Finally we give a couple examples to illustrate these ideas. First we use Proposition 8 to show that the connected sum of n copies of +3-surgery on the right-handed trefoil, #nS33(T2,3), for any n>0, does not bound a negative definite 4-manifold wit…

Let L be a nonunimodular definite lattice. Using a theorem of Elkies we show that whether L embeds in the standard definite lattice of the same rank …

In this section we deal with the second family of Lemma 1.6. We start by discussing an example in full detail, then we proceed with the necessary computations for the whole family. The main goal of this section is to prove Proposition LABEL:secondfa…

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief an…

Given a chord-generic horizontally displaceable Legendrian submanifold $\Lambda\subset P\times \mathbb R$ with the property that its characteristic a…

We study the discrete Voronoi game, where two players alternately claim vertices of a graph for t rounds. In the end, the remaining vertices are divi…

In this paper a random graph model $G_{\mathbb{Z}^2_N,p_d}$ is introduced, which is a combination of fixed torus grid edges in $(\mathbb{Z}/N \mathbb…

We have introduced several FOL axiom systems GenReln for general relativity and showed that they are complete with respect to Lorentzian manifolds having the corresponding smoothness properties, see Theorem 7.2. From [3], we recalled our FOL definit…

We introduce several axiom systems for general relativity and show that they are complete with respect to the standard models of general relativity, …

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$. Erd\H{o}s proved tha…

We consider special cases of the two tree degree sequences problem. We show that if two tree degree sequences do not have common leaves then they alw…

A set of nonnegative matrices $\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$ is called primitive if there exist indices $i_1, i_2, \ldots, i_m$ such that $M…

We consider the worst-case query complexity of some variants of certain \cl{PPAD}-complete search problems. Suppose we are given a graph $G$ and a ve…

Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-ha…

In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liou…

Given a real enumerative problem ideally we would like to calculate the following data:

We study a 2-parameter family of enumerative problems over the reals. Over the complex field, these problems can be solved by Schubert calculus. In t…

We derive constraints on Lagrangian concordances from Legendrian submanifolds of the standard contact sphere admitting exact Lagrangian fillings. Mor…

We studied the hybrid Babai–Černý problem, where the question is to find tight bounds on the reset threshold for automata with the full transition monoid. We presented a series of n-state automata \mathrsfsVn in this class with the reset threshold e…

Motivated by the Babai conjecture and the Cerny conjecture, we study the reset thresholds of automata with the transition monoid equal to the full mo…

We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsv\'ath and Szab\'o's correction terms in…

Let (A,∂) be a differential graded algebra with coefficients in a noncommutative algebra A over a commutative ring R. We further assume that A=TA(M) is a tensor algebra over a free A–A-bimodule M with a preferred basis {c1,…,ck}.

To a differential graded algebra with coefficients in a noncommutative algebra, by dualisation we associate an $A_\infty$-category whose objects are …

The analysis of this section has been an inspiration for some later works regarding counting manifolds, which have already been published [48, 28].

In the first paper of this series (arxiv.org/abs/1210.2961) we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral i…

Up to permutations of degree sequences and vertices, there are 14 tree degree sequence quartets on at most 10 vertices without common leaves. This appendix gives an example realization for all of them.

We consider packing tree degree sequences in this paper. We set up a conjecture that any arbitrary number of tree degree sequences without common lea…

This research is supported under the Royal Society International Exchanges Scheme (ref. IE110369). Németi’s research was supported by OTKA grant No 81188. This work was partially undertaken whilst Stannett was a visiting fellow at the Isaac Newton I…

Logicians at the R\'enyi Mathematical Institute in Budapest have spent several years developing versions of relativity theory (special, general, and …

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