Subharmonicity of the modulus of quasiregular harmonic mappings

In this note we determine all numbers $q\in \mathbf R$ such that $|u|^q$ is a subharmonic function, provided that $u$ is a $K-$quasiregular harmonic mappings in an open subset $\Omega$ of the Euclidean space $\mathbf R^n$.Comment: 6 page

Algebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin model

The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain, of arbitrary spin-$s$, in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.Comment: 39 pages. Typos corrected. arXiv admin note: substantial text overlap with arXiv:1405.7398, arXiv:1412.139

Oscillation theorems for nonlinear differential equations

We establish new oscillation theorems for the nonlinear differential equation $$[a(t)\psi(x(t))|x'(t)|^{\alpha-1}x'(t)]'+q(t)f(x(t))=0, \alpha>0$$ where $a,q:[t0,\infty)\rightarrow R, \psi,f:R\rightarrow R$ are continuous, $a(t)>0$ and $\psi(x)>0$, $xf(x)>0$ for $x\not=0$. These criteria involve the use of averaging functions

International Courts in The Hague: A Comparison Between The ICTY and The ICC and Their Respective Contributions to the Rule of International Law from the Perspective of Victims of International Crimes

The ICTY is a model directing to ad hoc approach to solving crime problems in war combat. The shortcomings of this system are : an open issue whether all circumstances, necessary for founding the ad hoc tribunal, will always coincide, enormous impact of permanent members of the UN Security Council, which would thereby attain their own political goals, while international crimes, committed under their patronage or approval, would remain beyond reach of any righteous but not feigned punishments. The ICC, founded on the basis of Roman Stat, is immune to the objections directed to the Tribunal for ex Yugoslavia. But, the project of the permanent international and universal criminal court becomes a victim of own ambition. The question of such project realization has been posed. Lasting assignment of competences, which was the exclusive right of SB so far, was seen by the most powerful States as the impeachment of their own interests. Without the most powerful countries in military sense of the word, taking part directly or indirectly in most military operations, we cannot speak about reaching a universal international criminal judiciary system. The shortcomings of ICTY result just from its imposed nature, i.e. the fact that it founded the SB as a coercive measure. The ICC, founded on the basis of Roman Stat, experienced difficulties in practice just because just because we have with it a voluntary transfer of sovereignty and not an imposed jurisdiction.. The shortcomings of one and difficulties of the other court result from defects and problems encountered by the international law.Međunarodni krivični tribunal za bivšu jugoslaviju jeste model koji upućuje na ad hok pristup pravnog rešavanja problema zločina u ratnim sukobima. Mane ovog sistema su: pitanje da li će se uvek poklopiti sve okolnosti potrebne za osnivanje ad hok suda, prevelik uticaj (stalnih) članica Saveta bezbednosti Ujedinjenih nacija koje bi time ostvarivale sopstvene političke interese, dok bi međunarodna krivična dela učinjena pod njihovim patronatom ili odobrenjem ostajala van domašaja bilo kakvog pravednog kažnjavanja. Međunarodi krivični sud osnovan Rimskim statutom je imun na većinu prigovora upućenih Tribunalu za bivšu Jugoslaviju. Projekat međunarodnog stalnog i univerzalnog krivičnog suda postaje, međutim, žrtva sopstvene ambicioznosti. Postavlja se pitanje ostvarljivosti ovakvog projekta.Trajno prenošenje ovlašćenja koje je do sada bilo eksluzivno pravo SB, najmoćnije države vide kao ugrožavanje svojih interesai odbijaju da učestvuju u njemu. Bez vojno najmoćnijih zemalja koje direktno ili indirektno učestvuju u najvećem broju vojnih operacija, ne možemo govoriti o uspostavljanju univerzalnog međunarodnog krivičnog sudstva. Mane Međunarodnog suda za bivšu Jugoslaviju proizilaze upravo iz njegovog nametnutog karaktera, činjenice da je SB osnovao kao prinudnu meru. Međunarodi krivični sud osnovan Rimskim statom praktične teškoće doživljva, međutim, baš zbog težnje univerzalnosti i dobrovoljnog prenosa suvereniteta, a ne nametnute nadležnosti. Mane jednog i poteškoće drugog suda prozilaze upravo iz mana I teškoća današnjeg međunarodnog prava

Ashtekar Formulation of 2+1 Gravity on a Torus

Pure (2+1)-dimensional Einstein gravity is analysed in the Ashtekar formulation, when the spatial manifold is a torus. We have found a set of globally defined observables, forming a closed algebra. This allowed us to solve the quantum constraints, and to show that the reduced phase space of the Ashtekar formulation is greater then the corresponding space of the Witten formulation. Furthermore, we have found a globally defined time variable which satisfies all the requiriments of an extrinsic time variable in quantum gravity.Comment: 15 page