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The Economic and Geopolitical Effects of the European Union

The European Union (EU) is one of the largest trade unions in the world and has a substantial impact on the global economic system. This paper discusses the creation of the EU, the establishment of the Euro as the dominant currency, and the diversity between the member nations, comparing their similarities and contrasting their differences. Organizations that influence and shape the European Union such as the World Trade Organization, the European Central Bank, and the International Monetary Fund are also discussed. As an illustration of this influence, Greece’s declining economy is examined. A brief outline of the migrant crisis that the EU is currently facing is also covered, demonstrating issues related to the cost of and transportation for war refugees, as well as criteria for processing them. In conclusion, an examination of the European Union’s current capabilities of integrating into a political union is conducted, evaluating its level of political and economic cooperation, the capacity and willingness to create a central military force, and the recent developments in negotiations between the EU and Great Britain

Pressure modulating valve

A modulating valve device, operated by fluid pressure in fluid motors to control work pressure, is described

Regularity of Eigenstates in Regular Mourre Theory

The present paper gives an abstract method to prove that possibly embedded eigenstates of a self-adjoint operator $H$ lie in the domain of the $k^{th}$ power of a conjugate operator $A$. Conjugate means here that $H$ and $A$ have a positive commutator locally near the relevant eigenvalue in the sense of Mourre. The only requirement is $C^{k+1}(A)$ regularity of $H$. Regarding integer $k$, our result is optimal. Under a natural boundedness assumption of the multiple commutators we prove that the eigenstate 'dilated' by $\exp(i\theta A)$ is analytic in a strip around the real axis. In particular, the eigenstate is an analytic vector with respect to $A$. Natural applications are 'dilation analytic' systems satisfying a Mourre estimate, where our result can be viewed as an abstract version of a theorem due to Balslev and Combes. As a new application we consider the massive Spin-Boson Model.Comment: 27 page

Self-adjoint difference operators and classical solutions to the Stieltjes--Wigert moment problem

The Stieltjes-Wigert polynomials, which correspond to an indeterminate moment problem on the positive half-line, are eigenfunctions of a second order q-difference operator. We consider the orthogonality measures for which the difference operator is symmetric in the corresponding weighted $L^2$-spaces. These measures are exactly the solutions to the q-Pearson equation.In the case of discrete and absolutely continuous measures the difference operator is essentially self-adjoint, and the corresponding spectral decomposition is given explicitly. In particular, we find an orthogonal set of q-Bessel functions complementing the Stieltjes-Wigert polynomials to an orthogonal basis for $L^2(\mu)$ when $\mu$ is a discrete orthogonality measure solving the q-Pearson equation. To obtain the spectral decomposition of the difference operator in case of an absolutely continuous orthogonality measure we use the results from the discrete case combined with direct integral techniques.Comment: 22 pages; section 2 rewritten, to appear in Journal of Approximation Theor