The norming set of a symmetric 3-linear form on the plane with the $l_1$-norm

An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and$|T(x_1, \ldots, x_n)|=\|T\|,$ where ${\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$For $T\in {\mathcal L}(^n E),$ we define $${Norm}(T)=\Big\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\Big\}.$$${Norm}(T)$ is called the {\em norming set} of $T$. We classify ${Norm}(T)$ for every $T\in {\mathcal L}_s(^3 l_{1}^2)$.

Embodied Experiential Learning

This paper presents a craft in experiential teaching and an experiment in embodied learning for peacebuilders and change-makers. The theories, practices and experiments are part of the postgraduate course in Peace of Mind. The intention is to invite the reader to see experiential learning and awareness-based practices as a tool that enables a possibility to evolve our humanness. Interdisciplinary abstract methodologies from Indigenous and phenomenological philosophies support the argument that granular and qualitative knowledge emerges through the embodiment of human expression. It addresses the concept of fragmentation of the self, the importance to pause to give voice to knowledge that words cannot convey. Through the arts, the paper shows non-linear forms of communication with visual experiments. The purpose of this collaborative work is in the craft, in the process, and beyond the authorship.

Networked configurations as an emergent property of transverse aeolian ridges on Mars

Transverse aeolian ridges – enigmatic Martian features without a proven terrestrial analog – are increasingly important to our understanding of Martian surface processes. However, it is not well understood how the relationships between different ridges evolve. Here we present a hypothesis for the development of complex hexagonal networks from simple linear forms by analyzing HiRISE images from the Mars Reconnaissance Orbiter. We identify variable morphologies which show the presence of secondary ridges, feathered transverse aeolian ridges and both rectangular and hexagonal networks. We propose that the formation of secondary ridges and the reactivation of primary ridge crests produces sinuous networks which then progress from rectangular cells towards eventual hexagonal cells. This morphological progression may be explained by the ridges acting as roughness elements due to their increased spatial density which would drive a transition from two-dimensional bedforms under three-dimensional flow conditions, to three-dimensional bedforms under two-dimensional flow conditions.

Universality of High-Strength Tensors

A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order.

Pietsch’s variants of s-numbers for multilinear operators

We study variants of s-numbers in the context of multilinear operators. The notion of an $$s^{(k)}$$ s ( k ) -scale of k-linear operators is defined. In particular, we shall deal with multilinear variants of the $$s^{(k)}$$ s ( k ) -scales of the approximation, Gelfand, Hilbert, Kolmogorov and Weyl numbers. We investigate whether the fundamental properties of important s-numbers of linear operators are inherited to the multilinear case. We prove relationships among some $$s^{(k)}$$ s ( k ) -numbers of k-linear operators with their corresponding classical Pietsch’s s-numbers of a generalized Banach dual operator, from the Banach dual of the range space to the space of k-linear forms, on the product of the domain spaces of a given k-linear operator.

ON AN INTEGRAL OF -BESSEL FUNCTIONS AND ITS APPLICATION TO MAHLER MEASURE

Cogdell et al. [‘Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space’, Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density $a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$ , where $J_0$ is the order-zero Bessel function of the first kind and a and $r_m$ are positive real numbers. To facilitate our proof we develop an alternative description of the integral’s asymptotic behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix.

Identification of the Domain of the Sturm–Liouville Operator on a Star Graph

This article is devoted to the unique recovering of the domain of the Sturm–Liouville operator on a star graph. The domain of the Sturm–Liouville operator is uniquely identified from the set of spectra of a finite number of specially selected canonical problems. In the general case, the domain of the definition of the original operator can be specified by integro-differential linear forms. In the case when the domain of the Sturm–Liouville operator on a star graph corresponds to the boundary value problem, it is sufficient to choose only finite parts of the spectra of canonical problems for a unique identification of the boundary form. Moreover, the above statement is valid only for a symmetric star graph.