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1. On the Effects of Pain on fNIRS Classification

   Shamsi, F.; Najafizadeh, L. (April 2020, The OSA Biophotonics Congress)

   We present the first study of the effects of pain on the classification of fNIRS recordings, and evaluate the performance of a model trained on the pain-free data for classifying the data with the presence of pain. 

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2. On Harmonic Hilbert Spaces on Compact Abelian Groups

   https://doi.org/10.1007/s00041-023-09992-4  

   Das, Suddhasattwa; Giannakis, Dimitrios (February 2023, Journal of Fourier Analysis and Applications)

   Abstract Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted$$L^2$$ $L^2$spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach$$^*$$ ${}^{\ast }$-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian groupGto have the same spectrum as the$$C^*$$ $C^{\ast }$-algebra of continuous functions onG. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on$$L^2(G)$$ $L^2\left(G\right)$, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation. 

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3. On the general dyadic grids on

   https://doi.org/10.4153/S0008414X22000360  

   Anderson, Theresa C.; Hu, Bingyang (July 2022, Canadian Journal of Mathematics)

   Abstract Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson, and Wei), we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here, we extend this work to all dimensions, which turns out to have many surprising difficulties due to the fact that $d+1$ , not $2^d$ , grids is the optimal number in an adjacent dyadic system in $$\mathbb {R}^d$$ . As a byproduct, we show that a collection of $d+1$ dyadic systems in $$\mathbb {R}^d$$ is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on $$\mathbb {R}$$ . The underlying geometric structures that arise in this higher-dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and n -adic, for any n ) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific exa. 

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4. Carleson embedding on tri-tree and on tri-disc

   Mozolyako, Pavel; Psaromiligkos, Georgios; Volberg, Alexander; Zorin-Kranich, Pavel (May 2022, Revista matemática iberoamericana)

   We prove multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces in bi-disc and tri-disc this proves the embedding theorem of those Dirichlet spaces of holomorphic function on bi- and tri-disc. We completely describe the Carleson measures for such embeddings. The result below generalizes embedding result of [AMPVZ] from bi- tree to tri-tree and from Carleson–Chang condition to Carleson box condition. One of our embedding description is similar to Carleson–Chang–Fefferman condition and involves dyadic open sets. On the other hand, the unusual feature is that embedding on bi-tree and tri-tree turned out to be equivalent to one box Carleson condition. This is in striking difference to works of Chang–Fefferman and well known Carleson quilt counterexample. Finally, we explain the obstacle that prevents us from proving our results on poly-discs of dimension four and higher. 

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5. On finite energy monopoles on $$C x \Sigma$$.

   Wang, Donghao (July 2019, ArXiv.org)

   Let $$X=\mathbb{C}\times\Sigma$$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $$X$$ with finite analytic energy. The spin bundle $$S^+\to X$$ splits as $$L^+\oplus L^-$$. When $$2-2g\leq c_1(S^+)[\Sigma]<0$$, the moduli space is in bijection with the moduli space of pairs $$((L^+,\bar{\partial}), f)$$ where $$(L^+,\bar{\partial})$$ is a holomorphic structure on $L^+$ and $$f: \mathbb{C}\to H^0(\Sigma, L^+,\bar{\partial})$$ is a polynomial map. Moreover, the solution has analytic energy $$-4\pi^2d\cdot c_1(S^+)[\Sigma]$$ if $$f$$ has degree $$d$$. When $$c_1(S^+)=0$$, all solutions are reducible and the moduli space is the space of flat connections on $$\bigwedge^2 S^+$$. We also estimate the decay rate at infinity for these solutions. 

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