Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds

Given (M, g) a smooth compact Riemannian N-manifold, we prove that for any fixed positive integer K the problem

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds

Given (M, g) a smooth compact Riemannian N-manifold, N ≥ 2, we show that positive solutions to the problem

Peak point theorems for uniform algebras on smooth manifolds

It was once conjectured that if A is a uniform algebra on its maximal ideal space X, and if each point of X is a peak point for A, then A = C(X). This peak point conjecture was disproved by Brian Cole in 1968. However, Anderson and Izzo showed that the peak point conjecture does hold for uniform algebras generated by smooth functions on smooth two-manifolds with boundary. The corresponding assertion for smooth three-manifolds is false, but Anderson, Izzo, and Wermer established a peak point theorem for polynomial approximation on real-analytic three-manifolds with boundary. Here we establish a more general peak point theorem for real-analytic three-manifolds with boundary analogous to the two-dimensional result. We also show that if A is a counterexample to the peak point conjecture generated by smooth functions on a manifold of arbitrary dimension, then the essential set for A has empty interior.     

Function classes for double exponential integration formulas

The double exponential (DE) formulas for numerical integration are known to be highly efficient, more efficient than the single exponential (SE) formulas in many cases. Function classes suited to the SE formulas have already been investigated in the literature through rigorous mathematical analysis, whereas this is not the case with the DE formulas. This paper identifies function classes suited to the DE formulas in a way compatible with the existing theoretical results for the SE formulas. The DE formulas are good for more restricted classes of functions, but more efficient for such functions. Two concrete examples demonstrate the subtlety in the behavior of the DE formulas that is revealed by our theoretical analysis.     

Eigenvalues of $$(-\triangle + \frac{R}{2})$$ on manifolds with nonnegative curvature operator

In this paper, we show that the eigenvalues of are nondecreasing under the Ricci flow for manifolds with nonnegative curvature operator. Then we show that the only steady Ricci breather with nonnegative curvature operator is the trivial one which is Ricci-flat.     

A remark on roots of polynomials with positive coefficients

A short proof of a theorem of Dubickas on roots of polynomials with positive rational coefficients is presented.     

Convergence analysis of sectional methods for solving breakage population balance equations-II: the cell average technique

This work presents the convergence of the cell average technique (Kumar et al. in Powder Technol 179:205–228, 2007) for solving breakage population balance equation. Similarly to our paper Kumar and Warnecke (Numerische Math, 2008) of this series, we study convergence on four different types of meshes. A second order convergence is proved for uniform, locally uniform and non-uniform smooth meshes. Finally the scheme is analyzed on random mesh and it is found that the scheme is only first order accurate. Nevertheless we obtain for locally uniform as well as for random mesh one order higher accuracy than the fixed pivot technique discussed by the authors in the first paper. All mathematical observations of convergence analysis are also validated numerically and numerical results are compared with the results of the first part.     

Smooth circle actions on highly symmetric manifolds

We construct for the first time smooth circle actions on highly symmetric manifolds such as disks, spheres, and Euclidean spaces which contain two points with the same isotropy subgroup whose representations determined on the tangent spaces at the two points are not isomorphic to each other. This allows us to answer negatively a question of Hsiang and Hsiang [Some Problems in Differentiable Transformation Groups, Springer, Berlin, Problem 16, p. 228, 1968]. Dedicated to Prof. Yasuhiko Kitada on the occasion of his 60th birthday. Krzysztof Pawałowski was supported in part by the KBN Research Grant N 201 008 31/0524.     

Categoricity of computable infinitary theories

Computable structures of Scott rank are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of , this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank whose computable infinitary theories are each -categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank , which guarantee that the resulting structure is a model of an -categorical computable infinitary theory. Work on this paper began at the Workshop on Model Theory and Computable Structure Theory at University of Florida Gainesville, in February, 2007. The authors are grateful to the organizers of this workshop. They are also grateful for financial support from National Science Foundation grants DMS DMS 05-32644, DMS 05-5484. The second author is also grateful for the support of grants RFBR 08-01-00336 and NSc-335.2008.1.     

Convergence analysis of sectional methods for solving breakage population balance equations-I: the fixed pivot technique

In this work we study the convergence of the fixed pivot techniques (Kumar and Ramkrishna Chem. Eng. Sci. 51, 1311–1332, 1996) for breakage problems. In particular, the convergence is investigated on four different types of uniform and non-uniform meshes. It is shown that the fixed pivot technique is second order convergent on a uniform and non-uniform smooth meshes. Furthermore, it gives first order convergence on a locally uniform mesh. Finally the analysis shows that the method does not converge on a non-uniform random mesh. The mathematical results of convergence analysis are also validated numerically.     

Invariants of knot diagrams

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams. J. Hass was partially supported by an NSF grant.     

Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers for self-maps to an equivariant K-homology class. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in this case. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Lück and Rosenberg.     

Determination of holomorphic modular forms by primitive Fourier coefficients

We prove that Siegel modular forms of degree greater than one, integral weight and level N, with respect to a Dirichlet character of conductor are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of . The cases of other major types of holomorphic modular forms are included. The author is supported by the Grant-in-Aid for JSPS fellows.     

The complexity of the Specht modules corresponding to hook partitions

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E _s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E _s and of Specht modules corresponding to hook partitions μ restricted to E _s where s is the p-weight of μ.      

Normal Hopf subalgebras in cocycle deformations of finite groups

Let G be a finite group and let π : G → G′ be a surjective group homomorphism. Consider the cocycle deformation L = H ^σ of the Hopf algebra H = k ^G of k-valued linear functions on G, with respect to some convolution invertible 2-cocycle σ. The (normal) Hopf subalgebra corresponds to a Hopf subalgebra . Our main result is an explicit necessary and sufficient condition for the normality of L′ in L. This work was partially supported by CONICET, Fundación Antorchas, Agencia Córdoba Ciencia, ANPCyT and Secyt (UNC).     

PAC fields over finitely generated fields

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K. Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. This work constitutes a part of the Ph.D. dissertation of L. Bary-Soroker done at Tel Aviv University under the supervision of Prof. Dan Haran.     

Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

In this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non-homogeneous obstacles using convolution quadrature (CQ) techniques for the time variable and coupled boundary element method/finite element method for the space variable. The applicability of CQ to waves requires polynomial type bounds for operators related to the operator Δ − s ² in the right half complex plane. We propose a new systematic way of dealing with this problem, both at the continuous and semidiscrete-in-space cases. We apply the technique to three different situations: scattering by a group of sound-soft and -hard obstacles, by homogeneous and non-homogeneous obstacles.     

Universal Laurent series in finitely connected domains

Universal Laurent series where the universal approximation is valid on the boundary of the multiple connected open Ω appears for the first time in [2]. We show that it is possible to demand universal approximation only to a part of the boundary while on the remaining part the universal function can be smooth. Received: 29 May 2007     

Almost complex and complex structures on real projective Stiefel manifolds

We give a complete list of real projective Stiefel manifolds which admit almost complex structures and show that many of them are in fact complex manifolds. The first named author was supported in part by Grants 1/1486/94 and 2/1225/96 of VEGA (Slovakia) during the preparation of this work.