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]

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Eigenvector centrality is a linear algebra based graph invariant used in various rating systems such as webpage ratings for search engines. A generalization of the eigenvector centrality invariant is defined which is motivated by the need to design rating systems for bipartite graph models of time-sensitive and other processes. The linear algebra connection and some applications are described.

The problem of finding upper bounds for minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic 2-groups. We show that for any natural n ≥ 2 there is an undirected graph having 2 n + 6 vertices and automorphism group cyclic of order 2 n . This confirms an upper bound claimed by other authors for minimal number of vertices of undirected graphs having automorphism group Z/2 n Z.

In the modern study process it is important to teach pupils critical thinking and involvement in decision making. Formulation of negations and construction of counterexamples is one of the ingredients of critical thinking which are stressed in the new project of the mathematical standard for primary school „Skola 2030” in Latvia. The goal of this study is to analyze experience and skills of primary school pupils and students of teacher study programs, which are related to the ability to formulate negations and counterexamples. A qualitative and quantitative analysis of questionaire answers given by pupils and future teachers is performed in this study. Results of this study show that pupils make mistakes constructing negations and counterexamples. Teachers also have problems constructing correct assertions. These observations should stimulate universities to pay attention to teacher preparation in this sense. Teachers should teach correct usage of the negation operation at different levels of difficulty and correct construction of counterexamples.