Volume 26, Issue 2

We present a formal power series ∑Ai⁢j⁢xi⁢yj{\sum A_{ij}x^{i}y^{j}} over the Lazard ring Λ and the formal group laws Fn{F_{n}}, n≥2{n\geq 2}, over the quotient rings of Λ. For each Fn{F_{n}}, we construct a complex cobordism theory with singularities with the coefficient ring ℚ⁢[p1,…,p2⁢n]{\mathbb{Q}[p_{1},\dots,p_{2n}]}, with parameters pi{p_{i}}, |pi|=2⁢i{|p_{i}|=2i}.

In [1] Arkowitz and Brown presented an axiomatization of the reduced Lefschetz number of self-maps of finite CW-complexes. By the results of McCord [8], finite simplicial complexes are closely related to finite T0{T_{0}}-spaces. This connection and the axioms given by Arkowitz and Brown suggest an axiomatization of the reduced Lefschetz number of maps of finite T0{T_{0}}-spaces. However, using the notion of the subdivision of a finite T0{T_{0}}-space, we consider the degree and the Lefschetz number of not only self-maps. We also present some properties of the degree of maps between finite models of the circle 𝕊1{\mathbb{S}^{1}}.

We present the category of alternative algebras as a category of interest. This kind of approach enables us to describe derived actions in this category, study their properties and construct a universal strict general actor of any alternative algebra. We apply the results obtained in this direction to investigate the problem of the existence of an actor in the category of alternative algebras.

We introduce the pseudo Maurer–Cartan perturbation algebra, establish a structural result and explore the structure of this algebra. That structural result entails, as a consequence, what we refer to as the pseudo perturbation lemma. This lemma, in turn, implies the ordinary perturbation lemma.

We present homotopy classification of morphisms of differential graded algebras f:A→A′{f:A\to A^{\prime}} in terms of induced A∞{A_{\infty}}-algebra morphisms of corresponding minimal models {fi}:(H⁢(A),{mi})→(H⁢(A′),{mi′}){\{f_{i}\}\kern-1.0pt:\kern-1.0pt(H(A),\{m_{i}\})\kern-1.0pt\to\kern-1.0pt(H(A% ^{\prime}),\{m^{\prime}_{i}\})}.

Let Gn{G_{n}} and Gn′{G_{n}^{\prime}} be two nonisomorphic graphs on n vertices with spectra (with respect to the adjacency matrix) λ1≥λ2≥⋯≥λn and λ1′≥λ2′≥⋯≥λn′,\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\quad\text{and}\quad\lambda% ^{\prime}_{1}\geq\lambda^{\prime}_{2}\geq\cdots\geq\lambda^{\prime}_{n}, respectively. Define the distance between the spectra of Gn{G_{n}} and Gn′{G_{n}^{\prime}} as λ⁢(Gn,Gn′)=∑i=1n(λi-λi′)2 (or use ⁢∑i=1n|λi-λi′|).\lambda(G_{n},G^{\prime}_{n})=\sum_{i=1}^{n}(\lambda_{i}-\lambda^{\prime}_{i})% ^{2}\quad\biggl{(}\text{or use }\sum_{i=1}^{n}\lvert\lambda_{i}-\lambda^{% \prime}_{i}\rvert\biggr{)}. Define the cospectrality of Gn{G_{n}} by cs⁡(Gn)=min⁡{λ⁢(Gn,Gn′):Gn′⁢ not isomorphic to ⁢Gn}.\operatorname{cs}(G_{n})=\min\bigl{\{}\lambda(G_{n},G^{\prime}_{n}):G^{\prime}% _{n}\text{ not isomorphic to }G_{n}\bigr{\}}. In this paper, we investigate cs⁡(Gn){\operatorname{cs}(G_{n})} for special classes of graphs with respect to normalized Laplacian spectra and we find cs⁡(Kn){\operatorname{cs}(K_{n})}, cs⁡(n⁢K1){\operatorname{cs}(nK_{1})} and cs⁡(K2+(n-2)⁢K1){\operatorname{cs}(K_{2}+(n-2)K_{1})} (n≥2{n\geq 2}). We also find an upper bound for csn{\operatorname{cs}_{n}}.

For higher-order linear hyperbolic equations the problem on periodic solutions is investigated. The concepts of associated problems, and α-well-posedness are introduced. Necessary and sufficient conditions of well-posedness in the two-dimensional case, as well as unimprovable sufficient conditions of well-posedness and α-well-posedness in the multi-dimensional case are established.

The notion of a dihedral ∞{\infty}-simplicial module is introduced. The homotopy invariance of the structure of a dihedral ∞{\infty}-simplicial module is proved. The concept of the dihedral homology of a dihedral ∞{\infty}-simplicial module is developed. The notion of an involutive homotopy unital A∞{A_{\infty}}-algebra is introduced. The dihedral ∞{\infty}-simplicial module is constructed using an involutive homotopy unital A∞{A_{\infty}}-algebra. The concept of the dihedral homology of an involutive homotopy unital A∞{A_{\infty}}-algebra is developed. For the dihedral homology of involutive homotopy unital A∞{A_{\infty}}-algebras, the analogue of the Krasauskas–Lapin–Solov’ev exact sequence is obtained.

In this note we are concerned with the question: When are the polynomial solutions to a system of linear constant coefficient PDEs dense in the set of all its solutions?

Let 𝒦{\mathcal{K}} be an abelian category that has enough injective objects, let T:𝒦→A{T\colon\mathcal{K}\to A} be any left exact covariant additive functor to an abelian category A and let T(i){T^{(i)}} be a right derived functor, u≥1{u\geq 1}, [S. Mardešić, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000]. If T(i)=0{T^{(i)}=0} for i≥2{i\geq 2} and T(i)⁢Cn=0{T^{(i)}C_{n}=0} for all n∈ℤ{n\in\mathbb{Z}}, then there is an exact sequence 0⟶T(1)⁢Hn+1⁢(C*)⟶Hn⁢(T⁢C*)⟶T⁢Hn⁢(C*)⟶0,0\longrightarrow T^{(1)}H_{n+1}(C_{*})\longrightarrow H_{n}(TC_{*})% \longrightarrow TH_{n}(C_{*})\longrightarrow 0, where C*={Cn}{C_{*}=\{C_{n}\}} is a chain complex in the category 𝒦{\mathcal{K}}, Hn⁢(C*){H_{n}(C_{*})} is the homology of the chain complex C*{C_{*}}, T⁢C*{TC_{*}} is a chain complex in the category A , and Hn⁢(T⁢C*){H_{n}(TC_{*})} is the homology of the chain complex T⁢C*{TC_{*}}. This exact sequence is the well known Künneth’s correlation. In the present paper Künneth’s correlation is generalized. Namely, the conditions are found under which the infinite exact sequence ⋯⟶T(2⁢i+1)⁢Hn+i+1⟶⋯⟶T(3)⁢Hn+2⟶T(1)⁢Hn+1⟶Hn⁢(T⁢C*)\displaystyle\cdots\longrightarrow T^{(2i+1)}H_{n+i+1}\longrightarrow\cdots% \longrightarrow T^{(3)}H_{n+2}\longrightarrow T^{(1)}H_{n+1}\longrightarrow H_% {n}(TC_{*})⟶T⁢Hn⁢(C*)⟶T(2)⁢Hn+1⟶T(4)⁢Hn+2⟶⋯⟶T(2⁢i)⁢Hn+i⟶⋯\displaystyle\longrightarrow TH_{n}(C_{*})\longrightarrow T^{(2)}H_{n+1}% \longrightarrow T^{(4)}H_{n+2}\longrightarrow\cdots\longrightarrow T^{(2i)}H_{% n+i}\longrightarrow\cdots holds, where T(2⁢i+1)⁢Hn+i+1=T(2⁢i+1)⁢Hn+i+1⁢(C*){T^{(2i+1)}H_{n+i+1}=T^{(2i+1)}H_{n+i+1}(C_{*})}, T(2⁢i)⁢Hn+i=T(2⁢i)⁢Hn+i⁢(C*){T^{(2i)}H_{n+i}=T^{(2i)}H_{n+i}(C_{*})}. The formula makes it possible to generalize Milnor’s formula for the cohomologies of an arbitrary complex, relatively to the Kolmogorov homology to the Alexandroff–Čech homology for a compact space, to a generative result of Massey for a local compact Hausdorff space X and a direct system {U}{\{U\}} of open subsets U of X such that U¯{\overline{U}} is a compact subset of X .

Let Y=|X|{Y=\lvert X\rvert} be the geometric realization of a path-connected simplicial set X , and let G=π1⁢(X){G=\pi_{1}(X)} be the fundamental group. Given a subgroup H⊂G{H\subset G}, let G/H{G/H} be the set of cosets. Using the combinatorial model 𝛀⁢X→𝐏⁢X→X{\boldsymbol{\Omega}X\to\mathbf{P}X\to X} of the path fibration Ω⁢Y→P⁢Y→Y{{\Omega}Y\to{P}Y\to Y} and a canonical action μ:𝛀⁢X×G/H→G/H{\mu\colon\boldsymbol{\Omega}X\times G/H\to G/H}, we construct a covering map G/H→YH→Y{G/H\to Y_{H}\to Y} as the geometric realization of the associated short sequence G/H→𝐏⁢X×μG/H→X{G/H\to\mathbf{P}X\times_{\mu}G/H\to X}. This construction, in particular, does not use the existence of a maximal tree in X . For a 2-dimensional X and H={1}{H=\{1\}}, it can also be viewed as a simplicial approximation of a Cayley 2-complex of G .

Here is a sketch of a few of the connections between Prof. Nodar Berikashvili and his school and higher homotopy structures.

A (Hausdorf) hereditarily normal (not perfectly normal) space X is constructed, which has the following properties: (a) there exists a proper open subspace of X which is homeomorphic to the whole X (i.e., the space X is topologically infinite); (b) the space is homeomorphic to none of its proper Fσ{F_{\sigma}}-subspaces (i.e., the space X is topologically finite relative to the class of all its proper Fσ{F_{\sigma}}-subspaces).