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Function encoders are a recent technique that learn neural network basis functions to form compact, adaptive representations of Hilbert spaces of functions. We show that function encoders provide a principled connection to feature learning…

We continue the construction of the $:\phi^4_4:$ quantum field theory. In this paper we consider the Wick kernel of the interacting quantum field. Using the complex structure and the Fock-Bargmann-Berezin-Segal integral representation we…

For a Hamiltonian ${\hat H}$ containing a position-dependent (disordered) potential, we introduce a sequence of landscape functions $u_n(\vec{r})$ obeying ${\hat H} u_n(\vec{r}) = u_{n-1}(\vec{r})$ with $u_0(\vec{r}) = 1$. For $n \to…

We generalize the Lebesgue-Hausdorff Theorem on the characterization of Baire-one functions for $\sigma$-strongly functionally discrete mappings defined on arbitrary topological spaces

Let $X$ be a Banach space, let $(\Omega,\mu)$ be a $\sigma$-finite measure space and let $A,B\colon\Omega\to B(X)$ be strongly measurable $\gamma$-bounded functions. We show that for all $x\in X$ and all $x^*\in X^*$, there exist a Hilbert…

By using the quantum Ising chain as a test bed and treating the spin polarization along the external transverse field as the "generalized density", we examine the performance of different levels of density functional approximations parallel…

A transfunction is a function which maps between sets of finite measures on measurable spaces. Push-forward operators form one important class of examples of transfunctions and are identified with their respective measurable functions. In…

In the paper we investigate the continuity properties of the mapping $\Phi$ which sends any non-empty compact connected hv-convex planar set $K$ to the associated generalized conic function $f_K$. The function $f_K$ measures the average…

Electromagnetism is the paradigm case of a theory that satisfies relativistic locality. This can be proven by demonstrating that, once the theory's laws are imposed, what is happening within a region fixes what will happen in the…

A proof that minimum uncertainty states of the simplest periodic quantum system exist in a state space that is represented by a Colombeau algebra of generalised functions but not in Hilbert space or in the space of Schwartz distributions is…

It is well-known that point-set topology (without additional structure) lacks the capacity to generalize the analytic concepts of completeness, boundedness, and other typically-metric properties. The ability of metric spaces to capture this…

In this note, we give several definitions of metric corona properties which could be of interest in Set Topology, Functional Analysis and Approximation Theory, and prove that there are complete metrizable t.v.s. which are nice in the sense…

Using the $\mathbb{R}((X))$-measure, we define and study certain $\mathbb{C}((X))$-valued functions on $\mathrm{GL}_n(F)$ for $F$ a two-dimensional local field. In particular, we define a convolution product on such suitable functions,…

We describe integral lifts K(L), indexed by local fields L of degree n = [L:\Q_p], of the extraordinary cohomology theories K(n), and apply the generalized character theory of Hopkins, Kuhn and Ravenel to identify K(L)(BG) \otimes \Q$, for…

The object of the present paper is to introduce and investigate two new general subclasses ${{S}^{*}}C(\alpha ,\beta ;\gamma )$ and $T{{S}^{*}}C(\alpha ,\beta ;\gamma )~~(\alpha, \beta \in [0,1),~\gamma \in [0,1])$ of the analytic…

In contrast to the seminal weak localization prediction of a non-critical Hall constant ($R_{H}$) at the Anderson metal-insulator transition (MIT), $R_{H}$ in quite a few real disordered systems exhibits both, a strong $T$-dependence and…

We define the algebra of Colombeau generalized functions on the space of generalized points of {\mathbb R}^d which naturally contains the tempered generalized functions. The subalgebra of \mathscr{S}-regular generalized functions of this…

Inspired by the theories of Kaplansky-Hilbert modules and probability theory in vector lattices, we generalise functional analysis by replacing the scalars $\mathbb{R}$ or $\mathbb{C}$ by a real or complex Dedekind complete unital…

We discuss the structural and topological properties of a general class of weighted $L^1$ convolutor spaces. Our theory simultaneously applies to weighted $\mathcal{D}'_{L^1}$ spaces as well as to convolutor spaces of the Gelfand-Shilov…

We prove that if $X$ is a paracompact space, $Y$ is a metric space and $f:X\to Y$ is a functionally fragmented map, then (i) $f$ is $\sigma$-discrete and functionally $F_\sigma$-measurable; (ii) $f$ is a Baire-one function, if $Y$ is weak…