Uniform regularization of the problem of calculating the values of an operator

Let X and Y be linear normed spaces, W a set in X, A an operator from W into Y, and \(\mathfrak{W}\) the set of all operators or the set ? of linear operators from X into Y. With δ>0 we put $$v\left( {\delta ,\mathfrak{M}} \right) = \mathop {\inf }\limits_{T \in \mathfrak{M}} \mathop {\sup }\limits_{x \in W} \mathop {\sup }\limits_{\left\| {\eta - x} \right\|_X \leqslant \delta } \left\| {Ax - T\eta } \right\|_Y $$ . We discuss the connection of \(v\left( {\delta , \mathfrak{M}} \right)\) with the Stechkin problem on best approximation of the operator A in W by linear bounded operators. Estimates are obtained for \(v\left( {\delta , \mathfrak{M}} \right)\) e.g., we write the inequality , where H(Y) is Jung's constant of the space Y, and Ω(t) is the modulus of continuity of A in W.     

On some filters and ideals of the Medvedev lattice

Let \(\mathfrak{M}\) be the Medvedev lattice: this paper investigates some filters and ideals (most of them already introduced by Dyment, [4]) of \(\mathfrak{M}\) . If \(\mathfrak{G}\) is any of the filters or ideals considered, the questions concerning \(\mathfrak{G}\) which we try to answer are: (1) is \(\mathfrak{G}\) prime? What is the cardinality of \({\mathfrak{M} \mathord{\left/ {\vphantom {\mathfrak{M} \mathfrak{G}}} \right. \kern-0em} \mathfrak{G}}\) ? Occasionally, we point out some general facts on theT-degrees or the partial degrees, by which these questions can be answered.     

Factorization of Lipschitz functions and absolute convergence of Vilenkin-Fourier series

LetG be a Vilenkin group (i.e., an infinite, compact, metrizable, zero-dimensional Abelian group). Our main result is a factorization theorem for functions in the Lipschitz spaces \(\mathfrak{L}\mathfrak{i}\mathfrak{p}\) (α,p; G). As colloraries of this theorem, we obtain (i) an extension of a factorization theorem ofY. Uno; (ii) a convolution formula which says that \(\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha , r; G) = \mathfrak{L}\mathfrak{i}\mathfrak{p} (\beta , l; G)*\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha - \beta , r; G)\) for 0<β<α<∞ and 1≤r≤∞; and (iii) an analogue, valid for allG, of a classical theorem ofHardy andLittlewood. We also present several results on absolute convergence of Fourier series defined onG, extending a theorem ofC. W. Onneweer and four results ofN. Ja. Vilenkin andA. I. Rubinshtein. The fourVilenkin-Rubinshtein results are analogues of classical theorems due, respectively, toO. Szász, S. B. Bernshtein, A. Zygmund, andG. G. Lorentz.     

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.     

A property of an autonomous minimax optimal control problem

A control system \(\dot x = f\left( {x,u} \right)\) ,u) with cost functional $$\mathop {ess \sup }\limits_{T0 \leqslant t \leqslant T1} G\left( {x\left( t \right),u\left( t \right)} \right)$$ is considered. For an optimal pair \(\left( {\bar x\left( \cdot \right),\bar u\left( \cdot \right)} \right)\) ,ū(·)), there is a maximum principle of the form $$\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right) = \mathop {\max }\limits_{u \in \Omega \left( t \right)} \eta \left( t \right)f\left( {\bar x\left( t \right),u} \right).$$ By means of this fact, it is shown that \(\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right)\) is equal to a constant almost everywhere.     

Rational approximations of real numbers

For anyx ∈ r put $$c(x) = \overline {\mathop {\lim }\limits_{t \to \infty } } \mathop {\min }\limits_{(p,q\mathop {) \in Z}\limits_{q \leqslant t} \times N} t\left| {qx - p} \right|.$$ . Let [x₀; x₁,..., x_n, ...] be an expansion of x into a continued fraction and let \(M = \{ x \in J,\overline {\mathop {\lim }\limits_{n \to \infty } } x_n< \infty \}\) .Forx∈M put D(x)=c(x)/(1?c(x)). The structure of the set \(\mathfrak{D} = \{ D(x),x \in M\}\) is studied. It is shown that $$\mathfrak{D} \cap (3 + \sqrt 3 ,(5 + 3\sqrt 3 )/2) = \{ D(x^{(n,3} )\} _{n = 0}^\infty \nearrow (5 + 3\sqrt 3 )/2,$$ where \(x^{(n,3)} = [\overline {3;(1,2)_n ,1} ].\) This yields for \(\mu = \inf \{ z,\mathfrak{D} \supset (z, + \infty )\}\) (“origin of the ray”) the following lower bound: μ?(5+3√3)/2=5.0n>(5 + 3/3)/2=5.098.... Suppose a∈n. Put \(M(a) = \{ x \in M,\overline {\mathop {\lim }\limits_{n \to \infty } } x_n = a\}\) , \(\mathfrak{D}(a) = \{ D(x),x \in M(a)\}\) . The smallest limit point of \(\mathfrak{D}(a)(a \geqslant 2)\) is found. The structure of (a) is studied completely up to the smallest limit point and elucidated to the right of it.     

Fibered incidence groups which are not kinematic

An incidence space \((\beta ,\mathfrak{L})\) which is obtained from an affine space \((\beta _a ,\mathfrak{L}_a )\) by omitting a hyperplane is calledstripe space. If \((\beta _a ,\mathfrak{L}_a )\) is desarguesian, then \(\beta \) can be provided with a group operation “ ○ ” such that \((\beta ,\mathfrak{L}, \circ )\) becomes a kinematic space calledstripe group. It will be shown that there are stripe groups \((\beta ,\mathfrak{L}, \circ )\) where the incidence structure \(\mathfrak{L}\) can be replaced by another incidence structure ? such that \((\beta ,\Re , \circ )\) is afibered incidence group which is not kinematic. An application on translation planes concerning the group of affinities is also given.     

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 .

     

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.     

Distribution of integral functionals of a Brownian motion process

In this paper one considers methods which enable one to determine the distribution of certain functionals of a Brownian motion process. Among such functionals we have: the positive continuous additive functional of a Brownian motion, defined by the formula $$A\left( t \right) = \int\limits_{ - \infty }^\infty {\hat t\left( {t, y} \right)dF\left( y \right),} $$ where \(\hat t\left( {t, y} \right)\) is the Brownian local time process while F(y) is a monotonically increasing right continuous function; the functional $$B\left( t \right) = \mathop {\mathop \smallint \limits_{ - \infty } }\nolimits^\infty f\left( {y,\hat t\left( {t, y} \right)} \right)dy,$$ where f(y, x) is a continuous function; and the functional $$C\left( t \right) = \mathop {\mathop \smallint \limits_0 }\nolimits^t f\left( {w\left( s \right),\hat t\left( {sr} \right)} \right)ds$$ As an application of these methods one considers some concrete functionals such that \(\hat t^{ - 1} \left( z \right) = \min \left\{ {s:\hat t\left( {s, o} \right) = z} \right\},\mathop {\mathop \smallint \limits_{ - \infty } }\nolimits^\infty \hat t^2 \left( {t, y} \right)dy,\mathop {\sup }\limits_{y \in R^1 } \hat t\left( {T, y} \right)\) , where T is an exponential random time, independent of \(\hat t\left( {t, y} \right)\) .     

\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.     

Einige aussagen über die reduktion kohärenter analytischer modulgarben

Let (X, ) be a complex space and \(\mathfrak{F}\) a coherent -module. In analogy to the reduction red one can define a reduction \(\mathfrak{F}\) _red= \(\mathfrak{F}\) / \(\mathfrak{F}\) ′, where \(\mathfrak{F}\) ′ ? \(\mathfrak{F}\) is the subsheaf of “nilvalent” elements of \(\mathfrak{F}\) . (Even if X is reduced, we may have \(\mathfrak{F}\) ′ ≠ 0.) We prove that \(\mathfrak{F}\) ′ is coherent. Therefore we can construct the sheaf \(\mathfrak{F}\) (²)=( \(\mathfrak{F}\) ′)′ of nilvalent elements with respect to \(\mathfrak{F}\) ′. Iterating this process, we get a sequence ( \(\mathfrak{F}\) ⁽ⁿ⁾)n∈N of subsheaves of \(\mathfrak{F}\) . We show that on every compact subset of X the sheaves \(\mathfrak{F}\) (ⁿ) vanish for n sufficiently large (Satz 2).     

The index of centralizers of elements in classical Lie algebras

The index of a finite-dimensional Lie algebra $\mathfrak{g}$ is the minimum of dimensions of the stabilizers $\mathfrak{g}_\alpha $ over all covectors $\alpha \in \mathfrak{g}^ * $ . Let $\mathfrak{g}$ be a reductive Lie algebra over a field $\mathbb{K}$ of characteristic ≠ = 2. Élashvili conjectured that the index of $\mathfrak{g}_\alpha $ is always equal to the index, or, which is the same, the rank of $\mathfrak{g}$ . In this article, Élashvili’s conjecture is proved for classical Lie algebras. Furthermore, it is shown that if $\mathfrak{g} = \mathfrak{g}\mathfrak{l}_n $ or $\mathfrak{g} = \mathfrak{s}\mathfrak{p}_{2n} $ and $e \in \mathfrak{g}$ is a nilpotent element, then the coadjoint action of $\mathfrak{g}_e $ has a generic stabilizer. For $\mathfrak{g}$ , we give examples of nilpotent elements $e \in \mathfrak{g}$ such that the coadjoint action of $\mathfrak{g}_e $ does not have a generic stabilizer.     

An Analog of the Cameron--Johnson Theorem for Linear ℂ-Analytic Equations in Hilbert Space

The well-known Cameron--Johnson theorem asserts that the equation $\dot x = \mathcal{A}\left( t \right)x$ with a recurrent (Bohr almost periodic) matrix $\mathcal{A}\left( t \right)$ can be reduced by a Lyapunov transformation to the equation $\dot y = \mathcal{B}\left( t \right)y$ with a skew-symmetric matrix $\mathcal{B}\left( t \right)$ , provided that all solutions of the equation $\dot x = \mathcal{A}\left( t \right)x$ and of all its limit equations are bounded on the whole line. In the note, a generalization of this result to linear $\mathbb{C}$ -analytic equations in a Hilbert space is presented.     

On a theorem of Jackson

We prove that $$\mathop {L_n \in Z_n }\limits^{\inf } \mathop \omega \limits^{sup^* } \mathop {f \in H_\omega }\limits^{\sup } \frac{{\left\| {f - L_n \left( f \right)} \right\|}}{{\omega \left( {\frac{\pi }{{n + 1}}} \right)}} = 1\left( {n = 0,1,2,...} \right)$$ (n=0,1,2,...), where \(\mathop {L_n \in Z_n }\limits^{\inf } \) is taken over all linear polynomial approximation methods of degree not higher than n and \(\mathop \omega \limits^{sup^* } \) over all convex moduli of continuity ω(δ).     

On initial segments of degrees of constructibility

Let \(\mathfrak{M}\) be a fixed countable standard transitive model of ZF+V=L. We consider the structure Mod of degrees of constructibility of real numbers x with respect to \(\mathfrak{M}\) such that \(\mathfrak{M}\) (x) is a model. An initial segment Q \( \subseteq \) Mod is called realizable if some extension of \(\mathfrak{M}\) with the same ordinals contains exclusively the degrees of constructibility of real numbers from Q (and is a model of Z FC). We prove the following: if Q is a realizable initial segment, then $$[y \in Q \to y< x]]\& \forall z\exists y[z< x \to y \in Q\& \sim [y< z]]]$$ .     

A commutant realization of Odake’s algebra

The bcβγ-system $ \mathcal{W} $ of rank 3 has an action of the affine vertex algebra $ {V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ , and the commutant vertex algebra $ \mathcal{C}=\mathrm{Com}\left( {{V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right),\mathcal{W}} \right) $ contains copies of V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and Odake’s algebra $ \mathcal{O} $ . Odake’s algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and $ \mathcal{O} $ form a Howe pair (i.e., a pair of mutual commutants) inside $ \mathcal{C} $ . More generally, any finite-dimensional representation of a Lie algebra $ \mathfrak{g} $ gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of $ \mathfrak{s}{{\mathfrak{l}}_2} $ .     

Good gradings of basic Lie superalgebras

We classify good ?-gradings of basic Lie superalgebras over an algebraically closed field $\mathbb{F}$ of characteristic zero. Good ?-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the construction of super W-algebras. We also describe the centralizer of a nilpotent even element and of an $\mathfrak{s}\mathfrak{l}_2$ -triple in $\mathfrak{g}\mathfrak{l}\left( {\left. m \right|n} \right)$ and $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {\left. m \right|2n} \right)$ .     

Infinite series representations for complex numbers

We introduce two new algorithms that lead to finite or infinite series expansions for complex number in terms of ‘integral digits’ within the complex quadratic fields ? \(\left( {\sqrt { - m} } \right)\) , form=1, 2,…, 11. In particular, we derive complex number representations as sums of reciprocal of Gaussion integers and as sums of reciprocals of algebraic integers in ? \(\left( {\sqrt { - m} } \right)\) , form=2, 3, 7 and 11. In addition to convergence of the various algorithms we investigate the representation of ‘rationals’ relative to the fields ? \(\left( {\sqrt { - m} } \right)\) , form=1, 2, 3, 7 and 11.     

Banach spaces of locally Schlicht functions with the Hornich operations

Let \(S_ \propto ( \propto \geqq 0)\) be the set of normalized (see (1.2)) functions f holomorphic in D:|z|<1 with \(f''(z)/f'(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\) , and let be the set of normalized (see (1.6)) functions f meromorphic in D with the Schwarzian derivative \(\left\{ {f,z} \right\} = 0((1 - \left| z \right|^2 )^{ - \propto } )\) . We shall show that some topological properties of \(S_ \propto\) and , and of subsets of them, follow from those of the weighted H^∞ space \(H_ \propto ^\infty\) , consisting of functions f holomorphic in D with \(f(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\) , and those of subsets of \(H_ \propto ^\infty\) . The set S₁ is denoted by X in [3] and [4].