We show that, if a continuous Zd or Zd+-action of a compact metric space has the almost product property (which is weaker than the specification property), then, for any continuous function and any invariant measure, the topological pressure of the set of all generic points coincides with the sum of the metric entropy and the mean of the continuous function.
We extend the definition of zeta function and zeta polynomial to codes definedover finite rings with respect to a specified weight function. Moreover, we also investigate the Riemann hypothesis analogue for Type IV codes over any of the rings Z4, F2 + uF2 and F2 + νF2. Although, for small lengths, there are only a few actual Type IV codes over Z4, F2 + uF2 or F2 + νF2 that satisfy the Hamming distance upper bound 2(1 + n/6), we will show that zeta polynomials corresponding to these weight enumerators that meet this bound satisfy the Riemann hypothesis analogue property.
The sheet number of a 2-knot is an analogous quantity to the crossing number of a 1-knot. We prove that (i) a 2-knot is trivial if and only if the sheet number is equal to one, and (ii) there is no 2-knot with the sheet number two.
We consider a class of characteristic initial value problems on the complex projective plane. We give as initial functions on a line the Gauss hypergeometric function, and investigate what kind of solutions are obtained.We make use of the theory of the classical Euler-Poisson-Darboux equation to get an integral representation of the solution. The family of solutions contains Appell’s two-dimensional hypergeometric functions F1 and F2.
Horocyclic surfaces are surfaces in hyperbolic 3-space that are foliated by horocycles. We construct horocyclic surfaces associated with spacelike curves in the lightcone and investigate their geometric properties. In particular, we classify their singularities using invariants of corresponding spacelike curves.
Let M be a real hypersurface of class C2 in Cn, n ≥ 2, and let δM(z) be the Euclidean distance from z ∈ Cn to M. In this paper we give an explicit representation in complex tangential direction for the Levi form of the function δM (or -log δM) by Hermitian and symmetric matrices determined by a local defining function of M. As its application, we also show that, if M is defined by a C2-function &rho with d&rho ≠ 0, then the function -log δM is strictly plurisubharmonic in complex tangential direction near M if and only if M is Levi-flat and the symmetric matrix (δ2&rho/δzi δzj ) of degree n has maximal rank n - 1 on the complex tangent subspace T z1,0 (M) ⊂ T z1,0 (Cn) for each z ∈ M as a linear map. Moreover, we can get directly the well-known Levi condition as the condition in order that -log δM is weakly plurisubharmonic in complex tangential direction near and in one side of M.
This paper is concerned with the regularity of weak solutions to the general nonlinear sub-elliptic systems related to Hörmander’s vector fields. Based on the reverse Hölder inequality established by Gianazza, the Lp estimates for weak solutions to the systems under the super-quadratic controllable growth condition and the natural growth condition are established, respectively. Then we obtain the Hölder continuity for weak solutions.
We present a solvable two-dimensional piecewise linear chaotic map that arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the non-trivial ultradiscrete limit of the solution in spite of a problem known as ‘the minus-sign problem.’
A Wiener integral approach to the non-commutative harmonic oscillator, i.e. an approach via Brownian motions and matrix-valued stochastic differential equations, will be given. Such an approach is a continuation and an extension of the one made by the author in an earlier paper (Kyushu J. Math. 62 (2008), 63-68).