Large-scale computation of pseudospectra using ARPACK and eigs

ARPACK and its MATLAB counterpart, eigs, are software packages that calculate some eigenvalues of a large non-symmetric matrix by Arnoldi iteration with implicit restarts. We show that at a small additional cost, which diminishes relatively as the matrix dimension increases, good estimates of pseudospectra in addition to eigenvalues can be obtained as a by-product. Thus in large-scale eigenvalue calculations it is feasible to obtain routinely not just eigenvalue approximations, but also information as to whether or not the eigenvalues are likely to be physically significant. Examples are presented for matrices with dimension up to 200,000

An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants

We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible SO(3) monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710032]. In this monograph, we prove --- modulo a gluing theorem which is an extension of our earlier work in [arXiv:math/9907107] --- that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten invariants of the four-manifold. This conclusion is analogous to the Kotschick-Morgan conjecture concerning the wall-crossing formula for Donaldson invariants of a four-manifold with $b_2^+=1$; that wall-crossing formula and the resulting structure of Donaldson invariants for four-manifolds with $b_2^+=1$ were established, assuming the Kotschick-Morgan conjecture, by Goettsche [arXiv:alg-geom/9506018] and Goettsche and Zagier [arXiv:alg-geom/9612020]. In this monograph, we reduce the proof of the Kotschick-Morgan conjecture to an extension of previously established gluing theorems for anti-self-dual SO(3) connections (see [arXiv:math/9812060] and references therein). Since the first version of our monograph was circulated, applications of our results have appeared in the proof of Property P for knots by Kronheimer and Mrowka [arXiv:math/0311489] and work of Sivek on Donaldson invariants for symplectic four-manifolds [arXiv:1301.0377].Comment: x + 229 page

Immunocompetence in Hydra. Epithelial cells recognize self-nonself and react against it

The evolution of effective immunologic defense mechanisms in multicellular organisms involves the ability of host cells to distinguish betweeen self and nonself and to react appropriately to eliminate foreign tissue. By producing interspecies grafts we have obtained evidence that immunorecognition followed by incompatibility reactions occur in Hydra. Our results demonstrate that epithelial cells of Hydra recognize and phagocytose foreign hydra cells, indicating that they are the effector cells in the incompatibility reactions. This observation is consistent with the idea that immunocompetence appeared early in the evolution of multicellular organisms

Cloned interstitial stem cells grow as contiguous patches in hydra

The migration of interstitial cells was analyzed during the growth of stem cell clones in vivo. The spatial distribution of cloned cells was analyzed at a time by which extensive migration of interstitial cells could have occurred. All interstitial cell clones were found to form large contiguous patches of cells. The results indicate that there is little migration of large interstitial cells in undisturbed tissue during normal growth. This finding is surprising since numerous grafting experiments have shown extensive migration of these cells. The implications of finding nonrandomly distributed stem cells are discussed

Male and female stem cells and sex reversal in Hydra polyps

Single interstitial stem cells of male polyps of Hydra magnipapillata give rise to clones that differentiate either male or female gametes. To test the sexual stability of these clones, stem cells were recloned. The results indicate that stem cells from female clones are stable in their sexual differentiation capacity; male stem cells, by comparison, switch sexual phenotype at the rate of 10-2 per cell per generation. As a result, female polyps contain only female stem cells; male polyps contain a mixture of male and female stem cells. A model is presented in which the sexual phenotype of Hydra polyps is controlled by (i) the switching rate of male and female stem cells and (ii) the repression of female differentiation by male stem cells

Growth regulation in Hydra. Relationship between epithelial cell cycle length and growth rate

The relationship between epithelial cell production and growth rate was investigated in Hydra attenuata under different feeding regimes. The increase of epithelial cell number was compared to the duration of the epithelial cell cycle using standard methods of cell cycle analysis. The results indicate that cell cycle changes accompanying changes in feeding regime are not sufficient to explain the altered growth rate. Under heavy feeding regimes, epithelial cell production equals tissue growth rate. At low feeding level or under starvation conditions the epithelial cell cycle lengthens and growth rate of epithelial cell population is slowed. However, the cell cycle changes are insufficient to account for the reduction in tissue growth and thus there is an effective overproduction of epithelial cells amounting to 10% per day. Evidence suggests that these excess cells are phagocytized by neighboring cells in the tissue. Thus phagocytosis is directly or indirectly involved in regulating the growth of hydra tissue