Yasar Sozen

Structure of fractional spaces generated by second order difference operators

Abstract A second order difference operator with nonlocal boundary conditions is considered. The positivity of this operator in Banach spaces is proved. Moreover, the structure of the fractional spaces generated by this operator is investigated. Furthermore, the positivity of this operator in Holder spaces is proved.

The structure of fractional spaces generated by a two-dimensional elliptic differential operator and its applications

We consider the two-dimensional differential operator Au(x1,x2)=−a11(x1,x2)ux1x1(x1,x2)−a22(x1,x2)ux2x2(x1,x2)+σu(x1,x2) defined on functions on the half-plane Ω=R+×R with the boundary conditions u(0,x2)=0, x2∈R, where aii(x1,x2), i=1,2, are continuously differentiable and satisfy the uniform ellipticity condition a112(x1,x2)+a222(x1,x2)≥δ>0, σ>0. The structure of the fractional spaces Eα(A,Cβ(Ω)) generated by the operator A is investigated. The positivity of A in Hölder spaces is established. In applications, theorems on well-posedness in a Hölder space of elliptic problems are obtained.MSC: 35J25, 47E05, 34B27.

Finite Difference Method for the Reverse Parabolic Problem

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Dirichlet condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example.

Finite difference method for the reverse parabolic problem with Neumann condition

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Neumann condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example.

A note on reidemeister torsion and period matrix of Riemann surfaces

We consider compact Riemann surfaces Σg with genus at least 2. We explain the relation between the Reidemeister torsion of Σg and its period matrix.

Positivity of Two-Dimensional Elliptic Differential Operators in Hölder Spaces

This paper considers the operator Au(t,x) = −a11(t,x)utt(t,x)−a22(t,x)uxx(t,x)+σu(t,x), defined over the region R+×R with the boundary condition u(0,x) = 0,x∈R. Here, the coefficients aii(t,x), i = 1,2 are continuously differentiable and satisfy the uniform ellipticity condition a112(t,x)+a222(t,x)≥δ>0, and σ > 0. It investigates the structure of the fractional spaces generated by this operator. Moreover, the positivity of the operator in Holder spaces is proved.

A note on the parabolic equation with an arbitrary parameter at the derivative

Abstract We consider the parabolic differential equation (0.1) e u ′ ( t ) + A u ( t ) = f ( t ) , − ∞ t ∞ , in a Banach space E with a strongly positive operator A and with an arbitrary positive parameter e . We establish the well-posedness in difference analogue of Holder space of the high order uniform difference scheme for (0.1) . Moreover, in applications, the convergence estimates for the solutions of uniform difference schemes of the multi-dimensional parabolic differential equations with an arbitrary positive parameter e are obtained.

A note on Reidemeister torsion and pleated surfaces

This paper uses the notion of ℂ-symplectic chain complex and proves an explicit formula for the Reidemeister torsion of an arbitrary ℂ-symplectic chain complex in terms of intersection forms of the homologies. In applications, the formula is applied to closed manifolds and also compact manifolds with boundary by using the homologies with coefficients in complex numbers field. Moreover, an explicit formula for the Reidemeister torsion of representations from the fundamental group of a closed oriented hyperbolic surface to PSL2(ℂ) is presented in terms of the cup product of twisted cohomologies, which is related with Weil–Petersson form and thus the Thurston symplectic form. The formula is also applied to pleated surfaces.

The Positivity of Differential Operator with Nonlocal Boundary Conditions

We study a structure of fractional spaces Eα(Lp[0,1],Ax) generated by the positive differential operator Ax defined by the formula Axu = −a(x)d2udx2+δu with domain D(Ax) = {u∈C(2)[0,1]:u(0) = u(μ),u′(0) = u′(1),1/2≤μ≤1}. Here, a(x) is a smooth function defined on the segment [0,1] and a(x)≥a>0,δ>0. It is established that for any 0 < α < 1/2, 1 ≤ p < ∞, the norms in the spaces Eα(Lp[0,1],Ax) and Wp2α[0,1] are equivalent. The positivity of the differential operator Ax in Wp2α[0,1],(0≤α≤1/2,1≤p<∞) is established. In applications, well-posedness of nonlocal boundary problems for elliptic equations is established.

A note on exceptional groups and Reidemeister torsion

Let Σ be a closed orientable surface of genus at least 2 and G be one of the exceptional groups G2, F4, and E6. The present article considers the set Rep(Σ, G) of G-valued representations from the fundamental group π1(Σ) of the surface Σ to the exceptional group G. It proves that for such representations the notion of Reidemeister torsion is well-defined. It also establishes a formula for computing Reidemeister torsion of such representations in terms of the well-known symplectic structure on Rep(Σ, G), namely, the Atiyah-Bott-Goldman symplectic form for the Lie group G. Moreover, it applies to G-valued Hitchin representations.