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The Nagata automorphism is shifted linearizable
A polynomial automorphism F is called shifted linearizable if there exists a linear map L such that LF is linearizable. We prove that the Nagata automorphism N:=(X−YΔ−ZΔ2,Y+ZΔ,Z) where Δ=XZ+Y2 is shifted linearizable. More precisely, defining L(a,b,c) as the diagonal linear map having a,b,c on its diagonal, we prove that if ac=b2, then L(a,b,c)N is linearizable if and only if bc≠1. We do this as part of a significantly larger theory: for example, any exponent of a homogeneous locally finite derivation is shifted linearizable. We pose the conjecture that the group generated by the linearizable automorphisms may generate the group of automorphisms, and explain why this is a natural question.
Quantities that frequency-dependent selection maximizes
We consider a model of frequency-dependent selection, to which we refer as the Wildcard Model, that accommodates as particular cases a number of diverse models of biologically specific situations. Two very different particular models (Lessard, 1984; Bürger, 2005; Schneider, 2006), subsumed by the Wildcard Model, have been shown in the past to have a Lyapunov functions (LF) under appropriate genetic assumptions. We show that the Wildcard Model: (i) in continuous time is a generalized gradient system for one locus, multiple alleles and for multiple loci, assuming linkage equilibrium, and its potential is a Lyapunov function; (ii) the LF of the particular models are special cases of the Wildcard Model's LF; (iii) the LF of the Wildcard Model can be derived from a LF previously identified for a model of density- and frequency- dependent selection due to Lotka-Volterra competition, with one locus, multiple alleles, multiple species and continuous-time dynamics (Matessi and Jayakar, 1981). We extend the LF with density and frequency dependence to a multilocus, linkage equilibrium dynamics.
A note on k[z]-automorphisms in two variables
We prove that for a polynomial f 2 k[x, y, z] equivalent are: (1)f is a k[z]-coordinate of k[z][x, y], and (2) k[x, y, z]/(f) = k[2] and f(x, y, a) is a coordinate in k[x, y] for some a 2 k. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate f 2 k[x, y, z] which is also a k(z)-coordinate, is a [z]-coordinate. We discuss a method for onstructing automorphisms of k[x, y, z], and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method essentially linking Nagata with a non-tame R-automorphism of R[x], where R = k[z]/(z2).
On the directionally Newton-non-degenerate singularities of complex hypersurfaces
We introduce a minimal generalization of Newton-non-degenerate singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called directionally Newton-non-degenerate if the local embedded topological singularity type can be restored from a collection of Newton diagrams. A singularity that is not directionally Newton-non-degenerate is called essentially Newton-degenerate . For plane curves we give an explicit and simple characterization of directionally Newton-non-degenerate singularities, for hypersurfaces we give some examples. Then we treat the question: is Newton-non-degenerate or directionally Newton-non-degenerate a property of singular types or of particular representatives. Namely, is the non-degeneracy preserved in an equi-singular family? This is proved for curves. For hypersurfaces we give an example of a Newton-non-degenerate hypersurface whose equi-singular deformation consists of essentially Newton-degenerate hypersurfaces. Finally, the classical formulas for the Milnor number (Kouchnirenko) and the zeta function (Varchenko) of the Newton-non-degenerate singularity are generalized to some classes of directionally Newton-non-degenerate singularities.
Heisenberg–Weyl algebra revisited: Combinatorics of words and paths
The Heisenberg–Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg–Weyl algebra, which offers novel perspectives, methods and applications.
On test sets for nonlinear integer maximization
A finite test set for an integer maximization problem enables us to verify whether a feasible point attains the global maximum. We estabish in the paper several general results that apply to integer maximization problems wthe monlinear objective functions.
Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay
We study positive bounded wave solutions u(t,x)=ϕ(ν⋅x+ct), ϕ(−∞)=0, of equation ut(t,x)=δu(t,x)−u(t,x)+g(u(t−h,x)), x∈Rm(*). It is supposed that Eq. (∗) has exactly two non-negative equilibria: u1≡0 and u2≡κ>9. The birth function g∈C(R+,R+) satisfies a few mild conditions: it is unimodal and differentiable at 0,κ. Some results also require the positive feedback of g:[g(maxg),maxg]→R+ with respect to κ. If additionally ϕ(+∞)=κ, the above wave solution u(t,x) is called a travelling front. We prove that every wave ϕ(ν⋅x+ct) is eventually monotone or slowly oscillating about κ. Furthermore, we indicate c∗∈R+∪+∞ such that (∗) does not have any travelling front (neither monotone nor non-monotone) propagating at velocity c>c∗. Our results are based on a detailed geometric description of the wave profile ϕ. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass G of ’asymmetric’ tent maps such that given g∈G, there exists exactly one travelling front for each fixed admissible speed.
Nonlinear optimization over a weighted independence system
We consider the problem of optimizing a nonlinear objective function over a weighted independence system presented by a linear-optimization oracle. We provide a polynomial-time algorithm that determines an r-best solution for nonlinear functions of the total weight of an independent set, where r is a constant that depends on certain Frobenius numbers of the individual weights and is independent of the size of the ground set. In contrast, we show that finding an optimal (0-best) solution requires exponential time even in a very special case of the problem.
The McKay conjecture for exceptional groups and odd primes
Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G ! G and G := GF , such that the root system is of exceptional type or G is a Suzuki-group or Steinberg’s triality group. We show that all irreducible characters of CG(S), the centraliser of S in G, extend to their inertia group in NG(S), where S is any F-stable Sylow torus of (G, F). Together with the work in [17] this implies that the McKay-conjecture is true for G and odd primes ` different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [15].
Geometric quantization of integrable systems with hyperbolic singularities
We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we compute the effect of hyperbolic singularities, which make an infinite-dimensional contribution to the quantization, thus showing that this quantization depends strongly on polarization.