Collatz conjecture becomes theorem

The Collatz hypothesis is a theorem of the algorithmic theory of natural numbers. We prove the (algorithmic) formula that expresses the halting property of Collatz algorithm. The observation that Collatz's theorem cannot be proved in any elementary number theory completes the main result.Comment: The subsection 4.2 is improved. Contains 30+ pages, 5 figures. The paper submitted to TCS Journal awaits decision since 4 June 202

The investigation of YAlO3-NdAlO3 system, synthesis and characterization

The binary phase diagram of the YAlO3 (YAP) - NdAlO3 (NAP) system was determined by differential thermal analysis (DTA) and X-ray powder diffraction (XRD) measurements. High purity nanocrystalline powders and small single crystals of Y_{1-x}Nd_{x}AlO_3 (0 \leq x \leq 1) have been produced successfully by modified sol-gel (Pechini) and micro-pulling-down methods, respectively. Both end members show high mutual solubility >25% in the solid phase, with a miscibility gap for intermediate compositions. A solid solution with x \approx 0.2 melts azeotropic ca. 20 degrees below pure YAP. Such crystals can be grown from the melt without segregation. The narrow solid/liquid region near the azeotrope point could be measured with a "cycling" DTA measurement technique.Comment: 12 pages, 8 figures, submitted to J. Alloys. Comp

Identification of Extracellular Segments by Mass Spectrometry Improves Topology Prediction of Transmembrane Proteins

Transmembrane proteins play crucial role in signaling, ion transport, nutrient uptake, as well as in maintaining the dynamic equilibrium between the internal and external environment of cells. Despite their important biological functions and abundance, less than 2% of all determined structures are transmembrane proteins. Given the persisting technical difficulties associated with high resolution structure determination of transmembrane proteins, additional methods, including computational and experimental techniques remain vital in promoting our understanding of their topologies, 3D structures, functions and interactions. Here we report a method for the high-throughput determination of extracellular segments of transmembrane proteins based on the identification of surface labeled and biotin captured peptide fragments by LC/MS/MS. We show that reliable identification of extracellular protein segments increases the accuracy and reliability of existing topology prediction algorithms. Using the experimental topology data as constraints, our improved prediction tool provides accurate and reliable topology models for hundreds of human transmembrane proteins

Representation Theorem for Stacks

In the paper the concept of stacks is formalized. As the main result the Theorem of Representation for Stacks is given. Formalization is done according to [13].Białystok Technical University, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. Filters - part II. Quotient lattices modulo filters and direct product of two lattices. Formalized Mathematics, 2(3):433-438, 1991.Grzegorz Bancerek. Reduction relations. Formalized Mathematics, 5(4):469-478, 1996.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Grażyna Mirkowska and Andrzej Salwicki. Algorithmic Logic. PWN-Polish Scientific Publisher, 1987.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313-319, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990

Concequences of climate changes for agriculture

Paper presents the review of scenarios concerning the influence of climate changes for agriculture, focusing on the Polish situation. Numerous projections of changes in environmental conditions based on GCM models are diverse however their common field are the conclusions related to the adaptive capabilities of Polish agriculture to the expected changes. Suggested as necessary undertakings concern: the improvements in the water system, introducing to the production thermophilic and more resistant on water deficit crops, researches on pests and diseases that will intensify or will appear, the improvement and extension of the insurance system in the agriculture

Czy teoria pokoleń wyjaśnia naszą przyszłość?

Teoria cyklu pokoleniowego Straussa i Howe’a opisuje historię jako powtarzające się regularnie, choć w różnych wariantach, cztery fazy przebudzeń i kryzysów związanych z typami pokoleń. Przesilenia, przez które jako społeczeństwa systematycznie przechodzimy oraz odpowiadające im etapy cyklu, nazwano wzlotem, przebudzeniem, dezintegracją i kryzysem społecznym. Zgodnie z tą koncepcją od roku 2007 znajdujemy się w końcowym etapie cyklu – fazie kryzysu społecznego, który jest zawsze momentem przełomowym, z konfliktami prowadzącymi do wojen, nawet o charakterze totalnym. To czasy poważnych wstrząsów, kiedy społeczeństwo koncentruje się na reorganizacji systemów i instytucji, zmierzając do nowego porządku. Współcześnie najgroźniejsze źródła konfliktów to: kryzys demokracji liberalnej, nasilające się nierówności społeczne oraz skutki zmian klimatu, z szybko rosnącą migracją ludności na czele. Ich efektem mogą być np. zmiany terytorialne czy nowe rozwiązania polityczno-ustrojowe