We discuss the problem on approximation by tight wavelet frames on the field $$\mathbb{Q}_p$$ of $$p$$ -adic numbers. For tight frames in the field $$\mathbb{Q}p$$ , constructed earlier by the authors, we obtain approximation estimates for functions from Sobolev spaces with logarithmic weight.

Abstract Federal Title IX policy requires institutions of higher education (IHEs) to prevent and respond to sexual misconduct. Based on 23 in‐depth interviews, this study explores California Title IX coordinator experiences at a critical policy juncture—the recension of the Obama‐era guidance and the implementation of the Trump administration Title IX rules and regulations—to understand how they responded to, implemented, or resisted federal mandates. Most California Title IX officers reported that the recension of Obama guidelines had limited impact on their daily operations because they lived in a state with more progressive social policy. Participants singled out California case law, state statues, the California Education Code and university system mandates as protective legislation to ensure survivors’ rights. By contrast, the new Title IX regulations that became law in August 2020 were perceived as more consequential. The tight implementation time frame, combined with the complicated and unfunded policy directives, imposed heavy burdens on their administrative offices. Title IX officers foresaw implementation consequences to campus safety, survivors’ willingness to report, and to the credibility of the Title IX office. Taken together, their experiences lend support to progressive state legislative action, restorative justice approaches and for a survivor bill of rights as important counterpoints to federal Title IX policy.

Finite tight frames play an important role in miscellaneous areas, including quantum information theory. Here we apply a class of tight frames, equiangular tight frames, to address the problem of detecting the entanglement of bipartite states. Here we derive some entanglement criteria based on positive operator-valued measurements built from equiangular tight frames. We also present a class of entanglement witnesses based on the equiangular tight frames. At last, we generalize the entanglement criterion for bipartite systems to multipartite systems.

Рассматриваются фреймы Габора, порожденные функцией Гаусса. С помощью констант неопределенности оценивается локализация функций двойственных фреймов в зависимости от соотношения параметров частотно-временного окна и степени переполненности. Общий вывод таков: при увеличении диспропорции окна локализация быстро ухудшается. С другой стороны, чем более переопределена исходная система функций, тем лучше локализованы функции двойственного фрейма. Для жесткого фрейма локализация при одном и том же наборе параметров существенно лучше, чем для двойственного фрейма. Рассматриваемая задача тесно связана с задачей интерполяции по равномерным сдвигам функции Гаусса. Построение узловой функции при интерполяции и функции окна двойственного фрейма осуществляется с помощью одних и тех же коэффициентов. Эти коэффициенты играют важную роль и при выводе формул для констант неопределенности. Поэтому в работе изучаются их свойства, связанные со знакочередуемостью и монотонностью убывания по модулю. Библиография: 38 названий.

Fast deep learning with tight frame wavelets

This study aims to reconstruct signals that are sparse with a tight frame from undersampled data by using the ℓq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{q}$\\end{document}-minimization method. This problem can be cast as a ℓq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{q}$\\end{document}-minimization problem with a tight frame subjected to an undersampled measurement with a known noise bound. We proved that if the measurement matrix satisfies the restricted isometry property with δ2s≤1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\delta _{2s}\\leq 1/2$\\end{document}, there exists a value q0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q_{0}$\\end{document} such that for any q∈(0,q0]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q\\in (0,q_{0}]$\\end{document}, any signal that is s-sparse with a tight frame can be robustly recovered to the true signal. We estimated q0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q_{0}$\\end{document} as q0=2/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q_{0} = 2/3$\\end{document} in the case of δ2s≤1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\delta _{2s}\\leq 1/2$\\end{document} and discussed that the value of q0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q_{0}$\\end{document} can be much higher. We also showed that when δ2s≤0.3317\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\delta _{2s}\\leq 0.3317$\\end{document}, for any q∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q\\in (0,1]$\\end{document}, robust recovery for signals via ℓq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{q}$\\end{document}-minimization holds, which is consistent with the case of ℓq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{q}$\\end{document}-minimization without a tight frame.

Tight frames are extremely useful in applications. A scalable frame was recently introduced as a frame with the property of generating a tight frame by rescaling its frame vectors. In this paper, we consider piecewise scalable frames. We obtain some characterizations for them, and demonstrate that scalability is stable under unitary operators and isomorphisms between two Hilbert spaces. We further obtain a relation between the piecewise scalable frames in Hilbert spaces, and their tensor product

A metaheuristic MAC protocol for safety applications in cognitive vehicular networks

On Kirkwood–Dirac quasiprobabilities and unravelings of quantum channel assigned to a tight frame

Quantum neural networks (QNNs) have become an important tool for understanding the physical world, but their advantages and limitations are not fully understood. Some QNNs with specific encoding methods can be efficiently simulated by classical surrogates, while others with quantum memory may perform better than classical classifiers. Here we systematically investigate the problem-dependent power of quantum neural classifiers (QCs) on multiclass classification tasks. Through the analysis of expected risk, a measure that weighs the training loss and the generalization error of a classifier jointly, we identify two key findings: first, the training loss dominates the power rather than the generalization ability; second, QCs undergo a U-shaped risk curve, in contrast to the double-descent risk curve of deep neural classifiers. We also reveal the intrinsic connection between optimal QCs and the Helstrom bound and the equiangular tight frame. Using these findings, we propose a method that exploits loss dynamics of QCs to estimate the optimal hyperparameter settings yielding the minimal risk. Numerical results demonstrate the effectiveness of our approach to explain the superiority of QCs over multilayer Perceptron on parity datasets and their limitations over convolutional neural networks on image datasets. Our work sheds light on the problem-dependent power of QNNs and offers a practical tool for evaluating their potential merit.

For the actual inverse synthetic aperture radar (ISAR) system, some received pulses may be invalid or lost in the long coherent processing interval (CPI) due to nonideal conditions, such as target noncooperation or noise interference. Hence, it is usually difficult to obtain well-focused images for such sparse aperture ISAR data. To address this issue, a novel general analysis model with a tight frame for sparse ISAR imaging is proposed in this article, which operates directly on the ISAR image instead of the redundant sparse coefficient. Furthermore, with the help of the projection operator and majorization minimization algorithmic framework, a simple and effective accelerated iterative algorithm with general analysis model (AIA-GAM) is designed to solve the proposed ISAR imaging model, which only performs the simple proximal mapping in each iteration. The experimental results of simulation data and real measured data demonstrate that the proposed method has better imaging performance and faster convergence speed.

Abstract. The current programme for the site selection process in Germany schedules 11 months for geomechanical laboratory tests per potential future site (BGE, 2022). For the laboratory analysis of rock salt, this time frame seems realistic as no hydromechanical coupling has to be taken into account. In the case of a crystalline basement, the intact rock matrix is low permeable, but transport processes are dominated by ubiquitous fracture networks, which are hard to capture in laboratory settings. Thus, the effect that hydromechanical coupling in low permeable rock has on the speed of laboratory measurements will probably be no problem. However, for claystones, the geomechanical processes within the rock matrix strongly depend on the hydromechanical coupling, which thus determines the speed with which, for example, loading ramps can be applied in the laboratory. The need to maintain the pore pressure equilibrium during the experiments in order to know the acting effective stresses slows down the speed of the measurements significantly. Given the tight time frame of 11 months and the numerous parameters which have to be analysed according to law (StandAG, 2017), the low speed with which the experiments can be performed represents a severe hindrance. It is thus important to establish measurement protocols which enable the simultaneous determination of several parameters in order to save time. Based on our experience with experiments on Opalinus Clay, we offer a first idea of how such a measurement protocol could look and would like to invite a discussion on how to optimise experiments on claystones further in order to have a chance to meet the very tight time frame required by the site selection process.

Provable deep video denoiser using spatial–temporal information for video snapshot compressive imaging: Algorithm and convergence analysis

We present a method for constructing tight wavelet frames on locally compact zero-dimensional groups. When constructing frames, we do not use the principle of unitary extension. We also consider the approximate properties of the resulting frames for functions from the Sobolev space with logarithmic weight.

Construction of Mercedes-Benz frames via QR factorization

In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory, and optimal transport, using the methods of the second to answer questions posed in the first. In particular, we construct gradient flows in the Wasserstein space for a new potential, the tightness potential, which is a modification of the probabilistic frame potential. It is shown that the potential is suited for the application of a gradient descent scheme from optimal transport that can be used as the basis of an algorithm to evolve an existing frame toward a tight probabilistic frame.

Hybrid character sums and near-optimal partial Hadamard codebooks

Ensuring quality control in a COVID-19 clinical trial during the pandemic: The experience of the Inserm C20–15 DisCoVeRy study

Finite tight frames are interesting in various topics including questions of quantum information. Each complex tight frame leads to a resolution of the identity in the Hilbert space. Symmetric informationally complete measurements are a special class of equiangular tight frames. Applications of such frames in quantum physics deserve more attention than they have obtained. We derive uncertainty relations for a quantum measurement assigned to an equiangular tight frame. Main results follow from estimation of the corresponding index of coincidence. State-dependent and state-independent formulations are both addressed. Also, we discuss applications of considered measurements to detect entanglement and other correlations.

Background: This article shares the reflection journey of a newly formed workplace wellness team, whose members documented their experience throughout an eight-week project using reflective journaling. Reflection was seen as a valuable tool for developing greater self-awareness and insight into team dynamics. Sharing reflections among the team also served to strengthen bonds and provided a chance to discuss challenges that were faced. Aim: To demonstrate the importance of promoting and prioritising the practice of reflection amid the high workload intensity of a busy healthcare environment. Conclusion: A diverse group of healthcare workers was able to identify the value of dedicated worktime for reflection while meeting the demands of project targets and tight time frames. Implications for practice: Personal and team reflective practice offers teambuilding opportunities for newly formed teams along with time for confidence building, self-improvement and connecting with others Timely documentation of these reflections can give immediate insight into challenges individuals or the entire team may be experiencing, thus allowing prompt response and adjustments, in addition to creating a record for personal and team learnings over time To ensure engagement in, and sustainability of, wellness-oriented reflective practice, both bottom-up and top-down commitment is required