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The Birkhoff orthogonality has been recently intensively studied in connection with the geometry of Banach spaces and operator theory. The main aim of this paper is to characterize the Birkhoff orthogonality in${\\mathcal{L}}(X;Y)$under the assumption that${\\mathcal{K}}(X;Y)$is an$M$-ideal in${\\mathcal{L}}(X;Y)$. Moreover, we survey the known results, as well as giving some new and more general ones. Furthermore, we characterize an approximate Birkhoff orthogonality in${\\mathcal{K}}(X;Y)$.

Let Gp(x,y,z)=∥x+y+z∥p+∥z∥p−∥x+z∥p−∥y+z∥p be defined on a normed space X. The special case G2(x,y,z)=0,∀z∈X, where X is a real normed linear space, coincides with the trapezoid orthogonality (T-orthogonality), which was originally proposed by Alsina et al. in 1999. In this paper, for the case where X is a complex inner product space endowed with the inner product ⟨·,·⟩ and induced norm ∥·∥, it is proved that Sgn(G2(x,y,z))=Sgn(Re⟨x,y⟩),∀z∈X, and a geometric explanation for condition Re⟨x,y⟩=0 is provided. Furthermore, a condition G2(x,iy,z)=0,∀z∈X is added to extend the T-orthogonality to the general complex normed linear spaces. Based on some characterizations, the T-orthogonality is compared with several other well-known types of orthogonality. The fact that T-orthogonality implies Roberts orthogonality is also revealed.

Associated with the irreducible crystallographic root system A3, discretizations of symmetric Weyl orbit functions emerging from generic types of invariant lattices are reviewed. Utilizing an overview of pertinent properties of the root system A3, its Weyl group orbit and corresponding Weyl group invariant lattices are presented. Generalized affine Weyl groups are introduced and described together with their fundamental domains. The symmetric Weyl orbit functions of A3 are recalled and their discrete orthogonality relations on fragments of rescaled non-shifted invariant lattices are inspected. The corresponding unitary transform matrices are provided in general form, including two explicit examples.

The article’s primary focus is on the study of the number of countable non-isomorphic models of linearly ordered theories. The orthogonality of 1-types and their convex closures is employed to analyse a class of theories with a specific type of monotonic non-orthogonality, which includes weakly o-minimal theories. For such theories, a theorem analogous to L. Mayer’s result on the independence of any pairwise independent family of 1-types in o-minimal theories is proven. The article provides conditions for the infinity and maximality of the countable spectrum of weakly o-minimal theories.

In this study, we consider the Hermite-Hadamard type of unitary Carlsson’s orthogonality (UHH-C-orthogonality) to characterize real inner product spaces. We give a necessary and sufficient condition weaker than the homogeneity of symmetric HH-C-orthogonalities which characterizes inner product spaces among normed linear spaces of dimension at least three. In conclusion, some more characterizations of real inner product spaces are provided.

This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of x in the y direction is zero when the norm is Gâteaux differentiable in the y direction at x and x and y satisfy certain generalized orthogonality conditions. A case where x and y are approximately orthogonal is also analyzed and the value range of the Gâteaux derivative in this case is given. Moreover, two concepts are introduced: the angle between vectors in normed linear space and the ⊥Δ coordinate system in a smooth Minkowski plane. Relevant examples are given at the end of the paper.

We have studied parabolic cycles and given their equivalent matrix representations. Invariant properties of matrices under similarity have contributed to the Möbius-invariant properties in cycles. We have further discussed the inner product in the cycle space. Also, the geometrical properties of orthogonality and reflection have been studied to obtain the irregular points in the cycle space.

Using a hypothetical graph, Masahiro Mori proposed in 1970 the relation between the human likeness of robots and other anthropomorphic characters and an observer’s affective or emotional appraisal of them. The relation is positive apart from a U-shaped region known as the uncanny valley. To measure the relation, we previously developed and validated indices for the perceptual-cognitive dimension humanness and three affective dimensions: interpersonal warmth, attractiveness, and eeriness. Nevertheless, the design of these indices was not informed by how the untrained observer perceives anthropomorphic characters categorically. As a result, scatter plots of humanness vs. eeriness show the stimuli cluster tightly into categories widely separated from each other. The present study applies a card sorting task, laddering interview, and adjective evaluation (N=30) to revise the humanness, attractiveness, and eeriness indices and validate them via a representative survey (N=1311). The revised eeriness index maintains its orthogonality to humanness (r=.04, p=.285), but the stimuli show much greater spread, reflecting the breadth of their range in human likeness and eeriness. The revised indices enable empirical relations among characters to be plotted similarly to Mori’s graph of the uncanny valley. Accurate measurement with these indices can be used to enhance the design of androids and 3D computer animated characters.

In this paper, we provide efficient estimators and honest confidence bands for a variety of treatment effects including local average (LATE) and local quantile treatment effects (LQTE) in data-rich environments. We can handle very many control variables, endogenous receipt of treatment, heterogeneous treatment effects, and function-valued outcomes. Our framework covers the special case of exogenous receipt of treatment, either conditional on controls or unconditionally as in randomized control trials. In the latter case, our approach produces efficient estimators and honest bands for (functional) average treatment effects (ATE) and quantile treatment effects (QTE). To make informative inference possible, we assume that key reduced-form predictive relationships are approximately sparse. This assumption allows the use of regularization and selection methods to estimate those relations, and we provide methods for postregularization and post-selection inference that are uniformly valid (honest) across a wide range of models. We show that a key ingredient enabling honest inference is the use of orthogonal or doubly robust moment conditions in estimating certain reducedform functional parameters. We illustrate the use of the proposed methods with an application to estimating the effect of 401(k) eligibility and participation on accumulated assets. The results on program evaluation are obtained as a consequence of more general results on honest inference in a general moment-condition framework, which arises from structural equation models in econometrics. Here, too, the crucial ingredient is the use of orthogonal moment conditions, which can be constructed from the initial moment conditions. We provide results on honest inference for (function-valued) parameters within this general framework where any high-quality, machine learning methods (e.g., boosted trees, deep neural networks, random forest, and their aggregated and hybrid versions) can be used to learn the nonparametric/high-dimensional components of the model. These include a number of supporting auxiliary results that are of major independent interest: namely, we (1) prove uniform validity of a multiplier bootstrap, (2) offer a uniformly valid functional delta method, and (3) provide results for sparsitybased estimation of regression functions for function-valued outcomes.

This paper explores the concept of approximate Birkhoff–James orthogonality in the context of operators on semi-Hilbert spaces. These spaces are generated by positive semi-definite sesquilinear forms. We delve into the fundamental properties of this concept and provide several characterizations of it. Using innovative arguments, we extend a widely known result initially proposed by Magajna [17]. Additionally, we improve a recent result by Sen and Paul [24] regarding a characterization of approximate numerical radius orthogonality of two semi-Hilbert space operators, such that one of them is A -positive. Here, A is assumed to be a positive semi-definite operator.