A model for the homotopy theory of homotopy theory

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory', or more precisely that the category of such models has a well-behaved internal hom-object.

     

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K⁰(G)=G, Kⁿ⁺¹(G)=K(Kⁿ(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that Kⁿ(G) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G?Kⁿ(G) for all n, moreover Kⁿ(G) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.     

The stack of formal groups in stable homotopy theory

We construct the algebraic stack of formal groups and use it to provide a new perspective onto a recent result of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras. This leads to a geometric understanding of their results as well as to a generalisation.     

A homotopy perturbation algorithm to solve a system of Fredholm–Volterra type integral equations

In this paper, an application of He’s homotopy perturbation (HPM) method is applied to solve the system of Fredholm and Volterra type integral equations, the results revealing that the HPM is very effective and simple.     

A fast-slow dynamical systems theory for the Kuramoto type phase model

We present a fast-slow dynamical systems theory for the Kuramoto type phase model. When the order parameters are frozen, the fast system consists of independent oscillator equations, whereas the slow system describes the evolution of order parameters. We average out the slow system over the fast manifold to derive a weak form of an amplitude-angle coupled system for the evolution of Kuramoto?s order parameters. This yields the slow evolution of order parameters to be constant values which gives a rigorous proof to Kuramoto?s original assumption in his self-consistent mean-field theory.     

Characterizations of modalities and lex modalities

A reflective subuniverse in homotopy type theory is an internal version of the notion of a localization in topology or in the theory of ∞-categories. Working in homotopy type theory, we give new characterizations of the following conditions on a reflective subuniverse L: (1) the associated subuniverse $L^{\prime }$ of L-separated types is a modality; (2) L is a modality; (3) L is a lex modality; and (4) L is a cotopological modality. In each case, we give several necessary and sufficient conditions. Our characterizations involve various families of maps associated to L, such as the L-étale maps, the L-equivalences, the L-local maps, the L-connected maps, the unit maps ${\eta }_X$, and their left and/or right orthogonal complements. More generally, our main theorem gives an overview of how all of these classes related to each other. We also give examples that show that all of the inclusions we describe between these classes of maps can be strict.     

A characterisation of elementary fibrations

In the categorical approach to logic proposed by Lawvere, which systematically uses adjoints to describe the logical operations, equality is presented in the form of a left adjoint to reindexing along diagonal arrows in the base. Taking advantage of the modular perspective provided by category theory, one can look at those Grothendieck fibrations which sustain just the structure of equality, the so-called elementary fibrations, aka fibrations with equality.The present paper provides a characterisation of elementary fibrations which is a substantial generalisation of the one already available for faithful fibrations. The characterisation is based on a particular structure in the fibres which may be understood as proof-relevant equality predicates equipped with a principle of indiscernibility of identicals à la Leibniz. We exemplify this structure for several classes of fibrations, in particular, for fibrations used in the semantics of the identity type of Martin-Löf type theory. We close the paper discussing some fibrations related to Hofmann and Streicher's groupoid model of the identity type and showing that one of them is elementary.     

Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type

In this paper, we study the global L^∞ solutions for the Cauchy problem of nonsymmetric system (1.1) of Keyfitz-Kranzer type. When n=1, (1.1) is the Aw-Rascle traffic flow model. First, we introduce a new flux approximation to obtain a lower bound ρε,δ?δ>0 for the parabolic system generated by adding “artificial viscosity” to the Aw-Rascle system. Then using the compensated compactness method with the help of L¹ estimate of wε,δx(⋅,t) we prove the pointwise convergence of the viscosity solutions under the general conditions on the function P(ρ), which includes prototype function , where γ∈(−1,0)∪(0,∞), A is a constant. Second, by means of BV estimates on the Riemann invariants and the compensated compactness method, we prove the global existence of bounded entropy weak solutions for the Cauchy problem of general nonsymmetric systems (1.1).     

A formal theory of generalized intermediate syllogisms

In this paper, we continue developing the formal theory of intermediate quantifiers (expressions such as most, few, almost all, a lot of, many, a great deal of, a large part of, a small part of). The theory is a fuzzy-logic formalization of the concept introduced by Peterson in his book. We will syntactically prove that 105 generalized Aristotle's syllogisms introduced in this book are valid in our theory. At the same time, we will also prove that syllogisms listed there as invalid are invalid also in our theory. Therefore, we believe that our theory provides a reasonable mathematical model of the generalized syllogistics.     

Iterated function systems of finite type and the weak separation property

We prove that any iterated function system of finite type possesses the weak separation property.

     

A numerical method of structure-preserving model updating problem and its perturbation theory

A method is presented to update a special finite element (FE) analytical model, based on matrix approximation theory with spectral constraint. At first, the model updating problem is treated as a matrix approximation problem dependent on the spectrum data from vibration test and modal parameter identification. The optimal approximation is the first modified solution of FE model. An algorithm is given to preserve the sparsity of the model by multiple correction. The convergence of the algorithm is investigated and perturbation of the modified solution is analyzed. Finally, a numerical example is provided to confirm the convergence of the algorithm and perturbation theory.     

A novel class of model constitutive laws in nonlinear elasticity: Construction via Loewner theory

Using a solitonic connection, we show that the class of infinitesimal Bäcklund transformations originally introduced by Loewner in 1952 in a gasodynamic context results in physically interesting nonlinear model constitutive laws. We obtain laws previously used to model a variety of hard and soft nonlinear elastic responses. A natural extension of the latter leads to a novel class of model constitutive laws where the stress and strain are given parametrically in terms of elliptic functions. Such models allow a change in the concavity of the stress-strain law. Such behavior can be observed in the compression of polycrystalline materials or in the unloading regimes of superelastic nickel-titanium.     

均衡约束数学规划问题的一类广义Mond-Weir型对偶理论

针对均衡约束数学规划模型难以满足约束规范及难于求解的问题,基于Mond和Weir提出的标准非线性规划的对偶形式,利用其S稳定性,建立了均衡约束数学规划问题的一类广义Mond-Weir型对偶,从而为求解均衡约束优化问题提供了一种新的方法.在Hanson-Mond广义凸性条件下,利用次线性函数,分别提出了弱对偶性、强对偶性和严格逆对偶性定理,并给出了相应证明.该对偶化方法的推广为研究均衡约束数学规划问题的解提供了理论依据.     

Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type

Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gröchenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and Rauhut (2005) [24] that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel spaces of various types of interest as coorbits. We obtain several old and new wavelet characterizations based on explicit smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed smoothness and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable.     

Stability analysis and approximate solution of SIR epidemic model with Crowley-Martin type functional response and Holling type-II treatment rate by using homotopy analysis method

In this paper, SIR epidemic model with Crowley-Martin type functional response and Holling type-II treatment rate is investigated. The analysis of the model shows that it has two equilibria, namely disease-free and endemic. We investigate the existence and stability results of equilibria by using LaSalle''s invariant principle and Lyapunov function. $\mathfrak{R}_{0}$ has been found to ensure the extinction or persistence of the infection. Furthermore, homotopy analysis method is employed to obtain the series solution of the proposed model. By using the homotopy solutions, firstly, several $\hbar$-curves are plotted to demonstrate the regions of convergence, then the residual and square residual errors are obtained for different values of these regions. Secondly, the numerical solutions are presented for various iterations and the absolute error functions are applied to show the accuracy of the applied homotopy analysis method.     

Existence and multiplicity of the solutions of the p(x)–Kirchhoff type equation via genus theory

In this paper, by using variational approach and Krasnoselskii's genus theory, we show the existence and multiplicity of the solutions of the p(x)‐Kirchhoff type equation. Copyright © 2011 John Wiley & Sons, Ltd.     

A fractal graph model of capillary type systems

Abstract

We consider blood flow in a vessel with an attached capillary system. The latter is modelled with the help of a corresponding fractal graph whose edges are supplied with ordinary differential equations obtained by the dimension-reduction procedure from a three-dimensional model of blood flow in thin vessels. The Kirchhoff transmission conditions must be satisfied at each interior vertex. The geometry and physical parameters of this system are described by a finite number of scaling factors which allow the system to have self-reproducing solutions. Namely, these solutions are determined by the factors’ values on a certain fragment of the fractal graph and are extended to its rest part by virtue of these scaling factors. The main result is the existence and uniqueness of self-reproducing solutions, whose dependence on the scaling factors of the fractal graph is also studied. As a corollary we obtain a relation between the pressure and flux at the junction, where the capillary system is attached to the blood vessel. This relation leads to the Robin boundary condition at the junction and this condition allows us to solve the problem for the flow in the blood vessel without solving it for the attached capillary system.