Representations and Positive Definite Functions on Hypergroups

Some relationships between representations of a hypergroup X, its algebras, and positive definite functions on X are studied. Also, various types of convergence of positive definite functions on X are discussed.

Abstract Subdifferential Calculus and Semi-Convex Functions

∗ The work is partially supported by NSFR Grant No MM 409/94.

Some Comments on Q-Irresolute and Quasi-Irresolute Functions

The aim of this paper is to continue the study of θ-irresolute and quasi-irresolute functions as well as to give an example of a function which is θ-irresolute but neither quasi-irresolute nor an R-map and thus give an answer to a question posed by Ganster, Noiri and Reilly. We prove that RS-compactness is preserved under open, quasi-irresolute surjections.

Practical Stability and Vector-Lyapunov Functions for Impulsive Differential Equations with "Supremum"

Stability of nonlinear impulsive differential equations with "supremum" is studied. A special type of stability, combining two different measures and a dot product on a cone, is defined. Perturbing cone-valued piecewise continuous Lyapunov functions have been applied. Method of Razumikhin as well as comparison method for scalar impulsive ordinary differential equations have been employed.

Association between mean residual life (MRL) and failure rate functions for continuous and discrete lifetime distributions

The purpose of this study was to correct some mistakes in the literature and derive a necessary and sufficient condition for the MRL to follow the roller-coaster pattern of the corresponding failure rate function. It was also desired to find the conditions under which the discrete failure rate function has an upside-down bathtub shape if corresponding MRL function has a bathtub shape. The study showed that if discrete MRL has a bathtub shape, then under some conditions the corresponding failure rate function has an upside-down bathtub shape. Also the study corrected some mistakes in proofs of Tang, Lu and Chew (1999) and established a necessary and sufficient condition for the MRL to follow the roller-coaster pattern of the corresponding failure rate function. Similarly, some mistakes in Gupta and Gupta (2000) are corrected, with the ensuing results being expanded and proved thoroughly to establish the relationship between the crossing points of the failure rate and associated MRL functions. The new results derived in this study will be useful to model various lifetime data that occur in environmental studies, medical research, electronics engineering, and in many other areas of science and technology.

n-alkane carbon isotope ratios in plants and recent lake sediments from Cameroon

Trees and shrubs in tropical Africa use the C3 cycle as a carbon fixation pathway during photosynthesis, while grasses and sedges mostly use the C4 cycle. Leaf-wax lipids from sedimentary archives such as the long-chain n-alkanes (e.g., n-C27 to n-C33) inherit carbon isotope ratios that are representative of the carbon fixation pathway. Therefore, n-alkane d13C values are often used to reconstruct past C3/C4 composition of vegetation, assuming that the relative proportions of C3 and C4 leaf waxes reflect the relative proportions of C3 and C4 plants. We have compared the d13C values of n-alkanes from modern C3 and C4 plants with previously published values from recent lake sediments and provide a framework for estimating the fractional contribution (areal-based) of C3 vegetation cover (fC3) represented by these sedimentary archives. Samples were collected in Cameroon, across a latitudinal transect that accommodates a wide range of climate zones and vegetation types, as reflected in the progressive northward replacement of C3-dominated rain forest by C4-dominated savanna. The C3 plants analysed were characterised by substantially higher abundances of n-C29 alkanes and by substantially lower abundances of n-C33 alkanes than the C4 plants. Furthermore, the sedimentary d13C values of n-C29 and n-C31 alkanes from recent lake sediments in Cameroon (-37.4 per mil to -26.5 per mil) were generally within the range of d13C values for C3 plants, even when from sites where C4 plants dominated the catchment vegetation. In such cases simple linear mixing models fail to accurately reconstruct the relative proportions of C3 and C4 vegetation cover when using the d13C values of sedimentary n-alkanes, overestimating the proportion of C3 vegetation, likely as a consequence of the differences in plant wax production, preservation, transport, and/or deposition between C3 and C4 plants. We therefore tested a set of non-linear binary mixing models using d13C values from both C3 and C4 vegetation as end-members. The non-linear models included a sigmoid function (sine-squared) that describes small variations in the fC3 values as the minimum and maximum d13C values are approached, and a hyperbolic function that takes into account the differences between C3 and C4 plants discussed above. Model fitting and the estimation of uncertainties were completed using the Monte Carlo algorithm and can be improved by future data addition. Models that provided the best fit with the observed d13C values of sedimentary n-alkanes were either hyperbolic functions or a combination of hyperbolic and sine-squared functions. Such non-linear models may be used to convert d13C measurements on sedimentary n-alkanes directly into reconstructions of C3 vegetation cover.

CaV channels and cancer: canonical functions indicate benefits of repurposed drugs as cancer therapeutics

The importance of ion channels in the hallmarks of many cancers is increasingly recognised. This article reviews current knowledge of the expression of members of the voltage-gated calcium channel family (CaV) in cancer at the gene and protein level and discusses their potential functional roles. The ten members of the CaV channel family are classified according to expression of their pore-forming α-subunit; moreover, co-expression of accessory α2δ, β and γ confers a spectrum of biophysical characteristics including voltage dependence of activation and inactivation, current amplitude and activation/inactivation kinetics. CaV channels have traditionally been studied in excitable cells including neurones, smooth muscle, skeletal muscle and cardiac cells, and drugs targeting the channels are used in the treatment of hypertension and epilepsy. There is emerging evidence that several CaV channels are differentially expressed in cancer cells compared to their normal counterparts. Interestingly, a number of CaV channels also have non-canonical functions and are involved in transcriptional regulation of the expression of other proteins including potassium channels. Pharmacological studies show that CaV canonical function contributes to the fundamental biology of proliferation, cell-cycle progression and apoptosis. This raises the intriguing possibility that calcium channel blockers, approved for the treatment of other conditions, could be repurposed to treat particular cancers. Further research will reveal the full extent of both the canonical and non-canonical functions of CaV channels in cancer and whether calcium channel blockers are beneficial in cancer treatment.

Enhancing SPH using moving least-squares and radial basis functions

In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length - the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem.

On the construction of new bent functions from the max-weight and min-weight functions of old bent functions

Given a bent function f (x) of n variables, its max-weight and min-weight functions are introduced as the Boolean functions f + (x) and f − (x) whose supports are the sets {a ∈ Fn2 | w( f ⊕la) = 2n−1+2 n 2 −1} and {a ∈ Fn2 | w( f ⊕la) = 2n−1−2 n 2 −1} respectively, where w( f ⊕ la) denotes the Hamming weight of the Boolean function f (x) ⊕ la(x) and la(x) is the linear function defined by a ∈ Fn2 . f + (x) and f − (x) are proved to be bent functions. Furthermore, combining the 4 minterms of 2 variables with the max-weight or min-weight functions of a 4-tuple ( f0(x), f1(x), f2(x), f3(x)) of bent functions of n variables such that f0(x) ⊕ f1(x) ⊕ f2(x) ⊕ f3(x) = 1, a bent function of n + 2 variables is obtained. A family of 4-tuples of bent functions satisfying the above condition is introduced, and finally, the number of bent functions we can construct using the method introduced in this paper are obtained. Also, our construction is compared with other constructions of bent functions.

Harnessing the power of Drosophila model to obtain new insights on MYC and ABCC1 cellular functions: from neuroblastoma to autism spectrum disorder

MYCN amplification is a genetic hallmark of the childhood tumour neuroblastoma. MYCN-MAX dimers activate the expression of genes promoting cell proliferation. Moreover, MYCN seems to transcriptionally repress cell differentiation even in absence of MAX. We adopted the Drosophila eye as model to investigate the effect of high MYC to MAX expression ratio on cells. We found that dMyc overexpression in eye cell precursors inhibits cell differentiation and induces the ectopic expression of Antennapedia (the wing Hox gene). The further increase of MYC/MAX ratio results in an eye-to-wing homeotic transformation. Notably, dMyc overexpression phenotype is suppressed by low levels of transcriptional co-repressors and MYCN associates to the promoter of Deformed (the eye Hox gene) in proximity to repressive sites. Hence, we envisage that, in presence of high MYC/MAX ratio, the “free MYC” might inhibit Deformed expression, leading in turn to the ectopic expression of Antennapedia. This suggests that MYCN might reinforce its oncogenic role by affecting the physiological homeotic program. Furthermore, poor neuroblastoma outcome associates with a high level of the MRP1 protein, encoded by the ABCC1 gene and known to promote drug efflux in cancer cells. Intriguingly, this correlation persists regardless of chemotherapy and ABCC1 overexpression enhances neuroblastoma cell motility. We found that Drosophila dMRP contributes to the adhesion between the dorsal and ventral epithelia of the wing by inhibiting the function of integrin receptors, well known regulators of cell adhesion and migration. Besides, integrins play a crucial role during synaptogenesis and ABCC1 locus is included in a copy number variable region of the human genome (16p13.11) involved in neuropsychiatric diseases. Interestingly, we found that the altered dMRP/MRP1 level affects nervous system development in Drosophila embryos. These preliminary findings point out novel ABCC1 functions possibly defining ABCC1 contribution to neuroblastoma and to the pathogenicity of 16p13.11 deletion/duplication

Calculus, with analytic geometry.

Includes index.

Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Special functions of Weyl groups and their continuous and discrete orthogonality

Cette thèse s'intéresse à l'étude des propriétés et applications de quatre familles des fonctions spéciales associées aux groupes de Weyl et dénotées $C$, $S$, $S^s$ et $S^l$. Ces fonctions peuvent être vues comme des généralisations des polynômes de Tchebyshev. Elles sont en lien avec des polynômes orthogonaux à plusieurs variables associés aux algèbres de Lie simples, par exemple les polynômes de Jacobi et de Macdonald. Elles ont plusieurs propriétés remarquables, dont l'orthogonalité continue et discrète. En particulier, il est prouvé dans la présente thèse que les fonctions $S^s$ et $S^l$ caractérisées par certains paramètres sont mutuellement orthogonales par rapport à une mesure discrète. Leur orthogonalité discrète permet de déduire deux types de transformées discrètes analogues aux transformées de Fourier pour chaque algèbre de Lie simple avec racines des longueurs différentes. Comme les polynômes de Tchebyshev, ces quatre familles des fonctions ont des applications en analyse numérique. On obtient dans cette thèse quelques formules de <>, pour des fonctions de plusieurs variables, en liaison avec les fonctions $C$, $S^s$ et $S^l$. On fournit également une description complète des transformées en cosinus discrètes de types V--VIII à $n$ dimensions en employant les fonctions spéciales associées aux algèbres de Lie simples $B_n$ et $C_n$, appelées cosinus antisymétriques et symétriques. Enfin, on étudie quatre familles de polynômes orthogonaux à plusieurs variables, analogues aux polynômes de Tchebyshev, introduits en utilisant les cosinus (anti)symétriques.

O ensino de funfições trigonometricas através do software geogebra

In this study, we sought to address the weaknesses faced by most students when they were studying trigonometric functions sine and cosine. For this, we proposed the use of software Geogebra in performing a sequence of activities about the content covered. The research was a qualitative approach based on observations of the activities performed by the students of 2nd year of high school IFRN - Campus Caicfio. The activities enabled check some diculties encountered by students, well as the interaction between them during the tasks. The results were satisfactory, since they indicate that the use of software contributed to a better understanding of these mathematical concepts studied

Semiclassical scaling functions of sine-Gordon model

We present an analytic study of the finite size effects in sine-Gordon model, based on the semi-classical quantization of an appropriate kink background defined on a cylindrical geometry. The quasi-periodic kink is realized as an elliptic function with its real period related to the size of the system. The stability equation for the small quantum fluctuations around this classical background is of Lame type and the corresponding energy eigenvalues are selected inside the allowed bands by imposing periodic boundary conditions. We derive analytical expressions for the ground state and excited states scaling functions, which provide an explicit description of the flow between the IR and UV regimes of the model. Finally, the semiclassical form factors and two-point functions of the basic field and of the energy operator are obtained, completing the semiclassical quantization of the sine-Gordon model on the cylinder. (C) 2004 Elsevier B.V. All rights reserved.