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1.
Let be a singular cardinal of regular uncountable cofinality. Let {(): < } be a continuous increasing sequence withlimit , and let =()+(), < be regular cardinals. Let I be a normal ideal on , and assume that the reduced product</I admits a cofinal -scale of ordinal functions. Then +, where =||||I is the I-norm of . 相似文献
2.
The semilinear elliptic eigenvalue problem with superlinearpure power nonlinearity is considered. This problem is treatedfrom the standpoint of L2-theory and the precise asymptoticformula for the eigenvalue parameter = () as is established,where is the L2-norm of the solution u associated with . 2000Mathematics Subject Classification 35P30 (primary), 35J60 (secondary). 相似文献
3.
For Sturm-Liouville problems on [a, ) with a regular -dependentboundary condition at a, and the limit point case at , a techniqueof W. N. Everitt [1] is employed to obtain asymptotic formulaefor the associated m()-functions on rays and lines in the complex-plane. The method relies on asymptotic formulae for solutionsof the initial value problem for u'+qu = u, as || ,which the author has given in [4]. For the case of the regularleft endpoint, the asymptotic formulae on vertical lines sufficeto provide a direct proof of the formula for the total variationof the associated spectral function, a question which the authorhad raised in [3; Remark 5.2]. 相似文献
4.
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
5.
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
6.
A Schwarz Lemma for the Symmetrized Bidisc 总被引:1,自引:0,他引:1
Let be an analytic function from D to the symmetrized bidisc
We show that if (0) = (0,0) and () = (s, p) in the interiorof , then
Moreover, the inequality is sharp: we give an explicit formulafor a suitable in the event that the inequality holds withequality. We show further that the inverse hyperbolic tangentof the left-hand side of the inequality is equal to both theCaratheodory distance and the Kobayashi distance from (0,0)to (s, p) in int 相似文献
7.
Uncountable Saturated Structures have the Small Index Property 总被引:1,自引:0,他引:1
We prove the following theorem. Let m be an uncountable saturatedstructure of cardinality = < and assume that G is a subgroupof Aut (m) whose index is less than or equal to . Then thereexists a subset A of cardinality strictly less than such thatevery automorphism of m leaving A pointwise fixed is in G. 相似文献
8.
Spaces Universal under Closed Embeddings for Finite-Dimensional Complete Metric Spaces 总被引:1,自引:0,他引:1
We shall prove that for every natural number n and every cardinalnumber there exists an n-dimensional complete metric spaceXn, of weight such that every n-dimensional complete metricspace of weight is embeddable in Xn, as a closed subset. 相似文献
9.
El Amin Kaidi; Campoy Antonio Morales; Palacios Angel Rodriguez 《Bulletin London Mathematical Society》2000,32(6):703-708
We prove that if A is a prime non-commutative JB*-algebra whichis neither quadratic nor commutative, then there exist a primeC*-algebra B and a real number with < 1 such that A =B as involutive Banach spaces, and the product of A is relatedto that of B (denoted by , say) by means of the equality xy= xy+(1 )yx. 2000 Mathematics Subject Classification46K70, 46L70. 相似文献
10.
Let = 2cos (/5) and let []. Denote the normaliser ofG0() of the Hecke group G5 in PSL2() by N(G0()). Then N(G0())= G0(/h), where h is the largest divisor of 4 such that h2 divides. Further, N(G0())/G0() is either 1 (if h = 1), 2 x 2 (if h= 2) or 4 x 4 (if h = 4). 相似文献
11.
Logarithmic Convexity for Supremum Norms of Harmonic Functions 总被引:1,自引:0,他引:1
We prove the following convexity property for supremum normsof harmonic functions. Let be a domain in Rn, 0 and E a subdomainand a compact sebset of ,respectively. Then there exists a constant = (E, 0, ) (0, 1) such that for all harmonic functions u on, the inequality
is valid.The case of concentric balls E plays a key role in the proof.For positive harmonic funcitons ono osuch balls, we determinethe sharp constant in the inequlity. 相似文献
12.
Soient F un corps commutatif localement compact non archimédienet un caractère additif non trivial de F. Soient unereprésentation du groupe de WeilDeligne de F,et sa contragrédiente. Nous calculons le facteur (, , ). De manière analogue, nous calculons le facteur (x, , ) pour toute représentationadmissible irréductible de GLn(F). En conséquence,si F est de caractéristique nulle et si et se correspondentpar la correspondance de Langlands construite par M. Harris,ou celle construite par les auteurs, alors les facteurs (, , s) et (x, , s) sont égaux pour tout nombre complexe s. Let F be a non-Archimedean local field and a non-trivial additivecharacter of F. Let be a representation of the WeilDelignegroup of F and its contragredient representation. We compute (, , ). Analogously, we compute (x, , ) for all irreducible admissible representations of GLn(F).Consequently, if F has characteristic zero, and , correspondvia the Langlands correspondence established by M. Harris orthe correspondence constructed by the authors, then we have(, , s) = (x, , s) for all sC. 1991 Mathematics Subject Classification22E50. 相似文献
13.
Let X be a compact space,µ a Borel probability measureon X, T: X X a measure preserving continuous transformationand g: X R a continuous function. Then for some yX,
This Lemma is used to give an alternative proof of a resultby Ruzsa [6], which implies the following extension of a resultof Bergelson [1]. If E N satisfies
then there exists a set N such that n1|[1,n]| (E) for all, n 1, and any finite subset{1, ... k} satisfies Ø.
7 Moria St., Ramat Hasharon, Israel 相似文献
14.
1. Definition of the A-polynomial The A-polynomial was introduced in [3] (see also [5]), and wepresent an alternative definition here. Let M be a compact 3-manifoldwith boundary a torus T. Pick a basis , µ of 1T, whichwe shall refer to as the longitude and meridian. Consider thesubset RU of the affine algebraic variety R = Hom (1M, SL2C)having the property that () and (µ) are upper triangular.This is an algebraic subset of R, since one just adds equationsstating that the bottom-left entries in certain matrices arezero. There is a well-defined eigenvalue map
given by taking the top-left entries of () and (µ).1991 Mathematics Subject Classification 57M25, 57M50. 相似文献
15.
Metric Entropy of Convex Hulls in Hilbert Spaces 总被引:2,自引:0,他引:2
We show in this note the following statement which is an improvementover a result of R. M. Dudley and which is also of independentinterest. Let X be a set of a Hilbert space with the propertythat there are constants , >0, and for each n N, the setX can be covered by at most n balls of radius n. Then,for each n N, the convex hull of X can be covered by 2n ballsof radius . The estimate is best possible for all n N, apart from the value c=c(, , X).In other words, let N(, X), >0, be the minimal number ofballs of radius covering the set X. Then the above result isequivalent to saying that if N(, X)=O(1/) as 0, thenfor the convex hull conv (X) of X, N(, conv (X)) =O(exp(2/(12))). Moreover, we give an interplay between several coveringparameters based on coverings by balls (entropy numbers) andcoverings by cylindrical sets (Kolmogorov numbers). 1991 MathematicsSubject Classification 41A46. 相似文献
16.
Packing, Tiling, Orthogonality and Completeness 总被引:3,自引:0,他引:3
Let Rd be an open set of measure 1. An open set DRd is calleda tight orthogonal packing region for if DDdoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of Rd. The main contribution of this paperis a new way of proving the following result: D tiles Rd whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L2(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in Rd, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about packing Rd withtranslates of a certain non-negative function and, additionally,we have completeness of E in L2() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70. 相似文献
17.
We prove that the crossed product C*-algebra C*r(, ) of a freegroup with its boundary sits naturally between the reducedgroup C*-algebra C*r and its injective envelope I(C*r). In otherwords, we have natural inclusion C*r C*r(, ) I(C*r) of C*-algebras. 相似文献
18.
Let T be a Riesz operator on a Banach space and let {n} be anenumeration of the nonzero eigenvalues of T, each repeated accordingto algebraic multiplicity. Then a sufficient condition thatT admit a West decomposition T = C + Q, where C is compact andQ is quasinilpotent, is that n[n] < . Furthermore, C andQ can be chosen to have a quasinilpotent commutator. 相似文献
19.
Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05. 相似文献
20.
The Symmetrized Bidisc and Lempert's Theorem 总被引:2,自引:0,他引:2
Let G C2 be the open symmetrized bidisc, namely G = {(1 + 2,12) : |1| < 1, |2| < 1}. In this paper, a proof is giventhat G is not biholomorphic to any convex domain in C2. By combiningthis result with earlier work of Agler and Young, the authorshows that G is a bounded domain on which the Carathéodorydistance and the Kobayashi distance coincide, but which is notbiholomorphic to a convex set. 2000 Mathematics Subject Classification32F45 (primary), 15A18 (secondary). 相似文献