[en] Cantor and generalized contunuum hypotheses are shown to be possibly factually false. Two new hyper-continuum hypotheses are posted to challenge pure mathematics and stir investigations of the abstract multispatial hyperspace, which is important for physical applications. Unlike the Cantorian hypotheses, however, the new ones conjecture existence of certain distinct, mixed kind of hyper-continuum that may serve as the manifold over which dual pairs of linear vector spaces with quite unrestricted abstract operations could be effectively constructed

[en] It is proved that an algebraic number field of type (q^s,q^s,...,q^s) has relative integral basis over any of its subfield under certain conditions. The conductor and discriminant are also determined using the construction of genus fields of abelian number fields. (author). 8 refs

[en] An elementary necessary condition is established for a prime integer of the form 8k + 1 to be congruent.

[en] By virtue of the normal ordering of vacuum projector we directly derive some new complicated operator identities, regarding to the generalized Stirling number. (general)

[en] The author of this memoir Dora Mueller (born 1920) belongs - as well as Kurt Goedel-to the German minority playing an important role in the past life of Brno. The marriage of his son included her among the Goedels collaterals. She was chemist, but also pianist, historician, participant of antinacist movement and iniciator of Czech-German understanding after war. Following her personal experiences, remembrances of Goedels relatives and documental materials, she evokes the atmosphere of broader family milieu of Kurt Goedel

[en] Let A denote the class of functions f that are analytic in the unit disc D and normalized by f (0) = f (0) - 1 = 0. In this paper conditions are determined for star likeness of the Libera integral of functions such that arg {f} < d in D. The results of Miller and Mocanu (2002) on Libera transform of functions with bounded turning are improved. (author)

[en] Though Godel did not succeed by his insight in politics, a diplomatic inspiration can be found in his work. The statement that not everything that is true can be proven, along with the impossibility of reaching completeness and consistency at the same time, is not only restricted to mathematics. In the context of diplomacy, the extended meaning of the Godel lesson is a benefit rather than a restriction

[en] The famous results of Goedel - semantic completeness of classical (predicate) logic and deductive incompleteness of any ''reasonable'' arithmetic - are briefly presented for a reader able of elementary mathematical thinking

No abstract available

[en] In terms of a sufficiently fine-grained theory we should distinguish between classes, properties and concepts. Since properties are best modeled as a kind of non-trivial intensions while mathematical objects are never non-trivial intensions we should not speak about properties of mathematical objects. When we do use the term property in mathematics (as Goedel did) we either mean classes, or the more fine-grained entities to be called concepts. In the latter case concepts have to be defined so that various distinct concepts could identify one and the same object. The notion of construction in transparent intensional logic makes it possible to construe concepts as abstract procedures. At the same time we have to distinguish between this notion and the notion of construction in constructivist systems: the former - unlike the latter - are objective and, therefore, acceptable for a realist