Unitary Elements Research Articles

Shared or separated representations for letters with diacritics?

Understanding the front end of visual word recognition requires us to identify the processes by which letters are identified. Since most of the work on letter recognition has been conducted in English, letter perception modeling has been limited to the 26 letters of the Latin alphabet. However, many writing systems include letters with diacritic marks. In the present study, we examined whether diacritic letters are a mere variant of their base letter, and thus share the same abstract representation, or whether they function as separate elements from any other letters, and thus have separate representations. In Experiments 1A and 1B, participants performed an alphabetical decision task combined with masked priming. Target letters were preceded by the same letter (e.g., a-A), by a diacritic letter (e.g., â-A), or by an unrelated letter (e.g., z-A). The results showed that the primes sharing nominal identity (e.g., a) facilitated target processing as compared to unrelated primes (e.g., z), but that primes that included a diacritic mark (e.g., â) did not, with reaction times being similar to those in the unrelated priming condition. In Experiment 2 we replicated these results in a lexical decision task. Overall, this demonstrates that as long as diacritics are used in scripts to distinguish between lexical entries, the diacritic letters are not mere variants of their base letters but constitute unitary elements of the script in their own right, with diacritics contributing to the overall visual shape of a letter.

Temporal Coupled-Mode Theory of Electromagnetic Components Described by Magnetic Groups of Symmetry

We consider peculiarities of application of the temporal coupled-mode theory to the electromagnetic components with magnetic symmetry. The coupled-mode theory is widely used for analysis of reciprocal devices described by symmetrical scattering matrices and by unitary elements of symmetry, such as for example, a plane of symmetry. However, nonreciprocity and low symmetry of components described by antiunitary elements lead to a necessity of modification of the method. Using as an example a photonic crystal-based W-circulator described by an antiplane of symmetry, we show applicability of the method to the components with antiunitary elements. The circulator characteristics are calculated and compared with those obtained by a numerical method, demonstrating good agreement between the two methods.

Multilayer system of films heated from above

The paper deals with the development of convective flows in a complex system of thin liquid layers occurring under the effect of thermal load. To study this process a mathematical model which is based on the decomposition of complex problem into unitary elements (modules) with some set of rules allowing their linkage with each other was constructed. Each module represents a monotypic local model. The fluid flow and heat transfer in each of these modules are described by the Navier-Stokes and thermal conductivity equations. The boundary conditions required to solve these systems of equations are represented explicitly and written in the form of conservation laws. The numerical analysis of effect of thermal load on the characteristics of the liquid layers movement depending on the Marangoni number is carried out. It is shown that physical and thermophysical properties of liquid layers play the decisive role. It is exactly they control the convective flows in layers and determine the position and shape of interfaces. The results of model problem solutions are presented.

MF-property for countable discrete groups

We say that a group has an MF-property if it can be embedded in the group of unitary elements of the C*-algebra ΠM n /⊕M n . In the present paper we prove the MF-property for the Baumslag group $$\left\langle {a,b|{a^{{a^b}}} = {a^2}} \right\rangle $$ and also some general assertions concerning this property.

Constitutional Status of Montserrat as United Kingdom Overseas Territory

The article explores that partnership as a form of relations between Britain and the Overseas Territoriesis of various and nominal nature. The political and territorial relations between the UK and the overseasterritory under consideration are examined is the paper as a unitary federalism elements. Having examinedthe provisions of the Constitution of Montserrat, 2010 in the part of the Governor of the status of theregulation, the Cabinet, the Legislature, it is emphasized that a special place in the constitutional systemof government of Montserrat is taken by a representative of the British Crown in view of the powers whichwill subject to the established by the Constitution to make key decisions in the sphere of the island internalpolicy (convening the Cabinet, its resignation, the dissolution of the Legislative Assembly, law-makingfunction and others). This provides mechanisms to prevent the concentration of powers in the jurisdictionof the Governor. In particular, a significant role in the formation and implementation of the constitutionalcourse is assigned to the Prime Minister and the Legislative Assembly the recommendations of which aresubject to implementation in ordinary conditions. The theses are formulated on the advisability of furtherregulation of the competence of local public authorities in Montserrat and Governor, respectively, specifyingthe content and procedure for organizing and conducting the proceedings based on a considerationof questions on the resignation of the Cabinet, early dissolution of the Legislative Assembly, clarify thecriteria for such decisions, the coordination of positions on the dissolution of the Legislative Assembly.The paper studies the question on the Cabinet of Montserrat in the UK Cabinet by including membersof cabinet representing particular territories. Besides, the article mentions submitting by the LegislativeAssembly to UK House of Commons data on its opinion on draft laws relating to the entire Kingdom andsubmitting by the House of Commons the drafts related to the interests of the Commonwealth to the overseasterritories. The paper specifies the advisability of further regulation in the Constitution as to organizingand holding consultations with public authorities of overseas territory on the matters concerning theirinterests, referendum mechanisms for the people of islands to participate in the decision-making processof public nature, including the early termination of certain categories of officials.

Perturbation semigroup of matrix algebras

In this article we analyze the structure of the semigroup of inner perturbations in noncommutative geometry. This perturbation semigroup is associated to a unital associative *-algebra and extends the group of unitary elements of this *-algebra. We compute the perturbation semigroup for all matrix algebras.

Quantum gauge symmetries in noncommutative geometry

We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite-dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M_n(\mathbb{R}) , M_n(\mathbb{C}) and M_n(\mathbb{H}) , describing the finite noncommutative space of the Einstein–Yang–Mills systems, and the algebras \mathcal{A}_F=\mathbb{C}\oplus \mathbb{H} \oplus M_3(\mathbb{C}) and \mathcal{A}^{\mathrm{ev}}=\mathbb{H}\oplus \mathbb{H} \oplus M_4(\mathbb{C}) , that appear in Chamseddine–Connes derivation of the Standard Model of particle physics coupled to gravity. As a byproduct, we identify a “free” version of the symplectic group \operatorname{Sp}(n) (quaternionic unitary group).

The Russo–Dye theorem in nest subalgebras of factors

Let N be a nest in a factor R. In this paper, we introduce the concept of the admissible nest for R and prove that the weak operator topology closure of the convex hull of all the unitary elements in the nest subalgebra of the factor is the unit ball if and only if N is an admissible nest in R.

Isomorphisms of Brin-Higman-Thompson groups

Let $m, m', r, r',t, t'$ be positive integers with $r, r' \ge 2$. Let $L_r$ denote the ring that is universal with an invertible $1 \times r$ matrix. Let $M_m(L_r^{\otimes t})$ denote the ring of $m \times m$ matrices over the tensor product of $t$ copies of $L_r$. In a natural way, $M_m(L_r^{\otimes t})$ is a partially ordered ring with involution. Let $PU_m(L_r^{\otimes t})$ denote the group of positive unitary elements. We show that $PU_m(L_r^{\otimes t})$ is isomorphic to the Brin-Higman-Thompson group $t V_{r,m}$; the case $t =1$ was found by Pardo, that is, $PU_m(L_r)$ is isomorphic to the Higman-Thompson group $V_{r,m}$. We survey arguments of Abrams, \'Anh, Bleak, Brin, Higman, Lanoue, Pardo, and Thompson that prove that $t' V_{r',m'} \cong tV_{r,m} $ if and only if $r' = r$, $t'=t$ and $ \gcd(m',r'-1) = \gcd(m,r-1)$ (if and only if $M_{m'}(L_{r'}^{\otimes t'})$ and $M_m(L_r^{\otimes t})$ are isomorphic as partially ordered rings with involution).

On genuine infinite algebraic tensor products

In this paper, we study genuine infinite tensor products of some algebraic structures. By a genuine infinite tensor product of vector spaces, we mean a vector space \bigotimes_{i\in I} X_i whose linear maps coincide with multilinear maps on an infinite family \{X_i\}_{i\in I} of vector spaces. After establishing its existence, we give a direct sum decomposition of \bigotimes_{i\in I} X_i over a set \Omega_{I;X} , through which we obtain a more concrete description and some properties of \bigotimes_{i\in I} X_i . If \{A_i\}_{i\in I} is a family of unital ^* -algebras, we define, through a subgroup \Omega^{\rm ut}_{I;A}\subseteq \Omega_{I;A} , an interesting subalgebra \bigotimes_{i\in I}^{\rm ut} A_i . When all A_i are C^* -algebras or group algebras, it is the linear span of the tensor products of unitary elements of A_i . Moreover, it is shown that \bigotimes_{i\in I}^{\rm ut} \mathbb{C} is the group algebra of \Omega^{\rm ut}_{I;\mathbb{C}} . In general, \bigotimes_{i\in I}^{\rm ut} A_i can be identified with the algebraic crossed product of a cocycle twisted action of \Omega^{\rm ut}_{I;A} . On the other hand, if \{H_i\}_{i\in I} is a family of inner product spaces, we define a Hilbert C^*(\Omega^{\rm ut}_{I;\mathbb{C}}) -module \overline{\bigotimes}^{\rm mod}_{i\in I} H_i , which is the completion of a subspace \bigotimes_{i\in I}^{\rm unit} H_i of \bigotimes_{i\in I} H_i . If \chi_{\Omega^{\rm ut}_{I;\mathbb{C}}} is the canonical tracial state on C^*(\Omega^{\rm ut}_{I;\mathbb{C}}) , then \overline{\bigotimes}^{\rm mod}_{i\in I} H_i\otimes_{\chi_{\Omega^{\rm ut}_{I;\mathbb{C}}}}\mathbb{C} coincides with the Hilbert space \overline{\bigotimes}^{\phi_1}_{i\in I} H_i given by a very elementary algebraic construction and is a natural dilation of the infinite direct product \prod {\otimes}_{i\in I} H_i as defined by J. von Neumann. We will show that the canonical representation of \bigotimes_{i\in I}^{\rm ut} \mathcal{L}(H_i) on \overline\bigotimes^{\phi_1}_{i\in I} H_i is injective (note that the canonical representation of \bigotimes_{i\in I}^{\rm ut} \mathcal{L}(H_i) on \prod {\otimes}_{i\in I} H_i is not injective). We will also show that if \{A_i\}_{i\in I} is a family of unital Hilbert algebras, then so is \bigotimes_{i\in I}^{\rm ut} A_i .

Unitary operators in real von Neumann algebras

We prove that the open unit ball of any von Neumann algebra A is contained in the sequentially convex hull of U A , the set of unitary elements of A . Therefore the closed unit ball of A coincides with the closed convex hull of U A . In the complex case this statement is actually valid for any unital C ⁎ -algebra (it is the well-known Russo–Dye Theorem). However, for real scalars, the results we are presenting provide new information even for the algebra L ( H ) of bounded linear operators from an infinite-dimensional Hilbert space H into itself. Let us say in this sense that the possibility of rebuilding the unit ball of L ( H ) through the closed convex hull of the unitary elements appears in the literature as an open problem. We also obtain some results about the extremal structure of these spaces.

Probability representation of spin states and inequalities for unitary matrices

We review the probability representation of spin states described by probability distributions (tomograms). We use the relation of a tomogram to unitary group elements to obtain some inequalities for unitary matrices. We present the Cirel’son bound and the entropic inequalities for entangled spin states in the form of relations for functions on the unitary group.

Kinetics of formation of unitary three-dimensional structural elements: numerical experiment

For the first time the results of numerical experiments (by the Monte Carlo method) on the kinetics of three-dimensional free radical polymerization of tetrafunctional monomers within the framework of the formation of a unitary three-dimensional structural element on the cubic lattice 50 × 50 × 50 have been presented.

Banach space characterizations of unitaries: A survey

We survey Banach space characterizations of unitary elements of C ∗ -algebras, JB ∗ -triples, and JB-algebras. In the case of the existence of a pre-dual, appropriate specializations of these characterizations are also reviewed.

Continuous inverse algebras with involution

A large part of the theory of Banach *-algebras is developed and generalized to continuous inverse *-algebras (i.e. complex locally convex unital *-algebras with open unit group and continuous inversion) which are (Mackey) complete. If the involution is continuous, the closed unit ball with respect to the greatest C*-semi-norm is the closed convex hull of the unitary elements. (This is originally due to Palmer.) For hermitian continuous inverse *-algebras, we generalize characterizations due to Raĭikov, Pták, and Palmer, we prove the Shirali–Ford Theorem, and we show that closed subalgebras are equispectrally embedded.

REDUCED DYNAMICS FROM THE UNITARY GROUP TO SOME FLAG MANIFOLDS: INTERACTING MATRIX RICCATI EQUATIONS

In this paper we treat the time evolution of unitary elements in the N level system and consider the reduced dynamics from the unitary group U(N) to flag manifolds of the second type (in our terminology). Then we derive a set of differential equations of matrix Riccati types interacting with one another and present an important problem on a nonlinear superposition formula that the Riccati equation satisfies. Our result is a natural generalization of the paper Chaturvedi et al. [1] (arXiv: 0706.0964 [quant-ph]).

Approximately isometric lifting in quasidiagonal extensions

Let 0 → I → A → A/I → 0 be a short exact sequence of C*-algebras with A unital. Suppose that the extension 0 → I → A → A/I → 0 is quasidiagonal, then it is shown that any positive element (projection, partial isometry, unitary element, respectively) in A/I has a lifting with the same form which commutes with some quasicentral approximate unit of I consisting of projections. Furthermore, it is shown that for any given positive number e, two positive elements (projections, partial isometries, unitary elements, respectively) \( \bar a,\bar b \) in A/I, and a positive element (projection, partial isometry, unitary element, respectively) a which is a lifting of \( \bar a \), there is a positive element (projection, partial isometry, unitary element, respectively) b in A which is a lifting of \( \bar b \) such that ∥a−b∥ < \( \left\| {\bar a - \bar b} \right\| + \varepsilon \). As an application, it is shown that for any positive numbers e and \( \bar u \) in U(A/I)0, there exists u in U(A)0 which is a lifting of \( \bar u \) such that cel(u) < cel\( (\bar u) + \varepsilon \).

Banach spaces whose algebras of operators have a large group of unitary elements

AbstractWe prove that a complex Banach spaceXis a Hilbert space if (and only if) the Banach algebra$\mathcal L (X)$(of all bounded linear operator onX) is unitary and there exists a conjugate-linear algebra involution • on$\mathcal L (X)$satisfyingT•=T−1for every surjective linear isometryTonX. Appropriate variants for real spaces of the result just quoted are also proven. Moreover, we show that a real Banach spaceXis a Hilbert space if and only if it is a realJB*-triple and$\mathcal L (X)$is$w_{op}'$-unitary, where$w'_{op}$stands for the dual weak-operator topology.

Neurostatistics: Applications, challenges and expectations

Brain function and its relations to cognition and behavior can be elucidated only by the use of various complementary methods. Over the past 20 years, we have been studying the brain mechanisms underlying spatial processes using different methods, including the recording of single cell activity in behaving monkeys, functional magnetic resonance imaging (fMRI) and magnetoencephalography (MEG) in human subjects, all performing the same tasks. These methods provide partially overlapping perspectives, resulting in a gain in knowledge beyond the province of the individual method. A common aspect in this endeavor is the statistical analysis of the data acquired by different methods, especially regarding the encoding of information in unitary elements (single cell activity in neurophysiology, blood oxygenation level-dependent (BOLD) activation of voxels in fMRI, magnetic field strength in MEG) and the decoding of information from ensembles. In this paper we illustrate the various approaches, their data analysis and possible applications to medicine in the context of operations in space.

Lifting Unitary Elements with Application to Approximately Inner Automorphisms

Let A be a separable unital nuclear simple C*-algebra with torsion K0(A), free K1(A) and with the UCT. Let τ : A→M(\({\fancyscript K}\))/\({\fancyscript K}\) be a unital homomorphism. We prove that every unitary element in the commutant of τ (A) is an exponent, thus it is liftable. We also prove that each automorphism α on E with \( \ifmmode\expandafter\bar\else\expandafter\=\fi{\alpha } \in {\text{Aut}}_{0} {\left( A \right)} \) is approximately inner, where E is a unital essential extension of A by \({\fancyscript K}\) and \( \ifmmode\expandafter\bar\else\expandafter\=\fi{\alpha } \) is the automorphism on A induced by α.