gaussian window  (MathWorks Inc)

Journal: Monatshefte Fur Mathematik

Article Title: The AGM of Gauss, Ramanujan’s corresponding theory, and spectral bounds of self-adjoint operators

doi: 10.1007/s00605-024-02051-0

Figure Lengend Snippet: The root systems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{A}_1 \times \textsf{A}_1$$\end{document} A 1 × A 1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{D}_2$$\end{document} D 2 are isomorphic as well as the root systems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{B}_2$$\end{document} B 2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{C}_2$$\end{document} C 2 . All of them generate a (scaled) von Neumann lattice, by considering all integer linear combinations. The other existing root systems are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{A}_2$$\end{document} A 2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{G}_2$$\end{document} G 2 . Both generate a hexagonal lattice. Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a A_1 \times b A_1$$\end{document} a A 1 × b A 1 , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a, b > 0$$\end{document} a , b > 0 is also a root system which gives a rectangular lattice

Article Snippet: (Janssen [ ]) For the Gaussian window \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (t) = 2^{1/4} e^{-\pi t^2}$$\end{document} φ ( t ) = 2 1 / 4 e - π t 2 and a lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda = \Lambda (\alpha ) \subseteq \mathbb {R}^2$$\end{document} Λ = Λ ( α ) ⊆ R 2 of density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 2N$$\end{document} α = 2 N , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \in \mathbb {N}$$\end{document} N ∈ N , consider the Gabor system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(\varphi , \Lambda )$$\end{document} G ( φ , Λ ) .

Techniques:

Journal: Monatshefte Fur Mathematik

Article Title: The AGM of Gauss, Ramanujan’s corresponding theory, and spectral bounds of self-adjoint operators

doi: 10.1007/s00605-024-02051-0

Figure Lengend Snippet: The von Neumann lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^2$$\end{document} Z 2 contains the root system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{A}_1 \times \textsf{A}_1$$\end{document} A 1 × A 1 . By adding new points in the center of the fundamental cell (deep hole) we obtain a lattice with twice the density. The original lattice is contained as a sub-lattice and the new lattice contains the root system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{C}_2$$\end{document} C 2 . The new lattice is merely a rotation of the scaled original by 45 degrees

Article Snippet: (Janssen [ ]) For the Gaussian window \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (t) = 2^{1/4} e^{-\pi t^2}$$\end{document} φ ( t ) = 2 1 / 4 e - π t 2 and a lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda = \Lambda (\alpha ) \subseteq \mathbb {R}^2$$\end{document} Λ = Λ ( α ) ⊆ R 2 of density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 2N$$\end{document} α = 2 N , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \in \mathbb {N}$$\end{document} N ∈ N , consider the Gabor system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(\varphi , \Lambda )$$\end{document} G ( φ , Λ ) .

Techniques:

Journal: Monatshefte Fur Mathematik

Article Title: The AGM of Gauss, Ramanujan’s corresponding theory, and spectral bounds of self-adjoint operators

doi: 10.1007/s00605-024-02051-0

Figure Lengend Snippet: The hexagonal lattice of density 1 contains the root system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{A}_2$$\end{document} A 2 . By adding new points in the center of the fundamental triangle (deep hole) we obtain a lattice with thrice the density. The original lattice is contained as a sub-lattice and the new lattice contains the root system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{G}_2$$\end{document} G 2 . The new lattice is merely a rotation of the scaled original by 30 degrees

Article Snippet: (Janssen [ ]) For the Gaussian window \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (t) = 2^{1/4} e^{-\pi t^2}$$\end{document} φ ( t ) = 2 1 / 4 e - π t 2 and a lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda = \Lambda (\alpha ) \subseteq \mathbb {R}^2$$\end{document} Λ = Λ ( α ) ⊆ R 2 of density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 2N$$\end{document} α = 2 N , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \in \mathbb {N}$$\end{document} N ∈ N , consider the Gabor system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(\varphi , \Lambda )$$\end{document} G ( φ , Λ ) .

Techniques:

Journal: Science Advances

Article Title: No replication of direct neuronal activity–related (DIANA) fMRI in anesthetized mice

doi: 10.1126/sciadv.adl0999

Figure Lengend Snippet: One DIANA trial composed of total 20 prestimulus and 180 poststimulus frames (i.e., 1000-ms interstimulus interval) (mouse #3, see table S1). ( A ) tSNR map calculated from the average of 50 trials. ( B ) Cross-correlation map obtained with an expected neuronal response curve from Toi et al. , which was thresholded by absolute cross-correlation values of 0.1. Square box, 5 × 5–voxel S1BF ROI. ( C ) Trial-wise time courses of the active S1BF ROI. To visualize individual trials, 5 trials were plotted per row. Vertical bar, stimulus. ( D ) The averaged time course of 50 repeated trials with an expanded view in inset (red). An expected DIANA response (peaked at 25 ms) based on electrophysiology was also plotted (blue). A statistically significant positive peak was detected at 10 ms after the stimulus. Shaded area, SEM. ( E ) A histogram of data points (50 trials × 200 time points) in (C). A Gaussian noise distribution was observed with an SD of 0.377%. ( F ) A histogram of 50 individual trials at a fixed time (10 ms after stimulus) of the statistically significant positive peak. DIANA-like peaks may be observed from noisy data with limited samples just by chance.

Article Snippet: The DIANA response function was modeled as a Gaussian window function in MATLAB, “gausswin,” with a peak position at 25 ms after the stimulus and the full width at half maximum (FWHM) of ~25 ms.

Techniques:

Journal: Science Advances

Article Title: No replication of direct neuronal activity–related (DIANA) fMRI in anesthetized mice

doi: 10.1126/sciadv.adl0999

Figure Lengend Snippet: ( A ) Averaged time courses of six individual animals in the active S1BF ROI without excluding trials (same as ). ( B ) A Gaussian reference outlier function with a full width of half minimum (FWHM) of ~25 ms peaked at 25 ms (blue) and 35 ms as a control (red). ( C ) An average of all animals’ time courses without excluding trials. This is the same as . In six animals, an individual trial’s time course was correlated with a reference function shown in (B), and ranked among all trials, on the basis of its cross-correlation value. Then, in each animal, trials were separated into included and excluded categories, based on an outlier threshold of top 6, 10, and 20% cross-correlation values. ( D and E ) Averaged time courses of the included and excluded trials in individual animals for the 25-ms (D) and 35-ms peak reference function (E). Averaged time courses of the included trials showed noisy positive responses, while those of the excluded trials showed a negative peak around the expected peak time. Each color time course, each animal. ( F ) Means of subject-wise included trials time courses with the 25-ms (blue) and 35-ms peak reference function (red). Clearly, an exclusion process leads to a spurious peak from noisy data. It is fundamentally important to note that these peaks are erroneous because such a preselection process is statistically circular, summing noise in a biased manner to produce precisely the peak that was preselected. Error bars, SEM.

Article Snippet: The DIANA response function was modeled as a Gaussian window function in MATLAB, “gausswin,” with a peak position at 25 ms after the stimulus and the full width at half maximum (FWHM) of ~25 ms.

Techniques: Control

Journal: Science Advances

Article Title: No replication of direct neuronal activity–related (DIANA) fMRI in anesthetized mice

doi: 10.1126/sciadv.adl0999

Figure Lengend Snippet: ( A ) Gaussian reference functions for 5 × 5–voxel regions of the active contralateral thalamus (red) and S1BF (blue). Neuronal responses in the thalamus and S1 are expected to peak at ~15 and ~ 25 ms after the onset of whisker stimulus [figure 2 in the study by Toi et al. ], respectively. In three mice with significant BOLD responses also in the contralateral thalamus, a Gaussian reference function with a FWHM of ~15 ms peaked at 25 ms in the S1BF (blue) and 15 ms in the thalamus (red) was used to cross-correlate time courses of individual trials. In each animal, trials were selected, on the basis of a threshold of top 20, 50, and 80% cross-correlation values in each region. Two sets of selected trials (i.e., cortical and thalamic selection) for each threshold were obtained. ( B ) Averaged time courses of the S1BF and thalamus ROI obtained from the S1BF-selected trials. Blue, S1BF time course; red, thalamus. Error bars, SEM. ( C ) Cross-correlation map of one mouse (mouse #5). For the top 20% trials selected for the S1BF, voxel-wise cross-correlation values with Gaussian neural response functions peaking between 10 and 30 ms were calculated, and the highest cross-correlation values were mapped with a correlation amplitude threshold of 0.4 and a minimum of three contiguous voxels (see also fig. S4). ( D ) Averaged time courses of the S1BF and thalamus ROI obtained from the thalamus-selected trials. Blue, S1BF time course; red, thalamus. Artifactual peak intensity in the selected region was closely dependent on the selection threshold, while no obvious peak was observed in the unselected region. Error bars, SEM.

Article Snippet: The DIANA response function was modeled as a Gaussian window function in MATLAB, “gausswin,” with a peak position at 25 ms after the stimulus and the full width at half maximum (FWHM) of ~25 ms.

Techniques: Whisker Assay, Selection

Journal: Science Advances

Article Title: No replication of direct neuronal activity–related (DIANA) fMRI in anesthetized mice

doi: 10.1126/sciadv.adl0999

Figure Lengend Snippet: A Gaussian reference function with a FWHM of ~15 ms peaked at 25 ms in the S1BF and 15 ms in the thalamus was used to cross-correlate time courses of individual trials (see ). In all six individual mice, trials were selected jointly from both ROIs, based on a threshold of top 20, 50, and 80% combined cross-correlation values. ( A and B ) Averaged (A) and individual animal’s time courses (B) of the S1BF and thalamus ROI obtained from the jointly selected trials in six mice. Artifactual peaks with a 10-ms difference between thalamus and S1BF were obviously observed as expected. Error bars, SEM. ( C and D ) Cross-correlation (C) and peak time map (D) of one mouse (mouse #6). For top 20% trials selected for both S1BF and thalamus, voxel-wise cross-correlation values with Gaussian neural response functions peaked between 10 and 30 ms were calculated, and the highest cross-correlation values and time shifts were then mapped with a cross-correlation threshold of 0.35 and a minimum of three contiguous voxels (see also fig. S5).

Article Snippet: The DIANA response function was modeled as a Gaussian window function in MATLAB, “gausswin,” with a peak position at 25 ms after the stimulus and the full width at half maximum (FWHM) of ~25 ms.

Techniques: