Svenonius’ definability theorem and its generalizations to the infinitary logic Lω1ω or to a second order logic with countable conjunctions and disjunctions have been studied by Kochen [1], Motohashi [2], [3] or Harnik and Makkai [4], independently. In this paper, we consider a (Svenonius-type) definability theorem for the intuitionistic predicate logic IL with equality.

In this paper we consider the initial value problems for the following quasilinear wave equations with dissipative terms

with initial conditions

where

Let f(z) be entire i.e. analytic in the finite whole plane Z. The order of f(z) is defined as

where A ray χ(θ) = {z = r·eiθ : 0 < r < + ∞} is called a Julia direction of f(z) if, in any open sector containing the ray, f(z) takes all values of Z, with at most one finite exceptional value, infinitely often.

Enriques and Fano [4], [5] classified all the maximal connected algebraic subgroups of Cr3. Our aim is to give modern and rigorous proofs to their results. In [10], we studied the primitive subgroups. In this paper, we deal with exceptional imprimitive groups. The imprimitivity is an analytic notion. The natural translation of the imprimitivity in algebraic geometry is the de Jonquières type operation (definition (2.1)). Every de Jonquières type operation is imprimitive. However, the difference of these notions is subtle.

We denote by Z + the semi-group of positive integers. For a subset E of Z +, we denote by |E| (≦ + ∞) its cardinal number and by E(n) the intersection of E and an interval [1, n) (n ≧ 1). We shall identify a subset E of Z + with a sequence, arranging elements of E according to their order.

The following two related problems in Commutative Algebra are certainly very popular:

1. 1. Study the permanence of properties from a ring A to an idealadic completion of A; and

2. 2. Study the lifting of properties from A/I to A where A is a ring which is complete and separated in the I-adic topology.

Let M be an n-dimensional, complete, connected and locally compact Riemannian manifold and g be its metric. Denote by ΔM the Laplacian on M.

Let M be an n-dimensional, complete, connected and non compact Riemannian manifold and g be its metric. ΔM denotes the Laplacian on M.

Let S(m), T(n) be positive definite integral matrices and suppose that T is represented by S over each p-adic integer ring Zp . We proved arithmetically in [3] that T is represented by S over Z provided that m ≥ 2n + 3 and the minimum of T is sufficiently large. This guarantees the existence of at least one representation but does not give any asymptotic formula for the number of representations. To get an asymptotic formula we must employ analytic methods.

Let P +(3) and P +(3) be the 3-dimensional space-time Poincaré group and the Poincaré subsemigroup, that is, P(3) = R 3 × s SU(1, 1) and P +(3) = V +(3)=S SU(1, 1) where The multiplication is defined by the formula (x, g)(x′, g′) = (x + g*−1x′g−1, gg′) for x, x′ ∈ R3 and g, g′ ∈ SU(l, 1). Here x = (x0, x1, x2 ) is identified with the matrix

The aim of this paper is to prove that irreducible unitary representations of the Poincaré group P = R 4 × SSL(2, C) are reducible as the representations of the Poincaré subsemigroup P + = V + × SSL(2, C) with

Let A be a commutative ring and D be a derivation of A into itself. If there exists a homomorphism E: A → A[[t]] such that

then we say that D is integrable. Integrable derivations have many good properties. In fact, most of unpleasant phenomena of derivations in characteristic p disappear if we consider integrable derivations only.

Let f(x1 …, xm ) be a quadratic form with integer coefficients and c ∈ Z. If f(x) = c has a solution over the real numbers and if f(x) ≡ c (mod N) is soluble for every modulus N, then at least some form h in the genus of f represents c. If m ≧ 4 one may further conclude that h belongs to the spinor genus of f. This does not hold when m = 3.

Let Ω be an open set in the upper half plane {y > 0}, whose boundary is denoted by ∂Ω. Let ∂Ω contain an open segment Γ lying on the x-axis.

We consider the following system of first order degenerating on y = 0:

where kj, μj are real constants and bjk are in L∞(Ω), further kj are non-negative integers.