RECONSTRUCTION OF GAUSSIAN RANDOM FUNCTIONS FROM SPECTRUM DATA

It is known that a stationary random process is represented as a superposition of harmonic oscillations with real frequencies and uncorrelated amplitudes. In the study of nonstationary processes, it is natural to have increasing or declining oscillationсs. This raises the problem of constructing algorithms that would allow constructing broad classes of nonstationary processes from elementary nonstationary random processes. A natural generalization of the concept of the spectrum of a nonstationary random process is the transition from the real spectrum in the case of stationary to a complex or infinite multiple spectrum in the nonstationary case. There is also the problem of describing within the correlation theory of random processes in which the spectrum has no analogues in the case of stationary random processes, namely, the spectrum point is real, but it has infinite multiplicity for the operator image of the corresponding operator, and when the spectrum itself is complex. Reconstruction of the complex spectrum of a nonstationary random function is a very important problem in both theoretical and applied aspects. In the paper the procedure of reconstruction of random process, sequence, field from a spectrum for Gaussian random functions is developed. Compared to the stationary case, there are wider possibilities, for example, the construction of a nonstationary random process with a real spectrum, which has infinite multiplicity and which can be distributed over the entire finite segment of the real axis. The presence of such a spectrum leads, in contrast to the case of a stationary random process, to the appearance of new components in the spectral decomposition of random functions that correspond to the internal states of «strings», i.e. generated by solutions of systems of equations in partial derivatives of hyperbolic type. The paper deals with various cases of the spectrum of a non-self-adjoint operator , namely, the case of a discrete spectrum and the case of a continuous spectrum, which is located on a finite segment of the real axis, which is the range of values of the real non-decreasing function a(x). The cases a(x)=0, a(x)=a0, a(x)=x and a(x) is a piecewise constant function are studied. The authors consider the recovery of nonstationary sequences for different cases of the spectrum of a non-self-adjoint operator promising since spectral decompositions are a superposition of discrete or continuous internal states of oscillators with complex frequencies and uncorrelated amplitudes and therefore have deep physical meaning.

CONDITIONS AND MECHANISMS FOR EFFECTIVE PREVENTION OF CORRUPTION IN UKRAINE

The content of systemic network structuring of corruption as an institution of absolute anti-social orientation is revealed. Attention is paid to the conceptual purity of the categorical apparatus, in particular, the concepts of “corruption” and “bribery” are distinguished. The characteristics of personalized and corporate-associated subjects of corruption are given, the content of modern forms and characteristic varieties of the object of corruption bribery is revealed. A differentiated list is given and the content of specific markets of “corruption services” operating within the general “black” market of Ukraine is revealed. The socio-economic origins are shown and the nature of personal and corporate interests of the subjects, the bearers of corruption relations, is revealed. The real spectrum of destructive anti-social consequences of the synergy of the binary union of the “institution of corruption” with the “institution of the criminal “black” economy” is determined. There are three basic blocks of criminal economic activity: i) criminal trade; ii) provision of criminal services; iii) criminal violation of the rules of economic and commercial activity. On the basis of the generalized experience of the USA the necessary conditions are defined and suggestions are given concerning formation of organizational-economic and economic-legal anti-corruption mechanisms. It is proven that overcoming corruption is impossible without abolishing the legal principle of “presumption of innocence” in the field of confirming the legitimacy of taxpayers' incomes, who are obliged to provide comprehensive information proving the legitimacy of the sources of personal and family real estate. The conditions for ensuring an effective order in the field of accounting and control of the completeness of citizens ’compliance with tax obligations on the basis of improving the mechanism of income declaration and state-market accounting of real estate of the population of Ukraine are determined. In the context of critical remarks on the mistakes made over the past five years, proposals are made to improve the organizational-economic framework for declaring annual income, as well as socially just and anti-corruption legalization of wealth, property and capital of individuals by providing fair compensation to the state treasury for losses caused to the budget during the period of independence of Ukraine.

Bose–Einstein Condensation Processes with Nontrivial Geometric Multiplicities Realized via 𝒫𝒯−Symmetric and Exactly Solvable Linear-Bose–Hubbard Building Blocks

It is well known that, using the conventional non-Hermitian but PT−symmetric Bose–Hubbard Hamiltonian with real spectrum, one can realize the Bose–Einstein condensation (BEC) process in an exceptional-point limit of order N. Such an exactly solvable simulation of the BEC-type phase transition is, unfortunately, incomplete because the standard version of the model only offers an extreme form of the limit, characterized by a minimal geometric multiplicity K = 1. In our paper, we describe a rescaled and partitioned direct-sum modification of the linear version of the Bose–Hubbard model, which remains exactly solvable while admitting any value of K≥1. It offers a complete menu of benchmark models numbered by a specific combinatorial scheme. In this manner, an exhaustive classification of the general BEC patterns with any geometric multiplicity is obtained and realized in terms of an exactly solvable generalized Bose–Hubbard model.

Fatal Streptococcus pseudoporcinus disseminated infection in decompensated liver cirrhosis: a case report

Background Streptococcus pseudoporcinus (S. pseudoporcinus) was first identified in 2006. It cross-reacts with Lancefield group B antigen agglutination reagents and has been misidentified as S. agalactiae. Sites of S. pseudoporcinus isolation include the female genitourinary tract, urine, wounds, and dairy products. The prevalence of vaginal colonization is reportedly between 1 and 5.4%. Two uneventful cases of soft tissue infection caused by S. pseudoporcinus were reported in the past. However, since late 2019, six cases of invasive S. pseudoporcinus infections have emerged in the literature, one of which was fatal. Case presentation We describe a fatal case of a Caucasian male with spontaneous bacterial peritonitis associated with bacteremia due to a multidrug-resistant S. pseudoporcinus strain in a patient with decompensated liver cirrhosis. Despite the patient’s good general condition and stable blood test results when he had visited the outpatient clinic for large-volume paracentesis a few days before admission, this time he presented to the emergency department with a rapidly worsening clinical condition and with laboratory features consistent with multiple-organ dysfunction syndrome, and succumbed within a short period. Conclusions Contrary to what was thought until recently, multidrug-resistant S. pseudoporcinus may cause invasive, disseminated, fatal disease in humans. According to current limited data, vancomycin, linezolid, daptomycin, levofloxacin, clindamycin, and tetracycline seem to be the most effective antimicrobial agents against multidrug-resistant strains, and should be the empirical choice in cases of disseminated S. pseudoporcinus infection until laboratory antimicrobial susceptibility results are available. Improvements and new approaches for bacterial identification in routine clinical microbiology laboratories may reveal the real spectrum of S. pseudoporcinus infections in humans, which is currently believed to be underestimated. SS. pseudoporcinus could emerge as a serious medical problem in the near future, similar to other β-hemolytic streptococci.

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

We study the distribution of the eigenvalue condition numbers $$\kappa _i=\sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}$$ κ i = ( l i ∗ l i ) ( r i ∗ r i ) associated with real eigenvalues $$\lambda _i$$ λ i of partially asymmetric $$N\times N$$ N × N random matrices from the real Elliptic Gaussian ensemble. The large values of $$\kappa _i$$ κ i signal the non-orthogonality of the (bi-orthogonal) set of left $${\mathbf{l}}_i$$ l i and right $${\mathbf{r}}_i$$ r i eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint density function (JDF) $${{\mathcal {P}}}_N(z,t)$$ P N ( z , t ) of $$t=\kappa _i^2-1$$ t = κ i 2 - 1 and $$\lambda _i$$ λ i taking value z, and investigate its several scaling regimes in the limit $$N\rightarrow \infty $$ N → ∞ . When the degree of asymmetry is fixed as $$N\rightarrow \infty $$ N → ∞ , the number of real eigenvalues is $$\mathcal {O}(\sqrt{N})$$ O ( N ) , and in the bulk of the real spectrum $$t_i=\mathcal {O}(N)$$ t i = O ( N ) , while on approaching the spectral edges the non-orthogonality is weaker: $$t_i=\mathcal {O}(\sqrt{N})$$ t i = O ( N ) . In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as $$N\rightarrow \infty $$ N → ∞ . In such a regime eigenvectors are weakly non-orthogonal, $$t=\mathcal {O}(1)$$ t = O ( 1 ) , and we derive the associated JDF, finding that the characteristic tail $${{\mathcal {P}}}(z,t)\sim t^{-2}$$ P ( z , t ) ∼ t - 2 survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.

Gibbs States, Algebraic Dynamics and Generalized Riesz Systems

In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita–Takesaki theory in our context.