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Ju. T. Lisica and the author have defined in [4] strong homology groups HP(X;G) of inverse systems of spaces X={XX, pxx-, A) over directed cofinite sets A (every element AeA has only finitelymany predecessors). It was shown in [5] that these groups are functors on the coherent prohomotopy category CPHTop, introduced in [2] and [3]. The notion of strong or Steenrod homology H£(X; G) of an arbitrary space X was then defined [1], [6] and shown to be a functor on the strong shape category SSh [2], [3]. The procedure consisted in choosing a cofiniteANR-resolution p:X->X of Z([7], [8], [9]) and of defining HSV{X; G) as HV{X;G). That the group HSP{X; G) does not depend on the choice of the resolution is a consequence of the following factorizationtheorem ([3], Theorem II.2.3). If p:X->Xi$ a resolution and f:X-*Y is a coherent map into a cofiniteANR-system, then there exists a unique coherent homotopy class of coherent maps g: X―> Y such that gp and / are coherently homotopic. The definition of composition in CPHTop and the proof of the factorization theorem essentiallyused the assumption that the index sets A be cofinite. On the other hand, the construction of the homology groups HP(X; G) did not require thisassumption. Therefore, it remained unclear whether one can use also non-cofmite ANR-resolutions to determine the homology groups Hp{X; G) of the space X. To prove that thisis indeed the case is the main purpose of this paper. Such an information can prove useful in situations where a non-cofmite ANR-resolution naturally arises. The main idea of the proof is to replace a given ANR-resolution p: X―>X by a cofinite ANR-resolution p*: X-*X* using the "trick" described in ([9], Theorem I, 1.2). What remains to be done is to exhibit a natural isomorphism m* : HP(X; G)-^HP(X*; G). The correct formula for u* is easily found. However, the formula for the inverse v* of u* is less obvious. Even more complicated is the verificationof the two equalities u^v^―1, v*u*=l. In order to simplify notations throughout the paper we omit the coefficient groups G, although all results hold for an arbitrary G.

The characterization of the harmonic, Einstein and super-Einstein spaces bymeans of the firstor the second mean values (or the relations between them) for small geodesic spheres in a Riemannian manifold is recently studied by many authors ([3], [7], [10], [17], etc.). Among them, 0. Kowalski [10] characterized the three spaces by the degree of concordance of the two mean values in some sense, proposing new classes of spaces which should be located between the harmonic and the super-Einstein spaces. On the other hand M. Pinsky [12] verified that the stochastic mean values are also useful for the characterization of the Einstein spaces. In this paper, we study the above three mean values more in detail and fillthe blanks in the previous works (Theorem 2 below). The main tool for our proof is Schauder's estimate, which enables us to treat C°°manifolds even more easily than Cauchy-Kowalewski's method for C° manifolds used in most of the previous papers. We also introduce some other new conditions which also characterize the above three spaces, that is, the conditions (M2), (M4) and (L2)-(L4) (see section 2 for the definitions). The condition (M2) is a variation of (M3) given in [3]. But it seems to be more natural than (M3) in connection with (Ml). The conditions (M4) and (L4) are closely related to Helgason's expansion in [8: p. 435]. Indeed, the above three mean values coincide with Helgason's, if the manifold is an Euclidean space or a globally symmetric space of rank one, and we are interested in to what extent the Laplacian determines the mean values. Our results include the assertion; one of the operators induced by the three mean values is expanded by means of a sequence of polynomials of Laplacian if and only if the manifold is harmonic (Theorem 2 (1)). As a by-product, we also obtain some sufficientconditions for a C6 manifold to be analytic (Theorem 1). In the course of our proof, we partially answer for smooth manifolds to Kowalski's conjecture given in [10] (Theorem 3).

The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study quasi-Einstein spacetimes. Some basic geometric properties of such a spacetime are obtained. The applications of quasi-Einstein spacetimes in general relativity and cosmology are investigated. Finally, the existence of such spacetimes are ensured by several interesting examples.

Let $\ell_{2}(\tau)$ be the Hilbert space with weight $\tau$ and $\ell_{2}^{f}$ be the linear span of the canonical orthonormal basis of the separable Hilbert space $\ell_{2}$. In this paper, we prove that if a metric space $X$ ishomeomorphic to $\ell_{2}(\tau)$ or $\ell_{2](\tau) \times \ell_{2}^{f}$ then the hyperspace $Fin_{H}(X)$ of non-empty finite subsets of $X$ with the Hausdorff metric is homeomorphic to $\ell_{2}(\tau) \times \ell_{2}^{f}$.

An Einstein-Weyl structure is a natural generalization of an Einstein structure within the framework of conformal geometry. We are interested in considering an Einstein-Weyl structure on a CR manifold. A CR manifold has a conformal structure only on its hyperdistribution. In this paper, on a CR manifold we naturally define an Einstein-Weyl structure closely related to the conformal structure on the hyperdistribution.

In this paper, we propose a new control method for two-wheeled non-holonomic mobile robots. Since the non-holonomic mobile robots has non-integrable constraint conditions, it is difficult to derive a control law so that the robots are convergent to the target state. The proposed method designs a nonlinear regulator carrying out asymptotic convergence of non-holonomic mobile robots to a given trajectory family. The trajectory family is chosen so that it passes through the origin and its gradient at the origin is equal to zero. As the result, the mobile robots are convergent to the origin as the target state while generating their path. We name this method Path-generating Regulator. Neither the coordinate transformation nor the input transformation is needed in this method unlike other methods. Global asymptotic regulation property of this method is proven by a Lyapunov-based stability analysis. By numerical simulations and experiments, we show the validity of the proposed method.

In this paper, we investigate an isometric immersion of a compact connected Riemannian manifold $M$ into a Euclidean space and a sphere such that every geodesic in $M$ is closed and viewed as a helix (of general order) in the ambient space.

where U, V are mXn real matrices, tr denotes the trace of a corresponding square matrix and G runs through allmXn integral matrices. We write simply 0F,u,v(Z) for the theta series 0f,u,v(Z;0) when 0 is of order 0. For congruence subgroups of SL^{Z) the transformation formulas for theta series of degree 1 associated with F are well known. For example, we can find transformation formulas for theta series of degree 1 in [7],[8],in which multipliers are explicitly determined. Transformation formulas for the theta series 0f,u,v{Z',O) of degree n>l are also established in [1] in the case where F is even and U, V are zero (the condition on U, V is not necessary if 0 is of order 0 [9]). Using these results we can get many examples of Siegel modular forms for congruence subgroups. In this paper we determine a transformation formula for the theta series 0f,u,v(Z;0) associated with a positive integral symmetric matrix F and any real matrices U, V and using this, we get some examples of cusp forms for some congruence subgroups F' of Spn(Z). Cusp forms of weight n + 1 for Fr induce differentialforms of the first kind on the nonsingular model of the modular function fieldwith respect to /''. Our result shows that the geometric genus of the nonsingular model of the modular function fieldwith respect to F' is positive.

Here, P1 is the complex projective line. The purpose of this paper is to give affirmative answers to the conjecture under various conditions. It is separated into 3 parts. In Part 1, we assume that n=p is a prime number and obtain a result. Recently, Kato [2] has improved this result extensively. In Part 2, we assume that f(x) is a polynomial of degree p with p a prime number, and obtain a result. In Part 3, we assume

Recently, Solomon has introduced a zeta function which counts sublattices of a given lattice over an order ([5]). Let us recall the definition of this zeta function. Let I be a (finitedimensional) semisimple algebra over the rational fieldQ or over the j!?-adicfieldQp, and let A be an order in I. A is a Z-order when I is a Q-algebra, while A is a Zp-order when 2 is a Qp-algebra, where Zp is the ring of p-adic integers. Throughout this paper, p stands for a rational prime and the subscript p indicates the p-dAic completion. Let V be a finitelygenerated left I'-module, and let L be a full//-lattice in V. Solomon's zeta function is defined as