Arnold Mathematical Journal最新文献

In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated with cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky’s cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a red number. In this paper, we generalized this notion to what we call unrestricted red numbers which are related to reddening sequences. In addition to establishing this more general framework, we completely determine the red numbers and unrestricted red numbers for all finite mutation type of quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general.

Let (pin {mathbb {Z}}^n) be a primitive vector and (Psi :{mathbb {Z}}^nrightarrow {mathbb {Z}}, zrightarrow min (pcdot z, 0)). The theory of husking allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to (Psi ) “at infinity”. We apply this result to sandpile models on ({mathbb {Z}}^n). We prove existence of so-called solitons in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope A without lattice points except its vertices. Namely, for each function

there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with (Psi ) “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of A, intersect (see Fig. 1).

We establish the existence of stationary solutions for certain systems of reaction–diffusion-type equations in the corresponding (H^{2}) spaces. Our method relies on the fixed point theorem when the elliptic problem contains second-order differential operators with and without the Fredholm property, which may depend on the outcome of the competition between the natality and the mortality rates involved in the equations of the systems.

We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the alternating group. We identify the Galois groups of these 149—each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and is a first step toward the inverse Galois problem for Schubert calculus.

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.