Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n>1$ and both $6n^2-1$ and $12n^2+1$ are odd primes, then the general elliptic curve $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$ has only the integral point $(x, y)=(2, 0)$. By this result we can get that the above elliptic curve has only the trivial integral point for $n=3, 13, 17$ etc. Thus it can be seen that the elliptic curve $y^2=x^3+27x-62$ really is an unusual elliptic curve which has large integral points.
The interval function (in the sense of H. M. Mulder) is an important tool for studying those properties of a connected graph that depend on the distance between vertices. An axiomatic characterization of the interval function of a connected graph was published by Nebeský in 1994. In Section 2 of the present paper, a simpler and shorter proof of that characterization will be given. In Section 3, a characterization of geodetic graphs will be established; this characterization will utilize properties of the interval function.
The problem was motivated by Borůvka’s definitions of the carrier and the associated carrier. The inverse carrier problem is precisely defined and partially solved. Examples are given.
The purpose of this review is to analyze the involvement of protein kinases in the cardioprotective mechanism induced by chronic hypoxia. It has been reported that chronic intermittent hypoxia contributes to increased expression of the following kinases in the myocardium: PKCδ, PKCα, p-PKCε, p-PKCα, AMPK, p-AMPK, CaMKII, p-ERK1/2, p-Akt, PI3-kinase, p-p38, HK-1, and HK-2; whereas, chronic normobaric hypoxia promotes increased expression of the following kinases in the myocardium: PKCε, PKCβII, PKCη, CaMKII, p-ERK1/2, p-Akt, p-p38, HK-1, and HK-2. However, CNH does not promote enhanced expression of the AMPK and JNK kinases. Adaptation to hypoxia enhances HK-2 association with mitochondria and causes translocation of PKCδ, PKCβII, and PKCη to the mitochondria. It has been shown that PKCδ, PKCε, ERK1/2, and MEK1/2 are involved in the cardioprotective effect of chronic hypoxia. The role of other kinases in the cardioprotective effect of adaptation to hypoxia requires further research.
Within the framework of discrete probabilistic uncertain reasoning a large literature exists justifying the maximum entropy inference process, \ME, as being optimal in the context of a single agent whose subjective probabilistic knowledge base is consistent. In particular Paris and Vencovská completely characterised the \ME inference process by means of an attractive set of axioms which an inference process should satisfy. More recently the second author extended the Paris-Vencovská axiomatic approach to inference processes in the context of several agents whose subjective probabilistic knowledge bases, while individually consistent, may be collectively inconsistent. In particular he defined a natural multi-agent extension of the inference process \ME called the social entropy process, \SEP. However, while \SEP has been shown to possess many attractive properties, those which are known are almost certainly insufficient to uniquely characterise it. It is therefore of particular interest to study those Paris-Vencovská principles valid for \ME whose immediate generalisations to the multi-agent case are not satisfied by \SEP. One of these principles is the Irrelevant Information Principle, a powerful and appealing principle which very few inference processes satisfy even in the single agent context. In this paper we will investigate whether \SEP can satisfy an interesting modified generalisation of this principle.
As a consequence of a unique historical fact – the dissolution of Czechoslovakia – the divergencebetween the two social systems of the successor states occurred and caused unequal treatment of those oldagepensioners, previous citizens of the federal state, whose employer had had, by chance, its seat in the territoryof the other successor state. The subsequent decision-making over the pension claims of these citizensresulted in controversial judgments not only at the national but also at the Union level. The article focuseson the recent evolution in European Union Court of Justice case “Landtova” (C-399-09) as reflected in theCzech Constitutional Court’s judgement (Pl. ÚS 5/12) stating that EU law is inapplicable in these cases andtherefore the above mentioned decision of CJEU is ultra vires. In this context the auricle deals with relationshipof national and EU law in light of the principle of conferral, division of powers and cooperation betweenthe national and European Union courts.
For a nontrivial connected graph G of order n, the detour distance D(u, v) between two vertices u and v in G is the length of a longest u − v path in G. Detour distance is a metric on the vertex set of G. For each integer k with 1 ≤ k ≤ n−1, a coloring c : V (G) → N is a k-metric coloring of G if |c(u) − c(v)| + D(u, v) ≥ k + 1 for every two distinct vertices u and v of G. The value χ k m(c) of a k-metric coloring c is the maximum color assigned by c to a vertex of G and the k-metric chromatic number χ k m(G) of G is the minimum value of a k-metric coloring of G. For every nontrivial connected graph G of order n, χ 1m(G) ≤ χ 2m(G) ≤ . . . ≤ χ n−1 m (G). Metric chromatic numbers provide a generalization of several well-studied coloring parameters in graphs. Upper and lower bounds have been established for χ k m(G) in terms of other graphical parameters of a graph G and exact values of k-metric chromatic numbers have been determined for complete multipartite graphs and cycles. For a nontrivial connected graph G, the anti-diameter adiam(G) is the minimum detour distance between two vertices of G. We show that the adiam(G)-metric chromatic number of a graph G provides information on the Hamiltonian properties of the graph and investigate realization results and problems on this parameter.