Bar Constructions Research Articles

Calculation of the formation of normal cracks in a reinforced concrete element based on the deformation theory of plasticity of concrete by G.A. Geniev

The authors present a refined method of determining the moment of cracking in reinforced concrete bar constructions using the diagram of deformation of concrete built on the basis of the deformation theory of plasticity by G.A. Geniev in which the stress and strain invariants of concrete are linked by nonlinear dependences. In the resulting defining equations, the hypothesis of flat sections, as well as the premise of reaching the limit values of concrete deformations on the stretched fibers of the cross-section are used. Stresses in concrete are determined through deformation values in accordance with the nonlinear deformation diagram of concrete. On the basis of the assumptions accepted, analytical dependences for determining the moment of cracking in the sections of bending elements with single and double reinforcement have been acquired. The formulas obtained were used in the analysis of various factors influence on crack resistance of bendable reinforced concrete elements. It was found out that the moment of crack formation practically does not change when percentage of reinforcement of longitudinal tensile or compressed reinforcement changes. The most effective method of crack resistance improvement is the increase of concrete strength. The proposed methodology is verified by comparison with experimental results on reinforced concrete prototypes. It is concluded that the use of the diagram of nonlinear deformation of concrete on the basis of the theory of plasticity by G.A. Geniev allows to estimate more strictly the crack resistance of reinforced concrete rod elements.

Plethysms and operads

We introduce the T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document}-construction, an endofunctor on the category of generalized operads, as a general mechanism by which various notions of plethystic substitution arise from more ordinary notions of substitution. In the special case of one-object unary operads, i.e. monoids, we recover the T-construction of Giraudo. We realize several kinds of plethysm as convolution products arising from the homotopy cardinality of the incidence bialgebra of the bar construction of various operads obtained from the T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document}-construction. The bar constructions are simplicial groupoids, and in the special case of the terminal reduced operad Sym\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {Sym}$$\\end{document}, we recover the simplicial groupoid of Cebrian (Algebraic Geom Topol 21(1):421–446, 2021), a combinatorial model for ordinary plethysm in the sense of Pólya, given in the spirit of Waldhausen S and Quillen Q constructions. In some of the cases of the T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document}-construction, an analogous interpretation is possible.

The incidence comodule bialgebra of the Baez–Dolan construction

Starting from any operad P, one can consider on one hand the free operad on P, and on the other hand the Baez–Dolan construction on P. These two new operads have the same space of operations, but with very different notions of arity and substitution. The main result of this paper is that the incidence bialgebras of the two-sided bar constructions of the two operads constitute together a comodule bialgebra. The result is objective: it concerns comodule-bialgebra structures on groupoid slices, and the proof is given in terms of equivalences of groupoids and homotopy pullbacks. Comodule bialgebras in the usual sense are obtained by taking homotopy cardinality. The simplest instances of the construction cover several comodule bialgebras of current interest in analysis. If P is the identity monad, then the result is the Faà di Bruno comodule bialgebra (dual to multiplication and substitution of power series). If P is any monoid Ω (considered as a one-coloured operad with only unary operations), the resulting comodule bialgebra is the dual of the near-semiring of Ω-moulds under product and composition, as employed in Écalle's theory of resurgent functions in local dynamical systems. If P is the terminal operad, then the result is essentially the Calaque–Ebrahimi-Fard–Manchon comodule bialgebra of rooted trees, dual to composition and substitution of B-series in numerical analysis (Chartier–Hairer–Vilmart). The full generality is of interest in category theory. As it holds for any operad, the result is actually about the Baez–Dolan construction itself, providing it with a new algebraic perspective.

To the Rods Bending Calculating Problem with Allowance for the Transverse Shear Deformation

A redetermined approach to bar systems calculation is considered in this work taking into account edge termination self-balanced components influence where the required components of stress condition must not violate the equilibrium condition of any body elementary volume. This approach provides transverse shear deformation consideration with the accuracy necessary for vessels bar constructions reliability assessment.

B ∞-Algebra Structure in Homology of a Homotopy Gerstenhaber Algebra

The minimality theorem states, in particular, that on cohomology H(A) of a dg algebra there exists sequence of operations mi : H(A)⊗i → H(A), i = 2, 3, . . . , which form a minimal A ∞ -algebra (H(A), {m i }). This structure defines on the bar construction BH(A) a correct differential dm so that the bar constructions (BH(A), d m ) and BA have isomorphic homology modules. It is known that if A is equipped additionally with a structure of homotopy Gerstenhaber algebra, then on BA there is a multiplication which turns it into a dg bialgebra. In this paper, we construct algebraic operations Ep,q : H(A) ⊗p ⊗H(A) ⊗q → H(A), p, q = 0, 1, 2, . . ., which turn (H(A), {m i }, {E p,q }) into a B ∞ -algebra. These operations determine on BH(A) correct multiplication, so that (BH(A), d m ) and BA have isomorphic homology algebras.

Cofibrancy of operadic constructions in positive symmetric spectra

We show that when using the underlying positive model structure on symmetric spectra one obtains cofibrancy conditions for operadic constructions under much milder hypothesis than one would need for general categories. Our main result provides such an analysis for a key operation, the "relative composition product" $\circ_{\mathcal{O}}$ between right and left $\mathcal{O}$-modules over a spectral operad $\mathcal{O}$, and as a consequence we recover (and usually strengthen) previous results establishing the Quillen invariance of model structures on categories of algebras via weak equivalences of operads, compatibility of forgetful functors with cofibrations and Reedy cofibrancy of bar constructions. Key to the results above are novel cofibrancy results for $n$-fold smash powers of positive cofibrant spectra (and the relative statement for maps). Roughly speaking, we show that such $n$-fold powers satisfy a (new) type of $\Sigma_n$-cofibrancy which can be viewed as "lax $\Sigma_n$-free/projective cofibrancy" in that it determines a larger class of cofibrations still satisfying key technical properties of "true $\Sigma_n$-free/projective cofibrancy."

Bar complexes and extensions of classical exponential functors

Nous calculons les groupes d’Ext entre foncteurs exponentiels classiques (i.e. puissances symétriques, extérieures ou divisées), et leur précomposition par le foncteur de torsion de Frobenius. Notre méthode repose sur les constructions bar, et relie ces calculs d’Ext avec l’homologie des espaces d’Eilenberg et Mac Lane.

Symmetric Bimonoidal Intermuting Categories and ω × ω Reduced Bar Constructions

A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for ω × ω-indexed family of iterated reduced bar constructions based on such a category.

Koszul duality in algebraic topology

The most prevalent examples of Koszul duality of operads are the self-duality of the associative operad and the duality between the Lie and commutative operads. At the level of algebras and coalgebras, the former duality was first noticed as such by Moore, as announced in his ICM talk at Nice (Moore in Actes du Congres International des Mathematiciens, Tome 1. Gauthier-Villars, Paris, pp. 335–339, 1971). This particular duality has typically been called Moore duality, and some prefer to call the general phenomenon Koszul–Moore duality. The second duality at the level of algebras was realized in the seminal work of Quillen on rational homotopy theory (Quillen in Ann Math 90(2):205–295, 1969). Our aim in these notes based on our talk at the Luminy workshop on Operads in 2009 is to try to provide some historical, topological context for these two classical algebraic dualities. We first review the original cobar and bar constructions used to study loop spaces and classifying spaces, emphasizing the less-familiar geometry of the cobar construction. Then, after some elementary topology, we state duality between bar and cobar complexes in that setting. Before explaining Quillen’s work, we also share some other ideas—calculations of Cartan–Serre and Milnor–Moore and philosophy of Eckmann–Hilton—which may have influenced him. After stating Quillen’s duality, we share some recent work which relates these constructions to geometry through Hopf invariants and in particular linking phenomena.

Bar constructions and Quillen homology of modules over operads

We show that topological Quillen homology of algebras and modules over operads in symmetric spectra can be calculated by realizations of simplicial bar constructions. Working with several model category structures, we give a homotopical proof after showing that certain homotopy colimits in algebras and modules over operads can be easily understood. A key result here, which lies at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits. We also prove analogous results for algebras and modules over operads in unbounded chain complexes.

Bar constructions for topological operads and the Goodwillie derivatives of the identity

We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.

The relationship between the diagonal and the bar constructions on a bisimplicial set

The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by the simplicial set W ¯ X , the bar construction on X. We stress the interest of this result by showing two relevant theorems which now become simple instances of it; namely, the Homotopy colimit theorem of Thomason, for diagrams of small categories, and the generalized Eilenberg–Zilber theorem of Dold–Puppe for bisimplicial Abelian groups. Among other applications, we give an algebraic model for the homotopy theory of (not necessarily path-connected) spaces whose homotopy groups vanish in degree 4 and higher.

On A Multiplicativity up to Homotopy of the Gugenheim Map

Abstract In the category of differential algebras with strong homotopy there is a Gugenheim's map {ρ 𝑖} : 𝐴* → 𝐶* from Sullivan's commutative cochain complex to the singular cochain complex of a space, which induces a differential graded coalgebra map of appropriate Bar constructions. Both (𝐵𝐴*, dBA , Δ,) and (𝐵𝐶*, dBC* , Δ,) carry multiplications. We show that the Gugenheim's map 𝐵{ρ 𝑖} : (𝐵𝐴*, dBA* , Δ,) → (𝐵𝐶*, dBC* , Δ,) is multiplicative up to homotopy with respect to these structures.

EXTENDED MODULI SPACES, THE KAN CONSTRUCTION, AND LATTICE GAUGE THEORY

Let Y be a CW-complex with asingle 0-cell, let K be its Kan group, a free simplicial group whose geometric realization is a model for the space ΩY of based loops on Y, and let G be a Lie group. By means of simplicial and cosimplicial techniques involving fundamental results of Kan’s and the standard W- and bar constructions, we obtain a weak G-equivariant homotopy equivalence from the geometric realization | Hom(K, G)| of the cosimplicial manifold Hom(K, G) of homomorphisms from K to G to the space Map o(Y, BG) of based maps from Y to the classifying space BG of G where G acts on BG by conjugation. Thereafter we carry out an explicit purely finite dimensional construction of generators of the equivariant cohomology of the geometric realization of Hom(K, G) and hence of the space Map o(Y, BG) of based maps from Y to the classifying space BG of G. For a smooth manifold Y, this may be viewed as a rigorous approach to lattice gauge theory, and we show that it then yields, (i) when dim( Y)=2, equivariant de Rham representatives of generators of the equivariant cohomology of twisted representation spaces of the fundamental group of a closed surface including generators for moduli spaces of semi-stable holomorphic vector bundles on complex curves so that, in particular, when G is compact, the known structure of stratified symplectic space on the twisted representation spaces results and (ii) when dim( Y)=3, equivariant cohomology generators including a rigorous combinatorial description of the Chern–Simons function for a closed 3-manifold. The latter is illustrated by a calculation of the Chern–Simons invariants for flat SU(2)-connections over 3-dimensional lens spaces.

The Method of Secondary Grid Deformation

The type of spherical grid plays a significant role in dome coverings realized as bar constructions. Triangular spherical grids should resemble the sphere surface as far as possible. For technological reasons the grid should be formed of the lowest number of edges of different lengths. The range of the necessary changes in the edge lengths should be as low as possible. It would be advantageous to have a grid consisting of the lowest number of triangular faces included between the grid segments. This paper presents a method which makes it possible to determine relatively regular spherical grids. The examples of the dimension relations of triangular spherical grids determined by this method on the basis of the selected group of regular polyhedrons were provided. The characteristics of the polyhedrons of a great number of faces, obtained in this way, were also given. In this paper is shown only the basic shape of the method of secondary grid deformation.

Bar and cobar constructions II

Bar and cobar constructions II

Mappings of bar constructions

Quillen’s famous Theorem B describes the homotopy fiber ofBF :B C/arB, wheref :C is a functor andB the classifying space functor. This is here generalized to a description of the homotopy fiber ofB(F,α,β) :B(Y C,X)/ar(Y/t' C/t',X/t') where (F,α,β) : (Y C,X)/ar(Y/t' C/t',X/t') is a mapping of 2-sided bar construction data.

Cálculo no lineal de estructuras reticuladas y laminares

By many of electronic calculus it is possible to approach the non linear analysis of bar constructions efficiently and without the limitations of the traditional plastic Calculus. In order to prove this statement, the author presents certains structural analysis that have been carried out by means of techniques and calculation programs of his own. Further, the non linear analysis of revolving sheets is presented to which algorithms of great formal similarity to those in the case of bar constructions are applicable. Both for these latter ones as weil as for laminated structures the programs elaborated by the author allow a great variety of anelastic analyses (and as a special case, of elastic analysis).

Adjoint functors and bar constructions

The notions of universal bundle, classifying space and bar are closely linked but not interchangeable. The relations between them are well understood in some contexts, less so in others. Our aim here is not to elucidate these relations but, merely, as a first step in such an elucidation, to clarify the notion of a bar construction. The literature on bar constructions is too large to survey here. The name is due to Eilenberg and MacLane. Their is extensively described in [4]. Milgram [8] gave a bar for topological monoids; the elegant description by Steenrod [10] motivated this paper. MacLane [5] gives yet another elegant description of Milgram's construction. A bar for topological categories was defined by Segal [9] and an abstract bar construction of very great generality was introduced by May [7]. The cobar of Adams [1] is a dual sort of thing which ought to be considered under the same rubric. The universal bundle of Milnor, on the other hand, is not a bar construction. All these bar constructions bear to one another what is evidently more than a mere family resemblance. But the means for saying precisely in what this resemblance consists appear to be lacking. It is this deficiency which we propose to remedy. We begin by defining the notion of bar in a broad categorical setting by means of a universal property. For these we can prove a rather general existence theorem (5.8). Finally we indicate how some of the bar constructions mentioned above, and the cobar as well, are instances of the notion introduced here. Our existence theorem for bar constructions actually results from an existence theorem for coproducts of monads (cf. MacLane [4]). As a