Zur Verteilung der Quadrate ganzer Zahlen in rationalen Quaternionenalgebren

For γ ∈ ?letQ 〈γ〉 = ?[i]+?[i]j. where j. is a hypercomplex number withj2 = γ, and define addition and multiplication formally with respect to $zj = j\overline z $ for all z ∈ ?[i], so thatQ〈γ〉 becomes a quaternion algebra over the rationals. Further fix γ s.t.Q 〈γ 〉 is a division algebra and define for real X ≥ 1 where |Re(α)|, |Im(α)|, |Re(β)|, |Im(ö)|≤ X and Generalizing former results concerning Hamilton’s quaternions (i.e. the case γ =- 1) we show that, as X → ∞, when γ < 0, when γ > 0, when γ < 0, wheny γ 0. Thereby δ(t) is any upper bound of the error term in Dirichlet’s divisor problem, e.g. δ(t) =t0.315, C_γ, D_γ > 0 are numerical constants, and c, d are given by c := π(1 + log 2 - 2η) and d := π2(1 + 4 log 4 - 4π)/8, where π = 0.577 … is Euler’s constant.     

Best one-sided approximation of some classes of smooth functions on the whole real axis by cardinal splines of higher order

The objective of this paper is to study the quantity . In this paper we establish a duality theorem for the quantity for m≥r. By virtue of the duality theroem we obtain the exact estimation for for all m≥r.     

On the best convergence of trigonometric integrals

ОсНОВНОИ РЕжУльтАт Ё тОИ стАтьИ жАклУЧАЕт сь В слЕДУУЩЕМ. ЕслИ с ФИксИРОВАННыМr=2, 3,..., тО Дль тОгО, ЧтОБы ИМЕлО М ЕстО (*) гДЕ ДОстАтОЧНО ВыпОлНЕН ИЕ слЕДУУЩИх УслОВИИ: 1) ЕслИ $$1< \frac{{n_\nu + 1}}{{n_\nu }} = g_\nu \in N, \nu = 0,1,$$ тО Дль кАжДОгОΝ=0,1,... ИМЕ Ет МЕстО (**) 2) ЕслИ тО (***) $$n_{\nu + 1} \geqslant (2r - 1)n_\nu , \nu = 0,1,...$$ . ЕслИ И тО УслОВИь (**), (***) ьВльУтсь тАкжЕ НЕОБхОДИМыМИ Д ль (*).     

Un teorema sulla rappresentazione degli integrali

Let be a De Possel differentiation basis in a complete measure space , with μ≥0, μ(X)<+∞; letB be a Banach space and . When the F′ exists μ-a.e. onX, we prove that for eachp>0, the μ-integral of‖F′‖ ^p is the lower-bound of a class ofp-variations.     

The space BMO and strong means of Fourier series

пУстьS _k (f,x) — ЧАстНАь сУ ММА РьДА ФУРьЕ ФУНкцИ Иf пО тРИгОНОМЕтРИЧЕскОИ сИстЕМЕ,s _k (f,x) — ЧАстНАь сУММА сО пРьжЕННОгО РьДА. Дль \(\Delta _k^n = \left[ {\frac{n}{n},\frac{{k + 1}}{n}} \right)\) , гДЕk=0, 1, ...,n?1, пОлОжИМ , ЕслИt?δ _k ⁿ И , ЕслИt?[0, 1)δ _k ⁿ . пОкАжАНО, ЧтО ОпЕРАтО Ры ИМЕУт слАБыИ тИп (1,1). РАссМОтР ЕН РьД слЕДстВИИ О пОВ ЕДЕНИИ сИльНых сРЕДНИх РьДА ФУРьЕ сУММИРУЕМОИ ФУНкцИИ.     

On some classes of contact metric manifolds

We define four new classes of contact metric manifoulds using Tanaka connection and Jacobi operators. We prove that a contact metric manifold with the structure vector field ξ belonging to thek-nullity distribution is contact metric locally ?-symmetric (in the sense of D. B. Blair) if and only if the manifold is a and space. Also, we prove that a 3-dimensional contact metric and is locally ?-symmetric (in the sense of D. E. Blair) and give counter-examples of the converse.     

On inductive limits of measure spaces

We introduce a category of (topological) measure spaces in which inductive limitis exist and where the Banach spaces and (1≤p≤+∞) are isometric for arbitrary inductive systems of (topological) measure spaces.     

On a lemma of Butzer and Kirschfink

In [1], Butzer and Kirschfink discussed the convergence rates for C[0,1]-valued dependent random functions on Donsker's weak invariance principle and introduced the concept of dependency from below to deal with the martingale difference sequence. They asserted in their Lemma 8 that Lemma A. A martingale difference sequence (X_n,F_n,n≥1) with is dependent from below, i.e., for each 1≤i≤n and each n≥1 . The purpose of this note is to prove that Lemma A is not always true and to improve the conditions of Butzer and Kirschfink. We shall apply the notations in [1].     

On a characterization of finite dimensional real linear spaces

LetX be a real linear space; if is a linear topology onX, let denote the class of all bounded with respect to subsets ofX. In this paper it is shown that the spaceX is algebraically finite dimensional if and only if the class of all convex, absorbing and radially bounded subsets ofX is included in the intersection of all , where runs the set of all linear topologies onX.     

A generalization of a theorem of Kruskal

LetA be a set ofk-subsets of {1,...,n} with . We determine the minimal cardinality of thel-shadow ofA. This theorem generalizes a theorem of Kruskal [10], Katona [8], Lindström [11] and other which gives a solution for the problem without the condition . Furthermore we give a necessary and sufficient condition for the uniqueness of the solution in the KKL theorem.     

Congruence modularity implies the Arguesian identity

It is shown that if is any variety of algebras all of whose congruence lattices are modular, then the congruence lattice of every algebra in satisfies the Arguesian law.     

Double Hilbert transform on

We introduce a space , where is the testing function space whose functions are infinitely differentiable and have bounded support, and is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from onto itself.     

Convolution of the Fourier coefficients of the Eisenstein-Maass series

The convolution is defined as the sum where for n≠0 and W₀,W₁ are arbitrary smooth functions. Question: how to express these sums in the form of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian? The answer is given in terms of the bilinear form of the Hecke series associated with the eigenfunctions of the automorphic Laplacian and with regular cusp forms. The final identity may lead to a new possibility for the solution of the moment problem of the Riemann zeta-function.     

Local convergence in measure on semifinite von Neumann algebras

Suppose that ? is a von Neumann algebra of operators on a Hilbert space $\mathcal{H}$ and τ is a faithful normal semifinite trace on ?. The set of all τ-measurable operators with the topology t _τ of convergence in measure is a topological *-algebra. The topologies of τ-local and weakly τ-local convergence in measure are obtained by localizing t _τ and are denoted by t _τ1 and t _wτ1, respectively. The set with any of these topologies is a topological vector space. The continuity of certain operations and the closedness of certain classes of operators in with respect to the topologies t _τ1 and t _wτ1 are proved. S.M. Nikol’skii’s theorem (1943) is extended from the algebra $\mathcal{B}(\mathcal{H})$ to semifinite von Neumann algebras. The following theorem is proved: For a von Neumann algebra ? with a faithful normal semifinite trace τ, the following conditions are equivalent: (i) the algebra ? is finite; (ii) t _wτ1 = t _τ1; (iii) the multiplication is jointly t _τ1-continuous from to ; (iv) the multiplication is jointly t _τ1-continuous from to ; (v) the involution is t _τ1-continuous from to .     

Algebraic extensions of finite corank of hilbertian fields

We consider here a hilbertian fieldk and its Galois group (k _s/k). For a natural numbere we prove that almost all (σ) ∈ (k_s/k)^e have the following properties. (1) The closedsubgroup 〈σ〉 which is generated by σ₁, …, σ_e is a free pro-finite group withe generators. (2) LetK be a proper subfield of the fixed fieldk _s (σ) of 〈σ〉, …, σ_e ink _s, which containsk. Then the group (k _s/K) cannot be topologically generated by less thene+1 elements. (3) There does not exist a τ ∈ (k/k), τ≠1, of finite order such that [k _s (σ):k _s (σ, τ)]<∞. (4) Ife=1, there does not exist a fieldk?K?k _s (σ) such that 1<[k _s (σ):K]<∞. Here “almost all” is used in the sense of the Haar measure of the compact group (k_s/k)^e.     

Интегро-дифференциа ляные формулы сферических средних

Реэуме. Установлена формула для средних $$S(t)f(\mu ): = \frac{1}{{\left| {\sigma ^\prime } \right|}}\smallint _{\sigma (\mu )} f(\mu cos t + \nu sin t)d\nu $$ функцииf, определеннои на единичнои сфере а пространства R ⁿ ,n>2; σ(μ) — Экватор с полусом в μ, |σ′| — мера Экватора. Для проиэводнои ? _t ^(r) дано представление.     

Un teorema di unicità per una equazione iperbolica,non lineare

We prove the uniqueness of weak solutions of initial boundary value problems Each functiona _i is required to be sufficiently smooth and must satisfy the following conditions: \(e) \sum\limits_1^n {ij} \partial _{\eta _j \eta _h }^2 a_i (x, \ldots , \eta 1, \ldots , \eta _n )\xi _i \xi _j \leqslant 0, h = 1, \ldots , n,\) for some positive constantsK ₀, α, some non negative constantsK _i , some positive functionsH(t)∈L ¹(0,T) and for all ξ≡(ξ _i ), η≡(η _i )∈R ⁿ      

Minimax filtering of homogeneous random fields with white noise

We consider mean-square optimal linear estimation of the transform $$A\xi = \sum\limits_{k,j = 0}^\infty {a(k,j)} \xi ( - k, - j)$$ of the homogeneous random field ξ(k, j) with density f(λ, Μ) from the observations ξ(k, j) + η(k, j) with k ≤ 0, j ≤ 0, where η(k, j) is a random field with orthogonal values uncorrelated with ξ(k, j) (white noise). We determine the worst spectral densitiesf ₀(λ, Μ) ∈ and the minimax (robust) spectral characteristics of the optimal estimator of the transform Aξ for various density classes .