Related papers: Low-like basis theorems for Ramsey's theorem for p…
To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for…
This thesis pertains to the study of elliptic and parabolic partial differential equations on "thin" structures. The first main objective is to establish the strong and weak low-dimensional counterparts of the parabolic Neumann problem. The…
First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant…
In this article, we study a degenerate version of Ramsey's theorem for pairs and two colors ($\mathsf{RT}^2_2$), in which the homogeneous sets for color 1 are of bounded size. By $\mathsf{RT}^2_2$, it follows that every such coloring admits…
In this paper, we show that for each lattice basis, there exists an equivalent basis which we describe as ``strongly reduced''. We show that bases reduced in this manner exhibit rather ``short'' basis vectors, that is, the length of the…
It is known that for binary codes one can use Gr\"obner bases to obtain a subset of codewords of minimal support that can be used to determine the second generalized Hamming weight of the code. In this paper we establish conditions on a…
The rainbow Ramsey theorem states that every coloring of tuples where each color is used a bounded number of times has an infinite subdomain on which no color appears twice. The restriction of the statement to colorings over pairs (RRT22)…
We find all linear orders $L,L_2$ such that $L\to(\mathbb Z,L_2)^2$. The key is the identification of the orders $L$ such that $L\to(\mathbb Z,\mathbb Z)^1$.
Ramsey Theorem [6] for pairs is intuitionistically but not classically provable: it is equivalent to a subclassical principle [2]. In this note we show that Ramsey may be restated in an intuitionistically provable form, which is informative…
Ramsey's theorem for $n$-tuples and $k$-colors ($\mathsf{RT}^n_k$) asserts that every k-coloring of $[\mathbb{N}]^n$ admits an infinite monochromatic subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and two…
Let $\lambda_k$ denote the $k$-th successive minimum of a lattice $L$. We study properties of the lengths of certain bases of $L$. If $v_1, \dots v_n$ is a basis which is reduced in the sense of Minkowski we show that $\lvert v_k \rvert^2…
Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. In this work, we study such lattices and their reduction algorithms. Firstly, when the lattice is spanned over a two…
In this paper, we show that $\mathrm{RT}^{2}+\mathsf{WKL}_0$ is a $\Pi^{1}_{1}$-conservative extension of $\mathrm{B}\Sigma^0_3$.
We establish a consistency result by comparing two independent notions of generalised solutions to a large class of linear hyperbolic first order PDE systems with constant coefficients, showing that they eventually coincide. The first is…
A lattice reduction is an algorithm that transforms the given basis of the lattice to another lattice basis such that problems like finding a shortest vector and closest vector become easier to solve. We define a class of bases called…
Sparsity promoting regularization is an important technique for signal reconstruction and several other ill-posed problems. Theoretical investigation typically bases on the assumption that the unknown solution has a sparse representation…
We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to…
A combinatorial principle CECA is formulated and its equivalence with GCH+ certain weakenings of Box_lambda for singular lambda is proved. CECA is used to show that certain ``almost point- < tau'' families can be refined to point- < tau…
Given a finite point set $P \subset \mathbb{R}^d$, a $k$-ary semi-algebraic relation $E$ on $P$ is the set of $k$-tuples of points in $P$, which is determined by a finite number of polynomial equations and inequalities in $kd$ real…
We discuss an algorithm for the approximate solution of Schrodinger's equation for lattice gauge theory, using lattice SU(3) as an example. A basis is generated by repeatedly applying an effective Hamiltonian to a ``starting state.'' The…