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We discuss dense coding with n copies of a specific preshared state between the sender and the receiver when the encoding operation is limited to the application of group representation. Typically, to act on multiple local copies of these preshared states, the receiver needs quantum memory, because in general the multiple copies will be generated sequentially. Depending on available encoding unitary operations, we investigate what preshared state offers an advantage of using quantum memory on the receiver’s side. [ABSTRACT FROM AUTHOR]

Let O (3) denote the group of orthogonal 3 × 3 real matrices, and M the 5-dimensional real vector space of all 3 × 3 real symmetric matrices with trace zero. Let λ 1 (A) ≤ λ 2 (A) ≤ λ 3 (A) be the eigenvalues of A ∈ M and Ξ - = { A ∈ M ∣ λ 1 (A) = λ 2 (A) } . M is an inner product space with the inner product ⟨ A , B ⟩ = trace (A B) . Let G 3 (M) be the set of all 3-dimensional subspaces of M , a 6-dimensional Grassman manifold. O (3) acts on M on the left by conjugation via inner product preserving linear isomorphisms, which map any 3-dimensional subspace into another 3-dimensional subspace; thus G 3 (M) also has a left action of O (3) . G 3 (M) becomes a category, an action groupoid, with morphisms (V , M , W) ∈ G 3 (M) × O (3) × G 3 (M) , where W = M V M T . Composition of morphisms is (V 1 , N , V 2) ∘ (V 0 , M , V 1) = (V 0 , N M , V 2) . Let C be a category whose objects (V, S) consist of a real inner product space V and S ⊂ V , and whose arrows (V , S) → (W , T) consist of f : V → W , an inner product preserving real linear mapping such that f (S) ⊂ T . We have the functor V ↓ (V , M , W) W G 3 (M) ⟶ F - (V , Ξ - ∩ V) ↓ F - (M) (W , Ξ - ∩ W) C where F - (M) : V → W : A ↦ M A M T . Suppose further that S 3 denotes the group of permutations of { 1 , 2 , 3 } , and ρ : S 3 → O (3) denotes a group homomorphism which is isomorphic as a group representation to the natural representation of S 3 on R 3 (which permutes the coordinates). Let Obj (L S) denote the set of all V ∈ G 3 (M) whose isotropy subgroup contains S = ρ (S 3) as a subgroup. This paper completely describes the full subcategory L S of G 3 (M) with object set Obj (L S) , as well as the details of the above functor restricted to L S . Thus all the members V ∈ Obj (L S) are determined, as well as the smooth manifold structure on Obj (L S) ; it is embedded as a one-dimensional submanifold of G 3 (M) . The isotropy subgroups of all V ∈ Obj (L S) are computed and all pairs V , W ∈ Obj (L S) which are isomorphic via some M ∈ O (3) are determined. The sets Ξ - ∩ V are all determined, and the functorial mappings on morphism sets are computed. However, L S is not a Lie groupoid. The image of Obj (L S) under the functor π 1 F - is the collection of fibres of the smooth manifold ∐ V ∈ Obj (L S) V , which is the total space of the canonical vector bundle over the base manifold Obj (L S) . The bifurcation points of the family of subsets Ξ - ∩ V as V ranges over Obj (L S) (within this total space) are seen to be the points of Obj (L S) with infinite isotropy subgroups. We also show how this mathematical problem arises naturally from a problem in mathematical chemistry. Hence certain features of numerical calculations of energy eigenvalue intersection patterns of the simple chemical system H3 are rationalized through linearization about the triple intersection point. [ABSTRACT FROM AUTHOR]

We introduce certain $C^*$ -algebras and k -graphs associated to k finite-dimensional unitary representations $\rho _1,\ldots ,\rho _k$ of a compact group G. We define a higher rank Doplicher-Roberts algebra $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ , constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this $C^*$ -algebra is isomorphic to a corner in the $C^*$ -algebra of a row-finite rank k graph $\Lambda $ with no sources. For G finite and $\rho _i$ faithful of dimension at least two, this graph is irreducible, it has vertices $\hat {G}$ and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate this with some examples when $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ is simple and purely infinite, and with some K -theory computations. [ABSTRACT FROM AUTHOR]

The paper considers a function introduced by the authors, which depends on one complex variable, two real variables, and one more argument, which defines a trivial or proper subgroup of a three-dimensional proper Lorentz group, which, therefore, is a real number or a pair of real numbers. In this case, the first three arguments define representation spaces and basis functions in these spaces. It is shown that its particular values can be expressed via the Coulomb wave functions or Appell's hypergeometric function . The resulting formula for the transformation of the function is used to derive a continual addition theorem for this function and calculate the one-dimensional Fourier–Mellin-type integral transforms of the product of two Coulomb functions; its result is expressed via the function . [ABSTRACT FROM AUTHOR]

The paper considers a function introduced by the authors, which depends on one complex variable, two real variables, and one more argument, which defines a trivial or proper subgroup of a three-dimensional proper Lorentz group, which, therefore, is a real number or a pair of real numbers. In this case, the first three arguments define representation spaces and basis functions in these spaces. It is shown that its particular values can be expressed via the Coulomb wave functions or Appell's hypergeometric function . The resulting formula for the transformation of the function is used to derive a continual addition theorem for this function and calculate the one-dimensional Fourier–Mellin-type integral transforms of the product of two Coulomb functions; its result is expressed via the function . [ABSTRACT FROM AUTHOR]

The author gives a definition of orbifold Stiefel-Whitney classes of real orbifold vector bundles over special q-CW complexes (i.e., right-angled Coxeter complexes). Similarly to ordinary Stiefel-Whitney classes, orbifold Stiefel-Whitney classes here also satisfy the associated axiomatic properties. [ABSTRACT FROM AUTHOR]

Large-scale data applications are becoming an increasingly integral part of how both public and private sector organisations function. The transition towards a data-driven society means that processes within organisations will be organised structurally differently than they used to be and that decision-making will be based on profiles and algorithms more often than not. This change requires several adjustments to the legal regime, both to make the best possible use of the opportunities this change has to offer and to lay down safeguards against dangers and risks. To facilitate this process, a number of changes is needed to the current, individual-centred legal paradigm, such as laying down a protective regime for non-personal data, providing protection to public interests and societal harms and granting a bigger role for representative and collective actions and public interest litigation. [ABSTRACT FROM AUTHOR]

We show that the first twisted cohomology group associated with closed 1-forms on differentiable manifolds is related to certain 2-dimensional representations of the fundamental group. In particular, we construct examples of nowhere-vanishing 1-forms with non-trivial twisted cohomology. [ABSTRACT FROM AUTHOR]

We study the balance of G-gain graphs, where G is an arbitrary group, by investigating their adjacency matrices and their spectra. As a first step, we characterize switching equivalence and balance of gain graphs in terms of their adjacency matrices in M n (C G) . Then we introduce a represented adjacency matrix, associated with a gain graph and a group representation, by extending the theory of Fourier transforms from the group algebra C G to the algebra M n (C G) . We prove that, anytime G admits a finite-dimensional faithful unitary representation π , a G-gain graph is balanced if and only if the spectrum of the represented adjacency matrix associated with π coincides with the spectrum of the underlying graph, with multiplicity given by the degree of the representation. We show that the complex adjacency matrix of complex unit gain graphs and the adjacency matrix of a cover graph are indeed particular cases of our construction. This enables us to recover some classical results and prove some new characterizations of balance in terms of spectrum, index or structure of these graphs. [ABSTRACT FROM AUTHOR]

Affine equivalent classes of Boolean functions have many applications in modern cryptography and circuit design. Previous publications have shown that affine equivalence on the entire space of Boolean functions can be computed up to 10 variables, but not on the quotient Boolean function space modulo functions of different degrees. Computing the number of equivalent classes of cosets of Reed-Muller code R (1 , n) is equivalent to classifying Boolean functions modulo linear functions, which can be computed only when n ≤ 7. Based on the linear representation of the affine group A G L (n , 2) on the quotient space R (s , n) / R (k , n) , we obtain a useful counting formula to compute the number of equivalent classes. Instead of computing the conjugacy classes and representatives directly in A G L (n , 2) , we reduce the computation complexity by introducing an isomorphic permutation group Pn and performing the computation in Pn. With the proposed algorithm, the number of equivalent classes of cosets of R(1,n) can be computed up to 10 variables. Furthermore, the number of equivalent classes on R (s , n) / R (k , n) can also be computed when − 1 ≤ k < s ≤ n ≤ 10, which is a major improvement and advancement comparing to previous methods. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let G be an arbitrary group. We define a gain-line graph for a gain graph (Γ , ψ) through the choice of an incidence G -phase matrix inducing ψ. We prove that the switching equivalence class of the gain function on the line graph L (Γ) does not change if one chooses a different G -phase inducing ψ or a different representative of the switching equivalence class of ψ. In this way, we generalize to any group some results proven by N. Reff in the abelian case. The investigation of the orbits of some natural actions of G on the set H Γ of G -phases of Γ allows us to characterize gain functions on Γ, gain functions on L (Γ) , their switching equivalence classes and their balance property. The use of group algebra valued matrices plays a fundamental role and, together with the matrix Fourier transform, allows us to represent a gain graph with Hermitian matrices and to perform spectral computations. Our spectral results also provide some necessary conditions for a gain graph to be a gain-line graph. [ABSTRACT FROM AUTHOR]

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the 'three-fold way'. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly 'complex' representations), those that are self-dual thanks to a symmetric bilinear pairing (which are 'real', in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are 'quaternionic', in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds-real, complex and quaternionic-can be seen as Hilbert spaces of the other kinds, equipped with extra structure. © 2011 Springer Science+Business Media, LLC.

By using the wave packet transformation on cyclic group, we study the wave packet transformation on finite abelian group. In the case of cyclic group, this is a transformation on Banach space formed by cyclic group representation. We generalise this result to transformation formed by group representation of an abelian group. [ABSTRACT FROM AUTHOR]

Machine learning models developed from real-world data can inherit potential, preexisting bias in the dataset. When these models are used to inform decisions involving human beings, fairness concerns inevitably arise. Imposing certain fairness constraints in the training of models can be effective only if appropriate criteria are applied. However, a fairness criterion can be defined/assessed only when the interaction between the decisions and the underlying population is well understood. We introduce two feedback models describing how people react when receiving machine-aided decisions and illustrate that some commonly used fairness criteria can end with undesirable consequences while reinforcing discrimination. [ABSTRACT FROM AUTHOR]

In the present article, we explicitly compute the spectrum of the Laplace operator on smooth real-valued and complex-valued functions on connected compact simple Lie groups of rank four with a bi-invariant Riemannian metrics that correspond to the root systems and . [ABSTRACT FROM AUTHOR]

In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense. [ABSTRACT FROM AUTHOR]

Using a straightforward approach, we derived the solution of the inverse problem for the generalized hypergeometric Appell polynomials. Also, we established the recurrence formulas for the solutions of the connection problem between them and the Bernoulli polynomials, as well as between them and the Gould-Hopper polynomials and between two different generalized hypergeometric Appell polynomial families. In addition, we present one new recurrence identity for the generalized hypergeometric Appell polynomials. [ABSTRACT FROM AUTHOR]

We propose a general framework to contract unitary dual of Lie groups via holomorphic quantization of their coadjoint orbits, using geometric quantization. The sufficient condition for the contractibility of a representation is expressed via cocycles on coadjoint orbits. This condition is verified explicitly for the contraction of SU 2 into H . We construct two types of contractions that can be implemented on every matrix Lie group with diagonal contraction matrix. [ABSTRACT FROM AUTHOR]

A characterization of all cardinal numbers that are cardinals of automorphism groups of mono-unary algebras is given. Some connections are also established between the notion of an automorphism of mono-unary algebras and the continuum hypothesis. Close connections of the obtained results with one problem of Ulam are also considered. [ABSTRACT FROM AUTHOR]

We observe a connection between the existence of square-integrable representations of a locally compact group G and the existence of nonzero translation invariant operators from its Fourier–Stieltjes algebra B ⁢ (G) {B(G)} into L 2 ⁢ (G) {L^{2}(G)} or, equivalently, from L 2 ⁢ (G) {L^{2}(G)} into its enveloping von Neumann algebra C * ⁢ (G) * * {C^{*}(G)^{**}}. [ABSTRACT FROM AUTHOR]

In this paper we analyze the structure of decoherence-free subalgebra N (T) of a uniformly continuous covariant semigroup with respect to a representation π of a compact group G on h. In particular, we obtain that, when π is irreducible, N (T) is isomorphic to (ℬ (k) ⊗ 1 m) d for suitable Hilbert spaces k and m, and an integer d related to the connected components of G. We extend this result when π is reducible and N (T) is atomic by the decomposition of h due to the Peter–Weyl theorem. [ABSTRACT FROM AUTHOR]

This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, a(x,y)dy2 + b(x,y)dxdy + c(x,y)dx2 = 0, for a,b,c smooth real functions defined on an open set of R2. Generically, solutions of a BDE are given as leaves of a pair of foliations, and the action of a symmetry must depend not only whether it preserves or inverts the plane orientation, but also whether it preserves or interchanges the foliations. The first main result reveals this dependence, which is given algebraically by a formula relating three group homomorphisms defined on the symmetry group of the BDE. The second main result adapts methods from invariant theory of compact Lie groups to obtain an algorithm to compute general expressions of equivariant quadratic 1-forms under each compact subgroup of the orthogonal group O(2). [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

An explanation of the origin of dark matter is suggested in this work. The argument is based on symmetry considerations about the concept of mass. In Wigner’s view, the rest mass and the spin of a free elementary particle in flat space-time are the two invariants that characterize the associated unitary irreducible representation of the Poincaré group. The Poincaré group has two and only two deformations with maximal symmetry. They describe respectively the de Sitter (dS) and anti-de Sitter (AdS) kinematic symmetries. Analogously to their shared flat space-time limit, two invariants, spin and energy scale for de Sitter and rest energy for anti-de Sitter, characterize the unitary irreducible representation associated with dS and AdS elementary systems, respectively. While the dS energy scale is a simple deformation of the Poincaré rest energy and so has a purely mass nature, AdS rest energy is the sum of a purely mass component and a kind of zero-point energy derived from the curvature. An analysis based on recent estimates on the chemical freeze-out temperature marking in Early Universe the phase transition quark–gluon plasma epoch to the hadron epoch supports the guess that dark matter energy might originate from an effective AdS curvature energy.

Abstract In this paper we study the characteristic polynomial of multiparameter pencil z 1 A 1 + z 2 A 2 + ⋯ + z s A s. The main theorem states that a unitary representation of a finitely generated group contains a one-dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor. It follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible. The result is of kin to the Dedekind and Frobenius theorem on finite group determinant. [ABSTRACT FROM AUTHOR]

Binary Parseval frames share many structural properties with real and complex ones. On the other hand, there are subtle differences, for example that the Gramian of a binary Parseval frame is characterized as a symmetric idempotent whose range contains at least one odd vector. Here, we study binary Parseval frames obtained from the orbit of a vector under a group representation, in short, binary Parseval group frames. In this case, the Gramian of the frame is in the algebra generated by the right regular representation. We identify equivalence classes of such Parseval frames with binary functions on the group that satisfy a convolution identity. This allows us to find structural constraints for such frames. We use these constraints to catalogue equivalence classes of binary Parseval frames obtained from group representations. As an application, we study the performance of binary Parseval frames generated with abelian groups for purposes of error correction. We show that if p is an odd prime, then the group Z p q is always preferable to Z p q when searching for best performing codes associated with binary Parseval group frames. [ABSTRACT FROM AUTHOR]

Let G be a finite group acting linearly on a vector space V. We consider the linear symmetry groups GL(Gv) of orbits Gv⊆V, where the linear symmetry groupGL(S) of a subset S⊆V is defined as the set of all linear maps of the linear span of S which permute S. We assume that V is the linear span of at least one orbit Gv. We define a set of generic points in V, which is Zariski open in V, and show that the groups GL(Gv) for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group GL(Gw) such that V is the linear span of Gw. If the underlying characteristic is zero, “isomorphic” can be replaced by “conjugate in GL(V).” Moreover, in the characteristic zero case, we show how the character of G on V determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (Geom Dedicata 6(3):331-337, 1977. 10.1007/BF02429904). [ABSTRACT FROM AUTHOR]

It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason's theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner's theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space H ${\mathcal H}$ correspond to direct product decompositions H ≃ A × A ⊥ $\mathcal{H}\simeq A\times A^\perp$ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner's theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner's theorem, namely that Fact(X) is atomistic in a very strong way. [ABSTRACT FROM AUTHOR]

Our study examines the extent to which parliamentary candidates believe that membership of a certain social group allows Members of Parliament (MPs) to more effectively represent that group. Using the 2009 German Candidate Survey, we look at four social groups: women, immigrants, religious people and East Germans. The descriptive results indicate that support for group representation is highest for women and lowest for East Germans. The explanatory analyses reveal that women are more likely than men to believe that women are better at representing the interests of women. The same holds for immigrants, religious people, and East Germans. Candidates’ belief that MPs from a certain social group are better at representing that group tends to be limited to their own social group. Our results thus indicate that the belief that one should belong to a social group in order to effectively represent the interests of that group is mainly based on identity, rather than an overall belief in the link between descriptive and substantive representation. They also show that attitudes towards representation change significantly as a result of parliamentary socialisation, with candidates with parliamentary experience being significantly less likely to support the idea of group representation compared with those without such experience. [ABSTRACT FROM AUTHOR]

It is shown that the natural monomial characters of the symmetric group introduced in the above article are well-behaved with respect to π-elements. [ABSTRACT FROM AUTHOR]

In the present paper, we are going to use geometric and topological concepts, entities and properties of the integral curves of linear vector fields, and the theory of differential equations, to establish a representation for some groups on Rn(n ⩾ 1). Among other things, we investigate the surjectivity and faithfulness of the representation. At the end, we give some applications. [ABSTRACT FROM AUTHOR]

In Harding (Trans. Amer. Math. Soc. 348(5), 1839-1862 1996) it was shown that the direct product decompositions of any non-empty set, group, vector space, and topological space X form an orthomodular poset Fact X. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. Here we develop dynamics and an abstract version of a time independent Schrödinger's equation in the setting of decompositions by considering representations of the group of real numbers in the automorphism group of the orthomodular poset Fact X of decompositions. [ABSTRACT FROM AUTHOR]

We show that any bounded analytic semigroup on Lp (with 1 < p < ∞) whose negative generator admits a bounded H∞(Σθ) functional calculus for some θ ∊ (0, π/2 ) can be dilated into a bounded analytic semigroup (Rt)t≥ 0 on a bigger Lp-space in such a way that Rt is a positive contraction for any t≥0. We also establish a discrete analogue for Ritt operators and consider the case when Lp-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations of amenable groups on Banach spaces and establish various generalizations of Dixmier's unitarization theorem. [ABSTRACT FROM AUTHOR]