The Cauchy problem for the semilinear complex Ginzburg-Landau type equation is considered in homogeneous and isotropic spacetime. Global solutions and their asymptotic behaviours for small initial data are obtained. The non-existence of non-trivial global solutions is also shown. The effects of spatial expansion and contraction are studied through the problem.
Let (X, L) denote a polarized manifold of dimension five. This study considers the dimension of the global sections of KX + mL with m ≥ 6. In particular, we prove that h0(KX+ mL) ≥ (m−15) for any polarized 5-fold (X, L) with h0(L) > 0. Furthermore, we also consider (X, L) with h0 (KX + mL) = (m−15) for some m ≥ 6 with h0(L) > 0.
Let X be a compact metric space and E be a Banach space. Then lip (X, E) denotes the Banach space of all E-valued little Lipschitz functions on X. We show that lip(X, E)∗∗ is isometrically isomorphic to the Banach space of E∗∗-valued Lipschitz functions Lip(X, E∗∗) under several conditions. Moreover, we describe the isometric isomorphism from lip(X, E)∗∗ to Lip(X, E∗∗).
Let Ek (z) be the normalized Eisenstein series of weight k for the full modular group SL(2, ℤ). Let a > 0 be an even integer. In this paper we completely determine when the zeros of Ek interlace with the zeros of Ek+a. This generalizes a result of Nozaki on the interlacing of zeros of Ek and Ek+12.
We show that a knot whose minimum crossing number c(K) is even and greater than 30 is not fertile; there exists a knot K′with crossing number less than c such that K′is not obtained from a minimum crossing number diagram of K by suitably changing the over-under information.
We study the Hartree-Fock equation and the Hartree-Fock energy functional universally used in many-electron problems. We prove that the set of all critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold is finite. Since the Hartree-Fock equation, which is the corresponding Euler-Lagrange equation, is a system of nonlinear eigenvalue problems, the spectral theory for linear operators is not applicable. The present result is obtained by establishing the finiteness of the critical values associated with orbital energies less than a negative constant and combining the result with Koopmans' well-known theorem. The main ingredients are the proof of convergence of the solutions and the analysis of the Fréchet second derivative of the functional at the limit point.
We define a new kind of classical digamma function, and establish some of its fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler-type sums. The main results of Flajolet and Salvy's paper (Expo. Math. 7(1) (1998), 15-35) are immediate corollaries of the main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
We study the cohomology groups of vector bundles on neighborhoods of non-pluriharmonic loci in q-complete Kähler manifolds and in compact Kähler manifolds. Applying our results, we show variants of the Lefschetz hyperplane theorem.
We show that, based on Grabner's recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth r that have all Fourier coefficients integral for each of r = 1, 2, 3 and 4, and partly classifies them, where the classification is complete for r = 2, 3 and 4. In fact, we show that there exists no normalized extremal quasimodular forms of depth four with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.