Related papers: Computational Complexity of Smooth Differential Eq…

Let $H$ be a self-adjoint isotropic elliptic pseudodifferential operator of order $2$. Denote by $u(t)$ the solution of the Schr\"odinger equation $(i\partial_t - H)u = 0$ with initial data $u(0) = u_0$. If $u_0$ is compactly supported the…

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by $ R\le S \iff \ex f \, \forall x, y \, [xRy…

In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data $\phi$ possesses certain…

Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to…

In the first part we present a generalized implicit function theorem for abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all $\lambda$, but,…

We investigate the computational complexity of the discrete logarithm, the computational Diffie-Hellman and the decisional Diffie-Hellman problems in some identity black-box groups G_{p,t}, where p is a prime number and t is a positive…

We study the solvability of the equation for the smooth function F, H=-k F g, on a geodesically complete pseudo-Riemannian manifold (M,g), H being the covariant Hessian of F. A similar equation was considered by Obata and Gallot in the…

We study the $\mathrm{C}^2$ estimates for $p$-Hessian equations with general left-hand and right-hand terms on closed Riemannian manifolds of dimension $n$. To overcome the constraints of closed manifolds, we advance a new kind of…

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak Konig's Lemma within the…

Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$. We derive an $L^\infty$-estimate for bounded solutions to the complex $m$-th Hessian equations on $X$, assuming a positive right-hand side in the Orlicz space…

We consider the problem of smoothing algebraic cycles with rational coefficients on smooth projective complex varieties up to homological equivalence. We show that a solution to this problem would be incompatible with the validity of the…

This article presents a technique for proving problems hard for classes of the polynomial hierarchy or for PSPACE. The rationale of this technique is that some problem restrictions are able to simulate existential or universal quantifiers.…

Let V $\subset$ C n be an equidimensional algebraic set and g be an n-variate polynomial with rational coefficients. Computing the critical points of the map that evaluates g at the points of V is a cornerstone of several algorithms in real…

When the eigenvalues of the coefficient matrix for a linear scalar ordinary differential equation are of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The…

The discrete logarithm problem in Jacobians of curves of high genus $g$ over finite fields $\FF_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$. We present an algorithm for a family of plane curves whose…

By using the method developed in the paper [G.Pantsulaia, G.Giorgadze, On some applications of infinite-dimensional cellular matrices, {\it Georg. Inter. J. Sci. Tech., Nova Science Publishers,} Volume 3, Issue 1 (2011), 107-129], it is…

We tackle the long-standing question of the computational complexity of determining homology groups of simplicial complexes, a fundamental task in computational topology, posed by Kaibel and Pfetsch 20 years ago. We show that this decision…

The classical Theorem of Mumford states that a topologically regular complex algebraic surface in $\mathbb{C}^3$ with an isolated singular point is smooth. We proof that any Lipschitz regular complex algebraic set is smooth. No restriction…

We show that there are simplex pivoting rules for which it is PSPACE-complete to tell if a particular basis will appear on the algorithm's path. Such rules cannot be the basis of a strongly polynomial algorithm, unless P = PSPACE. We…

We are interested in the harmonic analysis on $p$-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space $X$ of unitary hermitian matrices of size $m$ over a ${\mathfrak p}$-adic field $k$ and…