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In this work, we consider nonlocal boundary value problems for parabolic equations on manifold. We set up the first order of accuracy difference scheme for the numerical solution of nonlocal boundary value problems for parabolic equations on circle. For the solutions of the difference scheme, we establish the stability estimates and coercivity estimates in various Hölder norms for the solutions of such boundary value problems. Furthermore, numerical results are given. [ABSTRACT FROM AUTHOR]

The present extended abstract considers the differential equations on smooth closed manifolds, investigates and establishes the well-posedness of nonlocal boundary value problems (NBVP) in Hölder spaces. It also establishes new coercivity estimates in various Hölder norms for the solutions of such boundary value problems for parabolic equations. [ABSTRACT FROM AUTHOR]

In this extended abstract, considering the differential equations on hyperbolic plane, we investigate and establish the well-posedness of boundary value problem for hyperbolic equations in Hölder spaces. Furthermore, we establish new coercivity estimates in various Hölder norms for the solutions of such boundary value problems for hyperbolic equations. [ABSTRACT FROM AUTHOR]

This extended abstract proves that Reidemeister torsion is well defined for n = 2, PSL(n,R)-valued Anasov representations. Moreover, it present a formula for computing Reidemeister torsion of these kind of representations. [ABSTRACT FROM AUTHOR]

This extended abstract presents formulas for computing Reidemeister torsion of three-holed sphere and one-holed torus. [ABSTRACT FROM AUTHOR]

We define Reidemeister torsion associated to a closed oriented Riemann surface Σ of genus at least 2 and with coefficients in the adjoint representation of Hitchin representations of PSp2n(R), n ≥ 1. Using symplectic chain complex, we present a volume element for Hitchin component of Hom (π1(Σ),PSp2n(R))/PSp2n(R) for n ≥ 1. [ABSTRACT FROM AUTHOR]

This paper considers the difference operator Aht,x approximation of the following differential operator At,xu(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 a112(t,x)+a222(t,x)≥δ>0, and σ > 0. It investigates the structure of the fractional spaces generated by this difference operator. Moreover, the positivity of Aht,x operator in difference analogy Ho¨lder spaces is proved. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we define C-symplectic chain complex. We also present a formula for the Reidemeister torsion of a C-symplectic chain complex. We apply the formula to closed manifolds. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the present paper, the differential operator A defined by ... with domain ... is considered. The Green function for A is constructed. The estimates for the Green function are obtained and the positivity of A in the Banach space C ([0,1] x R) is established. The structure of fractional spaces generated by A is investigated, the equivalence of the norm of these fractional spaces and Holder spaces is established. The positivity of A in the Holder space ... is obtained. [ABSTRACT FROM AUTHOR]

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 Ho¨lder spaces is proved. [ABSTRACT FROM AUTHOR]