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We study the evolution of the one dimensional periodic cubic Schr\"odinger equation (NLS) with bounded variation data. For the linear evolution, it is known that for irrational times the solution is a continuous, nowhere differentiable…

In this paper we give a combinatorial description of the renormlization limits of infinitely renormalizable unimodal maps with {\it essentially bounded} combinatorics admitting quadratic-like complex extensions. As an application we…

These notes provide a self-contained introduction to Schauder theory on manifolds. First, we derive Schauder estimates for a fourth-order parabolic linear problem with a first- and third-order boundary condition on a smooth compact manifold…

These informal notes, not intended for publication, provide an approach to the Borsuk--Ulam theorem via Stokes' theorem, in a similar spirit to Lima's proof of the Brouwer fixed point theorem. They are intended to be accessible to anyone…

We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al, Appl. Math. Comput. (2018) by introducing a unifying abstract framework, and derive a finite-dimensional…

Recently S. Chmutov introduced a generalization of the dual of a ribbon (or embedded) graph and proved a relation between Bollobas and Riordan's ribbon graph polynomial of a ribbon graph and its generalized duals. Here I show that the…

This paper is concerned with the concept of linear repetitivity in the theory of tilings. We prove a general uniform subadditive ergodic theorem for linearly repetitive tilings. This theorem unifies and extends various known (sub)additive…

We show that with a small modification, the formulation of the Einstein equations of Uggla et al, which uses tetrad variables normalised by the expansion, is a mixed symmetric hyperbolic/parabolic system. Well-posedness of the Cauchy…

The solution of equations from the title is well known since the Euler's time. However, its proof in the case of multiple roots of the characteristic polynomial is rather long and technical and even appearance of the factors $x^m$ looks…

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral…

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form $$u_t = a^{i'j'}u_{i'j'} + 2 x_n^{\gamma/2} a^{i'n} u_{i'n} + x_n^{\gamma} a^{nn} u_{nn} + b^{i'} u_{i'} + x_n^{\gamma/2} b^n…

We prove a comparison theorem for the averages of the solutions of two exterior parabolic problems, the second being the "symmetrization" of the first one, by using approximation of the Schwarz symmetrization by polarizations, as it was…

We provide counterexamples to uniqueness of solutions as well as a priori Calder\'on-Zygmund estimates for solutions below $L^2$ using convex integration argument for equations of the type $$ \text{div} (A (\nabla u)) = 0 \quad \text{in }…

We explore how the analysis of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman…

This is a follow-up of a previous article where we proved local stability estimates for a potential in a Schr\"odinger equation on an open bounded set in dimension $n=3$ from the Dirichlet-to-Neumann map with partial data. The region under…

We adapt the alternating linearization method for proximal decomposition to structured regularization problems, in particular, to the generalized lasso problems. The method is related to two well-known operator splitting methods, the…

In this paper, we prove a discrete embedding inequality for the Raviart--Thomas mixed finite element methods for second order elliptic equations, which is analogous to the Sobolev embedding inequality in the continuous setting. Then, by…

Physical ageing phenomena far from equilibrium naturally lead to dynamical scaling. It has been proposed to consider the consequences of an extension to a larger Lie algebra of local scale-transformation. The best-tested applications of…

We investigate the structure of the set of $T$-periodic solutions to periodically perturbed coupled delay differential equations on differentiable manifolds. By using fixed point index and degree-theoretic methods we prove the existence of…

An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with…