Arnold Mathematical Journal最新文献

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current). We follow up with a dynamical criterion for discreteness of the spectrum. In case of atomic spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.

We study the intersection between a smooth algebraic surface with an umbilic point and a plane parallel and close to the tangent plane at the umbilic. The problem has its origin in the study of isophote (equal illumination) curves in a 2-dimensional image. In particular, we study the circles which have exceptional tangency to this intersection curve: ordinary tangency at one point and osculating at another; ordinary tangency at three points; and 4-point tangency at a vertex. The centres of circles having ordinary tangency at two points trace out a curve whose closure is the symmetry set of the intersection curve, and the exceptional circles above give respectively cusps, triple crossings and endpoints of this set. We analyse the curves traced out by the contact points and centres of the exceptional circles as the plane approaches the tangent plane at the umbilic. We also briefly discuss the global structure of the symmetry set by means of a typical example.

To discuss the validity of the Kummer–Vandiver conjecture, we consider a generalized problem associated to the conjecture. Let p be an odd prime number and (zeta _p) a primitive p-th root of unity. Using new programs, we compute the Iwasawa invariants of ({textbf{Q}}(sqrt{d},zeta _p)) in the range (|d|<200) and (200<p <1{,}000{,}000). From our data, the actual numbers of exceptional cases seem to be near the expected numbers for (p<1{,}000{,}000). Moreover, we find a few rare exceptional cases for (|d|<10) and (p>1{,}000{,}000). We give two partial reasons why it is difficult to find exceptional cases for (d=1) including counter-examples to the Kummer–Vandiver conjecture.

We design a homotopy continuation algorithm, that is based on Viro’s patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients that satisfy certain concavity conditions, it tracks optimal number of solution paths, and it operates entirely over the reals. In more technical terms, we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba. We provide a detailed exposition of connections between Viro’s patchworking method, convex geometry of A-discriminant amoeba complements, and computational real algebraic geometry.

Let (lambda ) be a partition of an integer n and ({mathbb F}_q) be a finite field of order q. Let (P_lambda (q)) be the number of strictly upper triangular (ntimes n) matrices of the Jordan type (lambda ). It is known that the polynomial (P_lambda ) has a tendency to be divisible by high powers of q and (Q=q-1), and we put (P_lambda (q)=q^{d(lambda )}Q^{e(lambda )}R_lambda (q)), where (R_lambda (0)ne 0) and (R_lambda (1)ne 0). In this article, we study the polynomials (P_lambda (q)) and (R_lambda (q)). Our main results: an explicit formula for (d(lambda )) (an explicit formula for (e(lambda )) is known, see Proposition 3.3 below), a recursive formula for (R_lambda (q)) (a similar formula for (P_lambda (q)) is known, see Proposition 3.1 below), the stabilization of (R_lambda ) with respect to extending (lambda ) by adding strings of 1’s, and an explicit formula for the limit series (R_{lambda 1^infty }). Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.

The paper concerns a result in linear algebra motivated by ideas from tropical geometry. Let A(t) be an (n times n) matrix whose entries are Laurent series in t. We show that, as (t rightarrow 0), the logarithms of singular values of A(t) approach the invariant factors of A(t). This leads us to suggest logarithms of singular values of an (n times n) complex matrix as an analog of the logarithm map on ((mathbb {C}^*)^n) for the matrix group ({text {GL}}(n, mathbb {C})).

The Kepler orbits form a 3-parameter family of unparametrized plane curves, consisting of all conics sharing a focus at a fixed point. We study the geometry and symmetry properties of this family, as well as natural 2-parameter subfamilies, such as those of fixed energy or angular momentum. Our main result is that Kepler orbits is a ‘flat’ family, that is, the local diffeomorphisms of the plane preserving this family form a 7-dimensional local group, the maximum dimension possible for the symmetry group of a 3-parameter family of plane curves. These symmetries are different from the well-studied ‘hidden’ symmetries of the Kepler problem, acting on energy levels in the 4-dimensional phase space of the Kepler system. Each 2-parameter subfamily of Kepler orbits with fixed non-zero energy (Kepler ellipses or hyperbolas with fixed length of major axis) admits (mathrm { PSL}_2(mathbb {R})) as its (local) symmetry group, corresponding to one of the items of a classification due to Tresse (Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre (y^{prime prime }= omega (x, y, y^{prime })), vol. 32, S. Hirzel, 1896) of 2-parameter families of plane curves admitting a 3-dimensional local group of symmetries. The 2-parameter subfamilies with zero energy (Kepler parabolas) or fixed non-zero angular momentum are flat (locally diffeomorphic to the family of straight lines). These results can be proved using techniques developed in the nineteenth century by Lie to determine ‘infinitesimal point symmetries’ of ODEs, but our proofs are much simpler, using a projective geometric model for the Kepler orbits (plane sections of a cone in projective 3-space). In this projective model, all symmetry groups act globally. Another advantage of the projective model is a duality between Kepler’s plane and Minkowski’s 3-space parametrizing the space of Kepler orbits. We use this duality to deduce several results on the Kepler system, old and new.