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1.
The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u2 ? v2)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u2A(v) ? v2A(u))/(uA(v)2 ? vA(u)2), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u2A(v) ? v2A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A1, and B″(0) = 2B2 the following relation holds: (2B(u) + 3A1u)2 = 4A(u)3 ? (3A12 ? 8B2)u2A(u)2. To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.  相似文献   

2.
In the present paper we introduce a new definition for the Fourier space A (K) of a locally compact Hausdorff hypergroup K and prove that it is a Banach subspace of B (K). This definition coincides with that of Amini and Medghalchi in the case where K is a tensor hypergroup, and also with that of Vrem which is given only for compact hypergroups. We prove that Ap (K)* = PMq (K), where q is the exponent conjugate to p, in particular A (K)* = VN (K). Also we show that for Pontryagin hypergroups, A (K) = L2(K) * L2(K) = F (L1( )), where F stands for the Fourier transform on . Furthermore there is an equivalent norm on A (K) which makes A (K) into a Banach algebra isomorphic with L1( ). (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

3.
LetA=(A 1,...,A n ),B=(B 1,...,B n L(ℓ p ) n be arbitraryn-tuples of bounded linear operators on (ℓ p ), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ε a,b on the Calkin algebraC(ℓ p )≡L(ℓ p )/K(ℓ p ); , where quotient elements are denoted bys=S+K(ℓ p ) forSεL(ℓ p ). It is shown among other results that the kernel Ker(ε a,b ) is a non-separable subspace ofC(ℓ p ) whenever ε a,b fails to be one-one, while the quotient is non-separable whenever ε a,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A 1,...,A n }, {B 1,...,B n } needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces. Supported by the Academy of Finland, Project 32837.  相似文献   

4.
The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a Γ‐limit of three‐dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v : U → ?n, U ? ?n. We show that the L2‐distance of ?v from a single rotation matrix is bounded by a multiple of the L2‐distance from the group SO(n) of all rotations. © 2002 Wiley Periodicals, Inc.  相似文献   

5.
Summary LetI be an interval of the real line, and letA andB ben×n complex-valued, Lebesgue measurable, matrix functions defined onI such thatAL loc 1(I) andBL loc (I). Ifx=[x1x2x n ] t andu=[u 1 u 2u n ] t are column vectors defined onI such thatxAC loc 1anduL loc 1(I) then the linear control problem considered isx(t)=A(t) x(t)+B(t)u(t) (tI) wherex is the response, andu is the control. This paper is concerned with the problem of determining necessary and sufficient conditions onA andB to make (*) fully controllable onI, without departing from the basic requirementsAL loc 1 (I) andBL loc (I)Dedicated to Professor H.-W. Knobloch on the occasion of his sixtieth birthday  相似文献   

6.
The linear equation Δ2u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?2 has the particularly interesting nonlinear generalization Δg2u = 1, where Δg = e?2u Δ is the Laplace‐Beltrami operator for the metric g = e2ug0, with g0 the standard Euclidean metric on ?2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+Kg(x) exp(2u(x)) = 0 and Δ Kg(x) + exp(2u(x)) = 0, with x ∈ ?2, describing a conformally flat surface with a Gauss curvature function Kg that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ Kg exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K+L1(B1(y)), uniformly w.r.t. y ∈ ?2. We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.  相似文献   

7.
Let A and B be two n×n non-negative matrices. We write A ? B iff
u1(A ? B)u ? 0
for all column vectors u in Cn. Here u1 is the conjugate transpose of u. In this paper are stated equivalent conditions under which Ak ? Bk for all natural numbers k. The result is then generalized to Hermitian operators in a Hilbert space.  相似文献   

8.
We study some geometric properties associated with the t-geometric means A ?tB:= A1/2(A?1/2BA?1/2)tA1/2 of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted.  相似文献   

9.
Let \begin{align*}n\in\mathbb{N}\end{align*}, 0 <α,β,γ< 1. Define the random Kronecker graph K(n,α,γ,β) to be the graph with vertex set \begin{align*}\mathbb{Z}_2^n\end{align*}, where the probability that u is adjacent to v is given by pu,v u ? v γ( 1‐u )?( 1‐v )βnu ? v ‐( 1‐u )?( 1‐v ). This model has been shown to obey several useful properties of real‐world networks. We establish the asymptotic size of the giant component in the random Kronecker graph.© 2011 Wiley Periodicals, Inc. Random Struct. Alg.,2011  相似文献   

10.
We shall show an exact time interval for the existence of local strong solutions to the Keller‐Segel system with the initial data u0 in Ln /2w (?n), the weak Ln /2‐space on ?n. If ‖u0‖ is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in Ln /2w (?n) stems from obtaining a self‐similar solution which does not belong to any usual Lp(?n). Furthermore, the characterization of local existence of solutions gives us an explicit blow‐up rate of ‖u (t)‖ for n /2 < p < ∞ as tTmax, where Tmax denotes the maximal existence time (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

11.
Let K ? L be a field extension. Given K-subspaces A, B of L, we study the subspace ?AB? spanned by the product set AB = {abaA, bB}. We obtain some lower bounds on dim K ?AB? and dim K ?B n ? in terms of dim K A, dim K B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson.  相似文献   

12.
Let FA(Cn) denote the Fock space associated with a real linear transformation A on Cn which is symmetric and positive definite relative to the real inner product Rez,w〉, z,wCn. Let BA denote the Bargmann transform, mapping L2(Rn) unitarily onto FA(Cn). In this note, we show that one can find a group G, whose unitary irreducible representation at its base vector coincides with up to a constant multiple, where denotes the adjoint of BA and Kw denotes the reproducing kernel of FA(Cn).  相似文献   

13.
In this paper we give a numerical method to construct a rankm correctionBF (where then ×m matrixB is known and them ×n matrixF is to be found) to an ×n matrixA, in order to put all the eigenvalues ofA +BF at zero. This problem is known in the control literature as deadbeat control. Our method constructs, in a recursive manner, a unitary transformation yielding a coordinate system in which the matrixF is computed by merely solving a set of linear equations. Moreover, in this coordinate system one easily constructs the minimum norm solution to the problem. The coordinate system is related to the Krylov sequenceA –1 B,A –2 B,A –3 B, .... Partial results of numerical stability are also obtained.Dedicated to Professor Germund Dahlquist: on the occasion of his 60th birthday  相似文献   

14.
In this paper, we consider the p‐Laplacian equations in with supercritical growth where △ pu = div( | ? u | p ? 2 ? u),1 < p < N is the p‐Laplacian operator. Under certain assumptions on V (x) and f(u) that will be given in Section 1, we prove that the problem has at least a nontrivial solution by using variational methods combined with perturbation arguments. The solutions to subcritical p‐Laplacian equations are estimated applying the L norm. Copyright © 2012 John Wiley & Sons, Ltd.  相似文献   

15.
We consider the following modified version of the Banach-Mazur distance of convex bodies in \(\mathbb{R}^n :d\left( {K,L} \right) = \inf \left\{ {\left| \lambda \right|:\lambda \in \mathbb{R},\tilde K \subset \tilde L \subset \lambda \tilde K} \right\}\), where the infimum is taken over all non-degenerate affine images \(\tilde K\) and \(\tilde L\) of K and L. Gordon, Litvak, Meyer and Pajor in 2004 showed that for any two convex bodies d(K,L) ≤ n, moreover, if K is a simplex and L = ?L then d(K,L) = n. The following question arises naturally: Is equality only attained when one of the sets is a simplex? Leichtweiss in 1959, and later Palmon in 1992 proved that if d(K,B 2 n ) = n, where B 2 n is the Euclidean ball, then K is the simplex. We prove the affirmative answer to the question in the case when one of the bodies is strictly convex or smooth, thus obtaining a generalization of the result of Leichtweiss and Palmon.  相似文献   

16.
We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equation (1) with u: ?2 → ?. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data vn, we obtain a sequence of spatially highly oscillatory classical solutions un. By considering the Young measures (YMs) ν and µ generated by the sequences vn and un, respectively, as n → ∞, we derive a macroscopic evolution equation for the YM solution µ, and show exemplarily how such a measure‐valued equation can be exploited in order to obtain classical evolution equations for effective (macroscopic) quantities of the microstructure for suitable initial data vn and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.  相似文献   

17.
Let ‖·‖ be the Euclidean norm on R n and γn the (standard) Gaussian measure on R n with density (2π)n/2e. It is proved that there is a numerical constant c>0 with the following property: if K is an arbitrary convex body in R n with γn(K)≥1/2, then to each sequence u1,…,um∈ R n with ‖u1‖,…,‖um‖≤c there correspond signs ε1,…,εm=±1 such that ∑mi=1εiuiK. This improves the well-known result obtained by Spencer [Trans. Amer. Math. Soc. 289 , 679–705 (1985)] for the n-dimensional cube. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 351–360, 1998  相似文献   

18.
LetL 1(I, n,n M(I, n,m ) be the space of all pairs (A, B), whereA andB are measurable functions from a compact intervalI to n,n and n,m , respectively, andA is Lebesgue integrable. Also, let this space be endowed with the topology of theL 1-norm with respect toA and the topology of convergence in measure with respect toB. Then, the set of all pairs (A, B), for which the corresponding linear control system
  相似文献   

19.
Our aim in this note is to deal with boundary limits of monotone Sobolev functions with ▽u∈ Lp(·)logLq(·)(B) for the unit ball BRn. Here p(·) and q(·) are variable exponents satisfyingthe log-Hlder and the log log-Hlder conditions, respectively.  相似文献   

20.
Summary Letu be a real valued function on ann-dimensional Riemannian manifoldM n. We consider an inequality between theL q-norm ofu minus its mean value overM n and theL p-norm of the gradient ofu.The best constant in such inequality is exhibited in the following cases: i)M n is an open ball inIR n andp=1, 0<qn/(n–1); ii)M n is a sphere inIR n +1 and eitherp=1, 0<qn/(n–1) orp>n,q=.
Sunto Siau una funzione a valori reali dafinita su una varietà riemannianan-dimensionaleM n. Si considera una disuguaglianza tra la normaL q diu meno il suo valor medio suM n e la normaL p del gradiente diu.Si determina la costante ottimale in tale disuguaglianza nei seguenti casi: i)M n è un disco aperto inIR n ep=1, 0<qn/(n–1); ii)M n è una sfera inIR n +1 ep=1, 0<qn/(n–1) oppurep>n,q=.
  相似文献   

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