The physical limits of computation inspire an open problem that concerns decidable sets X \subseteq N and cannot be formalized in ZFC as it refers to the current knowledge on X

Let f(1)=2, f(2)=4, f(n+1)=f(n)! for n>1. E.Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. This conjecture implies the following unproven statement F: card(P(n^2+1)) P(n^2+1) subset [2,f(7)]. Let B denote the system: {x_i!=x_k: i,k in {1,...,9}} cup {x_i cdot x_j=x_k: i,j,k in {1,...,9}}. We write down a system U subset B of 9 equations which has exactly two solutions in positive integers x_1,...,x_9: (1,...,1) and (f(1),...,f(9)). We write down a system A subset B of 8 equations. Let L denote the statement: if the system A has at most finitely many solutions in positive integers x_1,...,x_9, then each such solution (x_1,...,x_9) satisfies x_1,...,x_9 leq f(9). The statement L is equivalent to the statement F. This heuristically proves the statement F. This proof does not yield that card(P(n^2+1))=omega. We explain the distinction between existing algorithms (i.e. algorithms whose existence is provable in ZFC) and known algorithms (i.e. algorithms whose definition is constructive and currently known to us). Conditions (1)-(5) concern sets X subset N. (1) A known algorithm with no input returns an integer n satisfying card(X) X subset (-infty,n]. (2) A known algorithm for every k in N decides whether or not k in X. (3) No known algorithm with no input returns the logical value of the statement card(X)=omega. (4) There are many elements of X and it is conjectured that X is infinite. (5) X has the simplest definition among known sets Y subset N with the same set of known elements. ** The set X={k in N: (f(7) (f(7),k) cap P(n^2+1) neq emptyset} satisfies conditions (1)-(4). No set X subset N will satisfy conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The statement F implies that conditions (1)-(5) hold for X=P(n^2+1).Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1109.3826, arXiv:1404.597

The Supersymmetric Effective Action of the Heterotic String in Ten Dimensions

We construct the supersymmetric completion of quartic $R+R^4$-actions in the ten-dimensional effective action of the heterotic string. Two invariants, of which the bosonic parts are known from one-loop string amplitude calculations, are obtained. One of these invariants can be generalized to an $R+F^2+F^4$-invariant for supersymmetric Yang-Mills theory coupled to supergravity. Supersymmetry requires the presence of $B\wedge R\wedge R\wedge R\wedge R$-terms, ($B\wedge F\wedge F\wedge F\wedge F$ for Yang-Mills) which correspond to counterterms in the Green-Schwarz anomaly cancellation. Within the context of our calculation the $\zeta(3)R^4$-term from the tree-level string effective action does not allow supersymmetrization.Comment: 42 pages, UG-9/9

CrC^r-right equivalence of analytic functions

Let $f,g:(\mathbb{R}^n,0)\rightarrow (\mathbb{R},0)$ be analytic functions. We will show that if $\nabla f(0)=0$ and $g-f \in (f)^{r+2}$ then $f$ and $g$ are $C^r$-right equivalent, where $(f)$ denote ideal generated by $f$ and $r\in \mathbb{N}$.Comment: 9 pages. Main result has been significantly improve

A proof of the Baum-Connes conjecture for reductive adelic groups

Let F be a global field, A its ring of adeles, G a reductive group over F. We prove the Baum-Connes conjecture for the adelic group G(A).Comment: 9 page

Many HH-copies in graphs with a forbidden tree

For graphs $H$ and $F$, let $\operatorname{ex}(n, H, F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalises the well-studied Tur\'an numbers of graphs, was initiated recently by Alon and Shikhelman. We show that if $F$ is a tree then $\operatorname{ex}(n, H, F) = \Theta(n^r)$ for some integer $r = r(H, F)$, thus answering one of their questions.Comment: 9 pages, 1 figur

Sequences of irreducible polynomials over odd prime fields via elliptic curve endomorphisms

In this paper we present and analyse a construction of irreducible polynomials over odd prime fields via the transforms which take any polynomial $f \in \mathbf{F}_p[x]$ of positive degree $n$ to $\left(\frac{x}{k} \right)^n \cdot f(k(x+x^{-1}))$, for some specific values of the odd prime $p$ and $k \in \mathbf{F}_p$.Comment: 9 pages. Exposition revised. References update