Group Of Linear Transformations Research Articles

Quasidifferentiability and Uniform Observability of Linear Time-Varying Singularly Perturbed Systems

For linear time-varying singularly perturbed systems (LTVSPS) with quasidifferentiable coefficients and a small parameter multiplying some derivatives, the problem of uniform observability is considered. Necessary and sufficient conditions for the quasidifferentiability of the set of output functions independent of the small parameter are proved, observability matrices independent of the small parameter for the slow subsystem and the family of fast subsystems associated with the LTVSPS are constructed, and a connection is established between them and the observability matrix of the original system. On the basis of the complete decomposition of the original LTVSPS with respect to the action of the group of linear nonsingular transformations, we prove rank sufficient conditions for the uniform observability of the LTVSPS that are independent of the small parameter and valid for all sufficiently small values of it. The conditions are expressed in terms of the observability matrices of the slow subsystem and the family of fast subsystems of smaller dimensions than the original LTVSPS.

Continued fractions related to a group of linear fractional transformations

Abstract There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group G 3 , 3 {G}_{3,3} generated by the linear fractional transformations a = 1 − 1 ∕ z a=1-1/z and b = z + 2 b=z+2 . This group is the unique subgroup of the modular group PSL ( 2 , Z ) {\rm{PSL}}(2,{\mathbb{Z}}) with index 2. We calculate the cusp point of an element given as a word in generators. Conversely, we use the continued fraction expansion of a given rational number p ∕ q p/q , to obtain an element in G 3 , 3 {G}_{3,3} with cusp point p ∕ q p/q . As a result, we say that the action of G 3 , 3 {G}_{3,3} on rational numbers is transitive.

Rank of a tensor and quantum entanglement

The rank of a tensor is analysed in the context of quantum entanglement. A pure quantum state of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for the characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state is called entangled, if it cannot be written in the product form, , what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed.

On the normalizer of the reflexive cover of a unital algebra of linear transformations

Given a unital algebra A of linear transformations on a finite-dimensional complex vector space V, in this paper we study the set Col(A) consisting of those invertible linear transformations S on V which map every subspace M∈Lat(A) to a subspace SM∈Lat(A). We show that Col(A) is the normalizer of the group of invertible linear transformations in the reflexive cover of A. For the unital algebra (A) which is generated by a linear transformation A, we give the complete description of Col(A). By using the primary decomposition of A, we first reduce the problem of characterizing Col(A) to the problem of characterizing the group Col(N) of a given nilpotent linear transformation N. While Col(N) always contains all invertible linear transformations of the commutant (N)′ of N, it is always contained in the reflexive cover of (N)′. We prove that Col(N) is a proper subgroup of (AlgLat(N)′)−1 if and only if at least two Jordan blocks in the Jordan decomposition of N are of dimension 2 or more. We also determine the group Col(J2⊕J2).

Polarization multiplexed diffractive computing: all-optical implementation of a group of linear transformations through a polarization-encoded diffractive network

Research on optical computing has recently attracted significant attention due to the transformative advances in machine learning. Among different approaches, diffractive optical networks composed of spatially-engineered transmissive surfaces have been demonstrated for all-optical statistical inference and performing arbitrary linear transformations using passive, free-space optical layers. Here, we introduce a polarization-multiplexed diffractive processor to all-optically perform multiple, arbitrarily-selected linear transformations through a single diffractive network trained using deep learning. In this framework, an array of pre-selected linear polarizers is positioned between trainable transmissive diffractive materials that are isotropic, and different target linear transformations (complex-valued) are uniquely assigned to different combinations of input/output polarization states. The transmission layers of this polarization-multiplexed diffractive network are trained and optimized via deep learning and error-backpropagation by using thousands of examples of the input/output fields corresponding to each one of the complex-valued linear transformations assigned to different input/output polarization combinations. Our results and analysis reveal that a single diffractive network can successfully approximate and all-optically implement a group of arbitrarily-selected target transformations with a negligible error when the number of trainable diffractive features/neurons (N) approaches N_pN_iN_o, where Ni and No represent the number of pixels at the input and output fields-of-view, respectively, and Np refers to the number of unique linear transformations assigned to different input/output polarization combinations. This polarization-multiplexed all-optical diffractive processor can find various applications in optical computing and polarization-based machine vision tasks.

An index formula for groups of isometric linear canonical transformations

We define a representation of the unitary group $U(n)$ by metaplectic operators acting on $L^2(\mathbb{R}^n)$ and consider the operator algebra generated by the operators of the representation and pseudodifferential operators of Shubin class. Under suitable conditions, we prove the Fredholm property for elements in this algebra and obtain an index formula.

Normal subgroups in the group of column-finite infinite matrices

Abstract The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

Stationary orbits of linear time-varying observation systems

In terms of matrix observability, the necessary and sufficient conditions are obtained for the linear timevarying observation system to have stationary orbits with respect to the linear time-varying transformation group of class C1 . The full invariant of the action of a transformation group is described. It is proved that for any matrix function A c C(T, Rn×n ), there exists such an n-vector function c(t), t c T, that the pair (A, c) is uniformly observable. The algorithm for constructing a stationary system is described.

Reflections on the structural and constitutive approaches to the theory of defects

An attempt is made to bring into harmony two of the paradigms commonly used in the theory of continuous distributions of defects. It is shown that the common differential geometric apparatus is provided neatly by the theory of G-structures. In the case of a structural model, based on putative experimental observations at the microscopic level, a G-structure can be shown to emerge from the group of linear transformations that preserve a tensorial quantity. For the phenomenological (macroscopic) constitutive model, the G-structure arises from the notion of material isomorphism and the underlying local symmetry group of the constitutive law. A comparative example is presented in the framework of certain smectic liquid crystals.

<i>LS</i> (3)-equivalence conditions of control points and application to spatial Bézier curves and surfaces

Let $G$ be a transformation group and act on $X$. Any elements $x, y \in X$ are called the $G-$equivalent elements if there exist a transformation $g \in G$ such that $y = gx$ is satisfied. Similarly let $A = \left\{ x_{1}, x_{2}, ..., x_{n} \right\}$ and $B = \left\{ y_{1}, y_{2}, ..., y_{n} \right\}$ be any two subspaces of $X$ with $n-$elements. Then the subspaces $A$ and $B$ are called the $G-$equivalent subspaces if there exist a transformation $g \in G$ such that $y_{i} = gx_{i}$ is satisfied for every $i = 1, 2, ..., n$. The linear similarity transformations' group in 3 dimensional Euclidean space will be denoted by $LS(3)$. This paper presents the $G-$equivalence conditions of the subspaces $A$ and $B$ of 3-dimensional Euclidean space $E^{3}$ with $m-$elements where the transformation group $G = LS(3)$ is the linear similarity transformation group in $E^{3}$. Later the $G = LS(3)-$equivalence conditions of Bezier curves and surfaces are studied in terms of the rational $G = LS(3)$ invariants of their control points. Finally by using quadratic Bezier curves, a simple letter S is designed and two different shadow curves of this letter (composite curves) are obtained. Then it is emphasized that these shadow curves are $G = LS(3)-$ equivalent to designed letter S.

The Maxwell-Cassano Equations of an Electromagnetic-nuclear Field Yields the Fermion & Hadron Architecture

The Helmholtzian operator and factorization, via the Maxwell-Cassano equations yields a fermion architecture table equivalent to that of the standard model and lead to linear transformation groups of the mesons and baryons, respectively; plus a straightforward elementary description of quark colour based on integral indices: 1,0,1, rather than the subjective, correlative explanation using: {R,G,B;Y} indexes.

Affine connections on 3-Sasakian and manifolds

The space of invariant affine connections on every 3-Sasakian homogeneous manifold of dimension at least seven is described. In particular, the subspace of invariant affine metric connections and the subclass with skew torsion are also determined. To this aim, an explicit construction of all 3-Sasakian homogeneous manifolds is exhibited. It is shown that the 3-Sasakian homogeneous manifolds which admit nontrivial Einstein with skew torsion invariant affine connections are those of dimension seven, that is, $${\mathbb {S}}^7$$, $${\mathbb {R}}P^7$$ and the Aloff–Wallach space $${\mathfrak {W}}^{7}_{1,1}$$. On $${\mathbb {S}}^7$$ and $${\mathbb {R}}P^7$$, the set of such connections is bijective to two copies of the conformal linear transformation group of the Euclidean space, while it is strictly bigger on $${\mathfrak {W}}^{7}_{1,1}$$. The set of invariant connections with skew torsion whose Ricci tensor satisfies that its eigenspaces are the canonical vertical and horizontal distributions, is fully described on 3-Sasakian homogeneous manifolds. An affine connection satisfying these conditions is distinguished, by parallelizing all the Reeb vector fields associated with the 3-Sasakian structure, which is also Einstein with skew torsion on the 7-dimensional examples. The invariant metric affine connections on 3-Sasakian homogeneous manifolds with parallel skew torsion have been found. Finally, some results have been adapted to the non-homogeneous setting.

Moment Functions on Affine Groups

Moments of probability measures on a hypergroup can be obtained from so-called (generalized) moment functions of a given order. The aim of this paper is to characterize generalized moment functions on a non-commutative affine group. We consider a locally compact group G and its compact subgroup K. First we recall the notion of the double coset space G / / K of a locally compact group G and introduce a hypergroup structure on it. We present the connection between K-spherical functions on G and exponentials on the double coset hypergroup G / / K. The definition of the generalized moment functions and their connection to the spherical functions is discussed. We study an important class of double coset hypergroups: specyfing K as a compact subgroup of the group of inverible linear transformations on a finitely dimensional linear space V we consider the affine group {mathrm {Aff,}}K. Using the fact that in the finitely dimensional case ({mathrm {Aff,}}K,K) is a Gelfand pair we give a description of exponentials on the double coset hypergroup {mathrm {Aff,}}K//K in terms of K-spherical functions. Moreover, we give a general description of generalized moment functions on {mathrm {Aff,}}K and specific examples for K=SO(n), and on the so-called ax+b-group.

Intertwining solutions for magnetic relativistic Hartree type equations

We consider the magnetic pseudo-relativistic Schrödinger equationwhere , m > 0, is an external continuous scalar potential, is a continuous vector potential and is a convolution kernel, is a constant, , . We assume that A and V are symmetric with respect to a closed subgroup G of the group of orthogonal linear transformations of .If for any , the cardinality of the G-orbit of x is infinite, then we prove the existence of infinitely many intertwining solutions assuming that is either linear in x or uniformly bounded. The results are proved by means of a new local realization of the square root of the magnetic laplacian to a local elliptic operator with Neumann boundary condition on a half-space. Moreover we derive an existence result of a ground state intertwining solution for bounded vector potentials, if G admits a finite orbit.

On Hotelling’s T2 test in a special paired sample case

ABSTRACTIn a special paired sample case, Hotelling’s T2 test based on the differences of the paired random vectors is the likelihood ratio test for testing the hypothesis that the paired random vectors have the same mean; with respect to a special group of affine linear transformations it is the uniformly most powerful invariant test for the general alternative of a difference in mean. We present an elementary straightforward proof of this result. The likelihood ratio test for testing the hypothesis that the covariance structure is of the assumed special form is derived and discussed. Applications to real data are given.

Rationality of Hilbert series in noncommutative invariant theory

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has a rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tensor algebra that arise naturally in the theory of polynomial identities.

Hochschild cohomology of the Weyl algebra and Vasiliev’s equations

We propose a simple injective resolution for the Hochschild complex of the Weyl algebra. By making use of this resolution, we derive explicit expressions for nontrivial cocycles of the Weyl algebra with coefficients in twisted bimodules as well as for the smash products of the Weyl algebra and a finite group of linear symplectic transformations. A relationship with the higher-spin field theory is briefly discussed.

Automorphism groups of Gaussian Bayesian networks

In this paper, we extend earlier work on groups acting on Gaussian graphical models to Gaussian Bayesian networks and more general Gaussian models defined by chain graphs with no induced subgraphs of the form $i\to j-k$. We fully characterise the maximal group of linear transformations which stabilises a given model and we provide basic statistical applications of this result. This includes equivariant estimation, maximal invariants for hypothesis testing and robustness. In our proof, we derive simple necessary and sufficient conditions on vanishing subminors of the concentration matrix in the model. The computation of the group requires finding the essential graph. However, by applying Studeny’s theory of imsets, we show that computations for DAGs can be performed efficiently without building the essential graph.

Cherednik and Hecke Algebras of Varieties with a Finite Group Action

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

Unitary rotation and gyration of pixelated images on rectangular screens.

In the two space dimensions of screens in optical systems, rotations, gyrations, and fractional Fourier transformations form the Fourier subgroup of the symplectic group of linear canonical transformations: U(2)F⊂Sp(4,R). Here, we study the action of this Fourier group on pixelated images within generic rectangular Nx×Ny screens; its elements here compose properly and act unitarily, i.e., without loss of information.