Randomly weighted sums of dependent subexponential random variables*

We consider the randomly weighted sums $ \sum\nolimits_{k = 1}^n {{\theta_k}{X_k},n \geqslant 1} $ , where $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ are n real-valued random variables with subexponential distributions, and $ \left\{ {{\theta_k},1 \leqslant k \leqslant n} \right\} $ are other n random variables independent of $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ and satisfying $ a \leqslant \theta \leqslant b $ for some $ 0 < a \leqslant b < \infty $ and all $ 1 \leqslant k \leqslant n $ . For $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ satisfying some dependent structures, we prove that $$ {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant m \leqslant n} \sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant k \leqslant n} {\theta_k}{X_k} > x} \right)\sim \sum\limits_{k = 1}^m {{\text{P}}\left( {{\theta_k}{X_k} > x} \right)} $$ as x??????.     

On the regular summability of multiple function series and Lebesgue functions

Основной целью работ ы является обобщение одного результата Кратца и Т раутнера [4], известного для одном ерных функциональны х рядов, на кратные ряды. Этот рез ультат касается суммируемо сти функционального ряда почти всюду при слабых пред положениях. В частности, он примен им к суммируемости по Чезаро и по Риссу. Мы рассматриваемd-кр атный ряд $$\mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x), \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d }^2< \infty $$ и предполагается, что функции \(f_{k_1 ,...,k_d } (x)\) интегрируе мы по пространству с полож ительной мерой и имеют почти вс юду ограниченные фун кции Лебега для метода суммирова ния Т. Метод Т определяетсяd-мерной матрицей \(T = \{ a_{m_1 ,...,m_d ;k_1 ,...,k_d } \} \) сл едующим образом: $$t_{m_1 ,...,m_d } (x) = \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty a_{m_1 ,...,m_d ;k_1 ,...,k_d } c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x).$$ Эти средние существу ют, поскольку мы предп олагаем, что \(a_{m_1 ,...,m_d ;k_1 ,...,k_d } = 0\) ,если max(k ₁,...,k _d) достаточно вели к (в зависимости, конеч но, отm ₁,...,m _d). При некоторых дополнительных усло виях на матрицуТ (см. (7)– (9) в разделе 3) устанавлива ется почти всюду регулярная схо димость средних \(t_{m_1 ,...,m_d } (x) \user2{} \user2{(}m_1 \user2{,}...\user2{,}m_d \user2{)} \to \infty \) . Как вспомогательный результат, в работе об общается теорема Алексича [1] о сх одимости почти всюду некоторы х подпоследовательн остей частных сумм функцио нального ряда.     

Subspace Arrangements of Type B n and D n

Let ${\mathcal{D}}_{n,k} $ be the family of linear subspaces of ?ⁿ given by all equations of the form $\varepsilon _1 x_{i_1 } = \varepsilon _2 x_{i_2 } = \cdot \cdot \cdot \varepsilon _k x_{i_k } ,$ for 1 ≤ < ? ? ? < i _k ≤ i and $\left( {\varepsilon _1 ,...,\varepsilon _k } \right)\varepsilon \left\{ { + 1, - 1} \right\}^k $ Also let ${\mathcal{B}}_{n,k,h} $ be ${\mathcal{D}}_{n,k} $ enlarged by the subspaces $x_{j_1 } = x_{j_2 } = \cdot \cdot \cdot x_{j_h } = 0,$ for 1 ≤. The special cases ${\mathcal{B}}_{n,2,1} $ and ${\mathcal{D}}_{n,2} $ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B _nand D _n respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of ${\mathcal{B}}_{n,k,h,} 1 \leqslant h < k$ , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold $\begin{gathered} {\mathcal{D}}_{n,2} \\ M_{n,k,h,} = {\mathbb{R}}^n \backslash \bigcup {{\mathcal{B}}_{n,k,h,} } \\ \end{gathered} $ . For instance, it is shown that $H^d \left( {M_{n,k,k - 1} } \right)$ is torsion-free and is nonzero if and only if d = t(k ? 2) for some $t,0 \leqslant t \leqslant \left[ {{n \mathord{\left/ {\vphantom {n k}} \right. \kern-0em} k}} \right]$ . Torsion-free cohomology follows also for the complement in ?ⁿof the complexification ${\mathcal{B}}_{n,k,h}^C ,1 \leqslant h < k$ .     

Irrational Factor of Order k and ITS Connections With k-Free Integers

We introduce an irrational factor of order k defined by \({I_{k}(n) ={\prod_{i=1}^{l}} p_{i}^{\beta_{i}}}\) , where \({n = \prod_{i=1}^{l} p_{i}^{\alpha_{i}}}\) is the factorization of n and \({\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i < k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}\) . It turns out that the function \({\frac{I_{k} (n)}{n}}\) well approximates the characteristic function of k-free integers. We also derive asymptotic formulas for \({\prod_{v=1}^{n} I_{k}(v)^{\frac{1}{n}}, \sum_{n \leqq x} I_{k}(n)}\) and \({\sum_{n \leqq x} (1 - \frac{n}{x}) I_{k}(n)}\) .     

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.     

Estimates for the coefficients of univalent functions in terms of the second coefficient

For the coefficients b_n of an odd function \(f(z) = z + \sum\nolimits_{k = 1}^\infty {{}^bk^{z^{2k + 1} } } \) , regular in the unit disk, we obtain the estimate $$|b_n | \leqslant \frac{1}{{\sqrt 2 }}\sqrt {1 + |b_1 |^2 } \exp \frac{1}{2}\left( {\delta + \frac{1}{2}|b_1 |^2 } \right),where \delta = 0.312,$$ (1) from which it follows that ¦b_n¦≤1, if ¦b₁¦≤0.524. It follows from (1) that the coefficients c_n, n = 3, 4,..., of a regular function \(f(2) = z + \sum\nolimits_{k = 2}^\infty {{}^ck^{z^k } } \) , univalent in the unit desk, satisfy $$|c_n | \leqslant \frac{1}{2}\left( {1 + \frac{{|c_2 |^2 }}{4}} \right)n\exp \left( {\delta + \frac{{|c_2 |^2 }}{8}} \right),where \delta = 0.312,$$ in particular, ¦c_n¦≤n, if ¦c₂¦≤1.046.     

Orthogonality properties of linear combinations of orthogonal polynomials II

Let and be polynomials orthogonal on the unit circle with respect to the measures dσ and dμ, respectively. In this paper we consider the question how the orthogonality measures dσ and dμ are related to each other if the orthogonal polynomials are connected by a relation of the form , for , where . It turns out that the two measures are related by if , where and are known trigonometric polynomials of fixed degree and where the 's are the zeros of on . If the 's and 's are uniformly bounded then (under some additional conditions) much more can be said. Indeed, in this case the measures dσ and dμ have to be of the form and , respectively, where are nonnegative trigonometric polynomials. Finally, the question is considered to which weight functions polynomials of the form where denotes the reciprocal polynomial of , can be orthogonal. This revised version was published online in June 2006 with corrections to the Cover Date.     

The ergodicity of service systems with an infinite number of servomechanisms

Existence, uniqueness, and ergodicity are proved for a stationary distribution for a service system having countably many servomechanisms with input flow rate μ_k depending on the number k of servomechanisms occupied, and with arbitrary (identical) distribution of the service time with finite mean μ, under the condition \(\mu \mathop {\overline {\lim } }\limits_{k \to \infty } \frac{{\lambda _k }}{{k + 1}}< 1\) . For this system we have, in particular, Erlang's formula $$p_k (t)\mathop \to \limits_{k + \infty } p_k = \frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}p_0 ,k = 0,1,...,p_0^{ - 1} = \sum\nolimits_{k = 0}^\infty {\frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}} ,\lambda _{ - 1} = 1.$$      

Two inequalities for pseudo-differential operators

Изучается ограничен ность псевдодиффере нциальных операторов на \(L^2 (R^n )\) и на пр остранствах Харди в \(R^n \) . Пусть \(D_k = \{ \xi \in R^n :2^{k - 1} \leqq \left| \xi \right|< 2^k \} , k = 1,2,3, \ldots ,\) и \(D_0 = \{ \xi \in R^n :\left| \xi \right|< 1\} \) . Псевдодиффер енциальный операторP с символом p определяется соотно шением $$Pf(x) = \int\limits_{R^n } {e^{ix \cdot \xi } p(x,\xi )\hat f(\xi )d\xi ,x \in R^n .} $$ Будем говорить, что p пр инадлежит классу \(\bar S_{\varrho ,} {}_\delta (M,N), 0 \leqq \delta ,\varrho \leqq 1\) , ес ли $$\left| {D_x^a p(x,\xi )} \right| \leqq C_a (1 + \left| \xi \right|)^{\delta \left| a \right|} , x,\xi \in R^n ,\left| a \right| \leqq M,$$ и $$\int\limits_{D_k } {\left| {D_x^a D_\xi ^\beta p(x,\xi )} \right|d\xi \leqq C_{a\beta } 2^{kn} 2^{k(\delta |a| - \varrho |\beta |)} , x} \in R^n , k = 0,1,2, \ldots ;|a| \leqq M, |\beta | \leqq N.$$ Изучаются условия, ко торым должны удовлет ворять ?. δ,M иN, чтобы для каждого символа \(p \in \bar S_\varrho , {}_\delta (M,N)\) соответствующий оп ераторP был ограниче н на \(L^2 (R^n )\) . Далее, пусть \(p \in S_\varrho , {}_\delta \) , если дл я всех мультииндексо в а и β выполнено условие $$|D_x^a D_\xi ^\beta p(x,\xi )| \leqq C_{a\beta } (1 + |\xi |)^{\delta |\alpha | - \varrho |\beta |} , x,\xi \in R^n .$$ Доказывается, что при 0≦δ<1 операторP отображ ает пространство Харди \(H^p (R^n )\) в локальное пространство Харди ? ^p , если символp принадл ежит классуS ₁, δ.     

Criterion of polynomial increase of a function and its derivatives

ДОкАжАНО, ЧтО Дль тОгО, ЧтОБы Дльr РАж ДИФФЕРЕНцИРУЕМОИ НА пРОМЕжУткЕ [А, + ∞) ФУНкцИИf сУЩЕстВОВА л тАкОИ МНОгОЧлЕН (1) $$P(x) = \mathop \Sigma \limits_{\kappa = 0}^{r - 1} a_k x^k ,$$ , ЧтО (2) $$\mathop {\lim }\limits_{x \to + \infty } (f(x) - P(x))^{(k)} = 0,k = 0,1,...,r - 1,$$ , НЕОБхОДИМО И ДОстАтО ЧНО, ЧтОБы схОДИлсь ИН тЕгРАл (3) $$\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{r - 1} }^{ + \infty } {f^{(r)} (t)dt.}$$ ЕслИ ЁтОт ИНтЕгРАл сх ОДИтсь, тО Дль кОЁФФИц ИЕНтОВ МНОгОЧлЕНА (1) ИМЕУт МЕс тО ФОРМУлы $$\begin{gathered} a_{r - m} = \frac{1}{{(r - m)!}}\left( {\mathop \Sigma \limits_{j = 1}^m \frac{{( - 1)^{m - j} f^{(r - j)} (x_0 )}}{{(m - j)!}}} \right.x_0^{m - j} + \hfill \\ + ( - 1)^{m - 1} \left. {\mathop \Sigma \limits_{l = 0}^{m - 1} \frac{{x_0^l }}{{l!}}\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{m - l - 1} }^{ + \infty } {f^{(r)} (t_{m - 1} )dt_{m - 1} } } \right),m = 1,2,...,r. \hfill \\ \end{gathered}$$ ДОстАтОЧНыМ, НО НЕ НЕОБхОДИМыМ Усл ОВИЕМ схОДИМОстИ кРА тНОгО ИНтЕгРАлА (3) ьВльЕтсь схОДИМОсть ИНтЕгРАл А \(\int\limits_a^{ + \infty } {x^{r - 1} f^{(r)} (x)dx}\)      

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.     

Some extremal properties of positive trigonometric polynomials

A class P_n of even positive trigonometric polynomials tn(?)=a₀ + a₁ cos ?+ ... + a_n cos · n?, satisfying the conditions: a_k ≥0 (k = 0,1, ..., n), a₀ < a₁ is considered. The behavior of the sequence of functionals $$v_n = _{t_n \mathop { \in P_n }\limits^{\inf } } \frac{{t_n \left( 0 \right) - a_o }}{{\left( {\sqrt {a_1 } - \sqrt {a_o } } \right)}}$$ , is studied; two-sided estimations are given for V_n and \(V_\infty = \mathop {\lim }\limits_{n \to \infty } V_n \) .     

A conjecture of Erdős and Reddy and the rational approximation of Mittag-Leffler type functions

Пусть \(f(z) = \mathop \sum \limits_{k = 0}^\infty a_k z^k ,a_0 \ne 0, a_k \geqq 0 (k \geqq 0)\) — целая функци я,π _n — класс обыкновен ных алгебраических мног очленов степени не вы ше \(n,a \lambda _n (f) = \mathop {\inf }\limits_{p \in \pi _n } \mathop {\sup }\limits_{x \geqq 0} |1/f(x) - 1/p(x)|\) . П. Эрдеш и А. Редди высказали пр едположение, что еслиf(z) имеет порядок ?ε(0, ∞) и $$\mathop {\lim sup}\limits_{n \to \infty } \lambda _n^{1/n} (f)< 1, TO \mathop {\lim inf}\limits_{n \to \infty } \lambda _n^{1/n} (f) > 0$$ В данной статье показ ано, что для целой функ ции $$E_\omega (z) = \mathop \sum \limits_{n = 0}^\infty \frac{{z^n }}{{\Gamma (1 + n\omega (n))}}$$ , где выполняется $$\lambda _n^{1/n} (E_\omega ) \leqq \exp \left\{ { - \frac{{\omega (n)}}{{e + 1}}} \right\}$$ , т.е. $$\mathop {\lim sup}\limits_{n \to \infty } \lambda _n^{1/n} (E_\omega ) \leqq \exp \left\{ { - \frac{1}{{\rho (e + 1)}}} \right\}< 1, a \mathop {\lim inf}\limits_{n \to \infty } \lambda _n^{1/n} (E_\omega ) = 0$$ . ФункцияE _ω (z) имеет порядок ?.     

Bases formed of successive primitives

Necessary and sufficient conditions are found in order for the system of successive primitives $$\left\{ {F_n (z) = \sum\nolimits_{k = 0}^\infty {\frac{{a_{k - n} }}{{k!}}z^k } } \right\}, n = 0,1,2, ...,$$ generated by the integer-valued function \(F_n (z) = \sum\nolimits_{k = 0}^\infty {\frac{{a_k }}{{k!}}zk} \) of growth no higher than first order of the normal typeσ(F₀(z) ε [1;σ] to form a quasi-power basis in the class [1; σ].     

Inequalities for zeros of Jacobi polynomials via Sturm’s theorem: Gautschi’s conjectures

Let \(x_{n,k}^{(\alpha ,\beta )}\) , \(k=1,\ldots ,n\) , be the zeros of Jacobi polynomials \(P_{n}^{(\alpha ,\beta )}(x)\) arranged in decreasing order on \((-1,1)\) , where \(\alpha ,\beta >-1\) , and \(\theta _{n,k}^{(\alpha ,\beta )}=\arccos x_{n,k}^{(\alpha ,\beta )}\) . Gautschi, in a series of recent papers, conjectured that the inequalities $$n\theta_{n,k}^{(\alpha,\beta)}<(n+1)\theta_{n+1,k}^{(\alpha,\beta)} $$ and $$(n+(\alpha+\beta+3)/2)\theta_{n+1,k}^{(\alpha,\beta)}<(n+(\alpha+\beta+1)/2)\theta_{n,k}^{(\alpha,\beta)}, $$ hold for all \(n\geq 1\) , \(k=1,\ldots ,n\) , and certain values of the parameters \(\alpha \) and \(\beta \) . We establish these conjectures for large domains of the \((\alpha ,\beta )\) -plane by using a Sturmian approach.     

On the Fourier--Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation

In this paper, we study the behavior of the Fourier--Haar coefficients $a_{m_1 , \ldots ,m_n } \left( f \right)$ of functions $f$ Lebesgue integrable on the $n$ -dimensional cube $D_n = \left[ {0,1} \right]^n $ and having a bounded Vitali variation $V_{D_n } f$ on it. It is proved that $$\sum\limits_{m_1 = 2}^\infty \cdots \sum\limits_{m_n = 2}^\infty {\left| {a_{m_1 , \ldots ,m_n } \left( f \right)} \right|} \leqslant \left( {\frac{{2 + \sqrt 2 }}{3}} \right)^n {\text{ }}.{\text{ }}V_{D_n } f$$ and shown that this estimate holds for some function of bounded finite nonzero Vitali variation.     

The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

A general LIL for trimmed sums of random fields in banach spaces

Given a field of independent identically distributed (i.i.d.) random variables $ \left\{ {X_{\bar n} ;\bar n \in \aleph ^d } \right\} $ indexed by d-tuples of positive integers and taking values in a separable Banach space B, let $ X_{\bar n}^{(r)} = X_{\bar m} $ is the r-th maximum of $ \left\{ {\left\| {X_{\bar k} } \right\|;\bar k \leqq \bar n} \right\} $ and let $ ^{(r)} S_{\bar n} = S_{\bar n} - \left( {X_{\bar n}^{(1)} + \cdots + X_{\bar n}^{(r)} } \right) $ be the trimmed sums, where $ S_{\bar n} = \sum\nolimits_{\bar k \leqq \bar n} {X_{\bar k} } $ . This paper aims to obtain a general law of the iterated logarithm (LIL) for the trimmed sums which improves previous works.     

Height of a Superposition

We observe that, given a poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ and a finite covering ${\user1{\mathcal{R}}} = {\user1{\mathcal{R}}}_{1} \cup \cdots \cup {\user1{\mathcal{R}}}_{n} $ of its ordering, the height of the poset does not exceed the natural product of the heights of the corresponding sub-relations: $$\mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}} \right)} \leqslant \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{1} } \right)} \otimes \cdots \otimes \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{n} } \right)}.$$ Conversely for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, every poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ of height at most $\xi_1\otimes\cdots\otimes\xi_n$ admits a partition ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ such that each ${\left( {E,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at most $\xi_k$ . In particular for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, the ordinal $$\xi _{1} \underline{ \otimes } \cdots \underline{ \otimes } \xi _{n} : = \sup {\left\{ {{\left( {\xi ^{\prime }_{1} \otimes \cdots \otimes \xi ^{\prime }_{n} } \right)} + 1:\xi ^{\prime }_{1} < \xi _{1} , \cdots ,\xi ^{\prime }_{n} < \xi _{n} } \right\}}$$ is the least $\xi$ for which the following partition relation holds $$\mathfrak{H}_{\xi } \to {\left( {\mathfrak{H}_{{\xi _{1} }} , \cdots ,\mathfrak{H}_{{\xi _{n} }} } \right)}^{2} $$ meaning: for every poset ${\left( {A,{\user1{\mathcal{R}}}} \right)}$ of height at least $\xi$ and every finite covering ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ , there is a $k$ for which the relation ${\left( {A,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at least $\xi_k$ . The proof will rely on analogue properties of vertex coverings w.r.t. the natural sum.     

Randomized large distortion dimension reduction

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\) . Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\) -dimensional affine subspaces of \({\mathbb {R}}^{n}\) . We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\) , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\) , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\) .