Algebras Of Type Research Articles

The centaur-algebra of observables

This letter explores a transition in the type of von Neumann algebra for asymptotically AdS spacetimes from the implementations of the different gravitational constraints. We denote it as the centaur-algebra of observables. In the first part of the letter, we employ a class of flow geometries interpolating between AdS2 and dS2 spaces, the centaur geometries. We study the type II∞ crossed product algebra describing the semiclassical gravitational theory, and we explore the algebra of bounded sub-regions in the bulk theory following TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document} deformations of the geometry and study the gravitational constraints with respect to the quasi-local Brown-York energy of the system at a finite cutoff. In the second part, we study arbitrary asymptotically AdS spacetimes, where we implement the boundary protocol of an infalling observer modeled as a probe black hole proposed by [1] to study modifications in the algebra. In both situations, we show how incorporating the constraints requires a type II1 description.

Pre-Leibniz algebras

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras.

Algebraic Localization–Delocalization Phase Transition in Moving Potential Wells on a Lattice

AbstractThe localization and scattering properties of potential wells or barriers uniformly moving on a lattice are strongly dependent on the drift velocity, owing to a violation of the Galilean invariance of the discrete Schrödinger equation. Here a type of localization–delocalization phase transition of algebraic type is unraveled, which does not require any kind of disorder and arises when a power‐law potential well drifts fast on a lattice. While for an algebraic exponent α lower than the critical value dynamical delocalization is observed, for asymptotic localization, corresponding to asymptotic frozen dynamics, is instead realized. At the critical phase transition point an oscillatory dynamics is found, corresponding to Bloch oscillations. An experimentally accessible photonic platform for the observation of the predicted algebraic phase transition, based on light dynamics in synthetic mesh lattices, is suggested.

A Formalization of Core Why3 in Coq

Intermediate verification languages like Why3 and Boogie have made it much easier to build program verifiers, transforming the process into a logic compilation problem rather than a proof automation one. Why3 in particular implements a rich logic for program specification with polymorphism, algebraic data types, recursive functions and predicates, and inductive predicates; it translates this logic to over a dozen solvers and proof assistants. Accordingly, it serves as a backend for many tools, including Frama-C, EasyCrypt, and GNATProve for Ada SPARK. But how can we be sure that these tools are correct? The alternate foundational approach, taken by tools like VST and CakeML, provides strong guarantees by implementing the entire toolchain in a proof assistant, but these tools are harder to build and cannot directly take advantage of SMT solver automation. As a first step toward enabling automated tools with similar foundational guarantees, we give a formal semantics in Coq for the logic fragment of Why3. We show that our semantics are useful by giving a correct-by-construction natural deduction proof system for this logic, using this proof system to verify parts of Why3's standard library, and proving sound two of Why3's transformations used to convert terms and formulas into the simpler logics supported by the backend solvers.

The Essence of Generalized Algebraic Data Types

This paper considers direct encodings of generalized algebraic data types (GADTs) in a minimal suitable lambda-calculus. To this end, we develop an extension of System F ω with recursive types and internalized type equalities with injective constant type constructors. We show how GADTs and associated pattern-matching constructs can be directly expressed in the calculus, thus showing that it may be treated as a highly idealized modern functional programming language. We prove that the internalized type equalities in conjunction with injectivity rules increase the expressive power of the calculus by establishing a non-macro-expressibility result in F ω , and prove the system type-sound via a syntactic argument. Finally, we build two relational models of our calculus: a simple, unary model that illustrates a novel, two-stage interpretation technique, necessary to account for the equational constraints; and a more sophisticated, binary model that relaxes the construction to allow, for the first time, formal reasoning about data-abstraction in a calculus equipped with GADTs.

Unboxed Data Constructors: Or, How cpp Decides a Halting Problem

We propose a new language feature for ML-family languages, the ability to selectively unbox certain data constructors, so that their runtime representation gets compiled away to just the identity on their argument. Unboxing must be statically rejected when it could introduce confusion, that is, distinct values with the same representation. We discuss the use-case of big numbers, where unboxing allows to write code that is both efficient and safe, replacing either a safe but slow version or a fast but unsafe version. We explain the static analysis necessary to reject incorrect unboxing requests. We present our prototype implementation of this feature for the OCaml programming language, discuss several design choices and the interaction with advanced features such as Guarded Algebraic Datatypes. Our static analysis requires expanding type definitions in type expressions, which is not necessarily normalizing in presence of recursive type definitions. In other words, we must decide normalization of terms in the first-order λ-calculus with recursion. We provide an algorithm to detect non-termination on-the-fly during reduction, with proofs of correctness and completeness. Our algorithm turns out to be closely related to the normalization strategy for macro expansion in the cpp preprocessor.

Izrazite odlike i validacija d⋇-algebri - analitičko istraživanje

Introduction/purpose: This research introduces the concept of a d⋇-algebra, a unique structure in the field of abstract algebra. The study aims to explore the defining features and distinct properties of d⋇-algebras, distinguishing them from other algebraic systems and examining their interrelations with other types of algebras. Methods: The methodology includes the formal definition and characterization of d⋇-algebras, a comparative analysis with the existing algebraic structures, and an exploration of their interconnections. An algorithm is developed to verify whether a given structure meets the conditions of a d⋇-algebra. Results: The results reveal that d⋇-algebras possess unique properties not found in other algebraic systems. The comparative study clarifies their distinctive place within the algebraic landscape and highlights significant interrelations with other structures. The verification algorithm proves effective in identifying d⋇-algebras, providing a systematic approach for further study. Conclusions: In conclusion, d⋇-algebras represent a significant addition to abstract algebra, offering new theoretical insights and potential for future research. The study's findings enhance the understanding of algebraic systems and their interconnections, opening new avenues for exploration in the field.

On Schur-type theorem for Leibniz 3-algebras

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group G/ζ(G) of a group G is finite, then its derived subgroup [G,G] is also finite. This theorem was proved by B. Neumann in 1951. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, n-groups, associative algebras, Lie algebras, Lie n-algebras. In 2016, L.A. Kurdachenko, J. Otal and O.O. Pypka proved an analogue of Schur theorem for Leibniz algebras: if central factor-algebra L/ζ(L) of Leibniz algebra L has finite dimension, then its derived ideal [L,L] is also finite-dimensional. Moreover, they also proved a slightly modified analogue of Schur theorem: if the codimensions of the left ζ^l (L) and right ζ^r (L) centers of Leibniz algebra L are finite, then its derived ideal [L,L] is also finite-dimensional. One of the generalizations of Leibniz algebras is the so-called Leibniz n-algebras. It is worth noting that Leibniz n-algebra theory is currently much less developed than Leibniz algebra theory. One of the directions of development of the general theory of Leibniz n-algebras is the search for analogies with other types of algebras. Therefore, the question of proving analogs of the above results for this type of algebras naturally arises. In this article, we prove the analogues of the two mentioned theorems for Leibniz n-algebras for the case n=3. The obtained results indicate the prospects of further research in this direction.

Вариации жесткости

One of the main derived objects of a given structure is its automorphism group, which shows how freely elements of the structure can be related to each other by automorphisms. Two extremes are observed here: the automorphism group can be transitive and allow any two elements to be connected to each other, or can be one-element, when no two different elements are connected by automorphisms, i.e., the structure is rigid. The rigidity given by a one-element group of automorphisms is called semantic. It is of interest to study and describe structures that do not differ much from semantically rigid structures, i.e., become semantically rigid after selecting some finite set of elements in the form of constants. Another, syntactic form of rigidity is based on the possibility of getting all elements of the structure into a definable closure of the empty set. It is also of interest here to describe “almost” syntactically rigid structures, i.e., structures covered by the definable closure of some finite set. The paper explores the possibilities of semantic and syntactic rigidity. The concepts of the degrees of semantic and syntactic rigidity are defined, both with respect to existence and with respect to the universality of finite sets of elements of a given cardinality. The notion of a rigidity index is defined, which shows an upper bound for the cardinalities of algebraic types, and its possible values are described. Rigidity variations and their degrees are studied both in the general case, for special languages, including the one-place predicate signature, and for some natural operations with structures, including disjunctive unions and compositions of structures. The possible values of the degrees for a number of natural examples are shown, as well as the dynamics of the degrees when taking the considered operations.

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is “disjoint” in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.

Lie-Poisson gauge theories and κ-Minkowski electrodynamics

We consider gauge theories on Poisson manifolds emerging as semiclassical approximations of noncommutative spacetime with Lie algebra type noncommutativity. We prove an important identity, which allows to obtain simple and manifestly gauge-covariant expressions for the Euler-Lagrange equations of motion, the Bianchi and the Noether identities. We discuss the non-Lagrangian equations of motion, and apply our findings to the κ-Minkowski case. We construct a family of exact solutions of the deformed Maxwell equations in the vacuum. In the classical limit, these solutions recover plane waves with left-handed and right-handed circular polarization, being classical counterparts of photons. The deformed dispersion relation appears to be nontrivial.

Mutually Iso-Recursive Subtyping

Iso-recursive types are often taken as a type-theoretic model for type recursion as present in many programming languages, e.g., classes in object-oriented languages or algebraic datatypes in functional languages. Their main advantage over an equi-recursive semantics is that they are simpler and algorithmically less expensive, which is an important consideration when the cost of type checking matters, such as for intermediate or low-level code representations, virtual machines, or runtime casts. However, a closer look reveals that iso-recursion cannot, in its standard form, efficiently express essential type system features like mutual recursion or non-uniform recursion. While it has been folklore that mutual recursion and non-uniform type parameterisation can nicely be handled by generalising to higher kinds, this encoding breaks down when combined with subtyping: the classic “Amber” rule for subtyping iso-recursive types is too weak to express mutual recursion without falling back to encodings of quadratic size.We present a foundational core calculus of iso-recursive types withdeclaredsubtyping that can express both inter- and intra-recursion subtyping without such blowup, including subtyping between constructors of higher or mixed kind. In a second step, we identify a syntactic fragment of this general calculus that allows for more efficient type checking without “deep” substitutions, by observing that higher-kinded iso-recursive types can be inserted to “guard” against unwanted β-reductions. This fragment closely resembles the structure of typical nominal subtype systems, but without requiring nominal semantics. It has been used as the basis for a proposed extension of WebAssembly with recursive types.

Bit-Stealing Made Legal: Compilation for Custom Memory Representations of Algebraic Data Types

Initially present only in functional languages such as OCaml and Haskell, Algebraic Data Types (ADTs) have now become pervasive in mainstream languages, providing nice data abstractions and an elegant way to express functions through pattern matching. Unfortunately, ADTs remain seldom used in low-level programming. One reason is that their increased convenience comes at the cost of abstracting away the exact memory layout of values. Even Rust, which tries to optimize data layout, severely limits control over memory representation. In this article, we present a new approach to specify the data layout of rich data types based on a dual view: a source type, providing a high-level description available in the rest of the code, along with a memory type, providing full control over the memory layout. This dual view allows for better reasoning about memory layout, both for correctness, with dedicated validity criteria linking the two views, and for optimizations that manipulate the memory view. We then provide algorithms to compile constructors and destructors, including pattern matching, to their low-level memory representation. We prove our compilation algorithms correct, implement them in a tool called ribbit that compiles to LLVM IR, and show some early experimental results.

Wave trapping by freely floating obstacles in a discretely stratified fluid

In the context of the linear water wave theory, we model the interaction of time‐harmonic water waves with an infinite array of three‐dimensional freely floating bodies in a multilayer fluid through a spectral boundary value problem, consisting of a differential equation coupled with an algebraic system. We present a variational and operator formulation for the problem and derive sufficient conditions for the existence of trapped waves. We give examples of floating obstacles supporting trapped waves in a discretely stratified fluid in different obstacle configurations with a variable number of layers. Some conclusions regarding the dependence of the sufficient condition on the number of fluid layers are drawn, considering whether the sufficient condition is of the potential or algebraic type.

Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migration–consumption interaction

A no-flux initial-boundary value problem for ut=Δ(uϕ(v)),vt=Δv−uv,(⋆) is considered in smoothly bounded subdomains of with and suitably regular initial data, where φ is assumed to reflect algebraic type cross-degeneracies by sharing essential features with for some . Based on the discovery of a gradient structure acting at regularity levels mild enough to be consistent with degeneracy-driven limitations of smoothness information, in this general setting it is shown that with some measurable limit profile and some null set , a corresponding global generalized solution, known to exist according to recent literature, satisfies ρ(u(⋅,t))⇀⋆ρ(u∞)in L∞(Ω) and v(⋅,t)→0in Lp(Ω)for all p⩾1 as , where , . In the particular case when either and is arbitrary, or and , additional quantitative information on the deviation of trajectories from the initial data is derived. This is found to imply a lower estimate for the spatial oscillation of the respective first components throughout evolution, and moreover this is seen to entail that each of the uncountably many steady states of () is stable with respect to a suitably chosen norm topology.

TRIPLE TORTKEN IDENTITIES

We define a triple Tortken product in Novikov algebras. Using computer algebra calculations, we give a list of polynomial identities up to degree 5 satisfied by Tortken triple product in every Novikov algebra. It has applications in theoretical physics, specifically in the field of quantum field theory and topological field theory. A Novikov algebra is defined as a vector space equipped with a binary operation called the Novikov bracket. The Jacobi identity ensures that the Novikov bracket behaves analogously to the commutator in Lie algebras. However, unlike Lie algebras, Novikov algebras are non-associative due to the presence of the Jacobi identity rather than the associativity condition. Novikov algebras find applications in theoretical physics, particularly in the study of topological field theories and quantum field theories on noncommutative spaces. They provide a framework for describing and analyzing certain algebraic structures that arise in these areas of physics. It's worth noting that Novikov algebras are a specific type of non-associative algebra, and there are various other types of non-associative algebras studied in mathematics and physics, each with its own defining properties and applications.

Using the onto-semiotic approach to analyze novice algebra learners’ meaning-making processes with different representations

Representations are key to mathematical activities and meaning-making processes as they are part of modeling, connecting, communicating, and understanding mathematical ideas and concepts. The current study sought to examine a group of novice algebra learners’ interactions with different representations from an onto-semiotic approach. A case study method was employed to understand how different algebraic practices (abstracting, generalizing, justifying, and operating on symbols) and functional thinking types (recursive, covariational, and correspondence) were facilitated through working with multiple representations. Three 6th graders participated in the study by completing 12 algebra tasks and taking part in two interviews. The onto-semiotic approach guided the data analysis process that involved the identification of mathematical objects that emerged in the participating students’ mathematical practices. Then, the configuration of objects and semiotic functions established by the students in the functional situations was examined to understand the role of representations in the students’ development of algebraic thinking and practices. Findings showed that abstraction is an essential process for generalization. Thinking about far figures facilitated abstraction and generalization through helping students construct non-ostensive concrete/pictorial representations. Verbal representations interacted with all representations and preceded symbolic representations. Working with near figures promoted recursive and covariational thinking while examining the far figures usually resulted in correspondence thinking. Implications for the school curriculum are discussed in the paper.

Description of the automorphism groups of some Leibniz algebras

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all elements $a,b,c\in L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,b\in L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of the Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of nilpotent three-dimensional Leibniz algebras.

Groupoid and algebra of the infinite quantum spin chain

It is well known that certain features of a quantum theory cannot be described in the standard picture on a Hilbert space. In particular, this happens when we try to formally frame a quantum field theory, or a thermodynamic system with finite density. This forces us to introduce different types of algebras, more general than the ones we usually encounter in a standard course of quantum mechanics. We show how these algebras naturally arise in the Schwinger description of the quantum mechanics of an infinite spin chain. In particular, we use the machinery of Dirac-Feynman-Schwinger (DFS) states developed in recent works to introduce a dynamics based on the modular theory by Tomita-Takesaki, and consequently we apply this approach to describe the Ising model.

Parameterized Algebraic Protocols

We propose algebraic protocols that enable the definition of protocol templates and session types analogous to the definition of domain-specific types with algebraic datatypes. Parameterized algebraic protocols subsume all regular as well as most context-free and nested session types and, at the same time, replace the expensive superlinear algorithms for type checking by a nominal check that runs in linear time. Algebraic protocols in combination with polymorphism increase expressiveness and modularity by facilitating new ways of parameterizing and composing session types.