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1. On Single-Distance Graphs on the Rational Points in Euclidean Spaces

   https://doi.org/10.4153/S0008439520000181  

   Bau, Sheng; Johnson, Peter; Noble, Matt (March 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract For positive integers n and d > 0, let $$G(\mathbb {Q}^n,\; d)$$ denote the graph whose vertices are the set of rational points $$\mathbb {Q}^n$$ , with $$u,v \in \mathbb {Q}^n$$ being adjacent if and only if the Euclidean distance between u and v is equal to d . Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $$\mathbb {Q}^n$$ . In this paper, we show that a space $$\mathbb {Q}^n$$ has the property that all pairs of non-trivial distance graphs $$G(\mathbb {Q}^n,\; d_1)$$ and $$G(\mathbb {Q}^n,\; d_2)$$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $$G(\mathbb {Q}^n,\; d)$$ . 

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2. An Empirical Evaluation of the Calibration of Auditory Distance Perception under Different Levels of Virtual Environment Visibilities

   https://doi.org/10.1109/VR58804.2024.00089  

   Lin, Wan-Yi; Venkatakrishnan, Rohith; Venkatakrishnan, Roshan; Babu, Sabarish V; Pagano, Christopher; Lin, Wen-Chieh (March 2024, IEEE)

   The perception of distance is a complex process that often involves sensory information beyond that of just vision. In this work, we investigated if depth perception based on auditory information can be calibrated, a process by which perceptual accuracy of depth judgments can be improved by providing feedback and then performing corrective actions. We further investigated if perceptual learning through carryover effects of calibration occurs in different levels of a virtual environment’s visibility based on different levels of virtual lighting. Users performed an auditory depth judgment task over several trials in which they walked where they perceived an aural sound to be, yielding absolute estimates of perceived distance. This task was performed in three sequential phases: pretest, calibration, posttest. Feedback on the perceptual accuracy of distance estimates was only provided in the calibration phase, allowing to study the calibration of auditory depth perception. We employed a 2 (Visibility of virtual environment) ×3 (Phase) ×5 (Target Distance) multi-factorial design, manipulating the phase and target distance as within-subjects factors, and the visibility of the virtual environment as a between-subjects factor. Our results revealed that users generally tend to underestimate aurally perceived distances in VR similar to the distance compression effects that commonly occur in visual distance perception in VR. We found that auditory depth estimates, obtained using an absolute measure, can be calibrated to become more accurate through feedback and corrective action. In terms of environment visibility, we find that environments visible enough to reveal their extent may contain visual information that users attune to in scaling aurally perceived depth. 

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3. Examples of non-Dini domains with large singular sets

   https://doi.org/10.1515/ans-2022-0058  

   Kenig, Carlos; Zhao, Zihui (January 2023, Advanced Nonlinear Studies)

   Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\} , has finite ( d − 2 ) \left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d-2} -measures. 

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4. On the Strict Majorant Property in Arbitrary Dimensions

   https://doi.org/10.1093/qmath/haac021  

   Gressman, P T; Guo, S; Pierce, L B; Roos, J; Yung, P -L (July 2022, The Quarterly Journal of Mathematics)

   Abstract In this work we study d-dimensional majorant properties. We prove that a set of frequencies in $$\mathbb{Z}^d$$ satisfies the strict majorant property on $L^p([0,1]^d)$ for all p > 0 if and only if the set is affinely independent. We further construct three types of violations of the strict majorant property. Any set of at least d + 2 frequencies in $$\mathbb{Z}^d$$ violates the strict majorant property on $L^p([0,1]^d)$ for an open interval of $$p \not\in 2\mathbb{N}$$ of length 2. Any infinite set of frequencies in $$\mathbb{Z}^d$$ violates the strict majorant property on $L^p([0,1]^d)$ for an infinite sequence of open intervals of $$p \not\in 2\mathbb{N}$$ of length 2. Finally, given any p > 0 with $$p \not\in 2\mathbb{N}$$, we exhibit a set of d + 2 frequencies on the moment curve in $$\mathbb{R}^d$$ that violate the strict majorant property on $L^p([0,1]^d).$ 

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5. Rigidity and continuous extension for conformal maps of circle domains

   https://doi.org/10.1090/tran/8923  

   Ntalampekos, Dimitrios (July 2023, Transactions of the American Mathematical Society)

   We present sufficient conditions so that a conformal map between planar domains whose boundary components are Jordan curves or points has a continuous or homeomorphic extension to the closures of the domains. Our conditions involve the notions of cofat domains and C N E D CNED sets, i.e., countably negligible for extremal distances, recently introduced by the author. We use this result towards establishing conformal rigidity of a class of circle domains. A circle domain is conformally rigid if every conformal map onto another circle domain is the restriction of a Möbius transformation. We show that circle domains whose point boundary components are C N E D CNED are conformally rigid. This result is the strongest among all earlier works and provides substantial evidence towards the rigidity conjecture of He–Schramm [Invent. Math. 115 (1994), no. 2, 297–310], relating the problems of conformal rigidity and removability. 

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