We study the primitive integral representations of a quadratic form in n - 1 variables by a reduced quadratic form in n variables. Our aim is to describe the orbits of such representations under the action of the unit group of the reduced form. That description provides the mass of the representations, considered by Shimura. A formula for computation of the mass is also proved in the indefinite case.
Let P-2 = partial derivative/partial derivative t+ (-A)(2 )+ V-2 be the parabolic Schrodinger type operator on Rn+1(n >= 5), where the nonnegative potential V is independent oft and belongs to the reverse Holder class RHs(s >= n/2) and the Gaussian class associated with (-Delta)(2). In this paper, we will establish the Lp estimates for the Riesz transforms V2 alpha del jP( 2)(-beta) , where j = 0,1,2,3,0< alpha <= 1-j/4, j/4 < beta <= 1, and beta - alpha >= j/4.
Let g be a non-compact simple exceptional Lie algebra over R with an automorphism sigma of order four and h the fixed point set of sigma. Suppose that the dimension of the center of h is at most one and h(C) contains a Cartan subalgebra in g(C). In this paper we shall classify non-compact 4-symmetric pairs under the certain equivalence relation.
Applying a generalized do Carmo-Wallach theory based on a generalization of Theorem of Tsunero Takahashi, we classify harmonic totally real maps of the 3-sphere into the complex projective spaces. This means that we employ differential geometry of vector bundles with connections and construct the moduli spaces of those maps explicitly.
In this paper, we consider a positional numeration system in R-n called the rotational beta expansion. The expansion of an element z is an element of R-n is a sum of the form z = (beta M)(-1)d(1) + (beta M)(-2)d(2) + , where the radix is beta M for some fixed real number beta > 1 and matrix M is an element of O(n). We reformulate the rotational beta expansion where M is an element of SO(4) into the so-called q-expansion on the set H of real quaternions. In particular, we obtain necessary and sufficient conditions for the q-expansion of a quaternion to be periodic when the base q is a Pisot quaternion.
One of the open problems in Gorenstein homological algebra is: when is the class of Gorenstein injective modules closed under arbitrary direct limits? It is known that if the class of Gorenstein injective modules, $\mathcal{GI}$, is closed under direct limits, then the ring is noetherian. The open problem is whether or not the converse holds. We give equivalent characterizations of $\mathcal{GI}$ being closed under direct limits. More precisely, we show that the following statements are equivalent:\\ (1) The class of Gorenstein injective left $R$-modules is closed under direct limits.\\ (2) The ring $R$ is left noetherian and the character module of every Gorenstein injective left $R$-module is Gorenstein flat.\\ (3) The class of Gorenstein injective modules is covering and it is closed under pure quotients.\\ (4) $\mathcal{GI}$ is closed under pure submodules.
Let X and Y be horospherical Mori fibre spaces which are birational equivariantly with respect to the group action. Then, there is a horospherical Sarkisov program from X/S to Y /T .
We establish the L-P-L-q-boundedness of subelliptic pseudo-differential operators on a compact Lie group G. Effectively, we deal with the L-P-L-q-bounds for operators in the sub-Riemmanian setting because the subelliptic classes are associated to a Hormander sub-Laplacian. The Riemannian case associated with the Laplacian is also included as a special case. Then, applications to the L-P-L-q-boundedness of pseudo-differential operators in the Hormander classes on G are given in the complete range 0 <= delta <= rho <= 1, delta < 1. This also gives the L-P-L-q-bounds in the Riemannian setting, because the later classes are associated with the Laplacian on G. In both cases, in the Riemannian and the sub-Riemannian settings, necessary and sufficient conditions for the L-P-L-q-boundedness of operators are also analysed.
In this paper, we prove some new connectivity of the Julia sets J of the complex Henon maps H(x, y) = (x(2)+c+ay, ax) with sufficiently small |a|. We investigate the connectivity of J for the parameters near the boundary of the Mandelbrot set. We first give some conditions related to the connectivity of J for sufficiently small |a|, which are useful for considering the connectivity of J for the parameters near the boundary of the Mandelbrot set. We consider a perturbation {H-a,H-lambda t}(a is an element of D delta 0,0 <= tlambda(0)=exp(2 pi im/l) is an element of partial derivative D as t -> 0 and lambda(l)(t) can be represented by exp(L-t+i theta(t)) with L-t not equal 0 for 0
In this article, we will prove a finiteness theorem for nonconstant meromorphic functions satisfying the condition (C rho) on a complete Kahler manifold which share four distinct values.
By specializing regular polynomial with Galois group PGL(2, 7) and using Newton polygon technique, we construct PGL(2, 7)-extensions over Q unramified over their unique quadratic subfields. The Galois group over the quadratic field is a simple group PSL(2, 7).
We investigate the efficiencies and the relations of quandle invariants for knots. For example, we see that for any finitely generated connected quandle X, there exists a knot diagram which admits a surjective X-coloring. Also, we show the equivalence of shadow cocycle invariants and 3-cocycle invariants and the independence of homotopy invariants and non-abelian cocycle invariants.
This paper treats the initial-boundary value problem for a quasilinear parabolic system in a the construction process of parallel honeycombs in a beehive. After constructing the local strict solutions by using the theory of abstract parabolic equations, we will define the maximal strict solutions. Unfortunately, we cannot give any general sufficient conditions on the parameters or initial functions for global existence, but we can investigate asymptotic behaviors of the maximal solutions as t -> T-max. From numerical computations, we already have a number of examples which suggest the blowup of maximal solutions (i.e., T-max < infinity); at the end of the paper, we shall present one such numerical example.
We introduce random pre-branched Koch curves and construct loop-erased random walks on these graphs. We prove the existence of the scaling limit and show that the sample path of the limit process is almost surely self-avoiding, while it has box-counting dimension strictly greater than 1.
We show the existence of quadratic number fields possessing an everywhere unramified Galois extension with Galois group assumption of Bunyakovsky's conjecture. (A) over tilden, the double covering group of the alternating group, under the assumption of Bunyakovsky's conjecture.
In this paper, by utilizing some functional inequalities and combining with De Giorgi-Moser iteration technique, we obtain some estimates for the weighted L-infinity-norm of eigenfunctions of the Witten-Laplacian on smooth metric measure spaces, which enables us to establish the upper bounds for multiplicity of corresponding eigenvalues. We remark that those estimates are established without any assumption on curvature. As an important application, we prove the uniform convergence of the heat kernel of weighted parabolic equation on smooth metric measure spaces. In addition, our result remains ture for some isomorphism classes of weighted Riemannian manifold with boundary satisfying certain conditions. Furthermore, by the standard theory of elliptic and parabolic equations, we prove the regularity and uniqueness of parabolic heat kernel on smooth metric measure spaces.
Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in particular, to homology planes arising from arrangements of lines. In certain cases, the presentation is trivial, and we can obtain infinite new exotic algebraic and analytic structures on C(n )for n >= 3. We also find the first examples of homology planes of log-general type with an infinite fundamental group. The infiniteness can be obtained geometrically by orbifold morphisms to orbicurves.
The notion of F-Yang-Mills connections gives a generalization of Yang-Mills connections, p-Yang-Mills connections and exponential Yang-Mills connections. Here, F is a strictly increasing C2-function. In this paper, we study an instability for F-Yang-Mills connections on principal fiber bundles over irreducible symmetric R-spaces. In classical Yang-Mills theory, Simons showed that the non-existence theorem for non-flat, weakly stable Yang-Mills connections over the standard sphere with dimension more than four. Recently, a Simons type instability theorem for F-Yang-Mills connections over the standard sphere was given by Baba-Shintani. The purpose of this paper is to prove that the converse of this theorem does not hold in general. In fact, we give a concrete example of F-Yang-Mills instable, irreducible symmetric R-spaces except for the standard sphere. For this, we first give a sufficient condition for an irreducible symmetric R-space to be F-Yang-Mills instable. Next, by classifying the irreducible symmetric R-spaces satisfying this condition, we find that the standard sphere and the Cayley projective plane are only such irreducible symmetric R-spaces. In particular, the Cayley projective plane is F-Yang-Mills instable.
There are only finitely many alternating symmetric unions for a given partial knot. In this paper, we give a formula for the Q-polynomial of a knot with the symmetric union presentation D boolean OR D*(m) and show that, if 2degQ(D) > degQ(D boolean OR D*(infinity)), then there are only finitely many quasi-alternating knots with the symmetric union presentation D boolean OR D*(m) for any knot diagram D. We also give a formula for the Q-polynomial of a knot with the symmetric union presentation D boolean OR D*(m(1), m(2)).
On a contact Riemannian manifold which is compact and not assumed to be integrable, we intend to construct a parametrix for the Kohn-Rossi Laplacian. In particular, we will explicitly express the inverse of its principal part. Beals-Greiner constructed it in the case where the manifold is integrable. Our study depends heavily on theirs. We have some tools useful for the study in the non-integrable case, by means of which their results are extended to the general case and furthermore the inverse can be revealed more clearly.
