A fast algorithm for numerical solutions to Fortet's equation

A fast algorithm for computation of default times of multiple firms in a structural model is presented. The algorithm uses a multivariate extension of the Fortet's equation and the structure of Toeplitz matrices to significantly improve the computation time. In a financial market consisting of M1 firms and N discretization points in every dimension the algorithm uses O(nlogn·M·M!·N^M(M-1)/2) operations, where n is the number of discretization points in the time domain. The algorithm is applied to firm survival probability computation and zero coupon bond pricing.     

Isoperimetric inequalities,isometric actions and the higher Newman numbers

M will be a compact connected n-dimensional Riemannian manifold. If M contains a closed connected k-dimensional, 2 k < n, minimal immersed submanifold of M, we define the kth isoperimetric number of M, Ñ _k (M), as the infimum of the volumes of all such submanifolds. We obtain a number of interesting estimates for Ñ _k (M), for both general and special manifolds, which appear to be new.Next we turn to isometric actions and a 1931 theorem of M. H. A. Newman involving the size of orbits of group actions on manifolds. We introduce the higher Newman numbers N _k (M), 1 k n. Roughly speaking, if M admits isometric actions of compact connected Lie groups with k-dimensional principal orbits, N _k (M) is defined as the infimum over all such actions of the maximum volume of all maximal dimensional orbits. We observe that N _k (M) Ñ _k (M), 2 k < n, provided N _k (M) is defined; hence our prior estimates for the isoperimetric numbers of M apply directly to the higher Newman numbers.As a best possible candidate we conjecture that N _k (M) vol S ^k (i(M)/), 1 k n, where i(M) denotes the radius of injectivity of M and S ^k (i(M)/) denotes the standard k-sphere of radius i(M)/. We verify the conjecture for various special cases. We conclude the paper by studying Newman's theorem for compact connected Lie groups with invariant metrics and obtaining a lower bound for the size of small subgroups.     

On a conjecture by Erdös-Straus

Two algorithms have been constructed. The first is intended for obtaining such residue-classes represented by the numberN to the given modulusM, that for the primen N (modM) equation (1) is solvable in natural numbersx,y,z. Particularly, whenM=120120 (see Table 2) we obtain 198 suchN, i.e. the hypothesis indicated below is true with a probability greater than 0.99835. The second algorithm is intended for testing the conjecture by Erdös-Straus when 10⁷ <n10⁸.     

Polar actions on compact Euclidean hypersurfaces

Given an isometric immersion of a compact Riemannian manifold of dimension n ≥ 3 into Euclidean space of dimension n + 1, we prove that the identity component Iso ⁰(M ⁿ ) of the isometry group Iso(M ⁿ ) of M ⁿ admits an orthogonal representation such that for every . If G is a closed connected subgroup of Iso(M ⁿ ) acting polarly on M ⁿ , we prove that Φ(G) acts polarly on , and we obtain that f(M ⁿ ) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(G)-action. We also find several sufficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying warped product structure.      

Instability of the Rossby–Haurwitz wave in the invariant sets of perturbations

Stability of the Rossby–Haurwitz (RH) wave of subspace H₁H_n in an ideal incompressible fluid on a rotating sphere is analytically studied (H_n is the subspace of homogeneous spherical polynomials of degree n). It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant sets M₋ⁿ, M₊ⁿ, H_n and M₀ⁿ−H_n depending on the value of their mean spectral number χ(t)=η(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets M₋ⁿ and M₊ⁿ due to a hyperbolic dependence between K(t) and χ(t). A factor space with a factor norm of the perturbations is introduced using the invariant subspace H_n of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to H_n, the factor norm controls the perturbation part orthogonal to H_n. It is shown that in the set M₋ⁿ (χ(t)<n(n+1)), any nonzonal RH wave of subspace H₁H_n (n2) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M₀ⁿ−H_n. A necessary condition for this instability is given. The condition states that the spectral number χ(t) of the amplitude of each unstable mode must be equal to n(n+1), where n is the RH-wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave is shown. The instability in the invariant set M₊ⁿ of small-scale perturbations (χ(t)>n(n+1)) is still open problem.     

A metric on the manifold of immersions and its Riemannian curvature

E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n–1) dimensional manifold into ⁿ , and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation.     

Lefschetz Complex Conditions for Complex Manifolds

For any compact complex manifold M with a compatible symplectic form, we consider the homomorphisms L _1,0: H ^1,0(M) H ^{{n, n–1}(M) and L _{0, 1}: H ^{0, 1}(M) H ^{n – 1, n} (M) given by the cup product with [] ^{n – 1}, n being the complex dimension of M andH ^{*, *}(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L _{1, 0} (resp.L _{0, 1}) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology 27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L _{1, 0}characterizes those complex structures which are Abelian in the sense ofBarberis et al. [Ann. Global Anal. Geom. 13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1).     

Circular Chromatic Number and Mycielski Graphs

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by ^*(G). Later, Zhu called it circular chromatic number and denoted it by _c(G). Let (G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then _c(G)=(G). Denote by M(G) the Mycielski graph of G. Recursively define M^m(G)=M(M^m–1(G)). It was conjectured that if mn–2, then _c(M^m(K_n))=(M^m(K_n)). Suppose that G is a graph on n vertices. We prove that if , then _c(M(G))=(M(G)). Let S be the set of vertices of degree n–1 in G. It is proved that if |S| 3, then _c(M(G))=(M(G)), and if |S| 5, then _c(M²(G))=(M²(G)), which implies the known results of Chang, Huang, and Zhu that if n3, _c(M(K_n))=(M(K_n)), and if n5, then _c(M²(K_n))=(M²(K_n)).* Research supported by Grants from National Science Foundation of China and Chinese Academy of Sciences.     

The Maximum of a Critical Branching Wiener Process

Consider a critical branching Wiener process on ¹. Let M(n) be the location of the most right particle at time n. A limit distribution theorem is proved for n ^–1/2 M(n).     

Multipliers of a Family of Almost Modular Functions

The goal of this paper is to complete an investigation begun by Cohn and Knopp in their 1994 paper, Application of Dedekind eta-multipliers to modular equations. The paper concerned _k (z), a family of modular forms on ⁰(N) (N a positive integer) with possibly non-trivial multiplier systems. Cohn and Knopp defined new functions _k (z) and a new group containing ⁰(N) and proved that for all S in the larger group and for all k, _k (Sz) = M _k(S) _k (z), where M _k(S)²⁴ = 1. This yielded interesting invariance properties of _k , dependent on the values of M _k(S). Fixing a constant integer e, independent of k, Cohn and Knopp proved that for all k and all S in the larger group, M _k(S) ^e = (±1) ^e . They determined the sign of M _k(S) ^e in many, but not all, cases. In this paper, we give a complete determination of the values of M _k(S) ^e in the remaining cases.     

The non-cancellation group of a direct power of a (finite cyclic)-by-cyclic group

If M is a finitely generated group having a finite commutator subgroup, then the set (M) of all isomorphism classes of groups G such that G×M× is a finite set and coincides with the Mislin genus (M) of M if M is nilpotent. For such groups M, there is a group structure on (M) defined in terms of the indices of embeddings of G into M, for groups G representing elements of (M). Such embeddings do exist and their indices are necessarily finite. If M is nilpotent, then this group structure on (M) coincides with the Hilton-Mislin group structure on the genus of M. In this paper we calculate the group (H^k) where H^k is the direct product of k copies of a group the form H= a,b | aⁿ=1, bab^-1=a^u, for any relatively prime pair of natural numbers n,u. In particular we find that for each such group H we have an isomorphism (H²)(H^k) whenever k>2.The author wishes to acknowledge financial support from the National Research Foundation of South Africa.Mathematics Subject Classification (2000): 20E34, 20F28Revised version: 10 December 2003     

k-Weakly almost convex groups and π 1 ∞ \tilde M^3

We extend Cannon's notion ofk-almost convex groups which requires that for two pointsx, y on then-sphere in the Cayley graph which can be joined by a pathl ₁ of length k, there is a second pathl ₂ in then-ball, joiningx andy, of bounded length N(k). Ourk-weakly almost convexity relaxes this condition by requiring only thatl ₁ l ₂ bounds a disk of area C ₁(k)n ^{1 - (k)} +C ₂(k). IfM ³ is a closed 3-manifold with 3-weakly almost convex fundamental group, then ₁ .     

Uniqueness for the harmonic map flow in two dimensions

LetM be a two-dimensional Riemannian manifold with smooth (possibly empty) boundary. Ifu andv are weak solutions of the harmonic map flow inH ¹(M×[0,T]; S^N) whose energy is non-increasing in time and having the same initial data u₀ H¹(M,S^N) (and same boundary values H ^3/2(M; S^N) if M; S^N Ø) thenu=v.     

Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Let X _n1 ^* , ... X _nn ^* be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M _n ^* = \max\{X _n1 ^* , ... X _nn ^* }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M _n ^* and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let _n, _n N_n, , and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case _n , _n/_n > 0, as n .     

A note on quasi-periodic solutions of some elliptic systems

We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems u=f _x (u, y), whereu y ^M u(y) ^N ,f=f(x, y) is a real-analytic periodic function and is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT ^M +1. In the autonomous case,f=f(x), the above result holds for any .     

Many function-spaces are not normal if the domain is not compact

V. Neves [4] has proved that C(M, N) with Whitney's C-topology or Michor's extension of Schwartz's D-topology is not a normal topological space provided that M is not compact. This result was shown by giving a closed embedding of Van Douwen's non-normal space using means of non-standard analysis. In this paper we recover this theorem by standard-techniques and by working in the function-space itself instead of giving an embedding. A similar method is used to obtain the same result for various other function-spaces in the case that the domain is not compact: spaces of continuous functions and C ^k-functions with Whitney's topology and spaces of sections of arbitrary differentiability-classes. Even any subspace of these spaces with non-empty interior is not normal, for example the spaces of immersions, embeddings, Riemannian metrics and symplectic structures. This also answers an open problem posed by Hirsch [2].     

Reducible Dupin submanifolds

Pinkall's standard constructions for obtaining a Dupin hypersurface W in ^N from a Dupin hypersurface M in ⁿ , N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ^N with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ⁿ if and only if g=2.This research was supported by NSF Grant No. DMS 87-06015.     

Obstruction theory and coincidences of maps between nilmanifolds

Let f, g : M N be two maps between two compact nilmanifolds with dim M dim N = n. In this paper, we show that either the Nielsen coincidence number N(f, g) = 0 or N(f, g) = R(f, g) where R(f, g) denotes the Reidemeister number of f and g. Furthermore, we show that if N(f, g) > 0 then the primary obstruction o_n(f, g) to deforming f and g to be coincidence free on the n-th skeleton of M is non-trivial.Received: 30 April 2004; revised: 20 July 2004     

Non-linear Grassmannians as coadjoint orbits

For a given manifold M we consider the non-linear Grassmann manifold Gr _n (M) of n–dimensional submanifolds in M. A closed (n+2)–form on M gives rise to a closed 2–form on Gr _n (M). If the original form was integral, the 2–form will be the curvature of a principal S ¹ –bundle over Gr _n (M). Using this S ¹ –bundle one obtains central extensions for certain groups of diffeomorphisms of M. We can realize Gr _m–2 (M) as coadjoint orbits of the extended group of exact volume preserving diffeomorphisms and the symplectic Grassmannians SGr _2k (M) as coadjoint orbits in the group of Hamiltonian diffeomorphisms. Mathematics Subject Classification (2000):58B20Both authors are supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austrian Science Fund), project number P14195-MAT