Linear Map Research Articles

Completeness of the ZH-calculus

There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be elegantly described by the ZX-calculus, a graphical language for a class of string diagrams describing linear maps between qubits. The ZX-calculus has proven useful in a variety of areas of quantum information, but is less suitable for reasoning about operations outside its natural gate set such as multi-linear Boolean operations like the Toffoli gate. In this paper we study the ZH-calculus, an alternative graphical language of string diagrams that does allow straightforward encoding of Toffoli gates and other more complicated Boolean logic circuits. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over Z[12], which correspond to the approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an extended version of the ZH-calculus that is complete with respect to matrices over any ring R where 1+1 is not a zero-divisor.

Existence of Lie algebroids on the tangent bundle with a given anchor map of constant rank

We show that given a constant rank linear map, K:TM→TM, there exists a Lie algebroid with K as its anchor map, if and only if the image distribution, ImK, is involutive. As a byproduct, a new example of Lie algebroid bracket associated with a regular foliation is obtained through the projector onto the involutive distribution. The Lie algebroid bracket is not defined on the involutive distribution but on the whole space of vector fields of the manifold.

Deformations and Extensions of BiHom-alternative algebras

The aim of this paper is to deal with BiHom-alternative algebras which are a generalization of alternative and Hom-alternative algebras, their structure is defined with two commuting multiplicative linear maps. We study cohomology and one-parameter formal deformation theory of left BiHom-alternative algebras. Moreover, we study central and $T_\theta$-extensions of BiHom-alternative algebras and their relationship with cohomology. Finally, we investigate generalized derivations and give some relevant results.

Enriched purity and presentability in Banach spaces

The category of Banach spaces and linear maps of norm is locally -presentable but not locally finitely presentable. We prove, however, that is locally finitely presentable in the enriched sense over complete metric spaces. Moreover, in this sense, pure morphisms are just ideals of Banach spaces. We characterize classes of Banach spaces approximately injective with respect to sets of morphisms having finite-dimensional domains and separable codomains.

LDA enhanced one-bit compressive sensing method for high-throughput mass spectrometry data feature selection

Mass spectrometry (MS) is commonly used for the proteins detection of biological samples. However, it generates high-throughput data, posing huge computational challenges for signal processing. In previous studies, we have proposed a one-bit perturbed block CS (OB-PBCS) method to realize feature selection (FS) and obtain a feature set with reduced dimension for classification. However, its feature selection is realized by solving the reconstruction problem with simple greedy strategies, which fails to consider the final classification objective during reconstruction. Considering such problem, in this paper, we propose a hybrid linear discriminant analysis (LDA) enhanced one-bit CS method, called L-OBPBCS, which selects features by considering the subsequent classification and consists of three parts. The first part roughly filters out the features that do not favor the classification task with the proposed single feature decision coefficient. Then, OB-PBCS obtains significant features from the filtered feature set. Afterwards, LDA algorithm is further used for refining, where the amplitude of the optimal projection vector is gathered as the linear discrimination mapping coefficients to identify important features. To show the efficiency of our proposed L-OBPBCS method, classification based on the selected features for both synthesized dataset and real ovarian cancer MS dataset is evaluated. Results show that the feature set derived by L-OBPBCS ensures excellent classification performance, which surpasses the existing one-bit type FS method, and the computational complexity of L-OBPBCS is obviously reduced compared with the existing OB-PBCS method.

Nonlinear Mapping From EMG to Prosthesis Closing Velocity Improves Force Control With EMG Biofeedback.

When using EMG biofeedback to control the grasping force of a myoelectric prosthesis, subjects need to activate their muscles and maintain the myoelectric signal within an appropriate interval. However, their performance decreases for higher forces, because the myoelectric signal is more variable for stronger contractions. Therefore, the present study proposes to implement EMG biofeedback using nonlinear mapping, in which EMG intervals of increasing size are mapped to equal-sized intervals of the prosthesis velocity. To validate this approach, 20 non-disabled subjects performed force-matching tasks using Michelangelo prosthesis with and without EMG biofeedback with linear and nonlinear mapping. Additionally, four transradial amputees performed a functional task in the same feedback and mapping conditions. The success rate in producing desired force was significantly higher with feedback (65.4±15.9%) compared to no feedback (46.2±14.9%) as well as when using nonlinear (62.4±16.8%) versus linear mapping (49.2±17.2%). Overall, in non-disabled subjects, the highest success rate was obtained when EMG biofeedback was combined with nonlinear mapping (72%), and the opposite for linear mapping with no feedback (39.6%). The same trend was registered also in four amputee subjects. Therefore, EMG biofeedback improved prosthesis force control, especially when combined with nonlinear mapping, which showed to be an effective approach to counteract increasing variability of myoelectric signal for stronger contractions.

Optical Imaging Polarimetry of Comet 21P/Giacobini–Zinner during Its 2018 Apparition

We conducted polarimetric observations of comet 21P/Giacobini–Zinner in the optical wavelength region using the polarimetric imager PICO equipped with the 50 cm telescope for Public Outreach at the Mitaka Campus of the National Astronomical Observatory of Japan on UT 2018 September 16, at a solar phase angle of 77° for the comet. We used Gunn’s i′-band filter to cover the wavelength region where sunlight reflected by dust grains dominates cometary spectra. The intensity map of comet 21P/Giacobini–Zinner showed an elongated coma structure along the east–west direction. The linear polarization degree map was almost uniform and showed no evidence of jet or arc structures in the inner coma. The linear polarization degree was 22% in the coma, which is consistent with other optical polarization observations of 21P/Giacobini–Zinner for this and previous apparitions reported in the literature. The measured polarization angles (179°) were nearly perpendicular to the position angle of the scattering plane of 271°. There is no evidence of significant changes in the dust properties (composition, porosity, and size distribution) in the comet from its 1985 to 2018 apparitions. The spectral gradient of the linear polarization degree for the comet was nearly flat in the wavelength range of ∼510–830 nm at a solar phase angle of ∼77° during the 2018 apparition.

Stochastic linear quadratic optimal control problems with expectation-type linear equality constraints on the terminal states

A stochastic linear quadratic (LQ) optimal control problem with an expectation-type linear equality constraint on the terminal state is considered. Under the solvability condition on a stochastic Riccati equation and a surjectivity condition on the linear constraint mapping, the constrained stochastic LQ problem is solved completely by the Lagrangian duality theory. Some equivalent characterizations of the surjectivity condition are discussed by the controllability theory of linear control systems. Especially, the equivalence between the surjectivity condition and a Kalman-type rank condition is proved under proper conditions.

Lower bounds on the low-distortion embedding dimension of submanifolds of Rn

Let M be a smooth submanifold of Rn equipped with the Euclidean (chordal) metric. This note considers the smallest dimension m for which there exists a bi-Lipschitz function f:M↦Rm with bi-Lipschitz constants close to one. The main result bounds the best achievable embedding dimension m below in terms of the Lipschitz constants of f as well as the reach, volume, diameter, and dimension of M. This new lower bound is then applied to show that prior upper bounds for m via random matrices by Eftekhari and Wakin [6] are near-optimal in a wide range of settings. This supports random linear maps as being nearly as efficient as nonlinear maps at reducing the ambient dimension for manifold data in many situations. Along the way, we also prove results concerning the impossibility of achieving better nonlinear measurement maps with the Restricted Isometry Property (RIP) in compressive sensing applications.

Compatible ideals in ℚ-Gorenstein rings

Suppose R R is a F F -finite and F F -pure Q \mathbb {Q} -Gorenstein local ring of prime characteristic p > 0 p>0 . We show that an ideal I ⊆ R I\subseteq R is uniformly compatible ideal (with all p − e p^{-e} -linear maps) if and only if exists a module finite ring map R → S R\to S such that the ideal I I is the sum of images of all R R -linear maps S → R S\to R . In other words, the set of uniformly compatible ideals is exactly the set of trace ideals of finite ring maps.

Circular SAR Incoherent 3D Imaging with a NeRF-Inspired Method

Circular synthetic aperture radar (CSAR) has the potential to form 3D images with single-pass single-channel radar data, which is very time-efficient. This article proposes a volumetric neural renderer that utilizes CSAR 2D amplitude images to reconstruct the 3D power distribution of the imaged scene. The innovations are two-fold: Firstly, we propose a new SAR amplitude image formation model that establishes a linear mapping relationship between multi-look amplitude-squared SAR images and a real-valued 4D (spatial location (x, y, z) and azimuth angle θ) radar scattered field. Secondly, incorporating the proposed image formation model and SAR imaging geometry, we extend the neural radiance field (NeRF) methods to reconstruct the 4D radar scattered field using a set of 2D multi-aspect SAR images. Using real-world drone SAR data, we demonstrate our method for (1) creating realistic SAR imagery from arbitrary new viewpoints and (2) reconstructing high-precision 3D structures of the imaged scene.

Generalized iterated-sums signatures

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Király and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map on the tensor algebra by considering a deformed quasi-shuffle product of words. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.

An unsupervised learning-based generalization of Data Envelopment Analysis

In this paper, we introduce an unsupervised machine learning method for production frontier estimation. This new approach satisfies fundamental properties of microeconomics, such as convexity and free disposability (shape constraints). The new method generalizes Data Envelopment Analysis (DEA) through the adaptation of One-Class Support Vector Machines with piecewise linear transformation mapping. The new technique aims to reduce the overfitting problem occurring in DEA. How to measure technical inefficiency through the directional distance function is also introduced. Finally, we evaluate the performance of the new technique via a computational experience, showing that the mean squared error in the estimation of the frontier is up to 83% better than the standard DEA in certain scenarios.

Multiple mixed state variable incremental integration for reconstructing extreme multistability in a novel memristive hyperchaotic jerk system with multiple cubic nonlinearity

Memristor-based chaotic systems with infinite equilibria are interesting because they generate extreme multistability. Their initial state-dependent dynamics can be explained in a reduced-dimension model by converting the incremental integration of the state variables into system parameters. However, this approach cannot solve memristive systems in the presence of nonlinear terms other than the memristor term. In addition, the converted state variables may suffer from a degree of divergence. To allow simpler mechanistic analysis and physical implementation of extreme multistability phenomena, this paper uses a multiple mixed state variable incremental integration (MMSVII) method, which successfully reconstructs a four-dimensional hyperchaotic jerk system with multiple cubic nonlinearities except for the memristor term in a three-dimensional model using a clever linear state variable mapping that eliminates the divergence of the state variables. Finally, the simulation circuit of the reduced-dimension system is constructed using Multisim simulation software and the simulation results are consistent with the MATLAB numerical simulation results. The results show that the method of MMSVII proposed in this paper is useful for analyzing extreme multistable systems with multiple higher-order nonlinear terms.

Hybrid zonotopes: A new set representation for reachability analysis of mixed logical dynamical systems

This article presents a new set representation named the hybrid zonotope that is equivalent to the union of 2N constrained zonotopes – convex polytopes – through the addition of N binary zonotope factors. The major contribution of this manuscript is a closed-form solution for exact forward reachable sets of discrete-time, linear hybrid systems modeled as mixed logical dynamical systems. The proposed approach captures the worst-case exponential growth in the number of convex sets required to represent the nonconvex reachable set while exhibiting only linear growth in the complexity of the hybrid zonotope set representation. Redundancy removal techniques are provided that leverage binary trees to store the combinations of binary factors of the hybrid zonotope that map to nonempty convex subsets. Numerical examples show the hybrid zonotope’s ability to compactly represent nonconvex reachable sets with an exponential number of features. Furthermore, the hybrid zonotope is shown to be closed under linear mappings, Minkowski sums, generalized intersections, and halfspace intersections.

Unraveling Twisty Linear Polarization Morphologies in Black Hole Images

We investigate general relativistic magnetohydrodynamic simulations to determine the physical origin of the twisty patterns of linear polarization seen in spatially resolved black hole images and explain their morphological dependence on black hole spin. By characterizing the observed emission with a simple analytic ring model, we find that the twisty morphology is determined by the magnetic field structure in the emitting region. Moreover, the dependence of this twisty pattern on spin can be attributed to changes in the magnetic field geometry that occur due to the frame dragging. By studying an analytic ring model, we find that the roles of Doppler boosting and lensing are subdominant. Faraday rotation may cause a systematic shift in the linear polarization pattern, but we find that its impact is subdominant for models with strong magnetic fields and modest ion-to-electron temperature ratios. Models with weaker magnetic fields are much more strongly affected by Faraday rotation and have more complicated emission geometries than can be captured by a ring model. However, these models are currently disfavoured by the recent EHT observations of M87*. Our results suggest that linear polarization maps can provide a probe of the underlying magnetic field structure around a black hole, which may then be usable to indirectly infer black hole spins. The generality of these results should be tested with alternative codes, initial conditions, and plasma physics prescriptions.

Image encryption based on combined between linear feedback shift registers and 3D chaotic maps

Protecting information sent through insecure internet channels is a significant challenge facing researchers. In this paper, we present a novel method for image data encryption that combines chaotic maps with linear feedback shift registers in two stages. In the first stage, the image is divided into two parts. Then, the locations of the pixels of each part are redistributed through the random numbers key, which is generated using linear feedback shift registers. The second stage includes segmenting the image into the three primary colors red, green, and blue (RGB); then, the data for each color is encrypted through one of three keys that are generated using three-dimensional chaotic maps. Many statistical tests (entropy, peak signal-noise ratio (PSNR), mean square error (MSE) and correlation) were conducted on a group of images to determine the strength and efficiency of the proposed method, and the result proves that the proposed method provided a good level of safety. The obtained results were compared with those of other methods, and the result of comparing confirms the superiority of the proposed method.

Superbiderivations of Simple Modular Lie Superalgebras of Witt Type and Special Type

Let [Formula: see text] be a simple modular Lie superealgebra of Witt type or special type over an algebraically closed field of characteristic [Formula: see text]. In this paper, all the superbiderivations of [Formula: see text] are studied. By means of weight space decompositions with respect to a suitable torus and the standard [Formula: see text]-grading structures of [Formula: see text], we show that the supersymmetric superbiderivations of [Formula: see text] are zero. Generalizing a result on the skewsymmetric biderivation of Lie algebras to the super case, we find that all the superbiderivations of [Formula: see text] are inner. As applications, the linear supercommuting maps and the supercommutative post-Lie superalgebra structures on [Formula: see text] are described.

Linear maps preserving matrices annihilated by a fixed polynomial

Let Mn(F) be the algebra of n×n matrices over an arbitrary field F. We consider linear maps Φ:Mn(F)→Mr(F) preserving matrices annihilated by a fixed polynomial f(x)=(x−a1)⋯(x−am) with m≥2 distinct zeroes a1,a2,…,am∈F; namely,f(Φ(A))=0wheneverf(A)=0.Suppose that f(0)=0, and the zero set Z(f)={a1,…,am} is not an additive group. Then Φ assumes the form(†)A↦S(A⊗D1At⊗D20s)S−1, for some invertible matrix S∈Mr(F), invertible diagonal matrices D1∈Mp(F) and D2∈Mq(F), where s=r−np−nq≥0. The diagonal entries λ in D1 and D2, as well as 0 in the zero matrix 0s, are zero multipliers of f(x) in the sense that λZ(f)⊆Z(f).In general, assume that Z(f)−a1 is not an additive group. If Φ(In) commutes with Φ(A) for all A∈Mn(F), or if f(x) has a unique zero multiplier λ=1, then Φ assumes the form (†).The above assertions follow from the special case when f(x)=x(x−1)=x2−x, for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map Φ:Mn(F)→Mr(F) sending disjoint rank one idempotents to disjoint idempotents always assume the above form (†) with D1=Ip and D2=Iq, unless Mn(F)=M2(Z2).

Contribution of Linear Algebra and Matrix in Mathematics: An analytical Study

In this we are presenting a study on the linear algebra and matrix in mathematics. Linear Algebra is the branch of mathematics concerned with the study of vectors, vector spaces also called linear spaces, linear maps also called linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics: thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has applications in the natural sciences and the social sciences. Since nonlinear models can often be approximated by linear ones