Mathematical and numerical methods for inverse problems using fundamental solutions techniques

Fundação para a Ciência e a Tecnologia - SFRH/BD/27914/2006

The group inverse of a product

In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.

Further results on the inverse along an element in semigroups and rings

In this paper, we introduce a new notion in a semigroup $S$ as an extension of Mary's inverse. Let $a,d\in S$. An element $a$ is called left (resp. right) invertible along $d$ if there exists $b\in S$ such that $bad=d$ (resp. $dab=b$) and $b\leq_\mathcal{L}d$ (resp. $b\leq_\mathcal{R}d$). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) $\pi$-regularity and left (right) $*$-regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally we study the (left, right) inverse along a product in a ring, and, as an application, Mary's inverse along a matrix is expressed.

The inverse along a product and its applications

In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix.

Development of a subject-specific musculoskeletal shoulder model : inverse and forward dynamics applications

A prevalência de pessoas que referem dor no complexo articular do ombro, com concomitante limitação na capacidade para realizar atividades da vida diária, é elevada. Estes níveis de prevalência sobrecarregam quer os utentes, como a própria sociedade. A evidência científica atual indicia a existência de uma relação entre as alterações da articulação escápulo-torácica e as patologias associadas à articulação gleno-umeral. A capacidade de quantificar, cinemática e cineticamente, as disfunções ao nível das articulações escápulo-torácica e gleno-umeral, é algo de enorme importância, quer para a comunidade biomecânica, como para a clínica. No decorrer dos trabalhos desta tese foi desenvolvido, através do software OpenSim, um modelo tridimensional músculo-esquelético do complexo articular do ombro que inclui a representação do tórax/coluna, clavícula, omoplata, úmero, rádio, cúbito e articulações que permitem os movimentos relativos desses segmentos, assim como, 16 músculos e 4 ligamentos. Com um total de 11 graus de liberdade, incluindo um novo modelo articular escápulo-torácico, os resultados demonstram que este é capaz de reconstruir de forma precisa e rápida os movimentos escápulo-torácicos e glenoumerais, recorrendo para tal, à cinemática inversa, e à dinâmica inversa e direta. Conta ainda com um método de transformação inovador para determinar, com base nas especificidades dos sujeitos, os locais de inserção muscular. As principais motivações subjacentes ao desenvolvimento desta tese foram contribuir para o aprofundar do atual conhecimento sobre as disfunções do complexo articular do ombro e, simultaneamente, proporcionar à comunidade clínica uma ferramenta biomecânica de livre acesso com o intuito de melhor suportar as decisões clínicas e dessa forma concorrer para uma prática mais efetiva.

Inverse Problems for multiple invariant curves

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. In the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these.

Inverse problems for invariant algebraic curves: explicit computations

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are nondegenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

A musculoskeletal shoulder model based on pseudo-inverse and null-space optimization.

The goal of the present work was assess the feasibility of using a pseudo-inverse and null-space optimization approach in the modeling of the shoulder biomechanics. The method was applied to a simplified musculoskeletal shoulder model. The mechanical system consisted in the arm, and the external forces were the arm weight, 6 scapulo-humeral muscles and the reaction at the glenohumeral joint, which was considered as a spherical joint. The muscle wrapping was considered around the humeral head assumed spherical. The dynamical equations were solved in a Lagrangian approach. The mathematical redundancy of the mechanical system was solved in two steps: a pseudo-inverse optimization to minimize the square of the muscle stress and a null-space optimization to restrict the muscle force to physiological limits. Several movements were simulated. The mathematical and numerical aspects of the constrained redundancy problem were efficiently solved by the proposed method. The prediction of muscle moment arms was consistent with cadaveric measurements and the joint reaction force was consistent with in vivo measurements. This preliminary work demonstrated that the developed algorithm has a great potential for more complex musculoskeletal modeling of the shoulder joint. In particular it could be further applied to a non-spherical joint model, allowing for the natural translation of the humeral head in the glenoid fossa.

What is the Jacobian of a Riemann surface with boundary?

We define the Jacobian of a Riemann surface with analytically parametrized boundary components. These Jacobians belong to a moduli space of "open abelian varieties" which satisfies gluing axioms similar to those of Riemann surfaces, and therefore allows a notion of "conformal field theory" to be defined on this space. We further prove that chiral conformal field theories corresponding to even lattices factor through this moduli space of open abelian varieties.

Constitutive activity and inverse agonism at the α(₁a) and α(₁b) adrenergic receptor subtypes.

The α(1b)-adrenergic receptor (AR) was, after rhodopsin, the first G protein-coupled receptor (GPCR) in which point mutations were shown to trigger constitutive (agonist-independent) activity. Constitutively activating mutations have been found in other AR subtypes as well as in several GPCRs. This chapter briefly summarizes the main findings on constitutively active mutants of the α(1a)- and α(1b)-AR subtypes and the methods used to predict activating mutations, to measure constitutive activity of Gq-coupled receptors and to investigate inverse agonism. In addition, it highlights the implications of studies on constitutively active AR mutants on elucidating the molecular mechanisms of receptor activation and drug action.

Une posture inversée: le yoga global

L'article expose un comparatisme reposant sur la pratique des regards croisés dont nous postulons la pertinence pour comprendre l'échange d'idées et de pratiques entre l'Inde et l'«Occident». Durant les XIXe et XXe siècles naît l'un des phénomènes de mondialisation les plus intéressants, à savoir l'exportation et la globalisation du yoga indien. Pendant que l'«Occident» s'éveille au yoga spirituel inauguré par Vivekananda (dès 1893), des yogis indiens soumettent leur tradition à l'expérimentation scientifique moderne (années 20). Dans leur ouvrage commun Sport et Yoga (1941/48), Selvarajan Yesudian (le yogi chrétien indien) et Elisabeth Haich (l'ésotériste hongroise) illustrent d'une manière paradigmatique les synthèses créatives qui peuvent s'opérer dans les processus d'échanges pluri-dimensionels et pluri-directionnels entre les traditions indiennes et européennes que seule une posture comparative, capable de faire le va-et-vient entre les deux traditions, est à même de saisir.

Inverse relation between levels of anti-oxidized-LDL antibodies and eicosapentanoic acid (EPA).

Oxidative modification of LDL is thought to play an important role in the development of atherosclerosis. Susceptibility of LDL to peroxidation may partly depend on the compositional characteristics of the antioxidant and fatty acid content. The aim of this study was to examine the association between levels of antibodies to oxidized LDL and the various serum fatty acids in women. A total of 465 women aged 18-65 years were selected randomly from the adult population census of Pizarra, a town in southern Spain. Measurement of anti-oxidized-LDL was done by ELISA and the fatty acid composition of serum phospholipids was determined by GC. The levels of anti-oxidized-LDL antibodies were significantly related with age (r - 0.341, P < 0.001), BMI (r - 0.239, P < 0.001), waist:hip ratio (r - 0.285, P < 0.001), glucose (r - 0.208, P < 0.001), cholesterol (r - 0.243, P < 0.001), LDL-cholesterol (r - 0.185, P = 0.002), EPA (r - 0.159, P = 0.003), DHA (r - 0.121, P = 0.026), and the sum of the serum phospholipid n-3 PUFA (r - 0.141, P = 0.009). Multiple regression analysis showed that the variables that explained the behaviour of the levels of anti-oxidized-LDL antibodies were age (P < 0.001) and the serum phospholipid EPA (P < 0.001). This study showed that the fatty acid composition of serum phospholipids, and especially the percentage of EPA, was inversely related with the levels of anti-oxidized-LDL antibodies.

Inverse relation between FASN expression in human adipose tissue and the insulin resistance level.

BACKGROUND Adipose tissue is a key regulator of energy balance playing an active role in lipid storage and may be a dynamic buffer to control fatty acid flux. Just like PPARgamma, fatty acid synthesis enzymes such as FASN have been implicated in almost all aspects of human metabolic alterations such as obesity, insulin resistance or dyslipemia. The aim of this work is to investigate how FASN and PPARgamma expression in human adipose tissue is related to carbohydrate metabolism dysfunction and obesity. METHODS The study included eighty-seven patients which were classified according to their BMI and to their glycaemia levels in order to study FASN and PPARgamma gene expression levels, anthropometric and biochemical variables. RESULTS The main result of this work is the close relation between FASN expression level and the factors that lead to hyperglycemic state (increased values of glucose levels, HOMA-IR, HbA1c, BMI and triglycerides). The correlation of the enzyme with these parameters is inversely proportional. On the other hand, PPARgamma is not related to carbohydrate metabolism. CONCLUSIONS We can demonstrate that FASN expression is a good candidate to study the pathophysiology of type II diabetes and obesity in humans.