A Heuristic for the Prime Number Theorem

Conclusion  Might there be a chance of proving in a simple way thatx/π(x) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. LetL(x) be the upper convex hull of the full graph ofxπ(x) (precise definition to follow). The piecewise linear functionL(x) is increasing becausex/π(x) → ∞ asx → ∞. Moreover, using PNT, we can give a proof thatL(x) is indeed asymptotic tox/π(x). But the point of our work in this article is that for someone who wishes to understand why the growth of primes is governed by natural logarithms, a reasonable approach is to convince oneself via computation that the convex hull just mentioned satisfies the hypothesis of our theorem, and then use the relatively simple proof to show that this hypothesis rigorously implies the prime number theorem.     

Dead cores of singular Dirichlet boundary value problems with ϕ-Laplacian

The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem (ϕ(u′))′ = λf(t, u, u′), u(0) = u(T) = A. Here λ is the positive parameter, A > 0, f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable. This work was supported by grant no. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic and by the Council of Czech Government MSM 6198959214.     

The evidence

The Mathematical Intelligencer -     

Plasma cluster acceleration by means of external magnetic field

The possibility to employ external magnetic field for acceleration of plasma clusters is discussed in this paper both theoretically and experimentally. The optimum value of the magnetic field has been found for the “snow plough” model, in which case there is the maximum cluster acceleration. The obtained results show a real possibility of further increasing the plasma cluster velocity in the electromagnetic shock tube.     

Chemische Bestimmungsmethoden von Pharmaka in Lebensmitteln

Ohne Zusammenfassung

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Chemical methods for the determination of pharmaceuticals in food