Unit Norm Research Articles

Family Medicine Presence on Labor and Delivery: Effect on Safety Culture and Cesarean Delivery.

Currently, 40% of counties in the United States do not have an obstetrician or midwife, and in rural areas the likelihood of childbirth being attended to by a family medicine (FM) physician is increasing. We sought to characterize the effect of the FM presence on unit culture and a key perinatal quality metric in Iowa hospital intrapartum units. Using a cross-sectional design, we surveyed Iowa physicians, nurses, and midwives delivering intrapartum care at hospitals participating in a quality improvement initiative to decrease the incidence of cesarean delivery. We linked respondents with their hospital characteristics and outcomes data. The primary outcome was the association between FM physician, obstetrician (OB), or both disciplines' presence on labor and delivery and hospital low-risk, primary cesarean delivery rate. Unit culture was compared by hospital type (FM-only, OB-only, or Both). A total of 849 clinicians from 39 hospitals completed the survey; 13 FM-only, 11 OB-only, and 15 hospitals with both. FM-only hospitals were all rural, with <1,000 annual births. Among hospitals with <1,000 annual births, births at FM-only hospitals had an adjusted 34.3% lower risk of cesarean delivery (adjusted incident rate ratio = 0.66; 95% CI, 0.52-.0.98) compared with hospitals with both. Nurses endorsed unit norms more supportive of vaginal birth and stronger safety culture at FM-only hospitals (P <.05). Birthing hospitals staffed exclusively by FM physicians were more likely to have lower cesarean rates and stronger nursing-rated safety culture. Both access and quality of care provide strong arguments for reinforcing the pipeline of FM physicians training in intrapartum care.

A Power Method for Computing the Dominant Eigenvalue of a Dual Quaternion Hermitian Matrix

In this paper, we first study the projections onto the set of unit dual quaternions, and the set of dual quaternion vectors with unit norms. Then we propose a power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix. For a strict dominant eigenvalue, we show the sequence generated by the power method converges to the dominant eigenvalue and its corresponding eigenvector linearly. For a general dominant eigenvalue, we establish linear convergence of the standard part of the dominant eigenvalue. Based upon these, we reformulate the simultaneous localization and mapping problem as a rank-one dual quaternion completion problem. A two-block coordinate descent method is proposed to solve this problem. One block has a closed-form solution and the other block is the best rank-one approximation problem of a dual quaternion Hermitian matrix, which can be computed by the power method. Numerical experiments are presented to show the efficiency of our proposed power method.

Shape Sensing of Cantilever Column Using Hybrid Frenet-Serret Homogeneous Transformation Matrix Method.

The Frenet-Serret (FS) framework stands as a pivotal tool in shape sensing for various infrastructures. However, this tool suffers from accumulative errors, particularly at inflection points where the normal vector undergoes sign changes. To minimize the error, the traditional FS framework is modified by incorporating the homogeneous matrix transformation (HMT) method for segments containing inflection points. Additionally, inclination information is also used to calculate the unit tangent vector and the unit norm vector at the start point of each segment. This novel approach, termed the FS-HMT method, aims to enhance accuracy. To validate the effectiveness of the proposed method, a simulation of a cantilever column was conducted using finite element software ANSYS 19.2. The numerical results demonstrate the capability of the proposed method to accurately predict curves with inflection points, yielding a maximum error of 1.1%. Subsequently, experimental verification was performed using a 1 m long spring steel sheet, showcasing an error of 4.9%, which is notably lower than that of the traditional FS framework. Our proposed modified FS framework exhibits improved accuracy, especially in scenarios involving inflection points. These findings underscore its potential as a valuable tool for enhanced shape sensing in practical applications.

Disentangling Safety Culture's Role in Reducing Cesarean Overuse: Creating a Revised Labor Culture Survey.

To measure and assess the relationship of patient safety culture to reducing cesarean overuse. Maternity care hospitals in Michigan. Cross-sectional observational design, combining individual survey data with hospital characteristics using existing databases. Multivariate Poisson regression assessed the associations between survey scores and hospital nulliparous term singleton vertex cesarean rates. Factor analysis determined the scalability of survey items. Electronic survey distributed at the hospital site level. A total of 3091 clinicians from 54 out of 57 eligible hospitals completed the survey. Confirmatory factor analysis demonstrated best fit with a univariate model with two local factors. The new scale encompassing both local factors, including vaginal birth microculture and safety culture, is entitled "Unit Norms." The safety culture subdomain demonstrated an association with a reduction in hospital cesarean rate [-0.15; 95% CI: -0.27 to -0.04; incident rate ratio (IRR) 0.86], parallel to but lower in magnitude to vaginal birth microculture (-0.18; 95% CI: -0.35 to -0.02; IRR 0.84). Vaginal birth microculture remains the strongest predictor of cesarean delivery overuse; however, safety culture characteristics, including teamwork, psychological safety, and communication, correlate with lower cesarean delivery rates. Measuring these aspects of hospitals' culture may be important for other areas of quality improvement initiatives focused on quality and safety.

Designing incoherent unit norm tight frames via block coordinate descent-based alternating projection

In this paper, we introduce a novel methodology for constructing incoherent unit norm tight frames, termed block coordinate descent-based alternating projection. In the proposed approach, the underlying min-max problem was reformulated in a block-wise fashion by taking a fixed number of columns as a block and adding a spectral constraint, and it was approximately solved via an alternating projection method with Nesterov accelerated gradient technique. Numerical experiments demonstrate that the resulting algorithm is particularly efficient for the construction of incoherent unit norm tight frames with high redundancy and has great advantages in running efficiency.

Линейные интегральные операторы в пространствах непрерывных и существенно ограниченных вектор-функций

The well-established criterion for the action and boundedness of a linear integral operator K from the space L_∞ of essentially bounded functions to the space C of functions continuous on a compact set is extended to the case of functions taking values in Banach spaces. The study further shows that if the operator K is active and bounded in the space C, it is also active and bounded in the space L_∞, with the norms of K in C and L_∞ being identical. A precise expression for the general value of the norm of the operator K in these spaces, expressed in terms of its operator kernel, is provided. Addicionally, an example of an integral operator (for scalar functions) is given, active and bounded in each of the spaces C and L_∞, but not acting from L_∞ into C. Convenient conditions for checking the boundedness of the operator K in C and L_∞ are discussed. In the case of the Banach space Y of the image function values of K being finite-dimensional, these conditions are both necessary and sufficient. In the case of infinitedimensionality of Y, they are sufficient but not necessary (as proven). For "dim" Y&lt;∞, unimprovable estimates for the norm of the operator K are provided in terms of a 1-absolutely summing constant π_1 (Y), determined by the geometric properties of the norm in Y. Specifically, it is defined as the supremum over finite sets of nonzero elements of Y of the ratio of the sum of the norms of these elements to the supremum (over functionals with unit norm) of the sums of absolute values of the functional on these elements.

Analysis of the Current Situation of Big Data MOOCs in the Intelligent Era Based on the Perspective of Improving the Mental Health of College Students

A three-dimensional MOOC analysis framework was developed, focusing on platform design, organizational mechanisms, and course construction. This framework aims to investigate the current situation of big data MOOCs in the intelligent era, particularly from the perspective of improving the mental health of college students; moreover, the framework summarizes the construction experience and areas for improvement. The construction of 525 big data courses on 16 MOOC platforms is compared and analyzed from three aspects: the platform (including platform construction, resource quantity, and resource quality), organizational mechanism (including the course opening unit, teacher team, and learning norms), and course construction (including course objectives, teaching design, course content, teaching organization, implementation, teaching management, and evaluation). Drawing from the successful practices of international big data MOOCs and excellent Chinese big data MOOCs, and considering the requirements of authoritative government documents, such as the no. 8 document (J.G. [2019]), no. 3 document (J.G. [2015]), no. 1 document (J.G. [2022]), as well as the “Educational Information Technology Standard CELTS-22—Online Course Evaluation Standard”, recommendations about the platform, organizational mechanism, and course construction are provided for the future development of big data MOOCs in China.

A Note on the Hausdorff Distance Between Norm Balls and Their Linear Maps

We consider the problem of computing the (two-sided) Hausdorff distance between the unit ℓp1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p_{1}}$\\end{document} and ℓp2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p_{2}}$\\end{document} norm balls in finite dimensional Euclidean space for 1≤p1<p2≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1 \\leq p_{1} < p_{2} \\leq \\infty $\\end{document}, and derive a closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k_{1}$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k_{2}$\\end{document} unit D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$D$\\end{document}-norm balls, which are certain polyhedral norm balls in d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d$\\end{document} dimensions for 1≤k1<k2≤d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1 \\leq k_{1} < k_{2} \\leq d$\\end{document}. When two different ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p}$\\end{document} norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to different ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p}$\\end{document} unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting.

Dynamics of Ginzburg-Landau vortices for vector fields on surfaces

In this paper we consider the gradient flow of the following Ginzburg-Landau type energyFε(u):=12∫M|Du|g2+12ε2(|u|g2−1)2volg. This energy is defined on tangent vector fields on a 2-dimensional closed and oriented Riemannian manifold M (here D stands for the covariant derivative) and depends on a small parameter ε>0. If the energy satisfies proper bounds, when ε→0 the second term forces the vector fields to have unit length. However, due to the incompatibility for vector fields on M between the Sobolev regularity and the unit norm constraint, critical points of Fε tend to generate a finite number of singular points (called vortices) having non-zero index (when the Euler characteristic is non-zero). These types of problems have been extensively analyzed in the recent paper by R. Ignat & R. Jerrard [19]. As in Euclidean case (see, among the others [8]), the position of the vortices is ruled by the so-called renormalized energy.In this paper we are interested in the dynamics of vortices. We rigorously prove that the vortices move according to the gradient flow of the renormalized energy, which is the limit behaviour when ε→0 of the gradient flow of the Ginzburg-Landau energy.

Evaluation of a charge nurse leadership development program.

Evaluation of a charge nurse leadership development program.

MERCEDES-BENZ FRAMES IN $$\mathbb {R}^d$$ AS A DIRECT SUM OF A PAIR OF ORTHOGONAL TIGHT FRAMES

The well-known Mercedes-Benz frames in $$\mathbb {R}^d$$ are known to exist and they have been constructed too. They are presented here as an external direct sum of a pair of orthogonal frames. A necessary and sufficient condition for a unit norm tight frame to be a Mercedes-Benz frame in $$\mathbb {R}^d$$ is provided in terms of a vector from the null space of its synthesis operator. Also, a necessary and sufficient condition for a unit norm system of vectors to be a Mercedes-Benz frame is provided in terms of the frame potential and a vector from the null space of the synthesis operator.

Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems

Dynamical vectors characterizing instability and applicable as ensemble perturbations for prediction with geophysical fluid dynamical models are analysed. The relationships between covariant Lyapunov vectors (CLVs), orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors and finite-time normal modes (FTNMs) are examined for periodic and aperiodic systems. In the phase-space of FTNM coefficients, SVs are shown to equate with unit norm FTNMs at critical times. In the long-time limit, when SVs approach OLVs, the Oseledec theorem and the relationships between OLVs and CLVs are used to connect CLVs to FTNMs in this phase-space. The covariant properties of both the CLVs, and the FTNMs, together with their phase-space independence, and the norm independence of global Lyapunov exponents and FTNM growth rates, are used to establish their asymptotic convergence. Conditions on the dynamical systems for the validity of these results, particularly ergodicity, boundedness and non-singular FTNM characteristic matrix and propagator, are documented. The findings are deduced for systems with nondegenerate OLVs, and, as well, with degenerate Lyapunov spectrum as is the rule in the presence of waves such as Rossby waves. Efficient numerical methods for the calculation of leading CLVs are proposed. Norm independent finite-time versions of the Kolmogorov-Sinai entropy production and Kaplan-Yorke dimension are presented.

Counterexamples to the extendibility of positive unital norm-one maps

Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space.

Majorization-Minimization for Sparse Nonnegative Matrix Factorization With the $\beta$-Divergence

This article introduces new multiplicative updates for nonnegative matrix factorization with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -divergence and sparse regularization of one of the two factors (say, the activation matrix). It is well known that the norm of the other factor (the dictionary matrix) needs to be controlled in order to avoid an ill-posed formulation. Standard practice consists in constraining the columns of the dictionary to have unit norm, which leads to a nontrivial optimization problem. Our approach leverages a reparametrization of the original problem into the optimization of an equivalent scale-invariant objective function. From there, we derive block-descent majorization-minimization algorithms that result in simple multiplicative updates for either <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> -regularization or the more “aggressive” log-regularization. In contrast with other state-of-the-art methods, our algorithms are universal in the sense that they can be applied to any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -divergence (i.e., any value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> ) and that they come with convergence guarantees. We report numerical comparisons with existing heuristic and Lagrangian methods using various datasets: face images, an audio spectrogram, hyperspectral data, and song play counts. We show that our methods obtain solutions of similar quality at convergence (similar objective values) but with significantly reduced CPU times.

Extreme contractions on finite-dimensional Banach spaces

We study extreme contractions in the setting of finite-dimensional polyhedral Banach spaces. Motivated by the famous Krein–Milman Theorem, we prove that a rank $1$ linear operator of unit norm between such spaces can be expressed as a convex combination o

MOLECULES GENERATION TO INHIBIT COVID-19 DISEASE USING ENCODER-DECODER LSTM ARCHITECTURE AND PCA PROPERTIES

COVID-19 has become the world’s worst pandemic and has claimed over six million lives as of March 2022. The virus is now in alongside cancer as one of the most common causes of death. Likewise, there is no definitive or unique treatment for COVID-19 outside of a selected few drugs approved by the Food and Drug Administration (FDA). While Artificial Intelligence (AI) can be used to generate molecules that target Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), the virus responsible for COVID-19, such molecules are novel and do not yet exist in the market. With the emergence and availability of several drug datasets related to COVID-19 (tests, images, graphs, and ChEMBLs), recent works based on Deep Learning (DL) techniques have been employed to generate molecules and check the effectiveness of existing molecules on COVID-19. In our study, we investigated the benefits of an Encoder–Decoder (ED) architecture based on Long Short-Term Memory (LSTM) cells. As a result, the molecules were converted into a vector during the encoding phase, which was then decoded back into SMILES molecules strings. We propose an approach to incorporate four features of Principal Components Analysis (PCA) with Encoder–Decoder Long Short-Term Memory (ED-LSTM) for regularization, which means that, instead of avoiding linear mapping, we assumed that the data could be linearly separable. We concluded that ED-LSTM with unit norm constraint has the best reconstruction accuracy in the context of generating molecules. The resulting dataset was used with the aid of virtual screening and convolutional neural networks to check the drugs that have the best binding affinity with SARS-CoV-2. We achieved an accuracy of 87.35% on the test set.

Combining electro- and magnetoencephalography data using directional archetypal analysis.

Metastable microstates in electro- and magnetoencephalographic (EEG and MEG) measurements are usually determined using modified k-means accounting for polarity invariant states. However, hard state assignment approaches assume that the brain traverses microstates in a discrete rather than continuous fashion. We present multimodal, multisubject directional archetypal analysis as a scale and polarity invariant extension to archetypal analysis using a loss function based on the Watson distribution. With this method, EEG/MEG microstates are modeled using subject- and modality-specific archetypes that are representative, distinct topographic maps between which the brain continuously traverses. Archetypes are specified as convex combinations of unit norm input data based on a shared generator matrix, thus assuming that the timing of neural responses to stimuli is consistent across subjects and modalities. The input data is reconstructed as convex combinations of archetypes using a subject- and modality-specific continuous archetypal mixing matrix. We showcase the model on synthetic data and an openly available face perception event-related potential data set with concurrently recorded EEG and MEG. In synthetic and unimodal experiments, we compare our model to conventional Euclidean multisubject archetypal analysis. We also contrast our model to a directional clustering model with discrete state assignments to highlight the advantages of modeling state trajectories rather than hard assignments. We find that our approach successfully models scale and polarity invariant data, such as microstates, accounting for intersubject and intermodal variability. The model is readily extendable to other modalities ensuring component correspondence while elucidating spatiotemporal signal variability.

Properties of certain multi-angle tight frames in [formula omitted

Frames are now standard tools in signal processing, and have applications ranging from compressed sensing, to communication systems and quantum sensing. Designing frames with some special structure such as equiangularity and tightness is useful in frame theory and its applications. In practice, constructing such frames with a given size in a specific dimension can be difficult or impossible in some cases. This leads one to consider the construction of frames with few distinct angles among pairs of frame vectors. We look at a specific construction done previously, which, for a given dimension d and integer 1<k≤d, gives a unit norm tight frame such that the number of distinct angles among the vectors is bounded above by k. Here we give several properties of this multi-angle tight frame. In particular, we provide a precise value of the number of angles which depends on a relationship between d and k, and give a formula for the multiplicity of each angle. We also show how one can strategically choose subsets of such a multi-angle tight frame that will be equiangular or orthogonal. This property is meaningful in the context of erasures when one has to deal with subsets of a given frame that may or may not be a frame. Finally, we show a connection between certain unit norm tight frames with three angles and adjacency matrices of regular graphs.

Construction of unit norm tight frames inspired by the Paulsen problem

Unit norm tight frames (UNTFs) provide redundant Parseval-like decompositions of vectors, and are viewed as the extensions of the orthonormal bases with more vector number than the vector dimension. UNTFs are known to be robust against additive noise and erasures, and have great potential as encoding schemes. However, the construction of UNTFs has been proved difficult as the simultaneous constraints on the columns and the rows of the frame matrices do not commute. The famous Paulsen problem asks: “How close is a frame which is almost tight and almost unit norm to some UNTF?” In this paper, we present two methods for constructing UNTFs to provide explicit answers to the Paulsen problem. The first method intends to normalize the columns of the frame matrix while keeping its tightness property unchanged. This method iteratively updates the unitary factor (matrix) and the iteration number is less than the number of the frame matrix columns. The second method tries to make the frame tight under the (column) unit norm constraints. A parametrization process is employed to convert the constrained problem into an unconstrained one, and a gradient descent procedure is carried out to solve the corresponding unconstrained problem. Both methods show simple computation and rapid convergence, and can result in exact UNTFs of arbitrary dimensions.

Orthogonality of invariant vectors

Let G be a finite group with given subgroups H and K. Let π be an irreducible complex representation of G such that its space of H-invariant vectors as well as the space of K-invariant vectors are both one dimensional. Let vH (resp. vK) denote an H-invariant (resp. K-invariant) vector of unit norm in a given G-invariant inner product 〈,〉π on π. Our interest is in computing the square of the absolute value of 〈vH,vK〉π. This is the correlation constant c(π;H,K) defined by Gross [3]. In this paper, we study this question for G=GL2(Fq), where Fq is the finite field of q=pm elements of odd characteristic p, H is its split torus and K is a non-split torus. The first main theorem of this paper gives an explicit formula for |〈vH,vK〉π|2 modulo p. The key idea here is to analyse the mod p reduction of π. The second main theorem relates the behaviour of 〈vH,vK〉π under Shintani base change and gives a sufficient condition for 〈vH,vK〉π to vanish for an irreducible representation π=BC(τ) of PGL2(E), in terms of the epsilon factor of the base changing representation τ of PGL2(F), where E/F is a finite extension of finite fields.