A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.

Maximal Contractive Tuples

Maximality of a contractive tuple of operators is considered. A characterization for a contractive tuple to be maximal is obtained. The notion of maximality for a submodule of the Drury-Arveson module on the -dimensional unit ball is defined. For , it is shown that every submodule of the Hardy module over the unit disc is maximal. But for we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of maximal submodules is obtained.

On the Bounds of Certain Maximal Linear Codes in a Projective Space

The set of all subspaces of F-q(n) is denoted by P-q(n). The subspace distance d(S)(X, Y) = dim(X) + dim(Y)-2dim(X boolean AND Y) defined on P-q(n) turns it into a natural coding space for error correction in random network coding. A subset of P-q(n) is called a code and the subspaces that belong to the code are called codewords. Motivated by classical coding theory, a linear coding structure can be imposed on a subset of P-q(n). Braun et al. conjectured that the largest cardinality of a linear code, that contains F-q(n), is 2(n). In this paper, we prove this conjecture and characterize the maximal linear codes that contain F-q(n).

On an embedding property of generalized Carter subgroups

If E and F are saturated formations, we say that E is strongly contained in F if for any solvable group G with E-subgroup, E, and F-subgroup, F, some conjugate of E is contained in F. In this paper, we investigate the problem of finding the formations which strongly contain a fixed saturated formation E.

Our main results are restricted to formations, E, such that E = {G|G/F(G) ϵT}, where T is a non-empty formation of solvable groups, and F(G) is the Fitting subgroup of G. If T consists only of the identity, then E=N, the class of nilpotent groups, and for any solvable group, G, the N-subgroups of G are the Carter subgroups of G.

We give a characterization of strong containment which depends only on the formations E, and F. From this characterization, we prove:

If T is a non-empty formation of solvable groups, E = {G|G/F(G) ϵT}, and E is strongly contained in F, then

(1) there is a formation V such that F = {G|G/F(G) ϵV}.

(2) If for each prime p, we assume that T does not contain the class, S_p’, of all solvable p’-groups, then either E = F, or F contains all solvable groups.

This solves the problem for the Carter subgroups.

We prove the following result to show that the hypothesis of (2) is not redundant:

If R = {G|G/F(G) ϵS_r’}, then there are infinitely many formations which strongly contain R.

On the distribution of the signs of the conjugates of the cyclotomic units in the maximal real subfield of the qth cyclotomic field, q A prime

Let F = Ǫ(ζ + ζ ^–1) be the maximal real subfield of the cyclotomic field Ǫ(ζ) where ζ is a primitive qth root of unity and q is an odd rational prime. The numbers u₁=-1, u_k=(ζ^k-ζ^-k)/(ζ-ζ^-1), k=2,…,p, p=(q-1)/2, are units in F and are called the cyclotomic units. In this thesis the sign distribution of the conjugates in F of the cyclotomic units is studied.

Let G(F/Ǫ) denote the Galoi's group of F over Ǫ, and let V denote the units in F. For each σϵ G(F/Ǫ) and μϵV define a mapping sgn_σ: V→GF(2) by sgn_σ(μ) = 1 iff σ(μ) ˂ 0 and sgn_σ(μ) = 0 iff σ(μ) ˃ 0. Let {σ₁, ... , σ_p} be a fixed ordering of G(F/Ǫ). The matrix M_q=(sgn_σj(v_i) ) , i, j = 1, ... , p is called the matrix of cyclotomic signatures. The rank of this matrix determines the sign distribution of the conjugates of the cyclotomic units. The matrix of cyclotomic signatures is associated with an ideal in the ring GF(2) [x] / (x^p+ 1) in such a way that the rank of the matrix equals the GF(2)-dimension of the ideal. It is shown that if p = (q-1)/ 2 is a prime and if 2 is a primitive root mod p, then M_q is non-singular. Also let p be arbitrary, let ℓ be a primitive root mod q and let L = {i | 0 ≤ i ≤ p-1, the least positive residue of defined by ℓⁱ mod q is greater than p}. Let H_q(x) ϵ GF(2)[x] be defined by H_q(x) = g. c. d. ((Σ xⁱ/I ϵ L) (x+1) + 1, x^p + 1). It is shown that the rank of M_q equals the difference p - degree H_q(x).

Further results are obtained by using the reciprocity theorem of class field theory. The reciprocity maps for a certain abelian extension of F and for the infinite primes in F are associated with the signs of conjugates. The product formula for the reciprocity maps is used to associate the signs of conjugates with the reciprocity maps at the primes which lie above (2). The case when (2) is a prime in F is studied in detail. Let T denote the group of totally positive units in F. Let U be the group generated by the cyclotomic units. Assume that (2) is a prime in F and that p is odd. Let F₍₂₎ denote the completion of F at (2) and let V₍₂₎ denote the units in F₍₂₎. The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is non-singular. 2) U∩T = U². 3) U∩F²₍₂₎ = U². 4) V₍₂₎/ V₍₂₎² = ˂v₁ V₍₂₎²˃ ʘ…ʘ˂v_p V₍₂₎²˃ ʘ ˂3V₍₂₎²˃.

The rank of M_q was computed for 5≤q≤929 and the results appear in tables. On the basis of these results and additional calculations the following conjecture is made: If q and p = (q -1)/ 2 are both primes, then M_q is non-singular.

Efeito da terapia farmacológica anti-hipertensiva sobre densidade capilar e função endotelial microcirculatória

Hipertensos têm rarefação capilar e disfunção endotelial microcirculatória, tornando-se mais vulneráveis a lesões em órgãos-alvo. O estudo buscou avaliar o efeito de seis meses de tratamento farmacológico sobre densidade capilar e reatividade microvascular a estímulos fisiológicos e farmacológicos em hipertensos de baixo risco cardiovascular. Secundariamente testou-se a existência de diversidade nas respostas a diferentes estratégias anti-hipertensivas. Foram recrutados 44 pacientes, com 46,71,3 anos e 20 normotensos com 48,01,6 anos. Avaliaram-se dados antropométricos e laboratoriais e dosaram-se no soro o fator de crescimento vascular endotelial (VEGF), receptor Flt-1 para VEGF e óxido nítrico (NO). A contagem capilar foi por microscopia intravital, captando-se imagens da microcirculação no dorso da falange do dedo médio e contando os capilares com programa específico. Repetia-se o procedimento após hiperemia reativa pós-oclusiva (HRPO) para avaliar o recrutamento capilar. A reatividade vascular foi testada por fluxometria Laser Doppler, iontoforese de acetilcolina (Ach), HRPO e hiperemia térmica local (HTL). Os pacientes foram distribuídos aleatoriamente para dois grupos de tratamento: succinato de metoprolol titulado a 100 mg diários ou olmesartana medoxomila titulada a 40 mg diários, empregando-se, se necessário, a hidroclorotiazida. Os controles seguiram o mesmo protocolo inicial e após seis meses todos os testes foram repetidos nos hipertensos. As variáveis clínicas e laboratoriais basais eram semelhantes em comparação aos controles e entre os dois grupos de tratamento. Após seis meses, havia pequenas diferenças entre os grupos na relação cintura-quadril e HDL. A densidade capilar antes do tratamento era significativamente menor que no grupo controle (71,31,5 vs 80,61,8 cap/mm2 p<0,001 e HRPO 71,71,5 vs 79,52,6 cap/mm2 p<0,05) e, com o tratamento, aumentou para 75,41,1 cap/mm2 (p<0,01) no estado basal e para 76,81,1 cap/mm2 à HRPO (p<0,05). À reatividade vascular, a condutância vascular cutânea (CVC) em unidades de perfusão (UP)/mmHg era similar à HTL nos controles e hipertensos e aumentou com o tratamento nos dois subgrupos (metoprolol:1,730,2 a 1,900,2 p<0,001 e olmesartana:1,490,1 a 1,870,1 p<0,001). A CVC máxima à HRPO era menor nos hipertensos: 0,30(0,22-0,39) que nos controles: 0,39(0,31-0,49) com p<0,001. Após tratamento, aumentou para 0,41(0,29-0,51) com p<0,001. O aumento foi significativo apenas no grupo olmesartana (0,290,02 a 0,420,04 p<0,001). A diferença entre o tempo para atingir o fluxo máximo à HRPO aumentou no grupo metoprolol após tratamento 3,0 (-0,3 a 8,8) segundos versus olmesartana 0,4 (-2,1 a 2,4) segundos p<0,001. À iontoforese, a área sob a curva de fluxo (AUC) era similar nos grupos e aumentou com o tratamento, de 6087(3857-9137) para 7296(5577-10921) UP/s p=0,04. O VEGF e receptor não diferiam dos controles nem sofreram variações. A concentração de NO era maior nos hipertensos que nos controles: 64,9 (46,8-117,6) vs 50,7 (42-57,5) M/dl p=0,02 e não variou com tratamento. Em conclusão, hipertensos de baixo risco têm menor densidade e menor recrutamento capilar e ambos aumentam com tratamento. Apresentam também disfunção endotelial microcirculatória que melhora com a terapia.

Maximal Heat Generation in Nanoscale Systems

National Basic Research Programme of China G2009CB929300 National Natural Science Foundation of China 10404010 6052100160776061Supported by the National Basic Research Programme of China under Grant No G2009CB929300, and the National Natural Science Foundation of China under Grant Nos 10404010, 60521001 and 60776061.

An Algorithm for Group Formation and Maximal Independent Set in an Amorphous Computer

Amorphous computing is the study of programming ultra-scale computing environments of smart sensors and actuators cite{white-paper}. The individual elements are identical, asynchronous, randomly placed, embedded and communicate locally via wireless broadcast. Aggregating the processors into groups is a useful paradigm for programming an amorphous computer because groups can be used for specialization, increased robustness, and efficient resource allocation. This paper presents a new algorithm, called the clubs algorithm, for efficiently aggregating processors into groups in an amorphous computer, in time proportional to the local density of processors. The clubs algorithm is well-suited to the unique characteristics of an amorphous computer. In addition, the algorithm derives two properties from the physical embedding of the amorphous computer: an upper bound on the number of groups formed and a constant upper bound on the density of groups. The clubs algorithm can also be extended to find the maximal independent set (MIS) and $Delta + 1$ vertex coloring in an amorphous computer in $O(log N)$ rounds, where $N$ is the total number of elements and $Delta$ is the maximum degree.

Maximal L p -regularity for the Laplacian on Lipschitz domains

I.Wood: Maximal Lp-regularity for the Laplacian on Lipschitz domains, Math. Z., 255, 4 (2007), 855-875.

A 3-min all-out test to determine peak oxygen uptake and the maximal steady state

Burnley, M., Doust, J., Vanhatalo, A., A 3-min all-out test to determine peak oxygen uptake and the maximal steady state, Medicine & Science in Sports & Exercise. 38(11):1995-2003, November 2006. RAE2008