The theory of the <Emphasis Type="Italic">α</Emphasis> degrees is undecidable

Let α be an arbitrary Σ₁-admissible ordinal. For each n, there is a formula φ _n ([(x)\vec],[(y)\vec]\vec x,\vec y) such that for any relation R on a finite set F with n elements, there are α-degrees [(p)\vec]{\vec p} such that the relation defined by φ _n ([(x)\vec],[(p)\vec]\vec x,\vec p) is isomorphic to R. Consequently, the theory of α-degrees is undecidable.     

Hochschild cohomology of algebras of quaternionic type: The family Q(2ℬ)<Subscript>1</Subscript>(<Emphasis Type="Italic">k,s, a,c</Emphasis>) over a field of characteristic different from 2

In a previous joint paper of the author with A.I. Generalov and S.O. Ivanov, the Hochschild cohomology algebra of quaternionic-type algebras from the family Q(2ℬ)₁ over an algebraically closed field of characteristic 2 was calculated. In this paper, the Hochschild cohomology groups of algebras from this family over an algebraically closed field of characteristic different from 2 are calculated. As a corollary, the additive structure of the Hochschild cohomology of algebras of type Q(2 $ A $ A ) over a field of characteristic not 2 is described.     

The (-1/2λϕ2) model of quantum field theory

The model of a scalar field with interaction potential exp (-1/2²) ind-dimensional spacetime (d2) is considered. It is shown that the Green's functions of the model are the same as the Green's functions of the free field.Physicotechnical Institute, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 411–417, June, 1992.     

A blend of methods of recursion theory and topology: A Π<Subscript>1</Subscript><Superscript>0</Superscript> tree of shadow points

This paper is a sequel to our [7]. In that paper we constructed a ₁⁰ tree of avoidable points. Here we construct a ₁⁰ tree of shadow points. This tree is a tree of sharp filters, where a sharp filter is a nested sequence of basic open sets converging to a point. In the construction we assign to each basic open set on the tree an address in 2^<. One interesting fact is that while our ₁⁰ tree of sharp filters (a subtree of ^<) is isomorphic to the tree of addresses (a subtree of 2^<), the tree of addresses is recursively enumerable but not recursive. To achieve this end we use a finite injury priority argument.Mathematics Subject Classification (2000): 03D45, 03D80, 03C57, 54A20     

The s-Riesz transform of an s-dimensional measure in ℝ2 is unbounded for 1 < s < 2

In this paper, we prove that for s ∈ (1, 2) there exists no totally lower irregular finite positive Borel measure µ in ?² with such that ${\left\| {R\mu } \right\|_{{L^\infty }({m_2})}} < + \infty $ , where Rµ = µ*x/|x| ^s+1 and m ₂ is the Lebesgue measure in ?². Combined with known results of Prat and Vihtilä, this shows that for any s ∈ (0, 1) ∪ (1, 2) and any finite positive Borel measure in ?² with , we have ${\left\| {R\mu } \right\|_{{L^\infty }({m_2})}} = \infty $ .     

Discrete subgroups of <Emphasis Type="Italic">PU</Emphasis> (2, 1) acting on P<Stack><Subscript>ℂ</Subscript><Superscript>2</Superscript></Stack> and the Kobayashi metric

In this paper, we prove following: If G ⊂ PU (2, 1) is an infinite, discrete group, acting on P_ℂ² without complex invariant lines, then the component containing ℍP_ℂ² of the domain of discontinuity Ω(G) = PP_ℂ²∖ Λ (G), according to Kulkarni, is G-invariant complete Kobayashi hyperbolic. The authors were supported by the Universidad Autónoma de Yucatán and the Universidad Nacional Autónoma de México.     

Aut0(Λ<Superscript>2</Superscript>)-symmetrical 4-extensions of the 2-dimensional grid Λ<Superscript>2</Superscript>

There are different non-equivalent definitions of attractors in the theory of dynamical systems. The most common are two definitions: the maximal attractor and the Milnor attractor. The maximal attractor is by definition Lyapunov stable, but it is often in some ways excessive. The definition of Milnor attractor is more realistic from the physical point of view. The Milnor attractor can be Lyapunov unstable though. One of the central problems in the theory of dynamical systems is the question of how typical such a phenomenon is. This article is motivated by this question and contains new examples of so-called relatively unstable Milnor attractors. Recently I. Shilin has proved that these attractors are Lyapunov stable in the case of one-dimensional fiber under some additional assumptions. However, the question of their stability in the case of multidimensional fiber is still an open problem.     

The Auslander conjecture for groups leaving a form of signature (<Emphasis Type="Italic">n</Emphasis>−2, 2) invariant

The Auslander conjecture claims that every affine crystallographic group Γ is virtually solvable. We prove here this conjecture for the case when the linear part of Γ is contained in the orthogonal groupO(n−2,2). Dedicated to Hillel Furstenberg     

The British development of the theory of invariants (1841–1895)

The two main British exponents of the theory of invariants, Arthur Cayley and James Joseph Sylvester, first encountered the idea of an “invariant” in an 1841 paper by George Boole. In the 1850s, Cayley, Sylvester, and the Irish mathematician, George Salmon, formulated the basic concepts, developed the key techniques, and set the research agenda for the field. As Cayley and Sylvester continued to extend the theory off and on through the 1880s, first Salmon in 1859 Salmon, George. 1859. Lessons introductory to the modern higher algebra, Dublin: Hodges & Smith.  [Google Scholar] and later Edwin Bailey Elliott in 1895 Elliott, Edwin Bailey. 1895. An introduction to the algebra of quantics, Oxford: Oxford University Press.  [Google Scholar] codified it in high-level textbooks. This paper sketches the development of nineteenth-century invariant theory in British hands against a backdrop of personal, nationalistic, and internationalistic mathematical goals.     

Eisenstein series associated with Γ<Subscript>0</Subscript>(2)

In this paper, we define the normalized Eisenstein series ℘, e, and associated with Γ₀(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ₀(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.      

<Emphasis Type="Italic">OD</Emphasis>-Characterization of the automorphism groups of <Emphasis Type="Italic">O</Emphasis><Stack><Subscript>10</Subscript><Superscript>±</Superscript></Stack>(2)

The prime graph of a finite group was introduced by Gruenberg and Kegel. The degree pattern of a finite group G associated to its prime graph was introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions (1) |G| = |H| and (2) D(G) = D(H). Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Till now a lot of finite simple groups were shown to be OD-characterizable, and also some finite groups especially the automorphism groups of some finite simple groups were shown not being OD-characterizable but k-fold OD-characterizable for some k > 1. In the present paper, the authors continue this topic and show that the automorphism groups of orthogonal groups O ₁₀⁺(2) and O ₁₀⁻(2) are OD-characterizable.     

Existence of weak solutions of 2-D Euler equations with initial vorticityω 0∈L(log+ L) α (α>0)

It is proved that there exist global weak solutions of 2-D Euler equations inR ² under the assumption that the initial vorticity belongs to a kind of wider spaces,L ¹L(log⁺ L) (>0), which are Orlicz spaces containing spacesL ^p L ¹,L(log⁺ L) L (>1/2) and so on. This result improves on that of [2], [4], [11]. Moreover, these solutions are obtained by vanishing the viscosity term of Navier-Stokes equations.     

On Asymptotically Isometric Copies of l~β(0 < β < 1)

We get the characterizations of the family of all nonnegative,subadditive,β-absolutely homogeneous and continuous functionals defined on X,when the β-normed space X contains an asymptotically isometric copy of lβ.Moreover,it is proved that if a closed bounded β-convex subset K of a β-normed space contains an asymptotically isometric lβ-basis,then K contains a closed β-convex subset C which fails the fixed point property.     

Existence of Generalized Steiner Systems <Emphasis Type="Italic">GS</Emphasis>(2,4,ν,2)

Generalized Steiner systems GS(2,4,v,2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance five, in which each codeword has length v and weight four. Etzion conjectured its existence for any integer v 7 and v 1(mod 3). The conjecture has been verified for prime powers v > 7 and v 7(mod 12) by the latter two of the present authors. It has also been pointed out that there does not exist a GS(2,4,7,2). In this paper, constructions using frame generalized Steiner systems, two holey perfect bases and orthogonal Latin squares are discussed. With these constructions the conjecture is confirmed with the exception for v=7 and three possible exceptions for v 13, 52, 58.AMS classification: 05B05, 94B25     

The Δ<Superscript>0</Superscript><Subscript><Emphasis Type="Italic">α</Emphasis></Subscript>-Computable Enumerations of the Classes of Projective Planes

Studying computable representations of projective planes, for the classes K of pappian, desarguesian, and all projective planes, we prove that K ^c /? admits no hyperarithmetical Friedberg enumeration and admits a Friedberg Δ⁰_α+3-computable enumeration up to a Δ⁰ _α -computable isomorphism.     

Quincunx multiresolution analysis for <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℚ<Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack>)

With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.     

Isomorphic embedding of ℓ p n , 1<p<2, into ℓ 1 (1+ε)n

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that ℓ _p ⁿ , (C logn)^1/q(1+1/ε)-embeds into ℓ ₁ ^(1+ε)n , where 1<p<2 and 1/p+1/q=1. Supported by ISF.     

Analysis and geometry of Floer theory of Landau-Ginzburg model on ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

This article studies the Floer theory of Landau-Ginzburg (LG) model on C ⁿ . We perturb the Kähler form within a fixed Kähler class to guarantee the transversal intersection of Lefschetz thimbles. The C ⁰ estimate for solutions of the LG Floer equation can be derived then by our analysis tools. The Fredholm property is guaranteed by all these results.     

On averaging of rearrangement invariant ideals of the space L<Superscript>1</Superscript> (Ω, Σ, μ) by non-atomic <Emphasis Type=Italic>σ</Emphasis>-subalgebras of Σ

The problem of averaging = into mapping) is studied for conditional expectation operators which are considered on rearrangement invariant order ideals of measurable functions given on non-atomic probability supports.