Matrix methods for perfect signal recovery underlying range space of operators

The purpose of this work is to investigate perfect reconstruction underlying range space of operators in finite dimensional Hilbert spaces by a new matrix method. To this end, first we obtain more structures of the canonical $K$-dual. % and survey optimal $K$-dual problem under probabilistic erasures. Then, we survey the problem of recovering and robustness of signals when the erasure set satisfies the minimal redundancy condition or the $K$-frame is maximal robust. Furthermore, we show that the error rate is reduced under erasures if the $K$-frame is of uniform excess. Toward the protection of encoding frame (K-dual) against erasures, we introduce a new concept so called $(r,k)$-matrix to recover lost data and solve the perfect recovery problem via matrix equations. Moreover, we discuss the existence of such matrices by using minimal redundancy condition on decoding frames for operators. We exhibit several examples that illustrate the advantage of using the new matrix method with respect to the previous approaches in existence construction. And finally, we provide the numerical results to confirm the main results in the case noise-free and test sensitivity of the method with respect to noise.

The ChPT: top-down and bottom-up

In this work, higher-derivative corrections of the non-linear sigma model of both even and odd intrinsic-parity sectors are systematically studied, focusing on ordered amplitudes of flavor scalars in massless limit. It should correspond to a theory known as chiral perturbation theory (ChPT) without external sources and with only single-trace operators. We briefly overview its formal development and apply new S-matrix methods to its amplitude constructions. The bottom-up analysis of the tree-level amplitudes of different orders and multiplicities focuses on the formal structure of general ChPT. Possible theoretical simplifications based on the Kleiss-Kuijf and Bern-Carrasco-Johansson relations are presented. Finally, in the same context, the comparison with the so-called Z-function, which is connected with string theory, is also discussed.

Assessment method of the urban development level in the context of knowledge and innovation economy

The assessment method of the level of industrial city (Zaporozhіzhia) development influence affected by knowledge and innovation economy has been proposed in the article. Assessment of the sustainable development level is represented by three components: economic, social and environmental, each of which is characterized by certain indicators. The algorithm has been worked out to determine knowledge and innovation impact for urban sustainable development. It is comprised of three successive stages: indicators characterizing a certain sub-vector are determined in the first stage; sub-vector indexes are calculated in the second stage: the knowledge and innovation index; the knowledge and innovation index is calculated based on the additive and multiplicative model in the third stage. The knowledge and innovation index is defined as “stimulus” (booster) and “restriction” (depreciation) of urban sustainable development, that is, defined as specific ratios. To substantiate the importance of knowledge and innovation impact, not only quantitative approach to evaluation, but also qualitative analysis based on matrix methods have been applied. Changing specific ratios and the degree of specific ratios impact on urban sustainable development allows to analyze changes (increase or decrease) of sustainability level amid weights alterations. Changing specific ratios forms the priorities of economic policy and strategic landmarks.

Absolute and strong summability in degree $p \geqslant 1$ of $r$ times differentiated Fourier series and $r$ times differentiated conjugate Fourier series by matrix methods

We establish conditions of $|\gamma|_p$- and $[\gamma]_p$-summability in degree $p \geqslant 1$ of $r$ times differentiated Fourier series at the point where $\gamma = \| \gamma_{nk} \|$ is the matrix of transformation of series to sequence. Analogous conditions are considered also for $r$ times differentiated conjugate Fourier series.

Absolute summability of power $p$ of series, associated with conjugate Fourier series, by triangular matrix methods

We establish conditions of absolute summability of powers of series that are associated with conjugate Fourier series, by triangular matrix methods, and provide the application of the theorems proved to Voronoi-Nerlund method.

Tauberian theorems in the case of strong summability in degree $p$ of double series

We establish a Tauberian theorem in the case of strong summability in degree $p$ of double series by matrix methods, give its application to Abel methods.

Absolute and strong summability in degree $p \geqslant 1$ of series, associated with Fourier series, by matrix methods

We establish conditions of $|\gamma|_p$- and $[\gamma]_p$-summability in degree $p \geqslant 1$ of series, associated with Fourier series, at the point where $\gamma = \| \gamma_{nk} \|$ is the matrix of transformation of series to sequence.

Tauberian theorems in the case of absolute summability in degree $p$ of double series

We establish a Tauberian theorem in the case of absolute summability in degree $p$ of double series by matrix methods, give its application to Abel methods.

Fully Graph Sensitivity Solution of the Switched Circuits by Two-graph

The most general parameter of the electronic circuit is its sensitivity. Sensitivity analysis helps circuit designers to determine boundaries to predict the variations that a particular design variable will generate in a target specifications, if it differs from what is previously assumed. There are two basic methods for calculating the sensitivity: matrix methods and graph methods. The method described in this article is based on a graph, that contains separate input ad output nodes for each phase. This makes it possible to determine the transmission sensitivity even between partial switching phases. The described fully-graph method is suitable for switched current circuits and switched capacitors circuits, too