Cosymplectic quasi-Sasakian pseudo-Riemannian manifolds and coisotropic foliations

Si considera una varietà neutra \(\tilde M\) di dimensione 2m munita di una struttura conforme simplettica \(CS_p \left( {2m; R} \right) = \left( {\tilde \Omega , \tilde \upsilon } \right)\) . Vengono studiati i differenti problemi concernenti gli automorfismi infinitesimali della 2-forma quasi simplettica \(\tilde \Omega \) . Inoltre vengono formulate alcune proprietà di un fogliettamento con isotropoF _c su \(\tilde M\) .     

Hölderabschätzungen für Ableitungen von Lösungen der Gleichung\bar \partial u = f bei streng pseudokonvexem Rand

Using the local Kerzman kernel we prove regularity of solutions of \(\bar \partial \) u=f, where f is a \(\bar \partial \) -closed (0,1)-form in a strongly pseudoconvex domain G in ?^N. If f is in H^m,∞, then the solution is in \(\tilde C^{m,\mu } \) forμ<1, that is, the m-th derivatives are in C^o,μ/2 and in addition areμ-Hölder continuous on curves “parallel” to the holomorphic tangent bundle \(\tilde T\) ?G. If f is in C^m,α with o<α<1, then the solution is in \(\tilde C^{m,1 + \mu } \) forμ<α, that is, the m-th derivatives are in C^o,(1+μ/2, and they have first derivatives “parallel” to \(\tilde T\) ?G lying in \(\tilde C^{o,\mu } \) . We derive the same results for the global solution constructed by Grauert and Lieb, and similar estimates on complex manifolds.     

\mathfrak{E} of topological spaces II

In this paper, we obtain analogues, in the situation of \(\mathfrak{E}\) -extensions, of Magill's theorem on lattices of compactifications. We define an epireflective subcategory of the categoryT 2 of all Hausdorff spaces to be admissive (respectively finitely admissive) if for any \(\mathfrak{E}\) -regular spaceX, every Hausdorff quotient of \(\beta _\mathfrak{E} X\) which is Urysohn on \(\beta _\mathfrak{E} X - X\) (respectively which is finitary on \(\beta _\mathfrak{E} X - X\) ) and which is identity onX, has \(\mathfrak{E}\) . We notice that there are many proper epireflective subcategories ofT 2 containing all compact spaces and which are admissive; there are many such which are not admissive but finitely admissive. We prove that when \(\mathfrak{E}\) is a finitely admissive epireflective subcategory ofT 2, then the lattices of finitary \(\mathfrak{E}\) -extensions of two spacesX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic. Further if \(\mathfrak{E}\) is admissive, then the lattices of Urysohn \(\mathfrak{E}\) -extensions ofX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic.     

ОБ ИНтЕРпОлИРОВАНИИ В кОМплЕксНОИ ОБлАст И

LetD be a simply connected domain, the boundary of which is a closed Jordan curveγ; \(\mathfrak{M} = \left\{ {z_{k, n} } \right\}\) , 0≦k≦n; n=1, 2, 3, ..., a matrix of interpolation knots, \(\mathfrak{M} \subset \Gamma ; A_c \left( {\bar D} \right)\) the space of the functions that are analytic inD and continuous on \(\bar D; \left\{ {L_n \left( {\mathfrak{M}; f, z} \right)} \right\}\) the sequence of the Lagrange interpolation polynomials. We say that a matrix \(\mathfrak{M}\) satisfies condition (B _m ), \(\mathfrak{M}\) ∈(B _m ), if for some positive integerm there exist a setB _m containingm points and a sequencen _{p p=1} ^∞ of integers such that the series \(\mathop \Sigma \limits_{p = 1}^\infty \frac{1}{{n_p }}\) diverges and for all pairsn _i ,n _j ∈{n _p } _p=1 ^∞ the set \(\left( {\bigcap\limits_{k = 0}^{n_i } {z_{k, n_i } } } \right)\bigcap {\left( {\bigcup\limits_{k = 0}^{n_j } {z_{k, n_j } } } \right)} \) is contained inB _m . The main result reads as follows. {Let D=z: ¦z¦ \(\Gamma = \partial \bar D\) and let the matrix \(\mathfrak{M} \subset \Gamma \) satisfy condition (B_m). Then there exists a function \(f \in A_c \left( {\bar D} \right)\) such that the relation $$\mathop {\lim \sup }\limits_{n \to \infty } \left| {L_n \left( {\mathfrak{M}, f, z} \right)} \right| = \infty $$ holds almost everywhere on γ.     

Strong approximation of empirical process with independent but non-identically distributed random variables

Let{Y_t,t=1,2,…} be independent random variables with continuous distribution functionsF_i(y).For any y,dencte s=F_t(y)=1/t sum from i=1 to t F_i(y).The empirical process is defind by t~(-1/2)R(s,t) whereR(s,t)=t(1/t sum from i=1 to t I_((?)_t(Y_i)≤s)-s)=sum from i=1 to t I_(?)-ts=sum from i=1 to t I_(?)-(?)_t(y)=sum from i=1 to t I_(Y_(?)≤y)-sum from i=1 to t F_i(y).The purpose of this paper is to investigate the asymptotic properties of the empirical processR(s,t).We shall prove that for some integer sequence {t_k},there is a (?)-process (?)(s,t) such that(?)|R(s,t_k)-(?)(s,t_k)|=O(t_k~(1/2)(log t_k)~(-1/4)(log log t_k)~(1/2))a.s.where (?)(s,t) is a two-parameter Gaussian process defined in §1.     

Exact and asymptotic estimates forn-widths of some classes of periodic functions

Let \(\tilde W_p^r : = \left\{ {f\left| {f \in C^{r - 1} } \right.} \right.\left[ {0,2\pi } \right],f^{(i)} (0) = f^{(i)} (2\pi ),i = 0, \ldots ,r - 1,f^{(r - 1)}\) , abs. cont. on [0, 2π] andf ^(r)∈L ^p[0, 2π]}, and set \(\tilde B_p^r : = \left\{ {f\left| {f \in \tilde W_p^r ,} \right.\left\| {f^{(r)} } \right\|_p \leqslant 1} \right\}\) . We find the exact Kolmogrov, Gel'fand, and linearn-widths of \(\tilde B_p^r\) inL ^p forn even and allp∈(1, ∞). The strong asymptotic estimates forn-widths of \(\tilde B_p^r\) inL ^p are also obtained.     

On the limit superior of sequences of sets

qVЕРхНИИ пРЕДЕл пОслЕД ОВАтЕльНОстИ МНОжЕс тВA _n ОпРЕДЕльЕтсь сООтНО шЕНИЕМ \(\mathop {\lim sup}\limits_{n \to \infty } A_n = \mathop \cap \limits_{k = 1}^\infty \mathop \cup \limits_{n = k}^\infty A_n . B\) стАтьЕ РАссМАтРИВА Етсь слЕДУУЩИИ ВОпРО с: ЧтО МОжНО скАжАть О ВЕРхНИх пРЕДЕлАх \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) , еслИ ИжВЕстНО, ЧтО пРЕсЕЧЕНИь \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) «МАлы» Дль кАж-ДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) ? ДОкАжыВАЕтсь, Ч тО

1. ЕслИ \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — кОНЕЧНОЕ МНО жЕстВО Дль кАжДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , тО НАИДЕтсь тАкАь пОДпО слЕДОВАтЕльНОсть, Дл ь кОтОРОИ МНОжЕстВО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) сЧЕтНО;
2. ЕслИ \(2^{\aleph _0 } = \aleph _1\) , тО сУЩЕстВУЕ т тАкАь пОслЕДОВАтЕл ьНОсть (A_n), ЧтО \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — сЧЕтНОЕ МНОжЕстВО Дль лУБОИ п ОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , НО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) ИМЕЕт МОЩ-НОсть кОНтИНУУМА;
3. ЕслИA _n — БОРЕлЕ ВскИЕ МНОжЕстВА В НЕкОтОРО М пОлНОМ сЕпАРАБЕльНО М МЕтРИЧЕскОМ пРОстРАНстВЕ, И \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — сЧЕт НОЕ МНОжЕстВО Дль кАж ДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , тО сУЩЕстВУЕт тАкАь п ОДпОслЕДОВАтЕльНОсть, ЧтО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) — сЧЕтНОЕ МНОжЕстВО. кРОМЕ тОгО, ДОкАжАНО, Ч тО В слУЧАьх А) И В) В пОслЕДОВАтЕльНОстИ (A _n ) сУЩЕстВУЕт схОДьЩА ьсь пОДпОслЕДОВАтЕльНО сть.

кРОМЕ тОгО, ДОкАжАНО, Ч тО В слУЧАьх А) И В) В пОслЕДОВАтЕльНОстИ (А _n ) сУЩЕстВУЕт схОДьЩ Аьсь пОДпОслЕДОВАтЕльНО сть.     

Elliptische Differentialgleichungen mit variablen Koeffizienten in Gebieten mit unbeschränktem Rand

We discuss the spectrum of a symmetric elliptic differential operator A with domain \(\mathop {H^m }\limits^o (\Omega ) \cap H^{2m} (\Omega )\) in regions Ω with unbounded boundary \(\dot \Omega \) , where are \(\bar \Omega \) uniformely of class C^2m and on \(\dot \Omega \) the normal condition x·ν(x)≦μ for sufficient small positiveμ. We prove the A-priori-estimate \(\parallel u\parallel _{m,\Omega } \leqq c\parallel (l + r) (A - k)u\parallel _{o,\Omega } \) and show for all k>k, k≧0 suitable, there are no eigenvalues of A and by characterizing weighted Sobolev spaces with negative norm the existence of solutions \((l + r)_2 ^{ - 1} u \in \mathop H\limits^0{^m} (\Omega ) \cap H^{2m} (\Omega )\) of the equation (A?k)u=f, (1+r)f∈L²(Ω).     

Analytische Fortsetzbarkeit von zufälligen Potenzreihen bezüglich abhängiger Zufallsvariabler

Рассматриваются слу чайная величина \(\mathfrak{X} = (X_n (\omega ))\) , удовлетворяющая усл овиюE(X _n ⁴ )≦M, и соответствующ ий случайный степенн ой ряд \(f_x (z;\omega ) = \mathop \sum \limits_{n = 0}^\infty a_n X_n (\omega )z^n\) . Устанавливаются тео ремы непродолжимост и почти наверное:

1. дляf _x при условиях с лабой мультипликати вности на \(\mathfrak{X}\) ,
2. для \(f_{\tilde x}\) , где \(\mathop \mathfrak{X}\limits^ \sim = (\mathop X\limits^ \sim _n )\) есть подп оследовательность в \(\mathfrak{X}\) ,
3. для по крайней мере од ного из рядовf _x′ илиf _x″ , где \(\mathfrak{X}'\) и \(\mathfrak{X}''\) — некоторые п ерестановки \(\mathfrak{X}\) , выбираемые универс ально, т. е. независимо от коэффициентовa _n .

     

Absolute tauberian conditions for absolute hausdorff and quasi-hausdorff methods

The purpose of this paper is to prove that for a large set of absolute Hausdorff and quasi-Hausdorff methods the condition $$\sum\limits_{k = 1}^\infty {\left| {\lambda _n a_n - \lambda _{n - 1} a_{n - 1} } \right|< } \infty $$ is a Tauberian condition, i.e., its fulfillment together with the absolute summability of \(\sum\limits_{n = 0}^\infty {a_n } \) tos implies that \(\sum\limits_{n = 0}^\infty {\left| {a_n } \right|}< \infty \) and \(\sum\limits_{n = 0}^\infty {a_n } = s.\) a _n =s.     

Erratum et Addendum zu der Arbeit: Über exzeptionelle Mengen

Let X be a complex space and A?X a compact subspace. Let \(\tilde X\) be the blowing up of A in X and \(\tilde A\) ? \(\tilde X\) the resulting hyper-surface. Then the normal bundle of \(\tilde A\) in { \(\tilde X\) is weakly negative iff the normal bundle of the k-th infinitesimal neighborhood of A in X is weakly negative for all k?0. This corrects a theorem in [5].     

On coefficients of null-series and on sets of uniqueness of trigonometric and Walsh systems

В работе доказываютс я следующие утвержде ния. Теорема I.Пусть ? _n ↓0u \(\sum\limits_{n = 0}^\infty {\varepsilon _n^2 = + \infty } \) .Тогд а существует множест во Е?[0, 1]с μЕ=0 такое что:1. Существует ряд \(\sum\limits_{n = 0}^\infty {a_n W_n } (t)\) с к оеффициентами ¦а _n ¦≦{in¦n¦, который сх одится к нулю всюду вне E и ε∥a_n∥>0.2. Если b _n ¦=о(ε _n )и ряд \(\sum\limits_{n = 0}^\infty {b_n W_n (t)} \) сх одится к нулю всюду вн е E за исключением быть может некоторого сче тного множества точе к, то b _n =0для всех п. Теорема 3.Пусть ? _n ↓0u \(\mathop {\lim \sup }\limits_{n \to \infty } \frac{{\varepsilon _n }}{{\varepsilon _{2n} }}< \sqrt 2 \) Тогд а существует множест во E?[0, 1] с υ E=0 такое, что:

1. Существует ряд \(\sum\limits_{n = - \infty }^{ + \infty } {a_n e^{inx} ,} \sum\limits_{n = - \infty }^{ + \infty } {\left| {a_n } \right|} > 0,\) кот орый сходится к нулю в сюду вне E и ¦a_n≦¦n¦ для n=±1, ±2, ...
2. Если ряд \(\sum\limits_{n = - \infty }^{ + \infty } {b_n e^{inx} } \) сходится к нулю всюду вне E и ¦b_v¦=о(ε ¦n¦), то b_n=0 для всех я. Теорема 5. Пусть послед овательности S⁽¹⁾={ε ₀ ⁽¹⁾ , ε ₁ ⁽¹⁾ , ε ₂ ⁽¹⁾ , ...} u S²=ε ₀ ⁽²⁾ , ε ₁ ⁽²⁾ . ε ₂ ⁽²⁾ монотонно стремятся к нулю, \(\mathop {\lim \sup }\limits_{n \to \infty } \varepsilon ^{(i)} /\varepsilon _{2n}^{(i)}< 2,i = 1,2\) , причем \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n^{(2)} /\varepsilon _n^{(i)} = + \infty \) . Тогда для каждого ε>O н айдется множество Е? [-π,π], μE >2π — ε, которое является U(S¹), но не U(S¹) — множеством для тригонометричес кой системы. Аналог теоремы 5 для си стемы Уолша был устан овлен в [7].

     

Approximation to the transcendental relationship of two algebraic points of the function\wp \left( z \right) with complex multiplication

For fixed ?>0, the following inequality holds: $$\left| {\frac{u}{\upsilon } - \wp } \right| > Cexp\left( { - \left( {lnH} \right)^{2 + \varepsilon } } \right)$$ for all numbers β belonging to a field K of finite degree over Q. The constant C>0 does not depend on β. H is the height of β. \(\wp \) (u) and \(\wp \) (v) are algebraic numbers, and u/v is a transcendental number. \(\wp \) (z) is the Weierstrass function with complex multiplication and algebraic invariants. The proof is ineffective.     

Weak type estimates for the maximal operators associated to plane curves

For the plane curves Γ,the maximal operator associated to it is defined byMf(x)=sup|∫f(x-Γ(t))(r~(-1)t)r~(-1)dt|where is a Schwartz function.For a certain class of curves in R~2,M is shown to boundedon (H(R~2),Weak L~1(R~2).This extends the theorem of Stein & Wainger and the theo-rem of Weinberg.     

Invariant submanifolds and properCR foliations on a para-coKählerian manifold with concircular structure vector field

After recalling the basic properties of para-coKählerian manifolds \(\tilde M\) with concircular structure vector field ξ, the infinitesimal auto morphismsX of the structure 1-form \(\tilde \eta \) are considered. One of the results is that the Lie derivative of all powers of the structure 2-form \(\tilde \Omega ,\) i.e. \(\mathcal{L}x\tilde \Omega ^p ;p = 1,...,m,\) is exterior recurrent. Further two types of horizontal distributionsD _n which are normal to ξ. IfD _t (resp.D _n ) is involutive, the corresponding leafM _t (resp.M _n ) is a minimal submanifold of \(\tilde M\) . FurtherM _n is a symplectic submanifold and ξ is an umbilical normal section ofM _n . Finally proper immersion \(M \to \tilde M\) are discussed, whereM is aCR-sub-manifold whose horizontal distribution isD _t . It is shown that the vertical distribution is involutive, and the restriction of ξ toM is an symptotic direction. Some interesting special cases are treated.     

The value of two-person zero-sum repeated games with lack of information on both sides

We consider repeated two-person zero-sum games in which each player has only partial information about a chance move that takes place at the beginning of the game. Under some conditions on the information pattern it is proved that \(\mathop {\lim }\limits_{n \to \infty } v_n\) exists,v _n being the value of the game withn repetitions. Two functional equations are given for which \(\mathop {\lim }\limits_{n \to \infty } v_n\) is the only simultaneous solutions. We also find the least upper bound for the error term \(\left| {v_n - \mathop {\lim }\limits_{n \to \infty } v_n } \right|\) .     

Multivariate cardinal interpolation with radial-basis functions

For a radial-basis function?∶?→? we consider interpolation on an infinite regular lattice , tof∶? ⁿ→?, whereh is the spacing between lattice points and the cardinal function , satisfiesX(j)=δ _oj for allj∈? ⁿ. We prove existence and uniqueness of such cardinal functionsX, and we establish polynomial precision properties ofI _h for a class of radial-basis functions which includes \(\varphi (r) = r^{2q + 1} \) , \(\varphi (r) = r^{2q} \log r,\varphi (r) = \sqrt {r^2 + c^2 } \) , and \(\varphi (r) = 1/\sqrt {r^2 + c^2 } \) whereq∈? ₊. We also deduce convergence orders ofI _hf to sufficiently differentiable functionsf whenh→0.     

Die metrisierbaren linearen Teilräume des Raumes\mathfrak{O}_M von L. Schwartz

In Schwartz' terminology, a real or complex valued functionf, defined and infinitely differentiable on ? _n , belongs to \(\mathfrak{O}_M \) iff, as well as any of its derivatives, is at most of polynomial growth. The topology of \(\mathfrak{O}_M \) is defined by the seminorms sup{∣?(x)D ^p f(x)∣;x∈? ⁿ }, where ? belongs to \(\mathfrak{S}\) andD ^p is any derivative. It is well-known that \(\mathfrak{O}_M \) is non-metrisable. For any μ: ? ⁿ →?, let \(\mathfrak{B}_\mu \) be the space of all infinitely differentiable functionsf satisfying, for eachp, sup{∣(1+∣x∣²)^?μ(p) D ^p f(x)∣;x∈? ⁿ }<∞, with the obvious topology. These spaces, which are of very little use elsewhere in the theory of distributions, can be conveniently applied to characterise the metrisable linear subspaces of \(\mathfrak{O}_M \) : A linear subspace of \(\mathfrak{O}_M \) is metrisable if and only if it is, algebraically and topologically, a subspace of some \(\mathfrak{B}_\mu \) .     

Global analytic hypoellipticity of the\bar \partial-Neumann problem on circular domains

In this paper we show that if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded pseudoconvex circular domain with real analytic defining functionr(z) such that \(\sum\limits_{k = 1}^n {z_k \frac{{\partial r}}{{\partial z_k }}} \ne 0\) for allz near the boundary, then the solutionu to the \(\bar \partial\) -Neumann problem, $$square u = (\bar \partial \bar \partial * + \bar \partial *\bar \partial )u = f,$$ is real analytic up to the boundary, if the given formf is real analytic up to the boundary. In particular, if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then ? is analytic hypoelliptic.