Probleme deschise de fenomenologia lui Husserl prin prisma integrării sale în modelul ontologic informaţional propus de Mihai Drăgănescu

May another reality exist so as to generate our physical world? Both Mihai Drăgănescu and Edmund Husserl said yes. But there is a main difference between those two approaches. So, Romanian scholar Mihai Drăgănescu proposed an ontological model with strong phenomenological character in which information and material principles are at the same level. Instead Husserl proposed an idealist (transcendental) solution. In this respect Drăgănescu’s model seems to be more general and consistent. Also Mihai Drăgănescu says that Husserl’s Phenomenology can be integrated in his model. But for this some problems appeared. Our present work is dedicated to identifying such problems. In subsequent material we will analyze how these problems can be solved.

An Extremal Graph Problem with a Transcendental Solution

We prove that the number of multigraphs with vertex set {1, . . .,n} such that every four vertices span at most nine edges isan2+o(n2)whereais transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov, who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy–Schwarz arguments.Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.

Relations between Solutions of Differential Equations and Small Functions

We investigate relations between solutions, their derivatives of differential equationf(k)+Ak−1f(k−1)+⋯+A1f’+A0f=0, and functions of small growth, whereAj (j=0,1,…,k−1)are entire functions of finite order. By these relations, we see that every transcendental solution and its derivative of above equation have infinitely many fixed points.

LOWER ESTIMATES FOR THE GROWTH OF THE FOURTH AND THE SECOND PAINLEVÉ TRANSCENDENTS

Let $w(z)$ be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in $|z|\le r$, it is shown that $n(r,w)\gg r^2$ (respectively, $n(r,w)\gg r^{3/2}$), from which the growth estimate $T(r,w)\gg r^2$ (respectively, $T(r,w)\gg r^{3/2}$) immediately follows.AMS 2000 Mathematics subject classification: Primary 34M55; 34M10

Three Theorems on the Growth of Entire Transcendental Solutions of Algebraic Differential Equations

G. Polya [4] has posed the problem as to whether there are entire transcendental functions of order zero satisfying an algebraic differential equation with rational coefficients. G. Polya himself showed that this is impossible for a first order algebraic differential equation. The general problem is now completely solved. G. Valiron demonstrated an example of a third order algebraic differential equation with an entire transcendental solution of order zero (Theorem 1); V. V. Zimogljad (Theorem 2) proved that every entire transcendental solution of a second order algebraic differential equation is of a positive order. It seems to us expedient to bring these results all together. We give here a proof of Theorem 2 different from and in our view simpler than that of V. V. Zimogljad. Theorem 3 refines the results of G. Polya (and of others, see for example [10]) and establishes an exact lower bound for the order of an arbitrary entire transcendental solution satisfying a first order algebraic differential equation.