Search

Enriched motivic A -spaces are introduced and studied in this paper, where A is an additive category of correspondences. They are linear counterparts of motivic Γ-spaces. It is shown that rational special enriched motivic Cor ˜ -spaces recover connective motivic bispectra with rational coefficients, where Cor ˜ is the category of Milnor–Witt correspondences. [ABSTRACT FROM AUTHOR]

We give a quick proof of the fact that the relative de Rham cohomology groups H dR i (X / S) of a smooth and proper map X/S between schemes over Q are vector bundles on the base, replacing Hodge-theoretic and transcendental methods with A 1 -homotopy theory. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We introduce a new method for precisely relating algebraic structures in a presheaf category and judgements of its internal type theory. The method provides a systematic way to organise complex diagrammatic reasoning and generalises the well-known Kripke-Joyal forcing for logic. As an application, we prove several properties of algebraic weak factorisation systems considered in Homotopy Type Theory. [ABSTRACT FROM AUTHOR]

By making use of Halperin's local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by Gómez-Tato, Halperin and Tanré. In the latter half of this article, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with the pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau–de Rham cohomology algebra of the diffeological space. By using the pullback construction, we also discuss a local system model for a topological homotopy pushout. [ABSTRACT FROM AUTHOR]

Within the framework of Riehl–Shulman's synthetic (∞ , 1) -category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness condition à la Chevalley, Gray, Street, and Riehl–Verity, a two-sided Yoneda Lemma, as well as the proof of several closure properties. Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the two-sided case in a modular fashion. We also briefly discuss discrete two-sided cartesian fibrations in this setting, corresponding to (∞ , 1) -distributors. The systematics of our definitions and results closely follows Riehl–Verity's ∞ -cosmos theory, but formulated internally to Riehl–Shulman's simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic (∞ , 1) -categories correspond to internal (∞ , 1) -categories implemented as Rezk objects in an arbitrary given (∞ , 1) -topos. [ABSTRACT FROM AUTHOR]

We present a relatively elementary construction of a spin manifold with vanishing first rational Pontryagin class satisfying the conditions of Hirzebruch's prize question, using a modification of Sullivan's theorem for the realization of rational homotopy types by closed smooth manifolds. As such this is an alternative to the solutions of the problem given by Hopkins–Mahowald, though without the guarantee of the constructed manifold admitting a string structure. We present a particular solution which is rationally 7 connected with eighth Betti number equal to one; our approach yields many other solutions with complete knowledge of their rational homotopy type. [ABSTRACT FROM AUTHOR]

We show that if the universal cover of a closed smooth manifold admitting a metric with non-negative Ricci curvature is formal, then the manifold itself is formal. We reprove a result of Fiorenza–Kawai–Lê–Schwachhöfer, that closed orientable manifolds with a non-negative Ricci curvature metric and sufficiently large first Betti number are formal. Our method allows us to remove the orientability hypothesis; we further address some cases of non-closed manifolds. [ABSTRACT FROM AUTHOR]

We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable \infty-category of non-\mathbb {A}^1-invariant motivic spectra, which turns out to be equivalent to the \infty-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this \infty-category satisfies \mathbb {P}^1-homotopy invariance and weighted \mathbb {A}^1-homotopy invariance, which we use in place of \mathbb {A}^1-homotopy invariance to obtain analogues of several key results from \mathbb {A}^1-homotopy theory. These allow us in particular to define a universal oriented motivic \mathbb {E}_\infty-ring spectrum \mathrm {MGL}. We then prove that the algebraic K-theory of a qcqs derived scheme X can be recovered from its \mathrm {MGL}-cohomology via a Conner–Floyd isomorphism \[ \mathrm {MGL}^{**}(X)\otimes _{\mathrm {L}{}}\mathbb {Z}[\beta ^{\pm 1}]\simeq \mathrm {K}{}^{**}(X), \] where \mathrm {L}{} is the Lazard ring and \mathrm {K}{}^{p,q}(X)=\mathrm {K}{}_{2q-p}(X). Finally, we prove a Snaith theorem for the periodized version of \mathrm {MGL}. [ABSTRACT FROM AUTHOR]

We work within the framework of the variational bicomplex: we define A ∞ -algebra structures on horizontal and vertical cohomologies of (formally integrable) partial differential equations with the help of Merkulov's theorem. Since higher order A ∞ -algebra operations are related to Massey products, our observation implies the existence of invariants for differential equations that go beyond conservation laws. We also propose notions of formality for PDEs, and we present examples of formal equations. [ABSTRACT FROM AUTHOR]

By a theorem of Christensen and Hovey, the category of non-negatively graded chain complexes has a model structure, called the h-model structure or Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. The Dold–Kan correspondence induces a model structure on the category of simplicial modules. In this paper, we give a description of the two model categories and some of their properties, notably the fact that both are monoidal. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and uniqueness of solutions to a generalized nonlinear functional integral equation (G‐NLFIE) associated with certain epidemic models of infectious diseases, defined within the Banach space C[0,1]$$ C\left[0,1\right] $$. Our existence results include several specific cases of nonlinear functional integral equations that commonly occur in nonlinear sciences. We then introduce an iterative algorithm that combines Adomian's decomposition method (ADM) with the modified homotopy perturbation method (mHPM) to approximate solutions to the G‐NLFIE. The paper addresses the convergence properties and error analysis of this method. Finally, we present numerical examples to demonstrate the effectiveness and efficiency of our proposed approach. [ABSTRACT FROM AUTHOR]

We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring |$R$|⁠. The weak equivalences are given by: (1) an |$R$| -linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an |$R$| -homology equivalence between universal covers, and (3) |$R$| -homology equivalences. Analogously, for any field |${\mathbb{F}}$|⁠ , we construct three model structures on the category of connected simplicial cocommutative |${\mathbb{F}}$| -coalgebras. The weak equivalences in this context are (1 ′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2 ′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3 ′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3 ′), we prove that, when |${\mathbb{F}}$| is algebraically closed, the simplicial |${\mathbb{F}}$| -coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when |${\mathbb{F}}$| is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of |${\mathbb{F}}$|⁠. [ABSTRACT FROM AUTHOR]

We produce a fully faithful functor from finite type nilpotent spaces to cosimplicial binomial rings, thus giving an algebraic model of integral homotopy types. As an application, we construct an integral version of the Grothendieck–Teichmüller group. [ABSTRACT FROM AUTHOR]

The current study employed the innovative conformable fractional method to analyze the nonlinear Benney equations involving the conformable fractional derivative. Conformable fractional Benney equations have been examined by the conformable q‐Shehu analysis transform method. By including nonlinear factors, it offers a more precise depiction of wave propagation compared to linear models. Various natural phenomena, including ocean waves, plasma waves, and some forms of solitons, display nonlinear behavior that cannot be precisely explained by linear equations. The fractional Benney equation is important because it extends the classical Benney equation, which describes the evolution of weakly nonlinear and weakly dispersive long waves in shallow water. By incorporating fractional calculus operators, the fractional Benney equation provides a more accurate description of wave propagation phenomena in certain physical systems characterized by nonlocal or memory‐dependent behavior. The utilization of the Benney equation enables researchers to simulate these occurrences with greater realism. This study investigates the convergence and inaccuracy of the future scheme. The conformable q‐Shehu homotopy analysis transform method (Cq‐SHATM) generates h‐curves that demonstrate the convergence interval of the series solution obtained. In order to determine the effectiveness and suitability of the Cq‐SHATM, uniqueness and convergence theorems have been proven. This study presents an application that showcases the potential advantages and efficacy of the suggested method. Moreover, an error analysis is conducted to validate the precision of the scheme. Computational simulations are performed to verify the accuracy of the upcoming method. This study presents the results gained from the numerical and graphical analysis. The method presented in this work demonstrates a high level of computational accuracy and simplicity in analyzing and solving complex phenomena associated with conformable fractional nonlinear partial differential equations in the fields of science and technology. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we prove that the category of diffeological spaces does not admit a model structure transferred via the smooth singular complex functor from simplicial sets, resolving in the negative a conjecture of Christensen and Wu, in contrast to Kihara's model structure on diffeological spaces constructed using a different singular complex functor. Next, motivated by applications in quantum field theory and topology, we embed diffeological spaces into sheaves of sets (not necessarily concrete) on the site of smooth manifolds and study the proper combinatorial model structure on such sheaves transferred via the smooth singular complex functor from simplicial sets. We show the resulting model category to be Quillen equivalent to the model category of simplicial sets. We then show that this model structure is cartesian, all smooth manifolds are cofibrant, and establish the existence of model structures on categories of algebras over operads. Finally, we use these results to establish analogous properties for model structures on simplicial presheaves on smooth manifolds, as well as presheaves valued in left proper combinatorial model categories, and prove a generalization of the smooth Oka principle established by Berwick-Evans, Boavida de Brito and Pavlov. We apply these results to establish classification theorems for differential-geometric objects like closed differential forms, principal bundles with connection, and higher bundle gerbes with connection on arbitrary cofibrant diffeological spaces. [ABSTRACT FROM AUTHOR]

We develop an analogue of the theory of *-modules in the world of simplicial sets, based on actions of a certain simplicial monoid EM originally appearing in the construction of global algebraic K-theory. As our main results, we show that strictly commutative monoids with respect to a certain box product on these simplicial *-modules yield models of equivariantly and globally coherently commutative monoids, and we give a characterization of simplicial *-modules in terms of a certain mildness condition on the EM-action, relaxing the notion of tameness previously investigated by Sagave--Schwede and the frst author. [ABSTRACT FROM AUTHOR]

Let F → X p→ Y be a simply connected fibration with F and Y finite. Let Baut1X and Baut1p be the Dold--Lashof classifying spaces of X and p, respectively. In this paper, we study the relation between the coformality of Baut1F and that of Baut1p. [ABSTRACT FROM AUTHOR]

Pitts generalized nominal sets to finitely supported Cb-sets by utilizing the monoid Cb of name substitutions instead of the monoid of finitary permutations over names. Finitely supported Cb-sets provide a framework for studying essential ideas of models of homotopy type theory at the level of convenient abstract categories. Here, the interplay of two separate categories of finitely supported actions of a submonoid of End(D), for some countably infinite set D, over sets is first investigated. In particular, we specify the structure of free objects. Then, in the category of finitely supported Cb-sets, we characterize the finitely presentable objects and provide a generator in this category. [ABSTRACT FROM AUTHOR]

We introduce a definition and framework for internal topological symmetries in quantum field theory, including "noninvertible symmetries" and "categorical symmetries". We outline a calculus of topological defects which takes advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called "gauging" and "condensation defects"), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We give a procedure to construct trisections for closed manifolds generated by colored tensor models without restrictions on the number of simplices in the triangulation, therefore generalizing previous works in the context of crystallizations and PL-manifolds. We give a description of how trisection diagrams can arise from colored tensor model graphs for closed 4-manifolds. We further speculate on generalization of similar constructions for a class of singular-manifolds generated by simplicial colored tensor models. [ABSTRACT FROM AUTHOR]

This paper presents Leray–Schauder alternatives and a topological transversality (homotopy) theorem for compact upper semicontinuos maps having (strongly) convexly totally bounded ranges. [ABSTRACT FROM AUTHOR]

We define unbounded twisted complexes and bicomplexes generalising the notion of a (bounded) twisted complex over a DG category [6]. These need to be considered relative to another DG category B admitting countable direct sums and shifts. The resulting DG category of unbounded twisted complexes has a fully faithful convolution functor into Mod - B which factors through B if the latter is closed under twisting. As an application, we rewrite definitions of A ∞ -structures in terms of twisted complexes to make them work in an arbitrary monoidal DG category or a DG bicategory. [ABSTRACT FROM AUTHOR]

Given an algebra in a monoidal 2-category, one can construct a 2-category of right modules. Given a braided algebra in a braided monoidal 2-category, it is possible to refine the notion of right module to that of a local module. Under mild assumptions, we prove that the 2-category of local modules admits a braided monoidal structure. In addition, if the braided monoidal 2-category has duals, we go on to show that the 2-category of local modules also has duals. Furthermore, if it is a braided fusion 2-category, we establish that the 2-category of local modules is a braided multifusion 2-category. We examine various examples. For instance, working within the 2-category of 2-vector spaces, we find that the notion of local module recovers that of braided module 1-category. Finally, we examine the concept of a Lagrangian algebra, that is a braided algebra with trivial 2-category of local modules. In particular, we completely describe Lagrangian algebras in the Drinfeld centers of fusion 2-categories, and we discuss how this result is related to the classifications of topological boundaries of (3 + 1)d topological phases of matter. [ABSTRACT FROM AUTHOR]

In a 2005 paper, Casacuberta, Scevenels, and Smith construct a homotopy idempotent functor E$E$ on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map f$f$ is independent of the ZFC axioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe U$\mathcal {U}$. When specialized to an appropriate family, this produces a localization which when interpreted in the ∞$\infty$‐topos of spaces agrees with the localization corresponding to E$E$. Our approach generalizes the approach of Casacuberta et al. (Adv. Math. 197 (2005), no. 1, 120–139) in two ways. First, by working in homotopy type theory, our construction can be interpreted in any ∞$\infty$‐topos. Second, while the local objects produced by Casacuberta et al. are always 1‐types, our construction can produce n$n$‐types, for any n$n$. This is new, even in the ∞$\infty$‐topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about "small" types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice that implies that sets cover and that the law of excluded middle holds. [ABSTRACT FROM AUTHOR]

A d$d$‐dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal (∞,n)$(\infty,n)$‐category of d$d$‐bordisms (embedded into R∞$\mathbb {R}^\infty$ and equipped with a tangential (X,ξ)$(X,\xi)$‐structure) that lands in the Picard subcategory of the target symmetric monoidal (∞,n)$(\infty,n)$‐category. We classify these field theories in terms of the cohomology of the (n−d)$(n-d)$‐connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the (∞,n)$(\infty,n)$‐category of bordisms with Ω∞−nMTξ$\Omega ^{\infty -n}MT\xi$ as an E∞$E_\infty$‐space. This generalizes the celebrated result of Galatius–Madsen–Tillmann–Weiss (Acta Math. 202 (2009), no. 2, 195–239) in the case n=1$n=1$, and of Bökstedt–Madsen (An alpine expedition through algebraic topology, vol. 617, Contemp. Math., Amer. Math. Soc., Providence, RI, 2014, pp. 39–80) in the n$n$‐uple case. We also obtain results for the (∞,n)$(\infty,n)$‐category of d$d$‐bordisms embedding into a fixed ambient manifold M$M$, generalizing results of Randal–Williams (Int. Math. Res. Not. IMRN 2011 (2011), no. 3, 572–608) in the case n=1$n=1$. We give two applications: (1) we completely compute all extended and partially extended invertible TFTs with target a certain category of n$n$‐vector spaces (for n⩽4$n \leqslant 4$), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum (Forum Math. 25 (2013), no. 5, 1067–1106. arXiv:0912.4706). [ABSTRACT FROM AUTHOR]

In the early 1940s, P. A. Smith showed that if a finite p-group G acts on a finite dimensional complex X that is mod p acyclic, then its space of fixed points, XG, will also be mod p acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a G-space X is a equivariant homotopy retract of the p-localization of a based finite G-C.W. complex, given H < G and n, what is the smallest r such that if XH is acyclic in the (n+r)th Morava K-theory, then XG must be acyclic in the nth Morava K-theory? Barthel et. al. then answered this when G is abelian, by finding general lower and upper bounds for these "blue shift" numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equiv- alent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod p homology, and thus applies to all finite dimensional G-spaces. This unlocks new techniques and applications in chromatic fixed point theory. Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for K(n)*-homology disks and spheres. Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava K-theories of all real Grassmanians. [ABSTRACT FROM AUTHOR]

Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. Our framework is based on topology and discrete differential geometry, as well as gradient systems and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models. [ABSTRACT FROM AUTHOR]

We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal category are symmetric monoidal equivalences. As an application, we recover that the invariant of 2-dimensional manifolds given by the product of (extended) commutative Frobenius algebras in a symmetric tensor category is the multiplication of the invariants given by each of the algebras. [ABSTRACT FROM AUTHOR]

This article depicts an approximate solution of systems of nonlinear fractional biochemical reactions for the Michaelis-Menten enzyme kinetic model arising from the enzymatic reaction process. This present work is concerned with fundamental enzyme kinetics utilised to assess the efficacy of powerful mathematical approaches such as the homotopy perturbation method (HPM), homotopy analysis method (HAM), and homotopy analysis transform method (HATM) to get the approximate solutions of the biochemical reaction model with time-fractional derivatives. The Caputo-type fractional derivatives are explored. The proposed method is implemented to formulate a fractional differential biochemical reaction model to obtain approximate results subject to various settings of the fractional parameters with statistical validation at different stages. The comparison results reveal the complexity of the enzyme process and obtain approximate solutions to the nonlinear fractional differential biochemical reaction model. [ABSTRACT FROM AUTHOR]

Existing computer programs called proof assistants can verify the correctness of mathematical proofs but their specialized proof languages present a barrier to entry for many mathematicians. Large language models have the potential to lower this barrier, enabling mathematicians to interact with proof assistants in a more familiar vernacular. Among other advantages, this may allow mathematicians to explore radically new kinds of mathematics using an LLM-powered proof assistant to train their intuitions as well as ensure their arguments are correct. Existing proof assistants have already played this role for fields such as homotopy type theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a Cartan calculus on the homology of free loop spaces which is introduced by Kuribayashi, Wakatsuki, Yamaguchi and the author [ Cartan calculi on the free loop spaces , Preprint, arXiv:2207.05941, 2022]. In particular, it is proved that the Cartan calculus can be described by the loop product and the loop bracket in string topology. Moreover, by using the descriptions, we show that the loop product behaves well with respect to the Hodge decomposition of the homology of free loop spaces. [ABSTRACT FROM AUTHOR]

The present study aims to perform a comparative analysis of the plane Couette flow of a couple stress fluid under the influence of magnetohydrodynamics (MHD) using two different methods: the Optimal Auxiliary Function Method (OAFM) and the Homotopy Perturbation Method (HPM). The couple stress fluid is known for its non-Newtonian behavior, where the fluid's response to shear is influenced by the presence of internal microstructure. The OAFM and HPM are utlized to solve the governing equations of the couple stress fluid flow under MHD. The OAFM is a numerical technique that involves introducing an auxiliary function to simplify the equations, leading to an easier solution procedure. On the other hand, HPM is an analytical method that employs a series solution. The comparative analysis focuses on examining the accuracy, efficiency, and convergence behavior of the two methods. Various flow parameters such as the couple stress parameter, the magnetic parameter, and the velocity ratio are considered to investigate their influence on the flow behavior. Furthermore the HPM solution was compared with the OAFM solution using different graphs and tables. It reveals that the solution obtained by HPM is batter than OAFM solution. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new approach to construct monoidal Hom-Hopf algebras. We investigate monoidal Hom-Hopf algebra structure on a left (n, l)-Hom-crossed coproduct structure with a left n-Hom-smash product structure, obtaining Radford [ n , (n , l) ] -biproduct structure theorem. Then, we study a Hom-coaction admissible mapping system to characterize this Radford [ n , (n , l) ] -biproduct structure. Finally, we study the cosemisimplicity of a special Hom-smash coproduct and prove the related Maschke theorem. [ABSTRACT FROM AUTHOR]

Presheaf models of dependent type theory have been successfully applied to model HoTT, parametricity, and directed, guarded and nominal type theory. There has been considerable interest in internalizing aspects of these presheaf models, either to make the resulting language more expressive, or in order to carry out further reasoning internally, allowing greater abstraction and sometimes automated verification. While the constructions of presheaf models largely follow a common pattern, approaches towards internalization do not. Throughout the literature, various internal presheaf operators (√, Φ/extent, Ψ/Gel, Glue, Weld, mill, the strictness axiom and locally fresh names) can be found and little is known about their relative expressiveness. Moreover, some of these require that variables whose type is a shape (representable presheaf, e.g. an interval) be used affinely. We propose a novel type former, the transpension type, which is right adjoint to universal quantification over a shape. Its structure resembles a dependent version of the suspension type in HoTT. We give general typing rules and a presheaf semantics in terms of base category functors dubbed multipliers. Structural rules for shape variables and certain aspects of the transpension type depend on characteristics of the multiplier. We demonstrate how the transpension type and the strictness axiom can be combined to implement all and improve some of the aforementioned internalization operators (without formal claim in the case of locally fresh names). [ABSTRACT FROM AUTHOR]

In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT. [ABSTRACT FROM AUTHOR]

The discussion of this paper is to aim to examine application of the notion of C*-algebra valued fuzzy soft metric to homotopy theory using common coupled fixed point results from C*-class functions. We also tried to provide an illustration of our major findings. The results attained expand upon and apply to many of the findings in the literature. [ABSTRACT FROM AUTHOR]

This investigation centers on the analysis of an inverse hyperbolic partial differential equation, specifically addressing a coefficient inverse problem that emerges under the imposition of an over-determination condition. In order to address this challenging problem, we employ the well-established homotopy analysis technique, which has proven to be an effective and reliable approach in similar contexts. By utilizing this technique, our primary objective is to achieve an efficient and accurate solution to the inverse problem at hand. To substantiate the effectiveness and reliability of the proposed method, we present a numerical example as a practical illustration, demonstrating its applicability in real-world scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space |$X$| and a real parameter |$\ell \geq 0$|⁠. This space is roughly consisting of all paths of length |$\ell $| and has the reduced homology group that is isomorphic to the magnitude homology group of |$X$|⁠. To construct the magnitude homotopy type, we consider the poset structure on the spacetime |$X\times \mathbb{R}$| defined by causal (time- or light-like) relations. The magnitude homotopy type is defined as the quotient of the order complex of an intervals on |$X\times \mathbb{R}$| by a certain subcomplex. The magnitude homotopy type gives a covariant functor from the category of metric spaces with |$1$| -Lipschitz maps to the category of pointed topological spaces. The magnitude homotopy type also has a "path integral" like expression for certain metric spaces. By applying discrete Morse theory to the magnitude homotopy type, we obtain a new proof of the Mayer–Vietoris-type theorem and several new results including the invariance of the magnitude under sycamore twist of finite metric spaces. [ABSTRACT FROM AUTHOR]

We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $\mathrm {K}(n)$ - and $\mathrm {T}(n)$ -local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$ , then its semiadditive K-theory is of height $\leq n+1$. Under further hypothesis on $R$ , which are satisfied for example by the Lubin–Tate spectrum $\mathrm {E}_n$ , we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $\mathrm {T}(n+1)$ -localized K-theory, showing that they coincide for any $p$ -invertible ring spectrum and for the completed Johnson–Wilson spectrum $\widehat {\mathrm {E}(n)}$. [ABSTRACT FROM AUTHOR]

In this paper, we give a detailed introduction to the theory of (curved) L ∞ -algebras and L ∞ -morphisms, avoiding the concept of operads and providing explicit formulas. In particular, we recall the notion of (curved) Maurer–Cartan elements, their equivalence classes and the twisting procedure. The main focus is then the study of the homotopy theory of L ∞ -algebras and L ∞ -modules. In particular, one can interpret L ∞ -morphisms and morphisms of L ∞ -modules as Maurer–Cartan elements in certain L ∞ -algebras, and we show that twisting the morphisms with equivalent Maurer–Cartan elements yields homotopic morphisms. We hope that these notes provide an accessible entry point to the theory of L ∞ -algebras. [ABSTRACT FROM AUTHOR]

Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set‐based foundations. This expository article, written as lecture notes to accompany a three‐part mini course delivered at the Logic and Higher Structures workshop at CIRM‐Luminy, attempt to survey the state of the art, first presenting Voevodsky's simplicial model of univalent foundations and then touring Shulman's vast generalization, which provides an interpretation of homotopy type theory with strict univalent universes in any ∞$\infty$‐topos. As we will explain, this achievement was the product of a community effort to abstract and streamline the original arguments as well as develop new lines of reasoning. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions for coupled systems of fractional p-Laplacian quasilinear boundary value problems at resonance given by the Atangana–Baleanu–Caputo (shortly, ABC) derivatives formulations are based on the well-known Mittag-Leffler kernel utilizing Ge’s application of Mawhin’s continuation theorem. Examples are provided to demonstrate our findings. [ABSTRACT FROM AUTHOR]

We define a trace map for every cohomological correspondence in the motivic stable homotopy category over a general base scheme, which takes values in the twisted bivariant groups. Local contributions to the trace map give rise to quadratic refinements of the classical local terms, and some \mathbb {A}^1-enumerative invariants, such as the local \mathbb {A}^1-Brouwer degree and the Euler class with support, can be interpreted as local terms. We prove an analogue of a theorem of Varshavsky, which states that for a contracting correspondence, the local terms agree with the naive local terms. [ABSTRACT FROM AUTHOR]