Related papers: No bound for the first fixed point

It is well known that the entropy $H(X)$ of a finite random variable is always greater or equal to the entropy $H(f(X))$ of a function $f$ of $X$, with equality if and only if $f$ is one-to-one. In this paper, we give tights bounds on…

Let phi denote Euler's phi function. For a fixed odd prime we give an asymptotic series expansion in the sense of Poincare for the number E_q(x) of n<=x such that q does not divide phi(n). Thereby we improve on a recent theorem of B.K.…

We propose a new algorithm that finds an $\varepsilon$-approximate fixed point of a smooth function from the $n$-dimensional $\ell_2$ unit ball to itself. We use the general framework of finding approximate solutions to a variational…

The purpose of this article is twofold. The first aim is to prove that if there exist a sequence $\{\varphi_j\}\subset \mathrm{Aut}(\Omega)$ and $a\in \Omega$ such that $\lim_{j\to\infty}\varphi_j(a)=\xi_0$ and…

Addressing a question of Paul Larson we prove the following statement. If Chang's conjecture fails, Martin's axiom holds and the continuum is greater than $\aleph_2$, there are no weakly Laver ideals over $\aleph_1$. We also prove that…

We prove a version of the Lefschetz fixed point theorem for multivalued maps $F:X\multimap X$ in which $X$ is a finite $T_0$ space.

We introduce the forward limit set $\Lambda$ of a semigroup $S$ generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative…

We prove Los conjecture = Morley theorem in ZF, with the same characterization (of first order countable theories categorical in aleph_alpha for some (equivalently for every) ordinal alpha>0. Another central result here is, in this context:…

In [V. M. Abramov, \emph{Bull. Aust. Math. Soc.} \textbf{104} (2021), 108--117] the fixed point equation for an infinite nonnegative Toeplitz matrix has been studied. It was found the conditions for existence of a positive solution and…

We study the $2k$-th moment of the family of twisted modular $L$-functions to a fixed prime power modulus at the central values. We establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k…

We prove that no infinite field is definable in the theory of the free group

I define Goldbach counting function with N > 0 and square-free P > 0. Decomposition of this function is discovered and deduction formula is found. I propose a hypothesis on upper bound of Goldbach counting function and prove that Goldbach…

In this short paper we present an elementary proof of the infinitude of primes. Our proof is similar in spirit to Euler's proof that the reciprocals of primes diverges and only uses tools from elementary number theory and calculus. In…

Let f be an entire function with a bounded set of singular values, and suppose furthermore that the postsingular set of f is bounded. We show that every component of the escaping set I(f) is unbounded. This provides a partial answer to a…

We give an elementary proof of Noether's first Theorem while stressing the magical fact that the global quasi-symmetry only needs to hold for one fixed integration region. We provide sufficient conditions for gauging a global…

In this note, we prove a finiteness result for fibers that are canonical lifts in a given elliptic fibration. The question was motivated by the authors' construction of an arithmetic Euler top, and it highlights an interesting discrepancy…

In this work, we will present variants Fixed Point Theorem for the affine and classical contexts, as a consequence of general Brouwer's Fixed Point Theorem. For instance, the affine results will allow working on affine balls, which are…

Let $f$ be a $C^{2+\epsilon}$ expanding map of the circle and $v$ be a $C^{1+\epsilon}$ real function of the circle. Consider the twisted cohomological equation $v(x) = \alpha (f(x)) - Df(x) \alpha (x)$ which has a unique bounded solution…

We prove an upper bound for the asymptotics of counting functions of number fields with nilpotent Galois groups.

In this first of two papers, strong limits on the accuracy of physical computation are established. First it is proven that there cannot be a physical computer C to which one can pose any and all computational tasks concerning the physical…