Narrow Results

Let $G$ be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of $G$ on $C^{\ast }$-algebras $A$ and $B$ are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of $A$ and $B$ in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of $A$ and $B$.

There is an alternative formulation of the problem: an action of the dual group $Ĝ$ together with a suitably equivariant unitary homomorphism of $G$ give rise to a generalized fixed-point algebra via Landstad’s theorem, and a problem related to the above is to produce an action of $Ĝ$ and two such equivariant unitary homomorphisms of $G$ that give distinct generalized fixed-point algebras.

We present several situations where the condition on the images of $A$ and $B$ is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if $G$ is discrete, this will be the case for all actions of $G$.

We prove a version of Pedersen’s outer conjugacy theorem for coactions of compact groups, which characterizes outer conjugate coactions of a compact group in terms of properties of the dual actions. In fact, we show that every isomorphism of a dual action comes from a unique outer conjugacy of a coaction, which in this context should be called strong Pedersen rigidity. We promote this to a category equivalence.

The classical limit of wave quantum mechanics is analyzed. It is shown that the basic requirements of continuity and finiteness to the solution of the form ψ(x) = Ae^{i ϕ (x)} + Be^{-i ϕ (x)}, where and W(x) is the reduced classical action of the physical system, give the asymptote of the wave equation and general quantization condition for the action W(x), which yields the exact eigenvalues of the system.

This review is devoted to the Multiple Point Principle (MPP), according to which several vacuum states with the same energy density exist in Nature. The MPP is implemented to the Standard Model (SM), Family replicated gauge group model (FRGGM) and phase transitions in gauge theories with/without monopoles. Using renormalization group equations for the SM, the effective potential in the two-loop approximation is investigated, and the existence of its postulated second minimum at the fundamental scale is confirmed. Phase transitions in the lattice gauge theories are reviewed. The lattice results for critical coupling constants are compared with those of the Higgs monopole model, in which the lattice artifact monopoles are replaced by the point-like Higgs scalar particles with magnetic charge. Considering our (3+1)-dimensional space–time as, in some way, discrete or imagining it as a lattice with a parameter a = λ_P, where λ_P is the Planck length, we have investigated the additional contributions of monopoles to the β-functions of renormalization group equations for running fine structure constants α_i(μ) (i = 1, 2, 3 correspond to the U(1), SU(2) and SU(3) gauge groups of the SM) in the FRGGM extended beyond the SM at high energies. It is shown that monopoles have N_fam times smaller magnetic charge in the FRGGM than in the SM (N_fam is a number of families in the FRGGM). We have estimated also the enlargement of a number of fermions in the FRGGM leading to the suppression of the asymptotic freedom in the non-Abelian theory. We have reviewed that, in contrast to the case of the Anti-grand-unified-theory (AGUT), there exists a possibility of unification of all gauge interactions (including gravity) near the Planck scale due to monopoles. The possibility of the [SU(5)]³ or [SO(10)]³ unification at the GUT-scale ~10¹⁸GeV is briefly considered.

De Sitter space–time is considered to be represented by a D-dimensional hyperboloid embedded in (D+1)-dimensional Minkowski space–time. The string equation is derived from a string action which contains a Lagrange multiplier to restrict coordinates to de Sitter space–time. The string system of equations is equivalent to a type of generalized sinh–Gordon equation. The evolution equations for all the variables including the coordinates and their derivatives are obtained for D=2,3 and 4.

The issue of space–time gauge invariance for the bosonic string has been earlier addressed using the loop variable formalism. In this paper the question of obtaining a gauge invariant action for the open bosonic string is discussed. The derivative with respect to ln a (where a is a worldsheet cutoff) of the partition function — which is first normalized by dividing by the integral of the two-point function of a marginal operator — is a candidate for the action. Applied to the zero-momentum tachyon it gives a tachyon potential that is similar to those that have been obtained using Witten's background independent formalism. This procedure is easily made gauge invariant in the loop variable formalism by replacing ln a by Σ which is the generalization of the Liouville mode that occurs in this formalism. We also describe a method of resumming the Taylor expansion that is done in the loop variable formalism. This allows one to see the pole structure of string amplitudes that would not be visible in the original loop variable formalism.

Motion of a point particle (a particle without any internal structure) in the context of special relativity is investigated. The study is based on the action principle with either time or the proper time regarded as the evolution parameter. Actions corresponding to the motion of particle in general external tensor fields are obtained.

At a saddle-center bifurcation for Hamiltonian systems, the homoclinic orbit is cusp shaped at the nonlinear nonhyperbolic saddle point. Near but before the bifurcation, orbits are periodic corresponding to the unfolding of the homoclinic orbit, while after the bifurcation a double homoclinic orbit is formed with a local and global branch. The saddle-center bifurcation is dynamically unfolded due to a slowly varying potential. Near the unfolding of the homoclinic orbit, the period and action are analyzed. Asymptotic expansions for the action, period and dissipation are obtained in an overlap region near the homoclinic orbit of the saddle-center bifurcation. In addition, the unfoldings of the action and dissipation functions associated with zero energy orbits (periodic and homoclinic) near the saddle-center bifurcation are determined using the method of matched asymptotic expansions for integrals.

We continue our program of extending key techniques from geometric group theory to semigroup theory, by studying monoids acting by isometric embeddings on spaces equipped with asymmetric, partially defined distance functions. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Švarc–Milnor lemma to this setting.

Intelligent virtual agent behaviour is a crucial element of any virtual environment application as it essentially brings the environment to life, introduces believability and realism and enables complex interactions and evolution over time. However, the development of mechanisms for virtual agent perception and action is neither a trivial nor a straight-forward task. In this paper we present a model of perception and action for intelligent virtual agents that meets specific requirements and can as such be systematically implemented, can seamlessly and transparently integrate with knowledge representation and intelligent reasoning mechanisms, is highly independent of virtual world implementation specifics, and enables virtual agent portability and reuse.

Statements about relativistic effects are often subtle. In this essay we will demonstrate that the three classical tests of general relativity, namely perihelion precession, deflection of light and gravitational redshift, are passed perfectly by an extension of Newtonian gravity that includes gravitational time dilation effects while retaining a non-relativistic causal structure. This non-relativistic gravity theory arises from a covariant large speed of light expansion of Einstein’s theory of gravity that does not assume weak fields and which admits an action principle.

We show that a continuous action of a quantum semigroup on a finite quantum space (finite dimensional C^*-algebra) preserving a faithful state comes from a continuous action of the quantum Bohr compactification of . Using the classification of continuous compact quantum group actions on M₂, we give a complete description of all continuous quantum semigroup actions on this quantum space preserving a faithful state.

It is well known that injective objects play a fundamental role in many branches of mathematics. The question whether a given category has enough injective objects has been investigated for many categories. Also, quasi-injective modules and acts have been studied by many categorists. In this paper, we study quasi-injectivity in the category of actions of an ordered monoid on ordered sets ($Pos\text{-}S$) with respect to embeddings. Also, we give the relation between injectivity, quasi-injectivity (with respect to embeddings), and poset completeness in the category $Pos\text{-}S$ and some of its important subcategories.

In this paper, we introduced the concept of crossed modules for Hom–Lie antialgebras. It is proved that the category of crossed modules for Hom–Lie antialgebras and the category of ${\mathrm{\text{Cat}}}^1$-Hom–Lie antialgebras are equivalent to each other. The relationship between the crossed modules extension of Hom–Lie antialgebras and the third cohomology group is investigated.

Action of a pomonoid on partially ordered sets ($S$-posets) has beautiful aspects in practical subjects such as automata theory, projection algebra and theoretical computer science which makes it always capture the interest of mathematicians. On the other hand, the study of different kinds of weakly injectivity (which category theory inherited from homological and commutative algebra) is an interesting subject for mathematicians. One of the important kinds of weakly injectivity is quasi-injectivity. In this paper, we study quasi-injectivity in the category of $S$-posets with respect to special kind of order embeddings, namely, down-closed embeddings (dc-quasi-injectivity).

Let G be a nonabelian finite group. Then Irr(G/G') is an abelian group under the multiplication of characters and acts on the set of non-linear irreducible characters of G via the multiplication of characters. The purpose of this paper is to establish some facts about the action of linear character group on non-linear irreducible characters and determine the structures of groups G for which either all the orbit kernels are trivial orthe number of orbits is at most two. Using the established results on this action, it is very easy to classify groups G having at most three non-linear irreducible characters.

The problem of the representation of an action of a Lie–Rinehart algebra on a Lie 𝖠-algebra by means of a homomorphism of Lie–Rinehart algebras is studied. An eight-term exact sequence associated to an epimorphism of Lie–Rinehart algebras for the cohomology of Lie–Rinehart algebras developed by Casas, Ladra and Pirashvili is obtained. This sequence is applied to study the obstruction theory of Lie–Rinehart algebra extensions.

Let a finite group A act on a finite p-group P with |P| > p^e, where e is an integer with e ≥ 2. In this paper, we show that P is centralized by $O^p(A^{{\mathcal{u}}_{p-1}})$ if every non-maximal class p-group of order p^e in P is stabilized by O^p(A). As applications, some conditions are given for a finite group to be p-supersolvable.

Let $L$ be a Lie algebra crossed module. In this paper, we shall introduce a chain of crossed modules of $\mathrm{\text{Act}}(L)$ and show that if $M$ and $L$ are two isoclinic crossed modules, then ${\mathrm{\text{ID}}}^{\ast }\mathrm{\text{Act}}(L)\cong {\mathrm{\text{ID}}}^{\ast }\mathrm{\text{Act}}(M)$.

The strategic urban project (SUP) approach has become an important mode of urban renewal in both developed and developing contexts. In the last few decades, the application of this approach has been extended from flagship projects at key locations in Western countries, to informal settlement upgrading in developing countries. However, there is a lack of a clear conceptual framework for meaningful understanding of the achievements and limitations of SUPs in the context of informal settlement upgrading. Our understanding of how the SUP approach deals with the issues of informal settlements is limited. This paper fills the research gap by developing such a framework and applying it to compare three representative cases, namely the Favela–Bairro Programme in Rio de Janeiro, the Social Urbanism Strategy in Medellin, and the Kampung Improvement Programme (KIP) in Surabaya.