Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form(P_(a,b)){D~αu(x) + f(x, u(x)) = 0, x ∈(0, 1),u(0) = u(1) = 0, D~(α-3)u(0) = a, u(1) =-b,where 3 α≤ 4, Dαis the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) =-p(x)t~σ, with σ∈(-1, 1)and p being a nonnegative continuous function that may be singular at x = 0 or x = 1and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch¨auder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem(P_(0,0)).Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b 0 and f(x, t) = tφ(x, t), with φ(x, t) being a nonnegative continuous function in(0, 1)×[0, ∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green's function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem(P_(a,b)), which behaves like the unique solution of the homogeneous problem corresponding to(P_(a,b)). Some examples are given to illustrate the existence results.     

Existence of Renormalized Solutions for Nonlinear Parabolic Equations

We give an existence result of a renormalized solution for a class of nonlin- ear parabolic equations b（x,u）/ t-div （a（x,t,u, u））＋g（x,t,u,u ）＋H（x,t, u）=f,in QT, where the right side belongs to LP＇ （0,T;W-1,p＇（Ω）） and where b（x,u） is unbounded function of u and where - div （ a （ x, t, u, u） ） is a Leray-Lions type operator with growth ｜u ｜p- 1 in V u. The critical growth condition on g is with respect to u and no growth condition with re sp ect to u, while the function H （x, t, u） grows as｜ u ｜p - 1.     

A class of generalized Ornstein transformations with the weak mixing property

It is shown that there is a class of Ornstein transformations, distinct from the Ornstein mixing class, which are almost surely weak mixing     

A New Hybridized Mixed Weak Galerkin Method for Second-Order Elliptic Problems

In this paper, a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem. This method is designed by approximate some operators with discontinuous piecewise polynomials in a shape regular finite element partition. Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established. Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method.     

The Rain on Underground Porous Media Part I: Analysis of a Richards Model

The Richards equation models the water flow in a partially saturated underground porous medium under the surface. When it rains on the surface, boundary conditions of Signorini type must be considered on this part of the boundary. The authors first study this problem which results into a variational inequality and then propose a discretization by an implicit Euler’s scheme in time and finite elements in space. The convergence of this discretization leads to the well-posedness of the problem.     

PHASE TRANSITIONS IN COLLOIDAL SYSTEMS PART I: REFRACTOMETRIC STUDY OF THE BINARY SYSTEM SDS+WATER OR PENTANOL-1+WATER AND THE TERNARY SYSTEM WATER+SDS+PENTANOL-1 AT 298.15 K.

The monophasic ranges for the binary systems water+SDS and water+pentanoM and for the ternary system water+SDS+pentanoM has been studied by means of refractametric experiments at 298 K.

The curves of refractometric index versus the composition of the system, has shown the existence of several transitions labeled successive critic micellar concentrations. These transitions are detected by a change of the variation of the refractive index.

We have evidenced the very importance of the pentanol-1 in these amphiphilic solutions, acting as a surfactant or as an oil according the concentration.

For these systems it is passible to better understand the phase inversion phenomena.

At last, we present the phase diagram indicating the concentration range of the different microstructures.     

Synthesis and photoluminescent properties of a Schiff-base ligand and its mononuclear Zn(II), Cd(II), Cu(II), Ni(II) and Pd(II) metal complexes

Mononuclear Zn(II), Cd(II), Cu(II), Ni(II) and Pd(II) metal complexes of Schiff-base ligand(HL₁) derived from 8-acetyl-7-hydroxycoumarin and P-phenylenediamine were prepared and characterized by microanalytical, mass, UV–Vis, IR, ¹H NMR, ¹³C NMR, ESR, conductance and fluorescence studies. The measured low molar conductance values in DMSO indicate that the complexes are non-electrolytes. The structures of the solid complexes under study are established by using IR, electronic and ESR spectroscopy suggesting that Zn(II) and Ni(II) complexes are octahedral, Cd(II) complex is tetrahedral, Cu(II) and Pd(II) complexes are square planar. The ESR spectrum of the Cu(II) complex in DMSO at 298 and 150 K was recorded and its salient features are reported, it supports the mononuclear structure. The Schiff base exhibited photoluminescence originating from intraligand (π–π*) transitions. Metal-mediated enhancement is observed on complexation of HL with Zn(II) and Cd(II), whereas metal-mediated fluorescence quenching occurs in Cu(II), Ni(II) and Pd(II).     

Diode laser absorption measurements at the Hα-transition in laser induced plasmas on different targets

The diode laser atomic absorption spectroscopy (DLAAS) technique has been utilized to assess the degree of optical opacity of plasma at the wavelength of the H_α-line. The plasma is produced at atmospheric conditions by focusing a 6 ns Nd:YAG laser pulse at 1.064 μm on different solid target materials including aluminum, iron and titanium as major elements as well as flat pieces of plastic and wood characterized by a high content of hydrogen. The optical depth was investigated as a function of delay times ranging from 0 to 5 μs, and at laser fluences ranging from 7 to 19 J/cm², all at a fixed gate time of 1 μs. The results show that the plasma associated with metallic targets is almost optically thin at the H_α-line over all fluences and at delay times ≥ 1 μs, but rather thick for hydrogen-rich targets (plastic and wood) over all delay times and fluences.