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1.
A bifurcation problem governed by the boundary condition II   总被引:1,自引:0,他引:1  
In this work we consider the problem u = a(x)up in on , where is a smooth bounded domain, isthe outward unit normal to , is regarded as a parameter and0 < p < 1. We consider both cases where a(x) > 0 in or a(x) is allowed to vanish in a whole subdomain 0 of . Ourmain results include existence of non-negative non-trivial solutionsin the range 0 < < 1, where 1 is characterized by meansof an eigenvalue problem, uniqueness and bifurcation from infinityof such solutions for small , and the appearance of dead coresfor large enough .  相似文献   

2.
In this paper we find the multiplicities dim L() where is an arbitrary root and L() is an irreducible SLn-module withhighest weight . We provide different bases of the correspondingweight spaces and outline some applications to the symmetricgroups. In particular we describe certain composition multiplicitiesin the modular branching rule. 1991 Mathematics Subject Classification:20C05, 20G05.  相似文献   

3.
For a scattering system {A, A0} consisting of self-adjoint extensionsA and A0 of a symmetric operator A with finite deficiency indices,the scattering matrix {S()} and a spectral shift function arecalculated in terms of the Weyl function associated with a boundarytriplet for A*, and a simple proof of the Krein–Birmanformula is given. The results are applied to singular Sturm–Liouvilleoperators with scalar and matrix potentials, to Dirac operatorsand to Schrödinger operators with point interactions.  相似文献   

4.
We calculate the triple correlations for the truncated divisorsum R(n). The R(n) behave over certain averages just as theprime counting von Mangoldt function (n) does or is conjecturedto do. We also calculate the mixed (with a factor of (n)) correlations.The results for the moments up to the third degree, and thereforethe implications for the distribution of primes in short intervals,are the same as those we obtained (in the first paper with thistitle) by using the simpler approximation R(n). However, whenR(n) is used, the error in the singular series approximationis often much smaller than what R(n) allows. Assuming the GeneralizedRiemann Hypothesis (GRH) for Dirichlet L-functions, we obtainan ±-result for the variation of the error term in theprime number theorem. Formerly, our knowledge under GRH wasrestricted to -results for the absolute value of this variation.An important ingredient in the last part of this work is a recentresult due to Montgomery and Soundararajan which makes it possiblefor us to dispense with a large error term in the evaluationof a certain singular series average. We believe that our resultson the sums R(n) and R(n) can be employed in diverse problemsconcerning primes.  相似文献   

5.
Let F be a non-Archimedean local field, with the ring of integersoF. Let G = GLN(F), K = GLN (oF), and be a supercuspidal representationof G. We show that there exists a unique irreducible smoothrepresentation of K, such that the restriction to K of a smoothirreducible representation ' of G contains if and only if 'is isomorphic to ° det, where is an unramified quasicharacterof Fx. Moreover, we show that contains with the multiplicity1. As a corollary we obtain a kind of inertial local Langlandscorrespondence. 2000 Mathematics Subject Classification 22E50.  相似文献   

6.
Dynamics of projective morphisms having identical canonical heights   总被引:1,自引:0,他引:1  
Let , :N N be morphisms of degree at least 2 whose canonicalheights and are identical. We draw various conclusions aboutthe Green functions, Julia sets, and canonical local heightsof and . We use this information to completely characterize and in the following cases: (i) and are polynomial mapsin one variable; (ii) is the dth-power map; (iii) is a Lattèsmap.  相似文献   

7.
The main part of the paper deals with local existence and globalexistence versus blow-up for solutions of the Laplace equationin bounded domains with a non-linear dynamical boundary condition.More precisely, we study the problem consisting in: (1) theLaplace equation in (0, ) x ; (2) a homogeneous Dirichlet condition(0, ) x 0; (3) the dynamical boundary condition ; (4) the initial condition u(0, x) = u0 (x) on . Here is a regular and bounded domain in Rn, with n 1, and0 and 1 endow a measurable partition of . Moreover, m>1,2 p < r, where r = 2 (n – 1) / (n – 2) whenn 3, r = when n = 1,2, and u0 H1/2 , u0 = 0 on 0. The final part of the paper deals with a refinement of a globalnon-existence result by Levine, Park and Serrin, which is appliedto the previous problem. 2000 Mathematics Subject Classification35K55 (primary), 35K90, 35K77 (secondary).  相似文献   

8.
Let G be an almost simple algebraic group defined over Fp forsome prime p. Denote by G1 the first Frobenius kernel in G andlet T be a maximal torus. In this paper we study certain Jantzentype filtrations on various modules in the representation theoryof G1T. We have such filtrations on the baby Verma modules Z,where is a character of T. They are obtained via a certaindeformation of the natural homomorphism from Z into its contravariantdual Z. Using the same deformation we construct for each projectiveG1T-module Q a filtration of the vector space . We then prove that this filtration may also bedescribed in terms of the above-mentioned homomorphism Z() Z() and this leads us to a sum formula for our filtrations.When Q is indecomposable with highest weight in the bottom alcove(with respect to some special point) we are able to computethe filtrations on F(Q) explicitly for all . This is then thestarting point of an induction which proceeds via wall crossingsto higher alcoves. If our filtrations behave as expected undersuch wall crossings then we obtain a precise relation betweenthedimensions of the layers in the filtrations of F(Q) for an arbitraryindecomposable projective Q and the coefficients in the correspondingKazhdan–Lusztig polynomials. We conclude the paper byproving that the above results in the G1T theory have some analoguesin the representation theory of G (where, however, we have towork with representations of bounded highest weights) and thecorresponding theory for quantum groups at roots of unity. Theseresults extend previous work by the first author. 2000 MathematicsSubject Classification: 20G05, 20G10, 17B37.  相似文献   

9.
We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set RN. We use topological methods to establish the existenceof two connected sets D± of positive/negative solutionsin R x W2, p RN where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.  相似文献   

10.
We investigate the existence of a weak solution u to the quasilineartwo-point boundary value problem We assume that 1 < p < p ¬ = 2, 0 < a < , andthat f L1(0,a) is a given function. The number k stands forthe k-th eigenvalue of the one-dimensional p-Laplacian. Letp p x/a) denote the eigenfunction associated with 1; then p(kp x/a) is the eigenfunction associated with k. We show the existenceof solutions to (P) in the following cases. (i) When k=1 and f satisfies the orthogonality condition the set of solutions is bounded. (ii) If k=1 and ft L1(0,a) is a continuous family parametrizedby t [0,1], with then there exists some t* [0,1] such that (P) has a solutionfor f = ft*. Moreover, an appropriate choice of t* yields asolution u with an arbitrarily large L1(0,a)-norm which meansthat such f cannot be orthogonal to pp x/a. (iii) When k 2 and f satisfies a set of orthogonality conditionsto p(k p x/a) on the subintervals , again, the set of solutions is bounded. is a continuous family satisfying either or another related condition, then there exists some t* [0,1]such that (P) has a solution for f = ft*. Prüfer's transformation plays the key role in our proofs.2000 Mathematical Subject Classification: primary 34B16, 47J10;secondary 34L40, 47H30.  相似文献   

11.
Let (G)>0 be a family of ‘-thin’ Riemannian manifoldsmodeled on a finite metric graph G, for example, the -neighborhoodof an embedding of G in some Euclidean space with straight edges.We study the asymptotic behavior of the spectrum of the Laplace–Beltramioperator on G, as 0, for various boundary conditions. We obtaincomplete asymptotic expansions for the kth eigenvalue and theeigenfunctions, uniformly for kC–1, in terms of scatteringdata on a non-compact limit space. We then use this to determinethe quantum graph which is to be regarded as the limit object,in a spectral sense, of the family (G). Our method is a directconstruction of approximate eigenfunctions from the scatteringand graph data, and the use of a priori estimates to show thatall eigenfunctions are obtained in this way.  相似文献   

12.
Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball   总被引:1,自引:0,他引:1  
Weighted Triebel–Lizorkin and Besov spaces on the unitball Bd in d with weights wµ(x)=(1–|x|2)µ–1/2,µ0, are introduced and explored. A decomposition schemeis developed in terms of almost exponentially localized polynomialelements (needlets) {}, {} and it is shown that the membershipof a distribution to the weighted Triebel–Lizorkin orBesov spaces can be determined by the size of the needlet coefficients{f, } in appropriate sequence spaces.  相似文献   

13.
We prove that the Novikov assembly map for a group factorizes,in ‘low homological degree’, through the algebraicK-theory of its integral group ring. In homological degree 2,this answers a question posed by N. Higson and P. Julg. As adirect application, we prove that if is torsion-free and satisfiesthe Baum-Connes conjecture, then the homology group H1(; Z)injects in and in , for any ring A such that . If moreover B is of dimension lessthan or equal to 4, then we show that H2(; Z) injects in and in , where A is as before, and 2 is generated by the Steinberg symbols{,}, for . 2000 Mathematical Subject Classification: primary 19D55, 19Kxx,58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.  相似文献   

14.
The ideal space Id(A) of a Banach algebra A is studied as abitopological space Id(A), u, n, where u is the weakest topologyfor which all the norm functions I || a + I|| (with a A andI Id(A)) are upper semi-continuous, and n is the de Groot dualof u. When A is separable, nu is either a compact, metrizabletopology, or it is neither Hausdorff nor first countable. TAF-algebrasare shown to exhibit the first type of behaviour. Applicationsto Banach bundles (which motivate the study), and to PI-Banachalgebras, are given. 1991 Mathematics Subject Classification:46H10, 46J20.  相似文献   

15.
Unknotting Tunnels and Seifert Surfaces   总被引:2,自引:0,他引:2  
Let K be a knot with an unknotting tunnel and suppose thatK is not a 2-bridge knot. There is an invariant = p/q Q/2Z,with p odd, defined for the pair (K, ). The invariant has interesting geometric properties. It is oftenstraightforward to calculate; for example, for K a torus knotand an annulus-spanning arc, (K, ) = 1. Although is definedabstractly, it is naturally revealed when K is put in thinposition. If 1 then there is a minimal-genus Seifert surfaceF for K such that the tunnel can be slid and isotoped to lieon F. One consequence is that if (K, ) 1 then K > 1. Thisconfirms a conjecture of Goda and Teragaito for pairs (K, )with (K, ) 1. 2000 Mathematics Subject Classification 57M25,57M27.  相似文献   

16.
Grothendieck's Inequalities for Real and Complex JBW*-Triples   总被引:1,自引:0,他引:1  
We prove that, if and >0, if V and W are complex JBW*-triples (with preduals V* andW*, respectively), and if U is a separately weak*-continuousbilinear form on V x W, then there exist norm-one functionals1, 2 V* and 1, 2 W* satisfying for all (x, y) V x W. Here, for a norm-one functional on acomplex JB*-triple V, |·| stands for the prehilbertianseminorm on V associated to given by for all x W, where z V** satisfies z = |z| =1. We arrive at this form of ‘Grothendieck's inequality’through results of C.-H. Chu, B. Iochum, and G. Loupias, andan amended version of the ‘little Grothendieck's inequality’for complex JB*-triples due to T. Barton and Y. Friedman. Wealso obtain extensions of these results to the setting of realJB*-triples. 2000 Mathematical Subject Classification: 17C65,46K70, 46L05, 46L10, 46L70.  相似文献   

17.
For a finite abelian group G GL (n, ), we describe the coherent component Y of the moduli space of-stable McKay quiver representations. This is a not-necessarily-normaltoric variety that admits a projective birational morphism obtained by variation of GeometricInvariant Theory quotient. As a special case, this gives a newconstruction of Nakamura's G-Hilbert scheme HilbG that avoidsthe (typically highly singular) Hilbert scheme of |G|-pointsin . To conclude, we describe the toric fan of Y and hence calculate the quiver representationcorresponding to any point of Y.  相似文献   

18.
We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P1, ... , Pm} R[Y1, ... ,Y,X1, ... ,Xk], with degY(Pi) 2, degX(Pi) d, 1 i m. Let S R+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R+k Rk be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers Sx = –1(x) S is bounded by (2mkd)O(mk).  相似文献   

19.
We consider a Hamiltonian setup M, , H, L, , P, where M, isa symplectic manifold, L is a distribution of Lagrangian subspacesin M, P is a Lagrangian submanifold of M, H is a smooth time-dependentHamiltonian function on M, and :[a,b] M is an integral curveof the Hamiltonian flow starting at P. We do not require any convexity property of the Hamiltonianfunction H. Under the assumption that (b) is not P-focal, weintroduce the Maslov index imaslov of given in terms of thefirst relative homology group of the Lagrangian Grassmannian;under generic circumstances imaslov() is computed as a sortof algebraic count of the P-focal points along . We prove thefollowing version of the Index Theorem: under suitable hypotheses,the Morse index of the Lagrangian action functional restrictedto suitable variations of is equal to the sum of imaslov()and a convexity term of the Hamiltonian H relative to the submanifoldP. When the result is applied to the case of the cotangent bundleM = TM* of a semi-Riemannian manifold (M, g) and to the geodesicHamiltonian , we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesicswith variable endpoints in Riemannian geometry. 2000 MathematicalSubject Classification: 37J05, 53C22, 53C50, 53D12, 70H05.  相似文献   

20.
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   

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