Auswahl der wissenschaftlichen Literatur zum Thema „Spectral graph realizations“

The paper presents an analysis and overview of vertex–frequency analysis, an emerging area in graph signal processing. A strong formal link of this area to classical time–frequency analysis is provided. Vertex–frequency localization-based approaches to analyzing signals on the graph emerged as a response to challenges of analysis of big data on irregular domains. Graph signals are either localized in the vertex domain before the spectral analysis is performed or are localized in the spectral domain prior to the inverse graph Fourier transform is applied. The latter approach is the spectral form of the vertex–frequency analysis, and it will be considered in this paper since the spectral domain for signal localization is well ordered and thus simpler for application to the graph signals. The localized graph Fourier transform is defined based on its counterpart, the short-time Fourier transform, in classical signal analysis. We consider various spectral window forms based on which these transforms can tackle the localized signal behavior. Conditions for the signal reconstruction, known as the overlap-and-add (OLA) and weighted overlap-and-add (WOLA) methods, are also considered. Since the graphs can be very large, the realizations of vertex–frequency representations using polynomial form localization have a particular significance. These forms use only very localized vertex domains, and do not require either the graph Fourier transform or the inverse graph Fourier transform, are computationally efficient. These kinds of implementations are then applied to classical time–frequency analysis since their simplicity can be very attractive for the implementation in the case of large time-domain signals. Spectral varying forms of the localization functions are presented as well. These spectral varying forms are related to the wavelet transform. For completeness, the inversion and signal reconstruction are discussed as well. The presented theory is illustrated and demonstrated on numerical examples.

Network science has been extensively developed to characterize the structural properties of complex systems, including brain networks inferred from neuroimaging data. As a result of the inference process, networks estimated from experimentally obtained biological data represent one instance of a larger number of realizations with similar intrinsic topology. A modelling approach is therefore needed to support statistical inference on the bottom-up local connectivity mechanisms influencing the formation of the estimated brain networks. Here, we adopted a statistical model based on exponential random graph models (ERGMs) to reproduce brain networks, or connectomes, estimated by spectral coherence between high-density electroencephalographic (EEG) signals. ERGMs are made up by different local graph metrics, whereas the parameters weight the respective contribution in explaining the observed network. We validated this approach in a dataset of N = 108 healthy subjects during eyes-open (EO) and eyes-closed (EC) resting-state conditions. Results showed that the tendency to form triangles and stars, reflecting clustering and node centrality, better explained the global properties of the EEG connectomes than other combinations of graph metrics. In particular, the synthetic networks generated by this model configuration replicated the characteristic differences found in real brain networks, with EO eliciting significantly higher segregation in the alpha frequency band (8–13 Hz) than EC. Furthermore, the fitted ERGM parameter values provided complementary information showing that clustering connections are significantly more represented from EC to EO in the alpha range, but also in the beta band (14–29 Hz), which is known to play a crucial role in cortical processing of visual input and externally oriented attention. Taken together, these findings support the current view of the functional segregation and integration of the brain in terms of modules and hubs, and provide a statistical approach to extract new information on the (re)organizational mechanisms in healthy and diseased brains.

AbstractA real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix 𝒭 such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and 𝒭. This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of 𝒭.

The question of self-adjoint realizations of sign-indefinite second-order differential operators is discussed in terms of a model problem. Operators of the type [Formula: see text] are generalized to finite, not necessarily compact, metric graphs. All self-adjoint realizations are parametrized using methods from extension theory. The spectral and scattering theories of the self-adjoint realizations are studied in detail.

In recent years, Russian and international breeders have produced a great many of new varieties of Vitis vinifera grapes as well as interspecies hybrids, distinguished by a high quality of fruit and other useful economic and biological features. Having a big reserve of technologically important substances and hygienic factors of grapevine, the resistant varieties may prove especially efficient for the production of premium-class wines. The appearance of high-end Russian wines with protected geographical indication (PGI) and protected appellation of origin (PAO), first of all, fits in with the requirements of international markets. It is a necessary criterion for product quality and safety assurance at the highly competitive global market, and development of universally recognized brands. It also helps resolve a number of socio-economic issues, such as formation of winemaking culture, and production of wines of guaranteed quality from own grapes. This study is devoted to substantiating the necessity for development of methods of formation of single information databases on characteristic features of PGI and PAO wines, including their distinctive organoleptic, physical and chemical properties (extract components – the cation-anion composition, organic acids, total phenolic and anthocyanin content; unique colour characteristics), as well as the application of the system of organoleptic assessment of wines with the use of descriptive analysis of wine colour, flavour and taste. It is well-timed and relevant to determine the regularities of realization of the varietal potential of a grapevine plant in terms of climatic conditions of growing and geographical origin based on the study of the endogenous and exogenous components of wines with the use of the methods of high-performance capillary electrophoresis, spectral photometry, organoleptic analysis, and statistical techniques. This research generalizes and puts forth a contemporary view of varietal and geographical identification of wines. It is shown that the proposed research guideline is highly sought-after, and it is of fundamental and practical importance for the development of regional and national selection, genetic, viticultural and winemaking industries.

Platform-vibrator with shock is widely used in the construction industry for compacting and molding large concrete products. Its mathematical model, created in our previous work, meets all the basic requirements of shock-vibration technology for the precast concrete production on low-frequency resonant platform-vibrators. This model corresponds to the two-body 2-DOF vibro-impact system with a soft impact. It is strongly nonlinear non-smooth discontinuous system. This is unusual vibro-impact system due to its specific properties. The upper body, with a very large mass, breaks away from the lower body a very short distance, and then falls down onto the soft constraint that causes a soft impact. Then it bounces and falls again, and so on. A soft impact is simulated with nonlinear Hertzian contact force. This model exhibited many unique phenomena inherent in nonlinear non-smooth dynamical systems with varying control parameters. In this paper, we demonstrate the transient chaos in a vibro-impact system. Our finding of transient chaos in platform-vibrator with shock, besides being a remarkable phenomenon by itself, provides an understanding of the dynamical processes that occur in the platform-vibrator when varying the technological mass of the mold with concrete. Phase trajectories, Poincaré maps, graphs of time series and contact forces, Fourier spectra, the largest Lyapunov exponent, and wavelet characteristics are used in numerical investigations to determine the chaotic and periodic phases of the realization. We show both the dependence of the transient chaos on the control parameter value and the sensitive dependence on the initial conditions. We hope that this analysis can help avoid undesirable platform-vibrator behaviour during design and operation due to inappropriate system parameters, since transient chaos may be a dangerous and unwanted state of a vibro-impact system.

The aim of the work is synthesis and study on the properties of polyfunctional magnetosensitive nanocomposites (NC) and target-directed magnetic fluids (MF) based on physiological solution (PS), magnetite, gemcitabine (GEM) and HER2 antibodies (AB), promising for use in targeted antitumor therapy against MDA-MB-231 aggressive tumor cells of triple-negative human breast cancer (BC) with high proliferative and metastatic activity. The specific surface area (Ssp) of samples was determined by the method of thermal desorption of nitrogen using a device KELVIN 1042 of “COSTECH Instruments”. The size of nanoparticles (NP) has been estimated by the formula DBET = 6/(ρSBET), where ρ is the density of NC particle, SBET is the value of the specific surface area calculated by the polymolecular adsorption theory of Brunauer, Emmett and Teller (BET). The surface condition of nanodispersed samples was studied by IR spectroscopy (“Perkin Elmer” Fourier spectrometer, a model 1720X). To calculate the concentration of hydroxyl groups on the surface of nanostructures, the method of differential thermal analysis was used in combination with differential thermogravimetric analysis. The thermograms were recorded using a derivatograph Q-1500D of MOM firm (Hungary) in the temperature range of 20–1000 °C at a heating rate of 10 deg/min. X-ray phase analysis of nanostructures was performed using a diffractometer DRON-4-07 (CuKα radiation with a nickel filter in a reflected beam, the Bragg-Brentano focusing). The size and shape of NP were determined by electron microscopy (a transmission electron microscope (TEM) JEM-2100F (Japan)). The hysteresis loops of the magnetic moment of the samples were measured using a laboratory vibration magnetometer of Foner type at room temperature. Measurement of optical density, absorption spectra and GEM concentration in solutions was performed by spectrophotometric analysis (Spectrometer Lambda 35 UV/Vis Perkin Elmer Instruments). The amount of adsorbed substance on the surface of magnetite was determined using a spectrophotometer at λ = 268 nm from a calibration graph. The thickness of the adsorbed layer of GEM in the composition of Fe3O4@GEM NC was determined by magnetic granulometry. To study the direct cytotoxic/cytostatic effect of a series of experimental samples of MF based on PS, Fe3O4 NP, GEM, HER2 AB, as well as MF components in mono- or complex use, onto MDA-MB-231 cells in vitro, IC50 index was determined. MF were synthesized on the basis of single-domain Fe3O4 and PS, stabilized with sodium oleate (Ol.Na) and polyethylene glycol (PEG), containing GEM and HER2 (Fe3O4@GEM/Ol.Na/PEG/HER2+PS). The cytotoxic/cytostatic activity of MF against MDA-MB-231 cells was studied. It was found that as a result of application of synthesized MF composed of Fe3O4@GEM/Ol.Na/PEG/HER2+PS at the concentration of magnetite of 0.05 mg/mL, GEM - 0.004 mg/mL and HER2 AB - 0.013 μg/mL, a synergistic effect arose, with reduction of the amount of viable BC cells to 51 %. It has been proved that when using MF based on targeted Fe3O4/GEM/HER2 complex, the increased antitumor efficacy is observed compared to traditional use of the drug GEM, with a significant reduction (by four times) of its dose. The high cytotoxic/cytostatic activity of Fe3O4/GEM/HER2 complexes is explained by the fact that endogenous iron metabolism disorders play a significant role in the mechanisms of realization of the apoptotic program under the influence of nanocomposite. Thus, when the nanocomposite system contains Fe3O4/GEM/HER2 complexes in MDA-MB-231 cells, a significant increase is observed in the level of “free iron”, which favours formation of reactive oxygen species and causes oxidative stress (Fenton reaction). The consequences of oxidative stress are induction of apoptosis, enhancement of lipid peroxidation processes, as well as structural and functional rearrangement of biological membranes. The prospects have been shown of further studies of Fe3O4@GEM/Ol.Na/PEG/HER2+PS MF in order to create on their basis a magnetically carried remedy for use in targeted antitumor therapy.

This thesis deals with optimizing extremal eigenvalues of weighted graph Laplacian matrices. In general, the Laplacian matrix of a (weighted) graph is of particular importance in spectral graph theory and combinatorial optimization (e.g., graph partition like max-cut and graph bipartition). Especially the pioneering work of M. Fiedler investigates extremal eigenvalues of weighted graph Laplacians and provides close connections to the node- and edge-connectivity of a graph. Motivated by Fiedler, Göring et al. were interested in further connections between structural properties of the graph and the eigenspace of the second smallest eigenvalue of weighted graph Laplacians using a semidefinite optimization approach. By redistributing the edge weights of a graph, the following three optimization problems are studied in this thesis: maximizing the second smallest eigenvalue (based on the mentioned work of Göring et al.), minimizing the maximum eigenvalue and minimizing the difference of maximum and second smallest eigenvalue of the weighted Laplacian. In all three problems a semidefinite optimization formulation allows to interpret the corresponding semidefinite dual as a graph realization problem. That is, to each node of the graph a vector in the Euclidean space is assigned, fulfilling some constraints depending on the considered problem. Optimal realizations are investigated and connections to the eigenspaces of corresponding optimized eigenvalues are established. Furthermore, optimal realizations are closely linked to the separator structure of the graph. Depending on this structure, on the one hand folding properties of optimal realizations are characterized and on the other hand the existence of optimal realizations of bounded dimension is proven. The general bounds depend on the tree-width of the graph. In the case of minimizing the maximum eigenvalue, an important family of graphs are bipartite graphs, as an optimal one-dimensional realization may be constructed. Taking the symmetry of the graph into account, a particular optimal edge weighting exists. Considering the coupled problem, i.e., minimizing the difference of maximum and second smallest eigenvalue and the single problems, i.e., minimizing the maximum and maximizing the second smallest eigenvalue, connections between the feasible (optimal) sets are established.

A spectral graph realization is an embedding of a finite simple graph into Euclidean space that is constructed from the eigenvalues and eigenvectors of the graph's adjacency matrix. It has previously been observed that some polytopes can be reconstructed from their edge-graphs by taking the convex hull of a spectral realization of this edge-graph. These polytopes, which we shall call spectral polytopes, have remarkable rigidity and symmetry properties and are a source for many open questions. In this thesis we aim to further the understanding of this phenomenon by exploring the geometric and combinatorial properties of spectral polytopes on several levels. One of our central questions is whether already weak forms of symmetry can be a sufficient reason for a polytope to be spectral. To answer this, we derive a geometric criterion for the identification of spectral polytopes and apply it to prove that indeed all polytopes of combined vertex- and edge-transitivity are spectral, admit a unique reconstruction from the edge-graph and realize all the symmetries of this edge-graph. We explore the same questions for graph realizations and find that realizations of combined vertex- and edge-transitivity are not necessarily spectral. Instead we show that we require a stronger form of symmetry, called distance-transitivity. Motivated by these findings we take a closer look at the class of edge-transitive polytopes, for which no classification is known. We state a conjecture for a potential classification and provide complete classifications for several sub-classes, such as distance-transitive polytopes and edge-transitive polytopes that are not vertex-transitive. In particular, we show that the latter class contains only polytopes of dimension d <= 3. As a side result we obtain the complete classification of the vertex-transitive zonotopes and a new characterization for root systems.