**Abstract: **Let \(\mathbb{N}_0\) denote the set of all non-negative integers and \(X\) be any non-empty subset of \(\mathbb{N}_0\). Denote the power set of \(X\) by \(\mathcal{P}(X)\). An *integer additive set-labeling *(IASL) of a graph \(G\) is an injective function \(f : V (G) \to P(X)\) such that the image of the induced function \(f^+: E(G) \to \mathcal{P}(\mathbb{N}_0)\), defined by \(f^+(uv)=f(u)+f(v)\), is contained in \(\mathcal{P}(X)\), where \(f(u) + f(v)\) is the sumset of \(f(u)\) and \(f(v)\). If the associated set-valued edge function \(f^+\) is also injective, then such an IASL is called an integer additive set-indexer (IASI). An IASL \(f\) is said to be a topological IASL (TIASL) if \(f(V(G))\cup \{\emptyset\}\) is a topology of the ground set \(X\). An IASL is said to be an integer additive set-sequential labeling (IASSL) if \(f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}\). An IASL of a given graph \(G\) is said to be a topological integer additive set-sequential labeling of \(G\), if it is a topological integer additive set-labeling as well as an integer additive set-sequential labeling of \(G\). In this paper, we study the conditions required for a graph \(G\) to admit this type of IASL and propose some important characteristics of the graphs which admit this type of IASLs.

**Abstract: **Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\,d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.

**Abstract: **An account of the Schwartz space of rapidly decreasing functions as a topological vector space with additional special structures is presented in a manner that provides all the essential background ideas for some areas of quantum mechanics along with infinite-dimensional distribution theory.

**Abstract: **We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by a number of examples. The accuracy of the method is not worse, and in many cases better, than that of PadÃ© approximants, when the latter can be defined.

**Abstract: **This paper is a follow-up of a previous paper of the author published in Mathematics journal in 2015, which treats the so-called continuous fractional orthogonal derivative. In this paper, we treat the discrete case using the fractional orthogonal difference. The theory is illustrated with an application of a fractional differentiating filter. In particular, graphs are presented of the absolutel value of the modulus of the frequency response. These make clear that for a good insight into the behavior of a fractional differentiating filter, one has to look for the modulus of its frequency response in a log-log plot, rather than for plots in the time domain.

**Abstract: **Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286; 563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286; 563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).