Arithmetic, enumerative induction and size bias

Number theory abounds with conjectures asserting that every natural number has some arithmetic property. An example is Goldbach’s Conjecture, which states that every even number greater than 2 is the sum of two primes. Enumerative inductive evidence for such conjectures usually consists of small cases. In the absence of supporting reasons, mathematicians mistrust such evidence for arithmetical generalisations, more so than most other forms of non-deductive evidence. Some philosophers have also expressed scepticism about the value of enumerative inductive evidence in arithmetic. But why? Perhaps the best argument is that known instances of an arithmetical conjecture are almost always small: they appear at the start of the natural number sequence. Evidence of this kind consequently suffers from size bias. My essay shows that this sort of scepticism comes in many different flavours, raises some challenges for them all, and explores their respective responses.

Boundary observability and exact controllability of strongly coupled wave equations

<p style='text-indent:20px;'>In this paper, we study the exact controllability of a system of two wave equations coupled by velocities with boundary control acted on only one equation. In the first part of this paper, we consider the <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-d case. Then, using a multiplier technique, we prove that, by observing only one component of the associated homogeneous system, one can get back a full energy of both components in the case where the waves propagate with equal speeds (<i>i.e.</i> <inline-formula><tex-math id="M2">\begin{document}$ a = 1 $\end{document}</tex-math></inline-formula> in (1)) and where the coupling parameter <inline-formula><tex-math id="M3">\begin{document}$ b $\end{document}</tex-math></inline-formula> is small enough. This leads, by the Hilbert Uniqueness Method, to the exact controllability of our system in any dimension space. It seems that the conditions <inline-formula><tex-math id="M4">\begin{document}$ a = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ b $\end{document}</tex-math></inline-formula> small enough are technical for the multiplier method. The natural question is then : what happens if one of the two conditions is not satisfied? This consists the aim of the second part of this paper. Indeed, we consider the exact controllability of a system of two one-dimensional wave equations coupled by velocities with a boundary control acted on only one equation. Using a spectral approach, we establish different types of observability inequalities which depend on the algebraic nature of the coupling parameter <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> and on the arithmetic property of the wave propagation speeds <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula>.</p>

Quantitative destruction of invariant circles

<p style='text-indent:20px;'>For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> of an integrable system by a trigonometric polynomial of degree <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> perturbation <inline-formula><tex-math id="M3">\begin{document}$ R_N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ \|R_N\|_{C^r}&lt;\epsilon $\end{document}</tex-math></inline-formula>. We obtain a relation among <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> and the arithmetic property of <inline-formula><tex-math id="M8">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>, for which the area-preserving map admit no invariant circles with <inline-formula><tex-math id="M9">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>.</p>

Solving Problems with Smooth Sequences that are Subject to Local Restrictions

This article discusses the options that arise when solving problems with smooth sequences that are subject to local restrictions. It also continues the cycle of work on the study of smooth sequences and supplements the literature in the field of the study of this arithmetic property. The relevance of the current research is that the smooth sequences simulate the motion of the bodies taking into account the resistance of the medium. The methodology is in solving problems and proving theorems by calculating the formulas and building the graphs and providing the comments on them. It should be noted that when conducting a literature review about such problems and their solution, we noticed a lack of a detailed review and compactness of information. Thus, this work has a scientific novelty and, as a result, practical significance for the learning process. The paper presents a detailed description of the solutions of smooth sequences on the 1st, 2nd, 3rd differences; it also provides evidence with explanations of the theorems, gives illustrations of graphs of sequences under different conditions. The results of the study may be applicated to spaceship building and ballistics. Besides this, the article is supplemented with data in tables shown in the Appendices.

An Arithmetic Property of Moments of the $\beta$-Hermite Ensemble and Certain Map Enumerators

Moments of the $\beta$-Hermite ensemble are known to be related to the enumerative theory of topological maps. When $\beta \in \{ 1, 2 \}$, asymptotic information about these moments has been used to deduce asymptotics on the number of maps of given genus, and arithmetic information about these moments can sometimes be explained by underlying group actions on the set of maps. In this paper we establish a new arithmetic property about the $2q$-th moment of the $\beta$-Hermite ensemble, for any prime $q \ge 3$ and real number $\beta > 0$, that has a combinatorial interpretation in terms of maps but no known combinatorial explanation. In the process, we derive several additional results that might be of independent interest, including a general integrality statement and an efficient algorithm for evaluating expectations of multi-part elementary symmetric polynomials of bounded length.

Extensive parity analysis of broken 5-diamond partitions

In this paper, we investigate congruences for [Formula: see text] modulo 2 and characterize the parity of [Formula: see text] and [Formula: see text] according to the arithmetic property of [Formula: see text]. As a consequence, we obtain various Ramanujan type congruences for [Formula: see text]. We also extend these results to several infinite families of congruences.

Degrees, class sizes and divisors of character values

In the character table of a finite group there is a tendency either for the character degree to divide the conjugacy class size or the character value to vanish. There is also a partial divisibility where the determinant of the character is not 1. There are versions of these depending on a subgroup, based on an arithmetic property of spherical functions which generalizes the integrality of the values of the characters and the central characters.