ON possible Turán densities

The Turán density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π _∞ ^(k) consist of all possible Turán densities and let Π _fin ^(k) ? Π _∞ ^(k) be the set of Turán densities of finite k-graph families. Here we prove that Π _fin ^(k) contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that Π _fin ^(k) contains an irrational number for each k ≥ 3. Also, we show that Π _∞ ^(k) has cardinality of the continuum. In particular, Π _∞ ^(k) ≠ Π _fin ^(k) .     

Descriptive properties of density preserving autohomeomorphisms of the unit interval

We prove that density preserving homeomorphisms form a Π ₁ ¹ -complete subset in the Polish space ? of all increasing autohomeomorphisms of unit interval.     

The characterization of the Triebel-Lizorkin spaces forp=∞

We establish the characterization of the weighted Triebel-Lizorkin spaces for p=∞ by means of a “generalized” Littlewood-Paley function which is based on a kernel satisfying “minimal” moment and Tauberian conditions. This characterization completes earlier work by Bui et al. The definitions of the ? _∞,q ^α spaces are extended in a natural way to ? _∞,∞ ^α and it is proven that this is the same space as ? _∞,∞ ^α , which justifies the standard convention in which the two spaces are defined to be equal. As a consequence, we obtain a new characterization of the Hölder-Zygmund space ? _∞,∞ ^α .     

Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

The spectral problem in a bounded domain Ω?Rⁿ is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ _j ⁰ } _j=1 ^∞ and {λ _j ^∞ } _j=1 ^∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated.     

On the collineation groups of infinite projective and affine planes

A partial plane is a triple Π=(P,L,I) whereP is the set of points,L the set of lines andI?PXL the incidence relation satisfying the axiom that $$p_i {\rm I}\ell _j (i,j = 1,2) implies p_1 = p_2 or \ell _1 = \ell _2 .$$ Using methods of E. MENDELSOHN, Z. HEDRLIN and A. PULTR we prove the followingTHEOREM. Given a subgroup G ofthe collineation group Aut Π ofsome partial plane Π, there is a projective plane Π′such that Πis invariant under the automorphisms of Π′, Aut Π′Π^′=G,and we obtain an isomorphism of Aut Πonto Aut Π′by restriction. Moreover, any 3 points (lines) of Πare collinear (concurrent) in Π iff they are so in Π′. Corollaries of this result improve some of E. Mendelsohn's theorems [6,7].     

Finite separable field extensions with prescribed extensions of valuations. II

Let A¹,...,A^k be pairwise independent valuation rings of K. Prescribing extensions Δ _i ^j . of the value group Γ^j and extensions \(\mathfrak{L}_i^j\) of the residue field \(H^j\) of A^j (i=1,...,r^j) such that \(\sum\limits_{i = 1}^{r^j } {(\Delta _i^j :\Gamma ^j )} \cdot [\mathfrak{L}_i^j :H^j ] = n\) , we provide sufficient conditions for the existence of a separable field extension L of K of degree n with exactly r^j pairwise independent valuation rings B _i ^j lying over A^j, which have Δ _i ^j as value groups and \(\mathfrak{L}_i^j\) as residue fields.     

Multipliers for the Weyl transform and Laguerre expansions

Let P_k denote the projection of L²(R ^R ) onto the kth eigenspace of the operator (-δ+?x?² andS _N ^α =(1/A _N ^α )Σ _k ^N =0A _N?k ^α P _k . We study the multiplier transformT _N ^α for the Weyl transform W defined byW(T _N ^αf )=S _n ^αW(f) . Applications to Laguerre expansions are given.     

A method of constructing integrable linear equations and its application to Hill's equation

Starting with a given equation of the form $$\ddot x + [\lambda + \varepsilon f(t)] x = 0$$ , where λ > 0 and ? ? l is a small parameter [heref(t) may be periodic, and so Hill's equation is included], we construct an equation of the form y + [λ + ?f (t) + ?² _g (t)]y = 0, integrable by quadratures, close in a certain sense to the original equation. For x₀ = y₀ and x ₀ ^′ = y ₀ ^′ , an upper bound is obtained for ¦y—x¦ on an interval of length Δt.     

A short note on the Frobenius norm of the commutator

This note mainly aims to improve the inequality, proposed by Böttcher and Wenzel, giving the upper bound of the Frobenius norm of the commutator of two particular matrices in ? ^n×n . We first propose a new upper bound on basis of the Böttcher and Wenzel’s inequality. Motivated by the method used, the inequality ‖XY ? XY‖ _F ² ≤ 2‖X‖ _F ² ‖Y‖ _F ² is finally improved into $$ \left\| {XY - YX} \right\|_F^2 \leqslant 2\left\| X \right\|_F^2 \left\| Y \right\|_F^2 - 2[tr(X^T Y)]^2 . $$ . In addition, a further improvement is made.     

Faserungen,die aus Reguli mit gemeinsamer Berührprojektivität längs einer gemeinsamen Erzeugenden zusammengesetzt sind

Let Π_PP be a pappian projective 3-space and ? be a set of lines of Π_PP; we define:

u

a line g of Π_PP has the property R with respect to ?, if all lines of ? meeting g form a regulus

? has the property E ₃, if there exists a pencil \(\mathfrak{L}_0 \) of lines such that one line z of \(\mathfrak{L}_0 \) belongs to ? and all lines of { \(\mathfrak{L}_0 \backslash \left\{ z \right\}\) have the property R with respect to ?.

A spread with the property E ₃ (abbreviated E ₃-spread) is built up of reguli which have one line in common and the same tangent projectivity along their common line. We point out a method of constructing an E ₃-spread of Π_PP. This construction is applied to the real 3-space ?³ to generalize a result of D. Betten [2, S.327] and to prove that another result of D. Betten [3, S.140, Bsp. 2] yields E ₃-spreads. For each natural number n (∈?) we specify two E ₃-spreads \(\mathfrak{F}_n \) and \(\mathfrak{S}_n \) of ?³ such that two different elements of \(F: = \left\{ {\mathfrak{F}_n |n \in \mathbb{N}} \right\} \cup \left\{ {\mathfrak{S}_n |n \in \mathbb{N}} \right\}\) are not equivalent with respect to the collineation group of ?³ apart from \(\mathfrak{F}_1 \) each spread of F represents a 4-dimensional translation plane with a 6-dimensional collineation group. Finally, the properties R and E ₃ are used to characterize the elliptic linear line congruences of a pappian 3-space.     

Metrische Räume mit Dreiseitverbindbaren

This paper presents a system of axioms for n-dimensional metric geometry. For every group satisfying the axioms there exist a group-space and an embedding of into a projective-metric space Ω. We construct an isomorphism of onto a subgroup of a special orthogonal group O _n+1 ^* (K,f). This group belongs to a metric vector space (V,f) over a field K of characteristic ≠ 2 where dim rad V≦1. The (full) groups o _n+1 ^* (K,f) are models of the system of axioms.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

Interpolation in analytic Hölder classes in a neighborhood of an interior cusp

We consider the class of functions that are analytic in the domain G={z=x+iy:|z|<1, |y|>(?x)^β for ?11, and satisfy the Hölder condition with exponent α in the closure Λ _a ^α (G) of the domain G with interior cusps. As is proved, a nontangent set E condensed to the point O is an interpolation set for the pair of spaces Λ _a ^α (G), Λ^αβ(E) if and only if the set E is sparse. Thus, an increase in smoothness occurs in the trace space. Bibliography: 5 titles.     

Laplace-Transformation nichtnegativer und vektorwertiger Maße

Calling a function f: R ₊ ^p →R with $$\sum\limits_{i = 1}^n { \sum\limits_{j = 1}^n {\alpha _i \alpha _j } f(t_i + t_j )} \geqslant 0 for all (\alpha _1 ,...,\alpha _n ) \varepsilon R^n ,$$ , (t₁,...,t_n∈(R ₊ ^p )ⁿ, n∈? positiv semidefinit, the Laplace-transformations of finite nonnegative measures on R ₊ ^p are charac terised as the continuous bounded positiv semidefinit functions. Let H be a real Hilbertspace. A σ-additive mapping \(M: \mathfrak{B}_ + ^p \to H\) is called an orthogonal measure iff 〈M(A), M(B)?=0 for A∩B=ø. Exactly those mappings Y: R ₊ ^p →H are Laplacetransformations of H-valued orthogonal measures, which are continuous and bounded and for which ?Y(s), Y(t)? is only a function of s + t. Using this result one obtains a representation theorem for continuous semi-grouphomomorphisms defined on R ₊ ^p with values in the “unit intervall” of the selfadjoint operators on H.     

The approximation of a Hölder class of two variables by Riesz spherical means

For periodic functions of the Hölder class H ₂ ^α (0 < α≤1) defined in the two-dimensional space D₂, we find the asymptotic form as R → + ∞ of the quantity $$\mathop {\sup }\limits_{f \in H_2^\alpha } \parallel S_R^\delta (x,f) - f(x)\parallel _{C(E_2 )} \left( {\delta > \frac{1}{2} + \alpha } \right),$$ where S _R ^δ is the Riesz spherical mean of orderδ of the Fourier series of the functionf(x).     

Generalized monotone approximation inL p Space

Letf(x) ∈L _p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δ^k (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ _h ^k f(x)?0. Denote by ∏ _n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ ^k ∩ L _p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk.     

Connection between random curves,changes of time,and regenerative times of stochastic processes

The product of spaces Φ × D is considered, where Φ is the set of all continuous, nondecreasing functions ?:[0,∞)→(0,∞), ?(0)=0, ?(t)→∞(t→∞), and D is the set of all right continuous functions ξ:(0,∞)→X; here X is some metric space. Two mappings are defined: the first is the projection q(?,ξ)=ξ, and the second is the change of time U(?,ξ)=ξº?. The following equivalence relation is defined on D: $$\xi _1 \sim \xi _2 \Leftrightarrow \exists _{\varphi _1 , \varphi _1 } \in \Phi :\xi _1 ^\circ \varphi _1 = \xi _2 ^\circ \varphi _2 $$ . Let? be the set of all equivalence classes, and let L be the mapping ξ₄～ξ₂, Lξ is called the curve corresponding to ξ. The following theorem is proved: two stochastic processes with probability measures P¹ and P² on D possess identical random curves (i.e.,P¹ºL^?1=P²ºL^?1) if and only if there exist two changes of time (i.e., probability measures Q¹ and Q² on ?×D for which P¹=Q¹ºq^?1, P²=Q²ºq^?1 which take these two processes into a process with measure \(\tilde P\) (i.e., Q¹ºu^?1=Q²ºu^?1,=～P) If (P _x ¹ )_x∈X and (P _x ² )_x∈X are two families of probability measures for which P _x ¹ ºL^?1=P _x ² ºL^?1?x∈X then for each x ε X the corresponding measures Q _X ¹ andQ _X ² can be found in the following manner. The set of regenerative times of the family \(\left( {\tilde P_x } \right)_{x \in X} \) contains all stopping times which are simultaneously regenerative times of the families (p _x ¹ )_x∈X and (P _x ² )_x∈X and possess a certain special property of first intersection.