.

Generation of multiple vector beams through cascaded beam displacers and a segmented digital hologram

Complex vector light modes, characterized by a non-uniform transverse polarization distribution, have pervaded a wide range of research fields. In this study, we propose an approach that enables the simultaneous generation of multiple vector beams based on a spatially segmented digital hologram and two or more cascaded beam displacers. More precisely, an input beam is separated into multiple parallel copies spatially separated, which are then sent to the center of each segmented hologram, enabling independent modulation of each beam. The modulated beams are then judiciously recombined with a beam displacer to generate multiple vector modes in a simultaneous way. We demonstrated our technique with two arbitrary vector modes but the technique can be easily extended to more by inserting additional beam displacers. To assess the quality of the generated vector modes, we employed Stokes polarimetry to reconstruct their transverse polarization distribution and to measure their degree of non-separability. We envision that this technique will find significant applications in various fields, including optical communications, optical sensing, and optical tweezers to mention a few.

Acoustic metamaterials for realizing a scalable multiple phi-bit unitary transformation

The analogy between acoustic modes in nonlinear metamaterials and quantum computing platforms constituted of correlated two-level systems opens new frontiers in information science. We use an inductive procedure to demonstrate scalable initialization of and scalable unitary transformations on superpositions of states of multiple correlated logical phi-bits, classical nonlinear acoustic analog of qubits. A multiple phi-bit state representation as a complex vector in a high-dimensional, exponentially scaling Hilbert space is shown to correspond with the state of logical phi-bits represented in a low-dimensional linearly scaling physical space of an externally driven acoustic metamaterial. Manipulation of the phi-bits in the physical space enables the implementation of a non-trivial multiple phi-bit unitary transformation that scales exponentially. This scalable transformation operates in parallel on the components of the multiple phi-bit complex state vector, requiring only a single physical action on the metamaterial. This work demonstrates that acoustic metamaterials offer a viable path toward achieving massively parallel information processing capabilities that can challenge current quantum computing paradigms.

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z(N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian–Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase difference in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

What conjugate phase retrieval complex vectors can do in quaternion Euclidean spaces

Abstract Quaternion algebra ℍ {\mathbb{H}} is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space ℍ M {\mathbb{H}^{M}} with M ≥ 2 {M\geq 2} . Write ℂ η = { ξ : ξ = ξ 0 + β ⁢ η , ξ 0 , β ∈ ℝ } {\mathbb{C}_{\eta}=\{\xi:\xi=\xi_{0}+\beta\eta,\,\xi_{0},\,\beta\in\mathbb{R}\}} for η ∈ { i , j , k } {\eta\in\{i,\,j,\,k\}} . We remark that not only ℂ η M {\mathbb{C}_{\eta}^{M}} -vectors cannot allow traditional conjugate phase retrieval in ℍ M {\mathbb{H}^{M}} , but also ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors cannot allow phase retrieval in ℍ M {\mathbb{H}^{M}} . We are devoted to conjugate phase retrieval of ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors in ℍ M {\mathbb{H}^{M}} , where “conjugate” is not the traditional conjugate. We introduce the notions of conjugation, maximal commutative subset and conjugate phase retrieval. Using the phase lifting techniques, we present some sufficient conditions on complex vectors allowing conjugate phase retrieval. And some examples are also provided to illustrate the generality of our theory.

Monolayer Chiral Metasurface for Generation of Arbitrary Cylindrical Vector Beams

The cylindrical vector beam (CVB) has been widely studied and applied in recent years. However, many CVB generation methods suffer from complex systems, and large-size devices are required. Here, we propose a monolayer chiral metasurface composed of spin-sensitive unit cells which can generate different holograms for left- and right-circular polarization based on the combined modulation of geometric phase and detour phase. With a linearly polarized incident beam, the metasurface can generate CVBs with controllable polarization angles and orders, and even more complex vector beams. This work provides a new idea for the design of miniaturized optical devices for generating arbitrary vector beams.

Distribution of zeros of functions with exponential growth

For systems of equations with an infinite number of roots one can sometimes establish results of the type of the Kushnirenko-Bernstein-Khovanskii theorem by replacing the calculation of the number of the roots by the calculation of the asymptotic density of these roots. We consider systems of entire functions with exponential growth in $\mathbb C^n$ and calculate the asymptotic behaviour of the averaged distribution of their zeros in terms of the geometry of convex bodies in a complex vector space. Bibliography: 11 titles.