Narrow Results

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.

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 $G$.

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.

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.

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.

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.

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.

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 $L$ 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.

The Frame Problem is the problem of how to design a machine to use information so as to behave competently, with respect to the kinds of tasks a genuinely intelligent agent can reliably, effectively perform. I will argue that the way the Frame Problem is standardly interpreted, and so the strategies considered for attempting to solve it, must be updated. We must replace overly simplistic and reductionist assumptions with more sophisticated and plausible ones. In particular, the standard interpretation assumes that mental processes are identical to certain kinds of computational processes, and so solving the Frame Problem is a matter of finding a computational architecture that can effectively represent relations of semantic relevance. Instead, we must take seriously the possibility that the way in which intelligent agents use information is inherently different. Whereas intelligent agents are plausibly genuinely causally sensitive to semantic properties as such (to what they perceive, desire, believe intend, etc.), computational systems can only be causally sensitive to the formal features that represent these properties. Indeed, it is this very substitution of formal generalizations for genuinely semantic ones that is responsible for the way current AI systems are brittle, inflexible, and highly specialized. What we need is a more sophisticated way of investigating the relationship between computational information processing and genuinely semantic information use. I apply the generative methodology I have developed elsewhere for cognitive science and AI research to show how the Frame Problem can be appropriately updated.