Proper Extensions Research Articles

Proper extensions of the 2-sphere’s conformal group present entropy and are 4-transitive

AbstractIn this paper, we prove using elementary techniques that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. As an important consequence, we derive that any such group must always contain an element of positive topological entropy. We also provide a self-contained characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.

The quarks of attention: Structure and capacity of neural attention building blocks

Attention plays a fundamental role in both natural and artificial intelligence systems. In deep learning, attention-based neural architectures, such as transformer architectures, are widely used to tackle problems in natural language processing and beyond. Here we investigate the most fundamental building blocks of attention and their computational properties within the standard model of deep learning. We first derive a systematic taxonomy of all possible attention mechanisms within, or as extensions of the standard model into 18 classes depending on the origin of the attention signal, the target of the attention signal, and whether the interaction is additive or multiplicative. Second, using this taxonomy, we identify three key attention mechanisms: additive activation attention (multiplexing), multiplicative output attention (output gating), and multiplicative synaptic attention (synaptic gating). Output gating and synaptic gating are proper extensions of the standard model and all current attention-based architectures, including transformers, use either output gating or synaptic gating, or a combination of both. Third, we develop a theory of attention capacity and derive mathematical results about the capacity of basic attention networks comprising linear or polynomial threshold gates. For example, the output gating of a linear threshold gate of n variables by another linear threshold gate of the same n variables has capacity 2n2(1+o(1)), achieving the maximal doubling of the capacity for a doubling of the number of parameters. Perhaps surprisingly, multiplexing attention is used in the proofs of these results. Synaptic and output gating provide computationally efficient extensions of the standard model enabling sparse quadratic activation functions. They can also be viewed as primitives for collapsing several layers of processing in the standard model into shallow compact representations.

О трехзначных расширениях логики Клини

The paper is devoted to one of the most famous three-valued systems – Kleene's logic. The expressive capabilities of Kleene's logic and its three-valued expansions are described. We present two results. First, all possible three-valued expansions of Kleene's logic are found up to equivalence with respect to the mutual definability of connectives. It is shown that there are only twelve such expansions. This list includes both logics already known in the literature and completely new ones. For the found expansions, we describe the structure of the lattice ordered relative to the expressive power of its elements. Secondly, for Kleene's logic and its three-valued expansions we find how many extensions each of these logics has in the same language. Kleene's logic has only two proper extensions: the classical and the trivial ones. Generally, a three-valued logic in which Kleene's matrix is definable contains no more than three proper extensions: the classical one, the trivial one, and an intermediate logic, determined by the product of the the original logic's matrix and the matrix of classical logic in the same signature. Intermediate logics exist only for two types of three-valued expansions of Kleene's logic: in expansions equivalent to Łukasiewicz's logic, and in logics whose matrices contain both a bivalent submatrix, the universe of which consists of the classical truth values, and a submatrix, the universe of which consists the intermediate value alone. All three-valued expansions of Kleene's logic that do not preserve the classical values have only one extension of their own – the trivial one.

H β -Hausdorff Functions and Common Fixed Points of Multivalued Operators in a b -Metric Space and Their Applications

H β -Hausdorff functions for β ∈ 0 , 1 are introduced, and common fixed-point theorems for a pair of multivalued operators satisfying generalized contraction conditions are proven in a b -metric space. Our results are proper extensions and new variants of many contraction conditions existing in literature. In order to demonstrate applications of our result, we have proven an existence theorem for a unique common multivalued fractal of a pair of iterated multifunction systems and also an existence theorem for a common solution of a pair of Volterra-type integral equations.

Positive logics

Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

Minimum proper extensions in some lattices of subalgebras

Minimum proper extensions in some lattices of subalgebras

Detection of dead cores for reaction-diffusion equations with a non-smooth nonlinearity

This paper studies a class of nonlinear elliptic PDEs, arising for stationary reaction-diffusion models. The non-smooth nonlinearity gives rise to dead cores, that is, subdomains where the solution of the PDE vanishes. The paper gives a solid foundation for the numerical solution of the problem, including proper extensions of known maximum–minimum principles, Céa lemma, and convergence estimation of the FEM for locally Hölder continuous operators. Based on these, we finally detect dead cores numerically in various typical situations.

Diffusive model to assess the release of chemicals from a material under intermittent release conditions

We consider the archetype situation of a chemical species that diffuses in a material and irreversibly escapes through the interface. In our setup, the interface switches between two states corresponding to ‘release phase’ (absorbing boundary) during which the species is released to the exterior, and ‘pause phase’ (reflecting boundary) during which the species is not released and its concentration profile inside the material partially relaxes back to uniformity. By combining numerical solution of the diffusion equation and statistical analysis of the outcomes, we derive upper and lower bounds and an empirical approximation for the amount of species released up to a certain time, in which the only information about the release-pause alternation schedule is the number of release phases and the average duration of a release phase. The methodology is developed thinking especially to dermal exposure assessment in the case of a slab-like homogeneous material irreversibly releasing chemicals during a number of contacts. However, upon proper extensions, this approach might be useful for inspecting other situations that are encountered, for instance, when dealing with leakage of chemicals in environmental contexts and regulatory toxicology.

On minimum proper essential extensions in a category

In a concrete category A satisfying certain natural conditions, we investigate minimum proper essential extenstions (A-mpee’s) and how this relates to atoms in the complete lattice hoA of hull operators on A (if A # is the category with objects the A-objects but morphisms just the essential extensions in A, then hoA consists exactly of the reflections in A # ). A further hypothesis inserts itself: We call an A-mpee µ strong just in case the domain and codomain of µ are not isomorphic; [S] is the assumption that every A-mpee is strong. We show (1) if µ is strong, then an atom of hoA is determined by µ, and (2) if [S], then every id ≠ h ∈ hoA lies above an atom and every atom is determined by an A-mpee µ. Examples are presented in various categories of lattice-ordered groups.

Supernova Constraints on Dark Flavored Sectors

Proto-neutron stars forming a few seconds after core-collapse supernovae are hot and dense environments where hyperons can be efficiently produced by weak processes. By making use of various state-of-the-art supernova simulations combined with the proper extensions of the equations of state including Λ hyperons, we calculate the cooling of the star induced by the emission of dark particles X0 through the decay Λ → nX 0. Comparing this novel energy-loss process to the neutrino cooling of SN 1987A allows us to set a stringent upper limit on the branching fraction, BR(Λ → nX 0) ≤ 8 × 10−9, that we apply to massless dark photons and axions with flavor-violating couplings to quarks. We find that the new supernova bound can be orders of magnitude stronger than other limits in dark-sector models.

MOST SIMPLE EXTENSIONS OF ARE UNDECIDABLE

AbstractAll known structural extensions of the substructural logic $\textbf{FL}_{\textbf{e}}$ , the Full Lambek calculus with exchange/commutativity (corresponding to subvarieties of commutative residuated lattices axiomatized by $\{\vee , \cdot , 1\}$ -equations), have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.

Supernova constraints on dark flavored sectors

Proto-neutron stars forming a few seconds after core-collapse supernovae are hot and dense environments where hyperons can be efficiently produced by weak processes. By making use of various state-of-the-art supernova simulations combined with the proper extensions of the equations of state including $\Lambda$ hyperons, we calculate the cooling of the star induced by the emission of dark bosons $X^0$ through the decay $\Lambda\to n X^0$. Comparing this novel energy-loss process to the neutrino cooling of SN 1987A allows us to set stringent constraints on massless dark photons and axions with flavor-violating couplings to quarks. We find that this new supernova bound can be orders of magnitude stronger than other limits in dark-sector models.

Two pretabular linear extensions of relevance logic R

Pretabularity is the attribute of logics that are not characterised by finite matrices, but all of whose proper extensions are. Two of the first-known pretabular logics were Dummett’s famous super-intuitionistic logic LC and the well-known semi-relevance logic RM (= R-Mingle). In this paper, we investigate Anderson and Belnap’s relevance logic R with the extra axiom: (p → q) ∨ (q → p), which we name LR, and which is (much) weaker than RM, and so is not pretabular. This means that over LR, there may be some pretabular extensions other than RM, two of which this paper presents and with which it provides axiomatizations, characteristic algebras, and Routley-Meyer semantics.

Characterizing finite Boolean rings by using finite chains of subrings

Let R be a nonzero associative ring with identity. It is proved that R is a finite Boolean ring if (and only if) 1 is the only unit of R and there exists a finite maximal chain C each of whose n steps is a proper unital ring extension, R0:=F2 ⊂ ... ⊂ Rn=R, going from F2 to R. If these equivalent conditions hold and R has exactly n maximal ideals, then any such C has length n-1 and the number of unital subrings of R is Bn, the nth Bell number. It is also proved that if R has characteristic p for some prime number p, then R is isomorphic to a finite direct product of copies of Fp if (and only if) for some integer m ≥ 0, R has exactly m+1 maximal ideals and there exists a finite maximal chain of proper unital ring extensions, ℛ0:=Fp ⊂ ... ⊂ ℛm=R, going from Fp to R, such that ℛm-1 is a commutative ring and R is a unital ℛm-1-algebra. Additional characterizations, applications, examples and remarks are given.

Fixed points results in b-metric spaces

In this paper, we prove a common fixed point theorem for four pairs of mappings which satisfy the E.A- property in b-metric spaces. The proposed theorem expands and generalizes several well-known comparable results in the literature. Example is given showing that our result is proper extensions of the existing ones. Obtained results are useful in solving Mechanical problems as well as in image processing and deep learning. The established results satisfy the previous well known results.

Partition function and coherent states for the quantum multiverse

Abstract Motivated by a wider acceptance of the quantum cosmology and the idea of multiverse, we follow the second and third quantization formalism for the conceptual comprehension of quantum multiverse. The other motivating feature to pursue the present study is that at the different levels of quantization the equation of quantum state is similar to that of harmonic oscillator, which encourages us to give a glancing taste of the coherent states in the domain of quantum cosmology. Third quantization enables one to model the universe as a particle in a quantum field theory of the minisuperspace and the wave description prompts one to recognize interactions among the universes of the multiverse. This picture of the system further instigates us to investigate the partition function and the thermodynamic equations of the system. The path integral formalism with proper mathematical extensions are employed to find the partition function followed by the usual techniques of statistical mechanics to examine the thermodynamic properties.

Partial actions and proper extensions of two-sided restriction semigroups

We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We introduce and study several classes of partial actions of two-sided restriction semigroups that generalize partial actions of monoids and of inverse semigroups. We establish an adjunction between the category P ( S ) of proper extensions of a restriction semigroup S and a category A ( S ) of partial actions of S subject to certain conditions going back to the work of O'Carroll. In the category A ( S ) , we specify two isomorphic subcategories , one being reflective and the other one coreflective, each of which is equivalent to the category P ( S ) .

Some characterizations of ultrametrically injective spaces

In this article, we present a proof (in two different ways) of the fact that every hyperconvex ultrametric space is ultrametrically injective. The first proof will be constructive in nature and the second proof will appeal to Zorn’s Lemma. We show also that the ultrametrically injective hull $$T_X$$ , of an ultrametric space (X, m), as constructed by Bayod et al. and $${\mathcal {K}}(X)$$ (the space of all ultra-Katětov functions on an ultrametric space X) are both ultrametrically injective. Furthermore, it is shown that ultrametrically injective spaces do not contain proper essential extensions. In the end, we give a number of characterizations of the ultrametrically injective hull $$T_X$$ of an ultrametric space (X, m).

Fuzzy extensions of PROMETHEE: Models of different complexity with different ranking methods and their comparison

Models of Fuzzy Multi-Criteria Decision Analysis (FMCDA) are based, as a rule, on different approaches to fuzzy extension of source MCDA methods. For this, simplified models are used to approximate the functions of fuzzy variables with propagation of parametric fuzzy numbers (FNs) through all calculations. In this paper, authors suggest a novel approach to fuzzy extension of MCDA methods, for PROMETHEE-I/II, through development of fuzzy PROMETHEE-I/II (FPOMETHEE-I/II) models of different complexity: in addition to simplified models, the standard fuzzy arithmetic (SFA), and transformation methods (TMs) are implemented for assessing functions of FNs corresponding to these models. For ranking of alternatives, two defuzzification based, and one pairwise comparison ranking methods are implemented within the developed models. Special attention is paid to analysis of the overestimation problem, which can occur when using SFA in the presence of dependent variables in corresponding expressions, and to “proper fuzzy extensions” of PROMETHEE-I/II (i.e., results of all functions of FNs within the model are in accordance with the extension principle) based on TMs and, for some models, on the SFA. One of the key goals of this contribution is comparison of the distinctions in ranking alternatives by different FPROMETHEE-II models. It is demonstrated by evaluating a large number of scenarios based on Monte Carlo simulation that the probability of distinction in ranking alternatives by “proper” and “approximated” FPROMETHEE-II models may be considered as significant for ranking multicriteria problems. Another goal of this paper is analysis of the correctness of FPROMETHEE-I/II models with respect to the basic MCDA axiom related to ranking of dominated and dominating alternatives. Authors demonstrate that the basic axiom can be violated, in the general case, by all developed FPROMETHEE-I/II models and suggest an approach to fix this problem.

Collisionless kinetic theory for parametrically pumped magnons

We discuss collisionless kinetic equations describing the non-equilibrium dynamics of magnons in a ferromagnet exposed to an oscillating microwave field. Previously, this problem has been treated within the so-called "S-theory" where the collision integral in the kinetic equation for the magnon distribution is either neglected or taken into account phenomenologically via an effective relaxation time. However, the possibility of magnon condensation has not been included in S-theory. Moreover, the momentum integrations appearing in the magnon self-energies are usually decoupled by retaining only the term where the loop momentum is equal to the external momentum. In this work we critically examine the accuracy of these approximations and develop the proper extensions of S-theory. We show that these extensions can significantly modify the time evolution of the magnon distribution.