Spontaneous Meta-Arithmetic as the First Step Toward School Algebra

Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the “official” algebraic form of talk. In this paper we take a look at the very beginning of this process. Preliminary analyses of data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses

A modular description of X0(n)\mathscr{X}_0(n)

As we explain, when a positive integer $n$ is not squarefree, even over $\mathbb{C}$ the moduli stack that parametrizes generalized elliptic curves equipped with an ample cyclic subgroup of order $n$ does not agree at the cusps with the $\Gamma_0(n)$-level modular stack $\mathscr{X}_0(n)$ defined by Deligne and Rapoport via normalization. Following a suggestion of Deligne, we present a refined moduli stack of ample cyclic subgroups of order $n$ that does recover $\mathscr{X}_0(n)$ over $\mathbb{Z}$ for all $n$. The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: $\mathscr{X}_0(n)$ is also regular at the cusps. We also prove such regularity for $\mathscr{X}_1(n)$ and several other modular stacks, some of which have been treated by Conrad by a different method. For the proofs we introduce a tower of compactifications $\overline{Ell}_m$ of the stack $Ell$ that parametrizes elliptic curves---the ability to vary $m$ in the tower permits robust reductions of the analysis of Drinfeld level structures on generalized elliptic curves to elliptic curve cases via congruences.Comment: 67 pages; final version, to appear in Algebra and Number Theor

Algorithmic Statistics

While Kolmogorov complexity is the accepted absolute measure of information content of an individual finite object, a similarly absolute notion is needed for the relation between an individual data sample and an individual model summarizing the information in the data, for example, a finite set (or probability distribution) where the data sample typically came from. The statistical theory based on such relations between individual objects can be called algorithmic statistics, in contrast to classical statistical theory that deals with relations between probabilistic ensembles. We develop the algorithmic theory of statistic, sufficient statistic, and minimal sufficient statistic. This theory is based on two-part codes consisting of the code for the statistic (the model summarizing the regularity, the meaningful information, in the data) and the model-to-data code. In contrast to the situation in probabilistic statistical theory, the algorithmic relation of (minimal) sufficiency is an absolute relation between the individual model and the individual data sample. We distinguish implicit and explicit descriptions of the models. We give characterizations of algorithmic (Kolmogorov) minimal sufficient statistic for all data samples for both description modes--in the explicit mode under some constraints. We also strengthen and elaborate earlier results on the ``Kolmogorov structure function'' and ``absolutely non-stochastic objects''--those rare objects for which the simplest models that summarize their relevant information (minimal sufficient statistics) are at least as complex as the objects themselves. We demonstrate a close relation between the probabilistic notions and the algorithmic ones.Comment: LaTeX, 22 pages, 1 figure, with correction to the published journal versio

Dynamical Structure of Irregular Constrained Systems

Hamiltonian systems with functionally dependent constraints (irregular systems), for which the standard Dirac procedure is not directly applicable, are discussed. They are classified according to their behavior in the vicinity of the constraint surface into two fundamental types. If the irregular constraints are multilinear (type I), then it is possible to regularize the system so that the Hamiltonian and Lagrangian descriptions are equivalent. When the constraints are power of a linear function (type II), regularization is not always possible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent. It is shown that the inequivalence between the two formalisms can occur if the kinetic energy is an indefinite quadratic form in the velocities. It is also shown that a system of type I can evolve in time from a regular configuration into an irregular one, without any catastrophic changes. Irregularities have important consequences in the linearized approximation to nonlinear theories, as well as for the quantization of such systems. The relevance of these problems to Chern-Simons theories in higher dimensions is discussed.Comment: 14 pages, no figures, references added. Final version for J. Math. Phy

Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients

We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending some results first obtained by A. Figalli. Our main results are a very general equivalence between different descriptions for multidimensional diffusion processes, such as Fokker-Planck equations and martingale problems, under minimal regularity and integrability assumptions, and new existence and uniqueness results for diffusions having weakly differentiable coefficients, by means of energy estimates and commutator inequalities. Our approach relies upon techniques recently developed, jointly with L. Ambrosio, to address well-posedness for ordinary differential equations in metric measure spaces: in particular, we employ in a systematic way new representations and inequalities for commutators between smoothing operators and diffusion generators.Comment: Added references to further literature on the subjec

Phenotypic evolution and hidden speciation in Candidula unifasciata ssp. (Helicellinae, Gastropoda) inferred by 16S variation and quantitative shell traits

In an effort to link quantitative morphometric information with molecular data on the population level, we have analysed 19 populations of the conchologically variable land snail Candidula unifasciata from across the species range for variation in quantitative shell traits and at the mitochondrial 16S ribosomal (r)DNA locus. In genetic analysis, including 21 additional populations, we observed two fundamental haplotype clades with an average pairwise sequence divergence of 0.209 ± 0.009 between clades compared to 0.017 ± 0.012 within clades, suggesting the presence of two different evolutionary lineages. Integrating additional shell material from the Senckenberg Malacological Collection, a highly significant discriminant analysis on the morphological shell traits with fundamental haplotype clades as grouping variable suggested that the less frequent haplotype corresponds to the described subspecies C. u. rugosiuscula, which we propose to regard as a distinct species. Both taxa were highly subdivided genetically (FST = 0.648 and 0.777 P < 0.001). This was contrasted by the partition of morphological variance, where only 29.6% and 21.9% of the variance were distributed among populations, respectively. In C. unifasciata, no significant association between population pairwise FST estimates and corresponding morphological fixation indices could be detected, indicating independent evolution of the two character sets. Partial least square analysis of environmental factors against shell trait variables in C. u. unifasciata revealed significant correlations between environmental factors and certain quantitative shell traits, whose potential adaptational values are discussed