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Abstract. In this paper we present several ODE systems encoding the most essential observations and assumptions about the complex hierarchical interactive processes of tumor neo-vascularization (angiogenesis). From experimental results we infer that a significant marker of tumor aggressiveness is the oscillatory behavior of tumor size, which is driven by its vascularization dynamics. To study the forces underlying these oscillations we perform a Hopf point analysis of the angiogenesis models. In the analyzed models Hopf points appear if and only if a nontrivial set of time-delays is introduced into the tumor proliferation or the neo-vascularization process. We suggest to examine in laboratory experiments how to employ these results for containing cancer growth.1. Introduction. Growth of malignant tumors beyond the diameter of 1 − 2mm critically depends on their neo-vascularization, which provides vital nutrients and growth factors, and also clears toxic waste products of cellular metabolism [12]. Indeed, the role of angiogenesis -the formation of new blood vessels by budding from existing ones -as a target for cancer therapy, is currently a focus of intensive research [12], [8], [19].In order to establish successful anti-angiogenic treatment rationale, the dynamics of angiogenesis must be better understood. These dynamics are very complex, involving many interacting oscillatory processes, which operate on several scales of time and space. Their essential constituents are briefly described below.Having reached a certain size and, therefore, a certain critical volume/surface ratio, a shortage of oxygen (denoted hypoxia) and nutrients is created within the tumor. Under hypoxia the tumor produces proteins, notably Vascular Endothelial Growth Factor (V EGF ). Increasing V EGF levels lead to increased proliferation and mobility of endothelial cells, and, as a result, to increased formation of immature vessels by these cells. Consequently the blood supply of the tumor is augmented, encouraging tumor proliferation [22]

The ability of a few stem-cells to repopulate a severely damaged bone marrow (BM) guarantees the stability of our physical existence, and facilitates successful BM transplantations. What are the basic properties of stem cells that enable the maintenance of the system's homeostasis? In the present work we attempt to answer this question by investigating a discrete (in time and phase-space) dynamical system. The model we present is shown to retrieve the essential properties of homeostasis, as reflected in BM functioning, namely, (a) to produce a constant amount of mature cells, and (b) to be capable of returning to this production after very large perturbations. The mechanism guaranteeing the fulfillment of these properties is extrinsic--negative feedback control in the micro-environment--and does not need additional stochastic assumptions. Nevertheless, the existence of a simple intrinsic control mechanism, a clock which determines the switch to differentiation, ascertains that the system does not admit non-trivial extinction states. This result may help clarifying some of the controversy about extrinsic versus intrinsic control over stem cell fate. It should be stressed that all conclusions are valid for any system containing progenitor and maturing cells.

We suggest a cohomological framework to describe groups of I-type and involutive Yang-Baxter groups. These groups are key in the study of involutive non-degenerate set-theoretic solutions of the quantum Yang-Baxter equation. Our main tool is a lifting criterion for 1-cocycles, established here in a general non-abelian setting.

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