A method of multipliers for mathematical programming problems with equality and inequality constraints

This paper deals with the numerical solution of the general mathematical programming problem of minimizing a scalar functionf(x) subject to the vector constraints φ(x)=0 and ψ(x)≥0. The approach used is an extension of the Hestenes method of multipliers, which deals with the equality constraints only. The above problem is replaced by a sequence of problems of minimizing the augmented penalty function Ω(x, λ, μ,k)=f(x)+λ ^T φ(x)+kφ ^T (x)φ(x) ?μ ^T \(\tilde \psi \) (x)+k \(\tilde \psi \) ^T (x) \(\tilde \psi \) (x). The vectors λ and μ, μ ≥ 0, are respectively the Lagrange multipliers for φ(x) and \(\tilde \psi \) (x), and the elements of \(\tilde \psi \) (x) are defined by \(\tilde \psi \) _(j)(x)=min[ψ_(j)(x), (1/2k) μ_(j)]. The scalark>0 is the penalty constant, held fixed throughout the algorithm. Rules are given for updating the multipliers for each minimization cycle. Justification is given for trusting that the sequence of minimizing points will converge to the solution point of the original problem.     

Об операторе Кальдерона

The Calderon type operator $$Sx(t) = \int\limits_0^\infty {x(s)d\mathop {\min }\limits_{i = 0,1} \{ \varphi _i (s)/\Psi _\iota (t)\} } $$ is investigated from the point of view of its bounded action in symmetric spaces of measurable functions on [0, ∞), whereφ _i(t) andΨ _i(t) are concave positive functions on [0, ∞). The following assertions are proved. Theorem 1. Let 1) \(\alpha _{\varphi _1 } > \beta (E) \geqq \alpha (E) > \beta _{\varphi _0 } ,\) , 2)either \(\beta _{\psi _0 }< 1\) or \(\beta _{\psi 1}< 1\) ,where \(\alpha _{\varphi _1 } \) and \(\beta _{\psi _1 } \) denote exponents of the functions φ _i and Ψ_i, i=0, 1,and α(E),β(E) are indices of the space E. Then the Calderon operator acts boundedly from E into E _δ,where δ(t) and χ(t) stand for measurable solutions of the equations $$\psi _i (\delta (t)) = \chi (t)\varphi _i (t),i = 0,1,$$ and $$||x||_{E_{\delta ,\chi } } = ||x^{**} (\delta (t))\chi (t)||_E .$$ Theorem 2.If the ratio φ ₀/φ(t)/φ₁(t) is non-increasing, Ψ_i(t) are semimultiplicative and \(\alpha _{\psi _i } \) (i=0, 1),then a necessary condition for the Calderon operator to act boundedly from E into E _{δ, χ} $$\beta _{\varphi _1 } \geqq \beta (E) \geqq \alpha (E) \geqq \alpha _{\varphi _0 } .$$      

\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.     

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 \(\tilde M\) .     

Exactn-widths of hardy-sobolev classes

Let $\tilde h^r _{\infty ,\beta } $ and $\tilde H^r _{\infty ,\beta } $ denote those 2π-periodic, real-valued functions onR that are analytic in the strip |Imz|<β and satisfy the restrictions |Ref ^(r)(z)| ≤ 1 and |f ^(r)(z)| ≤ 1, respectively. We determine the Kolmogorov, linear, and Gel’fand widths of $\tilde h^r _{\infty ,\beta } $ inL _q[0, 2π], 1 ≤q ≤ ∞, and $\tilde H^r _{\infty ,\beta } $ inL _∞[0, 2π].     

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.     

The CF table

Letf be a continuous function on the circle ¦z¦=1. We present a theory of the (untruncated) “Carathéodory-Fejér (CF) table” of best supremumnorm approximants tof in the classes \(\tilde R_{mn} \) of functions $${{\tilde r(z) = \sum\limits_{k = - \infty }^m {a_k z^k } } \mathord{\left/ {\vphantom {{\tilde r(z) = \sum\limits_{k = - \infty }^m {a_k z^k } } {\sum\limits_{k = 0}^n {b_k } z^k ,}}} \right. \kern-\nulldelimiterspace} {\sum\limits_{k = 0}^n {b_k } z^k ,}}$$ , where the series converges in 1< ¦z¦ <∞. (The casem=n is also associated with the names Adamjan, Arov, and Krein.) Our central result is an equioscillation-type characterization: \(\tilde r \in \tilde R_{mn} \) is the unique CF approximant \(\tilde r^* \) tof if and only if \(f - \tilde r\) has constant modulus and winding numberω≥ m+ n+1?δ on ¦z¦=1, whereδ is the “defect” of \(\tilde r\) . If the Fourier series off converges absolutely, then \(\tilde r^* \) is continuous on ¦z¦=1, andω can be defined in the usual way. For general continuousf, \(\tilde r^* \) may be discontinuous, andω is defined by a radial limit. The characterization theorem implies that the CF table breaks into square blocks of repeated entries, just as in Chebyshev, Padé, and formal Chebyshev-Padé approximation. We state a generalization of these results for weighted CF approximation on a Jordan region, and also show that the CF operator \(K:f \mapsto \tilde r^* \) is continuous atf if and only if (m, n) lies in the upper-right or lower-left corner of its square block.     

Interpolation of states by vector states on certain operator algebras

LetT be an operator on an infinite dimensional Hilbert space \(\mathcal{H}\) with eigenvectorsv _i , ‖v _i ‖=1,i=1, 2, ..., andsp{v _i ?i∈n} dense in \(\mathcal{H}\) . Suppose that {v _i } is a Schauder basis for \(\mathcal{H}\) . We denote byA _T the ultraweakly closed algebra generated byT andI, the identity operator on \(\mathcal{H}\) . For any nonnegative sequence of scalars \(\left\{ {\alpha ,with = \sum\nolimits_1^\infty {\alpha _1 } = 1} \right\},\) , we associate an ultraweakly (normal) continuous linear functional \(\phi _\alpha = \sum\nolimits_1^\infty {\alpha _j } \omega _v\) where \(\phi _\alpha \left( A \right) = \lim _n \sum\nolimits_1^n {\alpha _j } \omega _v ,\) , and \(\omega _v ,\left( A \right) =< Av_1 ,v_1 >\) for allA∈A _T . We denote the set of all such linear functionals onA TbyF(T). The question that we investigate in this paper is whether each linear functional φ_α inF(T) is a vector state, i.e. does φ_α=ωx for some unit vectorx in \(\mathcal{H}\) ?     

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.     

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 positive definite measures

In this paper we study the Fourier transform of unbounded measures on a locally compact groupG. After a short introductory section containing background material, especially results established byL. Argabright andJ. Gil De Lamadrid we turn to the main subjects of the paper: first we characterize \(\Re \left( G \right), \mathfrak{J}\left( G \right)\) andB(G) cones in \(\mathfrak{W}\left( G \right)\) . After that we establish the subspace \(\mathfrak{W}_\Delta \left( G \right)\) of \(\mathfrak{W}\left( G \right)\) which contains \(\mathfrak{W}_p \left( G \right)\) , the linear span of all positive definite measures.     

Estimate of a polynomial in generalized exponential functions in the convex hull of two domains

Пусть $$f_n (z) = \exp \{ \lambda _n z\} [1 + \psi _n (z)], n \geqq 1$$ гдеψ _n (z) — регулярны в н екоторой односвязно й областиS, λ _n — нули целой функц ии экспоненциальног о ростаL(λ) с индикатрис ой ростаh(?), причем $$|L\prime (\lambda _n )| > C(\delta )\exp \{ [h(\varphi _n ) - \varepsilon ]|\lambda _n |\} \varphi _n = \arg \lambda _n , \forall \varepsilon > 0$$ . Предположим, что на лю бом компактеK?S $$|\psi _n (z)|< Aq^{|\lambda |_n } , a< q< 1, n \geqq 1$$ гдеA иq зависит только отK. Обозначим через \(\bar D\) со пряженную диаграмму функцииL(λ), через \(\bar D_\alpha \) — смещение. \(\bar D\) на векторα. Рассмотр им множестваD ₁ иD ₂ так ие, чтоD ₁ иD ₂ и их вьшуклая обо лочкаE принадлежатS. Пусть \(\bar D_{\alpha _1 } \subset D_1 , \bar D_{\alpha _2 } \subset D_2 \) Доказывается, что сущ ествует некоторая об ластьG?E такая, что \(\mathop \cup \limits_{\alpha \in [\alpha _1 ,\alpha _2 ]} \bar D_\alpha \subset G\) и дляz∈G верна оценка $$\sum\limits_{v = 1}^n {|a_v f_v (z)|} \leqq B\max (M_1 ,M_2 ), M_j = \mathop {\max }\limits_{t \in \bar D_j } |\sum\limits_{v = 1}^n {a_v f_v (t)} |$$ , где константаB не зав исит от {a _v }.     

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.     

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.     

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).     

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].     

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.     

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.     

L 2 existence theorems for the\bar \partial _b - Neumann problem on strongly pseudoconvex CR manifoldsproblem on strongly pseudoconvex CR manifolds

LetM be the boundary of a strongly pseudoconvex domain in \(\mathbb{C}^n \) ,n≥4 and ω be an open subset inM such that ?ω is the intersection ofM with a flat hypersurface. We establish theL ² existence theorems of the \(\bar \partial _b - Neumann\) problem on ω. In particular, we prove that the \(\bar \partial _b - Laplacian\) \(\square _b = \bar \partial _b \bar \partial _b^* + \bar \partial _b^* \bar \partial _b \) equipped with a pair of natural boundary conditions, the so-called \(\bar \partial _b - Neumann\) boundary conditions, has closed range when it acts on (0,q) forms, 1≤q≤n?3. Thus there exists a bounded inverse operator for \(\square _b \) , the \(\bar \partial _b - Neumann\) operatorN _b, and we have the following Hodge decomposition theorem on ω for \(\bar \partial _b \bar \partial _b^* N_b \alpha + \bar \partial _b^* \bar \partial _b N_b \alpha \) , for any (0,q) form α withL ²(ω) coefficients. The proof depends on theL ^p regularity of the tangential Cauchy-Riemann operators \(\bar \partial _b u = \alpha \) on ω?M under the compatibility condition \(\bar \partial _b \alpha = 0\) , where α is a (p, q) form on ω, where 1≤q≤n?2. The interior regularity ofN _b follows from the fact that \(\square _b \) is subelliptic in the interior of ω. The operatorN _b induces natural questions on the regularity up to the boundary ?ω. Near the characteristic point of the boundary, certain compatibility conditions will be present. In fact, one can show thatN _b is not a compact operator onL ²(ω).