Non-Integrability of Some Higher-Order Painlev\'e Equations in the Sense of Liouville

In this paper we study the equation $$w^{(4)} = 5 w" (w^2 - w') + 5 w (w')^2 - w^5 + (\lambda z + \alpha)w + \gamma,$$ which is one of the higher-order Painlev\'e equations (i.e., equations in the polynomial class having the Painlev\'e property). Like the classical Painlev\'e equations, this equation admits a Hamiltonian formulation, B\"acklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters $\gamma/\lambda = 3 k$, $\gamma/\lambda = 3 k - 1$, $k \in \mathbb{Z}$, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis approach. Then we study the integrability of the second and third members of the $\mathrm{P}_{\mathrm{II}}$-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability

On the integrability of a system describing the stationary solutions in Bose--Fermi mixtures

We study the integrability of a Hamiltonian system describing the stationary solutions in Bose--Fermi mixtures in one dimensional optical lattices. We prove that the system is integrable only when it is separable. The proof is based on the Differential Galois approach and Ziglin-Morales-Ramis method.Comment: 21 page

Bulgarian Housing. Status and Prospectives

The article analyses the current condition of Bulgarian multistorey apartment housing and reveals the main obstacles towards sustainable energy efficient retrofit of existing condominium housing. Housing is an outstanding phenomenon that influences strongly social economic and environmental development of a country. Securing of energy efficient, affordable housing is of great importance for sustainable urban and social development. The article formulates proposals for necessary improvements and changes in the legal, institutional, and financial framework that would allow sustainable renovation of existing Bulgarian housing

Qualitative Modelling of Quasi-homogeneous Effects in ERK and STAT Interaction Dynamics

On the basis of qualitative analysis of author's model published in previous paper, the stability and temporal behaviour of quasi-homogeneous distributions of ERK-protein concentrations are analyzed in terms of corresponding reaction-diffusion problem. The stable quasi-homogeneous distributions are treated as a dynamical basis of pathway compartmentalization. It is also shown, that a crowding effect exists in the form of loss of pathway stability. An experimentally verifiable issue for possible existence of protein scaffolding mechanism is derived on the basis of its qualitative correspondence with the pattern formation and molecular crowding effects inherent to the considered model. Moreover, it is demonstrated, that the predicted ERK and STAT pathway instability can be interpreted as traveling wave propagation of molecular concentration drop and jump from the nucleus membrane to the cell one and vice versa

Conceptualization and classification of balneo, SPA and wellness establichments in Bulgaria

This paper analyzes the origin, the process of development and consolidation of the health, SPA and wellness tourism in different regions of the world as well as in Bulgaria. Special attention is paid to the etymology of the words 'wellness' and 'SPA', used as a basis for differentiation of their structural elements as tourism segments. Under observations are some theoretical issues, including the significant need of term clarification of these forms of tourism. Other matters of high importance comprise of the classification of health, spa and wellness resorts, the vocational training of managerial and executive staff and etc. Based on the conducted research and analysis some recommendations are made in the aim to the further development of health, spa and wellness tourism