A class of Einstein (α, β)-metrics

In this paper, we study a special class of Finsler metrics, called (α, β)-metrics, which are defined by F = α?(β/α), where α is a Riemannian metric and β is a 1-form. We show that if ? = ?(s) is a polynomial in s, it is Einstein if and only if it is Ricci-flat. We also determine the Ricci-flat (α, β)-metrics which are not of the type F = (α + ?β)²/α.     

A new estimate of the Montgomery constant in the theory of zeros distribution of the Riemann zeta-function in the critical stripe

The new value α = 91/116 is obtained for the constant α such that for all σ ≥ α the “density hypothesis” is valid for the Riemann zeta-function in the critical stripe. The previous value α = 11/14 was obtained by M. Jutila in 1977.     

On a class of hyperbolic 3-orbifolds of small volume and small heegaard genus associated to 2-bridge links

We consider a class of hyperbolic 3-orbifoldsO(α/β); the underlying topological space of such an orbifold is the 3-sphere and the singular set is obtained by adding the two standard (upper and lower) unknotting tunnels to a 2-bridge linkL(α/β) (and associating branching order two to both unknotting tunnels). These 3-orbifolds are extremal with respect to the notion of Heegaard genus or Heegaard number of 3-orbifolds; it is to be expected that they are also extremal with respect to the volume, that is the smallest volume hyperbolic 3-orbifolds should belong to this or some closely related class. We show that an orbifoldO(α/β) has a uniqueD ₂-covering by an orbifold? _n(α/β) wose space is the 3-sphere and whose singular set is the same 2-bridge linkL(α/β) used for the construction ofO(α/β); moreoverO(α/β) is hyperbolic if and only if? _n(α/β) is hyperbolic. As the volumes of the orbifolds? _n(α/β) are known resp. can be computed, this allows to compute the volumes of the orbifoldsO(α/β). The problem of computation of volumes remains open for some closely related classes of 3-orbifolds which are also extremal with respect to the Heegaard genus (for example associating a branching order bigger than two to one or both unknotting tunnels).     

On uncountable cardinal sequences for superatomic Boolean algebras

The countable sequences of cardinals which arise as cardinal sequences of superatomic Boolean algebras were characterized by La Grange on the basis of ZFC set theory. However, no similar characterization is available for uncountable cardinal sequences. In this paper we prove the following two consistency results:

1. Ifθ = 〈κ _α :α <ω ₁〉 is a sequence of infinite cardinals, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.
2. Ifκ is an uncountable cardinal such thatκ ^<κ =κ andθ = 〈κ _α :α <κ ⁺〉 is a cardinal sequence such thatκ _α ≥κ for everyα <κ ⁺ andκ _α =κ for everyα <κ ⁺ such that cf(α)<κ, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.

     

Error bounds for high-resolution quantization with Rényi-α-entropy constraints

We consider the problem of optimal quantization with norm exponent r > 0 for Borel probability measures on ? ^d under constrained Rényi-α-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter α ∈ [1+r/d,∞]. For α ∈ [0,1 + r/d] we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error is decreasing exponentially fast with the entropy bound and the exact rate is determined for all α ∈ [0, ∞].     

Pseudo-holomorphic curves of constant curvature in complex Grassmannians

In this paper we consider pseudo-holomorphic curves in complex Grassmiannians. Let φ ₀, φ ₁, ?, $\varphi _{\alpha _0 } $ : S ² → G _k,n be a linearly full non-degenerate pseudo-holomorphic harmonic sequence, and let degφ_α and K _α be the degree and the Gauss curvature of φ_α (α = 0, 1, ?, α ₀) respectively. Assume that φ ₀, φ ₁, ?, $\varphi _{\alpha _0 } $ is totally unramified. Then we prove that (i) degφ_α for all α = 0, 1, ?, α ₀; (ii) $K_\alpha = \tfrac{4}{{k(\alpha _0 + 2\alpha (\alpha _0 - \alpha ))}}$ if K _α is constant for some α = 0, 1, ?, α ₀,. We also give some conditions for pseudo-holomorphic curves with constant Kähler angle in complex Grassmiannians to be of constant curvature.     

Complementing and exactly covering sequences

The following is proved (in a slightly more general setting): Let α₁, …, α_m be positive real, γ₁, …, γ_m real, and suppose that the system [nα_i + γ_i], i = 1, …, m, n = 1, 2, …, contains every positive integer exactly once (= a complementing system). Then $\text{α}{}_{\mathrm{i}}\text{α}{}_{\mathrm{j}}$ is an integer for some i ≠ j in each of the following cases: (i) m = 3 and m = 4; (ii) m = 5 if all α_i but one are integers; (iii) m ? 5, two of the α_i are integers, at least one of them prime; (iv) m ? 5 and α_n ? 2ⁿ for n = 1, 2, …, m ? 4.For proving (iv), a method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementing system of m ? 1 sequences.     

Existence for the α-patch model and the QG sharp front in Sobolev spaces

We consider a family of contour dynamics equations depending on a parameter α with 0<α?1. The vortex patch problem of the 2-D Euler equation is obtained taking α→0, and the case α=1 corresponds to a sharp front of the QG equation. We prove local-in-time existence for the family of equations in Sobolev spaces.     

Stability of solutions bifurcating from multiple eigenvalues

Let X and Y be real Banach spaces and G:X × $\text{R}$ be a twice continuously differentiate function which is not necessarily linear. Suppose G(u₀, α₀) = 0 and the dimension of the null space of G_u(u₀, α₀) is m, where 1 ? m < ∞. Usually, S = {(u, α):G(u, α) = 0}, in a neighborhood of (u₀, α₀), consists of a finite number of curves emanating from (u₀, α₀). We will determine the stability of points, (u, α), in S (i.e., the maximum of the real parts of the spectrum of G_u(u, α) for each (u, α) ∈ S) using a general perturbation theorem of Kato. Our results contain as a special case the stability theorems of Crandall and Rabinowitz for the case m = 1. We will also tie our stability theorems together with some bifurcation results of Decker and Keller. Finally we apply our results to systems of reaction diffusion equations.     

Modified Dyadic Integral and Fractional Derivative on ℝ+

For functions from the Lebesgue space L(?₊), we introduce the modified strong dyadic integral J _α and the fractional derivative D ^(α) of order α > 0. We establish criteria for their existence for a given function f ∈ L(?₊). We find a countable set of eigenfunctions of the operators D ^(α) and J _α, α > 0. We also prove the relations D ^(α)(J _α(f)) = f and J _α(D ^(α)(f)) = f under the condition that $\smallint _{\mathbb{R}_ + } f(x)dx = 0$ . We show the unboundedness of the linear operator $J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$ , where L _{J α} is its natural domain of definition. A similar assertion is proved for the operator $D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$ . Moreover, for a function f ∈ L(?₊) and a given point x ∈ ?₊, we introduce the modified dyadic derivative d ^(α)(f)(x) and the modified dyadic integral j _α(f)(x). We prove the relations d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) at each dyadic Lebesgue point of the function f.     

On vector-valued curves that allow a C 1,α parametrization

Let X be a Banach space and α ∈ (0, 1]. We find equivalent conditions for a function f: [0,1] → X to admit an equivalent parametrization, which is C ^1,α (i.e., has α-Hölder derivative). For X = ?, a characterization is well-known. However, even in the case X = ?² several new ideas are needed.     

Quasirecognition by prime graph of 2 D n (3α) where n = 4m + 1 ≥ 21 and α is odd

Let G be a finite group. The prime graph of G is denoted by Γ(G). In this paper, as the main result, we show that if G is a finite group such that Γ(G) = Γ(² D _n (3^α)), where n = 4m+ 1 and α is odd, then G has a unique non-Abelian composition factor isomorphic to ² D _n (3^α). We also show that if G is a finite group satisfying |G| = |² D _n (3^α)|, and Γ(G) = Γ(² D _n (3^α)), then G ? ² D _n (3^α). As a consequence of our result, we give a new proof for a conjecture of Shi and Bi for ² D _n (3^α). Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. Specifically, it is proved that ² D _n (3^α) is quasirecognizable by the spectrum.     

Normality and countable paracompactness of hyperspaces of ordinals

For an ordinal α, α² denotes the collection of all nonempty closed sets of α with the Vietoris topology and K(α) denotes the collection of all nonempty compact sets of α with the subspace topology of α². It is well known that α² is normal iff cfα=1. In this paper, we will prove that for every nonzero-ordinal α:

(1)

α² is countably paracompact iff cfα≠ω.

(2)

K(α) is countably paracompact.

(3)

K(α) is normal iff, if cfα is uncountable, then cfα=α.

In (3), we use elementary submodel techniques.     

On the <Emphasis Type="Italic">α</Emphasis>-superposition of functions of a <Emphasis Type="Italic">k</Emphasis>-valued logic

We reveal a relation between the operations of α-completion and closure for the systems of functions of a k-valued logic. For k = 3, 4 we construct the α-bases consisting of two binary operations. We prove that the complete system T of functions of a 4-valued logic containing all permutations of the set E ₄ = {0, 1, 2, 3} and the operation of addition modulo 4 is not α-complete, whereas its α-completion [T _α] will be an α-complete system.     

Pressing Down Lemma for λ-trees and its applications

For any ordinal λ of uncountable cofinality, a λ-tree is a tree T of height λ such that |T _α| < cf(λ) for each α < λ, where T _α = {x ∈ T: ht(x) = α}. In this note we get a Pressing Down Lemma for λ-trees and discuss some of its applications. We show that if η is an uncountable ordinal and T is a Hausdorff tree of height η such that |T _α | ? ω for each α < η, then the tree T is collectionwise Hausdorff if and only if for each antichain C ? T and for each limit ordinal α ? η with cf(α) > ω, {ht(c): c ∈ C} ∩ α is not stationary in α. In the last part of this note, we investigate some properties of κ-trees, κ-Suslin trees and almost κ-Suslin trees, where κ is an uncountable regular cardinal.     

On the sum ∑k=13 | 2πk − αk |, αk algebraic numbers

If α₁, α₂, α₃ are algebraic numbers satisfying (i) the height of α₁, α₂, α₃ do not exceed H (ii) the degree of the field generated by α₁, α₂, α₃ over the field of rational numbers do not exceed D, then a positive lower bound for

$\text{∑}\text{k=1}\text{3}\text{|2π}{}^{\mathrm{k}}\text{?αk|}$

is determined explicitly (except for an absolute constant) in terms of D and H.     

The notion of o-tightness and C-embedded subspaces of products

We consider the question: when is a dense subset of a space XC-embedded in X? We introduce the notion of o-tightness and prove that if each finite subproduct of a product X = Π_α?AX_α has a countable o-tightness and Y is a subset of X such that π_B(Y) = Π_α?BX_α for every countable B ? A, then Y is C-embedded in X. This result generalizes some of Noble and Ulmer's results on C-embedding.     

Note on the point character of l 1-spaces

We prove that for any ordinal α, any integer t ≥ 0, the point character of the space l ₁(ω _{α + t} ) is no more than ω _α . Combined with an earlier result from [5], this yields that for any infinite cardinal κ the point character of l ₁(κ) is the largest cardinal ω _α ≤ κ where α = 0 or a limit ordinal.     

Stability and bifurcation of equilibrium solutions of reaction-diffusion equations on an infinite space interval

Under the condition that f(x, y, z, α) and its partial derivatives decay sufficiently fast as $\text{¦x¦ → ∞}$ we will study the (linear) stability and bifurcation of equilibrium solutions of the scalar problem u_t = u_xx + f(x, u, u_x, α), u_x(?∞, t) = u_x(∞, t) = 0 (1) where α is a real bifurcation parameter. After introducing appropriate function spaces X and Y the problem (1) can be rewritten $\text{d}\text{dt}\text{u = G(u, α)}$, (7) where G:X×R → Y is given by G(u, α)(x) = u″(x) + f(x, u(x), u′(x), α). It will be shown, for each (u, α)?X × R, that the Fréchet derivative G_u(u,a): X → Y is not a Fredholm operator. This difficulty is due to the fact that the domain of the space variable x, is infinite and cannot be eliminated by making another choice of X and Y. Since G_u(u, α) is not Fredholm, the hypotheses of most of the general stability and bifurcation results are not satisfied. If (u₀, α₀?S = {(u, α): G(u, α) = 0}, (i.e., (u₀,α₀) is an equilibrium solution of (7)), a necessary condition on the spectrum of G_u(u₀, α₀) for a change in the stability of points in S to occur at G_u(u₀, α₀) will be given. When this condition is met, the principle of exchange of stability which means, in a neighborhood of (u₀, α₀), that adjacent equilibrium solutions for the same α have opposite stability properties in a weakened sense will be established. Also, when G_u or its first order partial derivatives, evaluated at (u₀, α₀), are not too degenerate, the shape of S in a neighborhood of (u₀, α₀) will be described and a strenghtened form of the principle of exchange of stability will be obtained.