Let be a finite group and let be a degree 1, -framed map such that and are simply connected, closed, oriented, smooth manifolds of dimension and such that the dimension of the singular set of the -space is at most . In the previous article, assuming is -connected, we defined the -equivariant surgery obstruction in a certain abelian group. There it was shown that if then is -framed cobordant to a homotopy equivalence . In the present article, we prove that the obstruction is a -framed cobordism invariant. Consequently, the -surgery obstruction is uniquely associated to above even if it is not -connected.
A commutative Banach algebra is said to have the property if the following holds: Let be a closed subspace of finite codimension such that, for every , the Gelfand transform has at least distinct zeros in , the maximal ideal space of . Then there exists a subset of of cardinality such that vanishes on , the set of common zeros of . In this paper we show that if is compact and nowhere dense, then , the uniform closure of the space of rational functions with poles off , has the property for all . We also investigate the property for the algebra of real continuous functions on a compact Hausdorff space.
Fix integers with k>0$"> and . Let be an integral projective curve with and a rank torsion free sheaf on which is a flat limit of a family of locally free sheaves on . Here we prove the existence of a rank subsheaf of such that . We show that for every there is an integral projective curve not Gorenstein, and a rank 2 torsion free sheaf on with no rank 1 subsheaf with . We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.
Representability of a finite relation algebra is determined by playing a certain two player game over `atomic -networks'. It can be shown that the second player in this game has a winning strategy if and only if is representable.
Let be a finite set of square tiles, where each edge of each tile has a colour. Suppose includes a special tile whose four edges are all the same colour, a colour not used by any other tile. The tiling problem we use is this: is it the case that for each tile there is a tiling of the plane using only tiles from in which edge colours of adjacent tiles match and with placed at ? It is not hard to show that this problem is undecidable.
From an instance of this tiling problem , we construct a finite relation algebra and show that the second player has a winning strategy in if and only if is a yes-instance. This reduces the tiling problem to the representation problem and proves the latter's undecidability.
We prove the existence of invariant projections from the Banach space of -pseudomeasures onto with for closed neutral subgroup of a locally compact group . As a main application we obtain that every closed neutral subgroup is a set of -synthesis in and in fact locally -Ditkin in . We also obtain an extension theorem concerning the Fourier algebra.
We examine the problem of factoring the th cyclotomic polynomial, over , and distinct primes. Given the traces of the roots of we construct the coefficients of in time . We demonstrate a deterministic algorithm for factoring in time when has precisely two irreducible factors. Finally, we present a deterministic algorithm for computing the sum of the irreducible factors of in time .
We consider real spaces only.
Definition. An operator between Banach spaces and is called a Hahn-Banach operator if for every isometric embedding of the space into a Banach space there exists a norm-preserving extension of to .
A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces such that there exists a Hahn-Banach operator of rank . The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman.
An -character of a group is the trace of an -representation of We show that all algebraic relations between -characters of can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space with We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of -representations of groups.
The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the -character variety of This paper provides a generalization of this result to all -character varieties.
In this paper we prove convergence and error estimates for the so-called 3-field formulation using piecewise linear finite elements stabilized with boundary bubbles. Optimal error bounds are proved in and in the broken norm for the internal variable , and in suitable weighted norms for the other variables and . 相似文献
We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.
(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .
(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .
The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.
We show that a simply connected homotopy associative and homotopy commutative mod -space with finitely generated mod cohomology is homotopy equivalent to a finite product of , , the three-connected cover and the homotopy fiber of the map for . Our result also shows that a connected -space in the sense of Sugawara with finitely generated mod cohomology has the homotopy type of a finite product of , and for .