Tips and Tricks in Molecular Imaging: A Practical Approach

A variety of imaging modalities include X-ray-based Computed Tomography (CT) scan, Ultrasound (US), Magnetic Resonance Imaging (MRI), Nuclear Medical Imaging (NMI), and Optical Imaging (OI) are used to help diagnose and treat diseases through anatomical, physiological, and functional representation. With the advent of molecular imaging using nanoparticles, detailed information about properties is provided and facilitates early detection of malignancies. Now, novel approaches in nanoparticle designing, the development of hybrid imaging modalities, and improvements in the sensitivity of instruments have raised the level of disease diagnosis. Regarding the fact that the molecular imaging fundamentals and basis of materials as contrast agents are different from each other, we have updated the brief synopsis of basic principles of imaging technique containing important points in detail with a practical approach.

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

A generalized Fourier transform by means of change of variables within multilinear approximation

The paper deals with approximations of periodic functions that play a significant role in harmonic analysis. The approach revisits the trigonometric polynomials, seen as combinations of functions, and proposes to extend the class of models of the combined functions to a wider class of functions. The key here is to use structured functions, that have low complexity, with suitable functional representation and adapted parametrizations for the approximation. Such representation enables to approximate multivariate functions with few eventually random samples. The new parametrization is determined automatically with a greedy procedure, and a low rank format is used for the approximation associated with each new parametrization. A supervised learning algorithm is used for the approximation of a function of multiple random variables in tree-based tensor format, here the particular Tensor Train format. Adaptive strategies using statistical error estimates are proposed for the selection of the underlying tensor bases and the ranks for the Tensor-Train format. The method is applied for the estimation of the wall pressure for a flow over a cylinder for a range of low to medium Reynolds numbers for which we observe two flow regimes: a laminar flow with periodic vortex shedding and a laminar boundary layer with a turbulent wake (sub-critic regime). The automatic re-parametrization enables here to take into account the specific periodic feature of the pressure.

Benchmarking of modernization of the economy of the world: experience for Ukraine

The article contains significant analytical information regarding the study of the experience of economic modernization reforms in the countries of the world and the possibility of applying this experience in Ukraine. A characteristic is given to the technological structures of the world economy and the actual change of structures, the distinctive features of such a change are given on the example of energy prices. The research of the types of modernization - pioneer and catching-up is given, the interdependence between the economic and social cycles of the modernization of society is stated. The examples of modernization both in highly developed economic states (USA, Great Britain, Germany, Sweden) and in the countries of the former socialist camp (Slovakia, Poland, Bulgaria) are given. It is stated that the modern conditions of globalization and the expansion of market relations create special precedents in which a conflict of interests is potentially possible - between objective and subjective factors, namely: the corporate nature of the functional representation system and the individual nature of decision-making. It is emphasized that the modernization of the economy, even if at the initial stage is aimed at satisfying the interests of individual institutional groups, in the course of the implementation of the modernization measures will be reoriented to the corporate interest groups built into the system. Provides effective recommendations for Ukraine in terms of entrepreneurial opening, increasing industrial production in the volume of GDP, completing the decentralization reform.

Working equation of LR-TD-DFT First-order polarization propagator approximation

We present the derivation of a new response method termed rst order po- larization propagator approximation. The electronic structure is given by a density functional representation. We provide a detailed derivation of the method along with explicit expressions for the relevant integrals and matrix elements.

Glucodensities: A new representation of glucose profiles using distributional data analysis

Biosensor data have the potential to improve disease control and detection. However, the analysis of these data under free-living conditions is not feasible with current statistical techniques. To address this challenge, we introduce a new functional representation of biosensor data, termed the glucodensity, together with a data analysis framework based on distances between them. The new data analysis procedure is illustrated through an application in diabetes with continuous-time glucose monitoring (CGM) data. In this domain, we show marked improvement with respect to state-of-the-art analysis methods. In particular, our findings demonstrate that (i) the glucodensity possesses an extraordinary clinical sensitivity to capture the typical biomarkers used in the standard clinical practice in diabetes; (ii) previous biomarkers cannot accurately predict glucodensity, so that the latter is a richer source of information and; (iii) the glucodensity is a natural generalization of the time in range metric, this being the gold standard in the handling of CGM data. Furthermore, the new method overcomes many of the drawbacks of time in range metrics and provides more in-depth insight into assessing glucose metabolism.

Functional integrals

This chapter presents an alternative approach to the quantization of fields, an approach that will be critically important for the development of quantum field theory in curved space, which is the subject of the second part of the book. It starts by providing a description of a functional integral in quantum mechanics, concentrating on the representation of an evolution operator. It then considers the functional representation of the Green functions and the generating functional in quantum field theory, including for fermionic theories. After that, perturbative calculations of the generating functionals and their general properties are formulated. The chapter ends with a brief description of ζ‎-regularization as a technique for defining functional determinants.

Functional Modeling of High-Dimensional Data: A Manifold Learning Approach

This paper introduces stringing via Manifold Learning (ML-stringing), an alternative to the original stringing based on Unidimensional Scaling (UDS). Our proposal is framed within a wider class of methods that map high-dimensional observations to the infinite space of functions, allowing the use of Functional Data Analysis (FDA). Stringing handles general high-dimensional data as scrambled realizations of an unknown stochastic process. Therefore, the essential feature of the method is a rearrangement of the observed values. Motivated by the linear nature of UDS and the increasing number of applications to biosciences (e.g., functional modeling of gene expression arrays and single nucleotide polymorphisms, or the classification of neuroimages) we aim to recover more complex relations between predictors through ML. In simulation studies, it is shown that ML-stringing achieves higher-quality orderings and that, in general, this leads to improvements in the functional representation and modeling of the data. The versatility of our method is also illustrated with an application to a colon cancer study that deals with high-dimensional gene expression arrays. This paper shows that ML-stringing is a feasible alternative to the UDS-based version. Also, it opens a window to new contributions to the field of FDA and the study of high-dimensional data.