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1.
For each d2 we construct a connected open set Rd such that = int (clos()), and for each k 1 and each p [1, ), the subsetWk, () fails to be dense in the Sobolev space Wk, p(), in thenorm of Wk, p(). 1991 Mathematics Subject Classification 46E35,46F05. 相似文献
2.
Quasi-Affinity in certain Classes of Operators 总被引:1,自引:0,他引:1
The family of operators S + V (, C, Re > 0), where V isan injective S-Volterra operator (that is, [S, V[ = V2) and A V1 generates a uniformly bounded C0-semigroup,is studied in the context of similarity and of the weaker quasi-affinityrelation. It is shown that S is similar to S + V for all , C,Re > 1, and is a quasi-affine transform of S + tV for allt 0 and 0 < < 1. 相似文献
3.
The Representation of Some Integers as a Subset Sum 总被引:1,自引:0,他引:1
Let A N. The cardinality (the sum of the elements) of A willbe denoted by |A| ((A)). Let m N and p be a prime. Let A {1, 2,...,p}. We prove thefollowing results. If |A| [(p+m2)/m]+m, then for every integer x such that0 x p 1, there is B A such that |B| = m and (B) x mod p. Moreover, the bound is attained. If |A| [(p+m2)/m]+m!, then there is B A such that |B| 0 mod m and (B) = (m 1)!p. If |A| [(p + 1)/3]+29, then for every even integer x such that4p s x p(p + 170)/48, there is S A such that x = (S). In particular,for every even integer a 2 such that p 192a 170, thereare an integer j 0 and S A such that (S) = aj+1. 相似文献
4.
Let X be a complex Banach space and let J:XX* be a duality sectionon X (that is, x,J(x)=||J(x)||||x||=||J(x)||2)=||x||2). Forany unit vector x and any (C0) contraction semigroup T={etA:t0}, Goldstein proved that if X is a Hilbert space and |T(t)x,j(x)|1 as t, then x is an eigenvector of A corresponding toa purel imaginary eigenvalue. In this article, we prove thata similar result holds if X is a strictly convex complex Banachspace. 相似文献
5.
Let be a bounded connected open set in RN, N 2, and let 0be the Dirichlet Laplacian defined in L2(). Let > 0 be thesmallest eigenvalue of , and let > 0 be its correspondingeigenfunction, normalized by ||||2 = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying
Let 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||2=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying
1991 Mathematics SubjectClassification 47F05. 相似文献
6.
Let B denote an infinite sequence of positive integers b1 <b2 < ..., and let denote the exponent of convergence ofthe series n = 1 1/bn; that is, = inf {s 0 : n = 1 1/bns <}. Define E(B) = {x [0, 1]: an(x) B (n 1) and an(x) asn }. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221–227]proved the inequality dimH E(B) /2 and conjectured (see ibid.,p. 225 and [T. W. Cusick, Quart. J. Math. Oxford (2) 41 (1990)p. 278]) that equality holds. In this paper, we give a positiveanswer to this conjecture. 相似文献
7.
Let be Fejér's sine polynomial. We prove the following statements.
- The inequality holds for all x, y (0, ) with x + y < if and only if 0 and + rß 1.
- The converse of the above inequality is valid for allx, y (0, ) with x + y < if and only if 0 and + rß 1.
- For all n N and x, y [0, ] we have . Both bounds are best possible.
8.
This paper treats the problem of minimizing the norm of vectorfields in L1 with prescribed divergence. The ridge of . playsan important role in the analysis, and in the case where R2is a polygonal domain, the ridge is thoroughly analysed andsome examples are presented. In the case where Rn is a Lipschitzdomain and the divergence is a finite positive Borel measure,the infimum is calculated, and it is shown that if an extremalexists, then it is of the form 1 = Fd, where F is a nonnegativefunction and d(x) is the distance from x to the boundary .Finally, if R2 is a polygonal domain and the measure is representedby a nonnegative continuous function, then an explicit expressionfor the extremal is given, and it is proven that this extremalis unique. 相似文献
9.
Let f be a continuous self-map of the unit circle, S1. The -limitpoints (x) of a point x are the set of all limit points of thesequence of iterates of f acting on x. We shall show that theset of all -limit points xS1(x) a closed set in S1. 相似文献
10.
Let be a finite-dimensional reductive Lie algebra of rank lover an algebraically closed field of characteristic zero. Givenx , we denote by x the centraliser of x in . It was conjecturedby Premet that the algebra (x)x of x-invariants is a gradedpolynomial algebra in l variables. In this note, we show thatthis conjecture does not hold for the minimal nilpotent orbitin the simple Lie algebra of type E8. As a consequence, a conjectureof Joseph on the semi-invariants of (bi)parabolics is not trueeither. 相似文献
11.
Bull London Math. Soc, 4 (1972), 370372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat vm1 ... = 1, where v is the canonical isomorphism(Am, Im, m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order pm for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that pm m = . Proof. If (b) is false for a given m, then there exists an element G, of order pr for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (pm) G pm G). There exists an opensubgroup Go of G such that (G0) = 0 and has order pr in G/G0.Choose H to be the subgroup of G generated by , and then aneasy application to G/G0 of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k1 F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'ab/k2) whose image $$\stackrel{\¯}{\sigma}$$ in Gal (k1/k2) generates this last group. The error occursin the statement that the canonical map v : AP A acts on pointsby sending ap a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that p = 1 (in Gal (F'/k2)). By applying Lemma 2 to G = Gal (F'ab/k2) and the map G Gal(k1/k2) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that pm = 1, for some m, andG = G' x H where G' acts trivially on k1 and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k1 k2is a cyclic Galois extension of degree pm. In the first case, we let K F'ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k2. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\¯}{\sigma}$$, I$$\stackrel{\¯}{\sigma }$$, $$\stackrel{\¯}{\sigma}$$) be an isomorphism defined over k1 and let v ... p1 = µ(R). If is replaced by for some Autk1((A, I, )) then is replacedby P. Thus, as µ(R) is finite, we may assume that pm1= 1 for some m. Choose K, as in (b), to be of degree pm overk2. Let m be a generator of Gal (K/k2) whose restriction tok1 is $$\stackrel{\¯}{\sigma }$$. Then : (A, I, ) (A$$\stackrel{\¯}{\sigma }$$, I$$\stackrel{\¯}{\sigma}$$, $$\stackrel{\¯}{\sigma }$$ = (A$$\stackrel{\¯}{\sigma}$$m, I$$\stackrel{\¯}{\sigma }$$m, $$\stackrel{\¯}{\sigma}$$m is an isomorphism defined over K and v mpm1, ... m =pm1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk2 which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of R*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F). 相似文献
12.
In this paper we study several kinds of maximal almost disjointfamilies. In the main result of this paper we show that forsuccessor cardinals , there is an unexpected connection betweeninvariants ae(), b() and a certain cardinal invariant md(+)on +. As a corollary we get for example the following result.For a successor cardinal , even assuming that < = and 2= +, the following is not provable in ZermeloFraenkelset theory. There is a +-cc poset which does not collapse andwhich forces a() = + < ae() = ++ = 2. We also apply the ideasfrom the proofs of these results to study a = a() and non(M).2000 Mathematics Subject Classification 03E17 (primary), 03E05(secondary). 相似文献
13.
Let f be a unit vector and T = {T(t) = etA: t 0} be a (C0)contraction semigroup generated by A on a complex Hilbert spaceX. If |T(t)f,f| 1 as t then f is an eigenvector of A correspondingto a purely imaginary eigenvalue. If one allows X to be a Banachspace, the same situation can be considered by replacing T(t)f,fby (T(t)f) where is a unit vector in X* dual to f. If |(T(t)f)| 1, as t , is f an eigenvector of A? The answer is sometimesyes and sometimes no. 相似文献
14.
On the Distribution of Denominators in Sylvester Expansions 总被引:1,自引:0,他引:1
For any x (0, 1], let the series be the Sylvester expansion of x. Galambos has shown that theLebesgue measure of the set [formula] is 1 when = e, the base of the natural logarithm. This paperprovides a proof that for any 1, A() has Hausdorff dimension1 when e. 2000 Mathematics Subject Classification 11K55 (primary),28A80 (secondary). 相似文献
15.
Banks W. D.; Garaev M. Z.; Heath-Brown D. R.; Shparlinski I. E. 《Bulletin London Mathematical Society》2008,40(1):88-96
We show that for any fixed > 0, there are numbers >0 and p0 2 with the following property: for every prime p p0 and every integer N such that p1/(4e )+ N p, the sequence1, 2, ..., N contains at least N quadratic non-residues modulop. We use this result to obtain strong upper bounds on the sizesof the least quadratic non-residues in Beatty and Piatetski-Shapirosequences. 相似文献
16.
A Relation between Hausdorff Dimension and a Condition on Time Sets for the Image by the Brownian Sheet to Possess Interior Points 总被引:2,自引:0,他引:2
We prove that if WN, d is a Brownian sheet mapping to Rd and E is a set in (0, )N of Hausdorff dimensiongreater than , then for almost every rotation about a point x and translation x such that x(E) (0, )N, the set x(E) is such that almost surely W(E) containsinterior points. The techniques are adapted from Kahane andRosen and generalize to higher dimensional time and range. 相似文献
17.
Let be a projective unitary representation of a countable groupG on a separable Hilbert space H. If the set B of Bessel vectorsfor is dense in H, then for any vector x H the analysis operatorx makes sense as a densely defined operator from B to 2(G)-space.Two vectors x and y are called -orthogonal if the range spacesof x and y are orthogonal, and they are -weakly equivalent ifthe closures of the ranges of x and y are the same. These propertiesare characterized in terms of the commutant of the representation.It is proved that a natural geometric invariant (the orthogonalityindex) of the representation agrees with the cyclic multiplicityof the commutant of (G). These results are then applied to Gaborsystems. A sample result is an alternate proof of the knowntheorem that a Gabor sequence is complete in L2( d) ifand only if the corresponding adjoint Gabor sequence is 2-linearlyindependent. Some other applications are also discussed. 相似文献
18.
Volume of a small Extrinsic Ball in a Submanifold 总被引:1,自引:0,他引:1
For a submanifold Mp R, we determine a two-term asymptoticformula for vol (Mp B(x)) for x Mp as 0. The second termis a quadratic curvature invariant of the second fundamentalform of the imbedding. Imbedded spheres are characterized amongcompact hypersurfaces by this term. 相似文献
19.
Let be an infinite cardinal and let G = 2. Now let β Gbe the Stone–ech compactification of G as a discrete semigroup,and let =<cβ G {xG\{0}:minsupp (x)}. We show that thesemigroup contains no nontrivial finite group. 相似文献
20.
Professor W. F. Hammond has kindly drawn my attention to a blunderin 4 of the above paper. He referred to the ( 2r) xß submatrix D of the skew-symmetric matrix displayednear the top of page 181, of which it is asserted that it issquare and non-singular, and pointed out that, from the factthat the matrix of which D forms part is regular, it may onlybe deduced that the columns of D are linearly independent; thatis, it only follows that 2r ß. The validity of the equation 2r = ß is essentialto the succeeding argument and, fortunately, may be establishedby alternative means. Using the nomenclature of the paper, wehave on F the set 1*, ..., 2r*, 1*, ..., ß* of independent3-cycles (independent because they cut independent 1-cycleson the curve C), which may be completed, to form a basis forsuch cycles on F, by a further set 1', ..., 2q2rpof independent 3-cycles, each of which meets C in a cycle homologousto zero on C. The cycles 1*, ..., * are invariant cycles andare independent on F so that, if > 2r + ß, thereis a non-trivial linear combination * of these having zero intersectionon C with each of the cycles 1*, ..., 2r*, 1*, ..., ß*.Thus we have. (* .k*)c = 0 = (* .i*)c i.e. (* .k*) = 0 = (* .i* on F (1 k 2r; 1 i ß). Furthermore, (j . C) 0 on C and we have (* .j .C)C = 0 i.e. (* .j) = 0 on F (1 j 2q 2r ß). It now follows that * 0 on F (for it has zero intersectionwith every member of a basic set of 3-cycles on F). But thiscondradicts the assumption that * is a non-trivial linear combinationof the independent cycles 1*, ...,*; and hence < 2r + ß. 相似文献