§0 Introduction §1 First and Second Variational Formulas for Area §2 Bishop Comparison Theorem §3 Bochner-Weitzenböck Formulas §4 Laplacian Comparison Theorem §5 Poincaré Inequality and the First Eigenvalue §6 Gradient Estimate and Harnack Inequality §7 Mean Value Inequality §8 Reilly’s Formula and Applications §9 Isoperimetric Inequalities and Sobolev Inequalities §10 Lower Bounds of Isoperime...

We prove that the sequence of MacLane key polynomials constructed in [7] and [3] for a valuation extension (K, ν) ⊂ (K(x), μ) is finite, provided that both ν and μ are divisorial and μ is centered over an analytically irreducible local domain (R,m) ⊂ K[x]. As a corollary, we prove Izumi’s theorem on comparison of divisorial valuations. We give explicit bounds for the Izumi constant in terms of ...

It is well known that Barr and Beck’s definition of comonadic homology makes sense also with a functor of coefficients taking values in a semi-abelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Here we focus on independence of the chosen comonad: conditions for homology to depend on the induce...

We consider a single particle which is bound by a central potential and obeys the Dirac equation. We compare two cases in which the masses are the same but Va < Vb, where V is the time-component of a vector potential. We prove generally that for each discrete eigenvalue E whose corresponding (large and small) radial wave functions have no nodes, it necessarily follows that Ea < Eb. As an illust...

In this paper we prove a categorification of the Grothendieck–Riemann–Roch theorem. Our result implies in particular theorem for Toën and Vezzosi's secondary Chern character. As main application, establish comparison between Toën–Vezzosi character classical character, show that categorified recovers de Rham realization.

The Gerstenhaber and Schack cohomology comparison theorem asserts that there is a cochain equivalence between the Hochschild complex of a certain algebra and the usual singular cochain complex of a space. We show that this comparison theorem preserves the brace algebra structures. This result gives a structural reason for the recent results establishing fine topological structures on the Hochsc...

In the present paper, we study some properties of fuzzy norm of linear operators. At first the bounded inverse theorem on fuzzy normed linear spaces is investigated. Then, we prove Hahn Banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. Finally the set of all compact operators on these spaces is studied.

The question of completeness of Hamiltonian systems is investigated for a class of potentials not necessarily bounded below. The result generalizes previous work of W. Gordon and D. Ebin. This paper extends the completeness theorem of Ebin [1] to include certain potential functions V not necessarily bounded below. The condition on V is essentially the same as a condition for a corresponding qua...

If a central vector potential V(r,a) in the Dirac equation is monotonic in a parameter a, then a discrete eigenvalue E(a) is monotonic in a. For such a special class of comparisons, this generalizes an earlier comparison theorem that was restricted to node free states. Moreover, the present theorem applies to every discrete eigenvalue.