Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere

We define Picard cycles on each smooth three-sheeted Galois cover C of the Riemann sphere. The moduli space of all these algebraic curves is a nice Shimura surface, namely a symmetric quotient of the projective plane uniformized by the complex two-dimensional unit ball. We show that all Picard cycles on C form a simple orbit of the Picard modular group of Eisenstein numbers. The proof uses a special surface classification in connection with the uniformization of a classical Picard-Fuchs system. It yields an explicit symplectic representation of the braid groups (coloured or not) of four strings.

Existence and Uniqueness of Solutions for Partial Differential-Functional Equations of the First Order with Deviating Argument of the Derivative of Unknown Function

We consider the existence and uniqueness problem for partial differential-functional equations of the first order with the initial condition for which the right-hand side depends on the derivative of unknown function with deviating argument.

Cauchy Problem for Differential Equation with Caputo Derivative

The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution of this problem. Two numerical examples are given.

Nonexistence Results of Solutions of Semilinear Differential Inequalities with Temperal Fractional Derivative on the Heinsenberg Group

2000 Mathematics Subject Classification: 26A33, 33C60, 44A15, 35K55

Impulsive Fractional Differential Inclusions Involving the Caputo Fractional Derivative

Mathematics Subject Classification: 26A33, 34A37.

Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey

2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10, 45K05, 74D05,

On q–Analogues of Caputo Derivative and Mittag–Leffler Function

Mathematics Subject Classification: 33D60, 33E12, 26A33

Integral Representations of the Logarithmic Derivative of the Selberg Zeta Function

AMS Subj. Classification: MSC2010: 11F72, 11M36, 58J37

Hermite Polynomials and the Zero-Distribution of Riemann's Zeta Function

AMS Subject Classification 2010: 11M26, 33C45, 42A38.

Riemann's Hypothesis

Riemann’s memoir is devoted to the function π(x) defined as the number of prime numbers less or equal to the real and positive number x. This is really the fact, but the “main role” in it is played by the already mentioned zeta-function.

Calculating the derivative of piecewise functions

Exercises involving the calculation of the derivative of piecewise defined functions are common in calculus, with the aim of consolidating beginners’ knowledge of applying the definition of the derivative. In such exercises, the piecewise function is commonly made up of two smooth pieces joined together at one point. A strategy which avoids using the definition of the derivative is to find the derivative function of each smooth piece and check whether these functions agree at the chosen point. Showing that this strategy works together with investigating discontinuities of the derivative is usually beyond a calculus course. However, we shall show that elementary arguments can be used to clarify the calculation and behaviour of the derivative for piecewise functions.

Second Order Optimality Conditions in Nonsmooth Unconstrained Optimization

AMS subject classification: 49J52, 90C30.

Riemann-Hilbert Problems with Canonical Normalization and Families of Commuting Operators

2010 Mathematics Subject Classification: 35Q15, 31A25, 37K10, 35Q58.

Fiber Scaffolds of Poly (glycerol-dodecanedioate) and its Derivative via Electrospinning for Neural Tissue Engineering

Peripheral nerves have demonstrated the ability to bridge gaps of up to 6 mm. Peripheral Nerve System injury sites beyond this range need autograft or allograft surgery. Central Nerve System cells do not allow spontaneous regeneration due to the intrinsic environmental inhibition. Although stem cell therapy seems to be a promising approach towards nerve repair, it is essential to use the distinct three-dimensional architecture of a cell scaffold with proper biomolecule embedding in order to ensure that the local environment can be controlled well enough for growth and survival. Many approaches have been developed for the fabrication of 3D scaffolds, and more recently, fiber-based scaffolds produced via the electrospinning have been garnering increasing interest, as it offers the opportunity for control over fiber composition, as well as fiber mesh porosity using a relatively simple experimental setup. All these attributes make electrospun fibers a new class of promising scaffolds for neural tissue engineering. Therefore, the purpose of this doctoral study is to investigate the use of the novel material PGD and its derivative PGDF for obtaining fiber scaffolds using the electrospinning. The performance of these scaffolds, combined with neural lineage cells derived from ESCs, was evaluated by the dissolvability test, Raman spectroscopy, cell viability assay, real time PCR, Immunocytochemistry, extracellular electrophysiology, etc. The newly designed collector makes it possible to easily obtain fibers with adequate length and integrity. The utilization of a solvent like ethanol and water for electrospinning of fibrous scaffolds provides a potentially less toxic and more biocompatible fabrication method. Cell viability testing demonstrated that the addition of gelatin leads to significant improvement of cell proliferation on the scaffolds. Both real time PCR and Immunocytochemistry analysis indicated that motor neuron differentiation was achieved through the high motor neuron gene expression using the metabolites approach. The addition of Fumaric acid into fiber scaffolds further promoted the differentiation. Based on the results, this newly fabricated electrospun fiber scaffold, combined with neural lineage cells, provides a potential alternate strategy for nerve injury repair.