Functions In L2 Research Articles

Complete topological asymptotic expansion for L2 and H1 tracking-type cost functionals in dimension two and three

In this paper, we study the topological asymptotic expansion of a topology optimisation problem that is constrained by the Poisson equation with the design/shape variable entering through the right hand side. Using an averaged adjoint approach, we give explicit formulas for topological derivatives of arbitrary order for both an L2 and H1 tracking-type cost function in both dimension two and three and thereby derive the complete asymptotic expansion. As the asymptotic behaviour of the fundamental solution of the Laplacian differs in dimension two and three, also the derivation of the topological expansion significantly differs in dimension two and three. The complete expansion for the H1 cost functional directly follows from the analysis of the variation of the state equation. However, the proof of the asymptotics of the L2 tracking-type cost functional is significantly more involved and, surprisingly, the asymptotic behaviour of the bi-harmonic equation plays a crucial role in our proof.

A peptide derived from sorting nexin 1 inhibits HPV16 entry, retrograde trafficking, and L2 membrane spanning

High risk human papillomavirus (HPV) infection is responsible for 99 % of cervical cancers and 5 % of all human cancers worldwide. HPV infection requires the viral genome (vDNA) to gain access to nuclei of basal keratinocytes of epithelium. After virion endocytosis, the minor capsid protein L2 dictates the subcellular retrograde trafficking and nuclear localization of the vDNA during mitosis. Prior work identified a cell-permeable peptide termed SNX1.3, derived from the BAR domain of sorting nexin 1 (SNX1), that potently blocks the retrograde and nuclear trafficking of EGFR in triple negative breast cancer cells. Given the importance of EGFR and retrograde trafficking pathways in HPV16 infection, we set forth to study the effects of SNX1.3 within this context. SNX1.3 inhibited HPV16 infection by both delaying virion endocytosis, as well as potently blocking virion retrograde trafficking and Golgi localization. SNX1.3 had no effect on cell proliferation, nor did it affect post-Golgi trafficking of HPV16. Looking more directly at L2 function, SNX1.3 was found to impair membrane spanning of the minor capsid protein. Future work will focus on mechanistic studies of SNX1.3 inhibition, and the role of EGFR signaling and SNX1-mediated endosomal tubulation, cargo sorting, and retrograde trafficking in HPV infection.

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

A Gabor orthonormal basis, on a locally compact Abelian (LCA) group A, is an orthonormal basis of L2(A) that consists of time-frequency shifts of some template f∈L2(A). It is well known that, on Rd, the elements of such a basis cannot have a good time-frequency localization. The picture is drastically different on LCA groups containing a compact open subgroup, where one can easily construct examples of Gabor orthonormal bases with f maximally localized, in the sense that the ambiguity function of f (i.e., the correlation of f with its time-frequency shifts) has support of minimum measure, compatibly with the uncertainty principle. In this note we find all the Gabor orthonormal bases with this extremal property. To this end, we identify all the functions in L2(A) that are maximally localized in the time-frequency space in the above sense – an issue that is open even for finite Abelian groups. As a byproduct, on every LCA group containing a compact open subgroup we exhibit the complete family of optimizers for Lieb's uncertainty inequality, and we also show previously unknown optimizers on a general LCA group.

A Peptide Derived from Sorting Nexin 1 Inhibits HPV16 Entry, Retrograde Trafficking, and L2 Membrane Spanning.

High risk human papillomavirus (HPV) infection is responsible for 99% of cervical cancers and 5% of all human cancers worldwide. HPV infection requires the viral genome (vDNA) to gain access to nuclei of basal keratinocytes of epithelium. After virion endocytosis, the minor capsid protein L2 dictates the subcellular retrograde trafficking and nuclear localization of the vDNA during mitosis. Prior work identified a cell-permeable peptide termed SNX1.3, derived from the BAR domain of sorting nexin 1 (SNX1), that potently blocks the retrograde and nuclear trafficking of EGFR in triple negative breast cancer cells. Given the importance of EGFR and retrograde trafficking pathways in HPV16 infection, we set forth to study the effects of SNX1.3 within this context. SNX1.3 inhibited HPV16 infection by both delaying virion endocytosis, as well as potently blocking virion retrograde trafficking and Golgi localization. SNX1.3 had no effect on cell proliferation, nor did it affect post-Golgi trafficking of HPV16. Looking more directly at L2 function, SNX1.3 was found to impair membrane spanning of the minor capsid protein. Future work will focus on mechanistic studies of SNX1.3 inhibition, and the role of EGFR signaling and SNX1- mediated endosomal tubulation, cargo sorting, and retrograde trafficking in HPV infection.

The Toxoplasma gondii F-Box Protein L2 Functions as a Repressor of Stage Specific Gene Expression.

Toxoplasma gondii is a foodborne pathogen that can cause severe and life-threatening infections in fetuses and immunocompromised patients. Felids are its only definitive hosts, and a wide range of animals, including humans, serve as intermediate hosts. When the transmissible bradyzoite stage is orally ingested by felids, they transform into merozoites that expand asexually, ultimately generating millions of gametes for the parasite sexual cycle. However, bradyzoites in intermediate hosts differentiate exclusively to disease-causing tachyzoites, which rapidly disseminate throughout the host. Though tachyzoites are well-studied, the molecular mechanisms governing transitioning between developmental stages are poorly understood. Each parasite stage can be distinguished by a characteristic transcriptional signature, with one signature being repressed during the other stages. Switching between stages require substantial changes in the proteome, which is achieved in part by ubiquitination. F-box proteins mediate protein poly-ubiquitination by recruiting substrates to SKP1, Cullin-1, F-Box protein E3 ubiquitin ligase (SCF-E3) complexes. We have identified an F-box protein named Toxoplasma gondii F-Box Protein L2 (TgFBXL2), which localizes to distinct perinucleolar sites. TgFBXL2 is stably engaged in an SCF-E3 complex that is surprisingly also associated with a COP9 signalosome complex that negatively regulates SCF-E3 function. At the cellular level, TgFBXL2-depleted parasites are severely defective in centrosome replication and daughter cell development. Most remarkable, RNAseq data show that TgFBXL2 conditional depletion induces the expression of stage-specific genes including a large cohort of genes necessary for sexual commitment. Together, these data suggest that TgFBXL2 is a latent guardian of stage specific gene expression in Toxoplasma and poised to remove conflicting proteins in response to an unknown trigger of development.

A robust computational framework for variational data assimilation of mean flows with sparse measurements corrupted by strong outliers

A variational framework for the reconstruction of time-averaged mean flows using a sparse set of observations with large magnitudes of noise (referred to as outliers), is presented. The observations constitute a set of point-wise measurements of the mean flow with outliers at certain measurement locations and are incorporated into a numerical simulation governed by the two-dimensional, incompressible Reynolds-averaged Navier-Stokes (RANS) equations with an unknown momentum forcing. This forcing, which corresponds to the divergence of the Reynolds stress tensor, is calculated from a direct-adjoint optimization procedure to reduce the deviation between the measured and estimated mean velocities. ℓ2, ℓ1, Huber, and hybrid loss functions are used to represent the discrepancy in the mean velocity field between the measurements and the predictions. A variety of algorithms are considered to solve the optimization problem with these loss functions and a performance comparison in terms of the quality and physical features of the recovered mean flow is presented. The Huber loss function performed best as it remained robust to strong outliers in the measurements with its ℓ1 contribution and also ensured the uniqueness of the optimal solution with its ℓ2 contribution. Huber loss functions restrict the effect of outliers at the local measurement locations, thereby not affecting the quality of the high-dimensional reconstructed mean flow field. The hybrid loss function, a modified form of the continuous Huber loss function, also recovered the mean flow with high accuracy. We demonstrate the performance of the data assimilation framework for the case of two-dimensional laminar flow around a circular cylinder at Re=100. We then extend the analysis to the case of two-dimensional laminar flow over a backward-facing step at Re=500, to further assess the efficacy and robustness of the data assimilation framework.

Hyperspectral image dynamic range reconstruction using deep neural network-based denoising methods

Hyperspectral (HS) measurement is among the most useful tools in agriculture for early disease detection. However, the cost of HS cameras that can perform the desired detection tasks is prohibitive-typically fifty thousand to hundreds of thousands of dollars. In a previous study at the Agricultural Research Organization’s Volcani Institute (Israel), a low-cost, high-performing HS system was developed which included a point spectrometer and optical components. Its main disadvantage was long shooting time for each image. Shooting time strongly depends on the predetermined integration time of the point spectrometer. While essential for performing monitoring tasks in a reasonable time, shortening integration time from a typical value in the range of 200 ms to the 10 ms range results in deterioration of the dynamic range of the captured scene. In this work, we suggest correcting this by learning the transformation from data measured with short integration time to that measured with long integration time. Reduction of the dynamic range and consequent low SNR were successfully overcome using three developed deep neural networks models based on a denoising auto-encoder, DnCNN and LambdaNetworks architectures as a backbone. The best model was based on DnCNN using a combined loss function of ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _{2}$$\\end{document} and Kullback–Leibler divergence on images with 20 consecutive channels. The full spectrum of the model achieved a mean PSNR of 30.61 and mean SSIM of 0.9, showing total improvement relatively to the 10 ms measurements’ mean PSNR and mean SSIM values by 60.43% and 94.51%, respectively.

Is hyperinterpolation efficient in the approximation of singular and oscillatory functions?

Singular and oscillatory functions play a crucial role in various applications, and their approximation is crucial for solving applied mathematics problems efficiently. Hyperinterpolation is a discrete projection method approximating functions with the L2 orthogonal projection coefficients obtained by numerical integration. However, this approach may be inefficient for approximating singular and oscillatory functions, requiring a large number of integration points to achieve satisfactory accuracy. To address this issue, we propose a new approximation scheme in this paper, called efficient hyperinterpolation, which leverages the product-integration methods to attain the desired accuracy with fewer numerical integration points than the original scheme. We provide theorems that explain the superiority of efficient hyperinterpolation over the original scheme in approximating such functions belonging to L1, L2, and continuous function spaces, respectively, and demonstrate through numerical experiments on the interval and the sphere that our approach outperforms the original method in terms of accuracy when using a limited number of integration points.

The Continuous Wavelet Transform in Sobolev Spaces Over Locally Compact Abelian Group and Its Approximation Properties

The Sobolev space over locally compact abelian group Hs(G) is defined and we extend the continuous wavelet transform to Sobolev space Hs(G) for arbitrary real s. This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces. Further, the asymptotic behaviour of the transforms of the L2 function for small scaling parameters is examined. In special cases, the wavelet transform converges to a generalized derivative of its argument. We also discuss the consequences for the discrete wavelet transform arising from this property. Numerical examples illustrate the main result.

The motivational aspect of feedback: A meta-analysis on the effect of different feedback practices on L2 learners’ writing motivation

With the growing body of research on writing feedback and the recognition of the critical role of writing motivation, controversy has emerged over the motivational function of feedback in second language (L2) writing contexts. To provide further evidence for the impact of different feedback practices on L2 learners’ writing motivation, the present meta-analysis synthesizes the results of 13 quantitative studies on the relationship between feedback and L2 writing motivation. It examines the effect of different feedback practices on L2 learners’ writing motivation and the variables moderating the effectiveness of those feedback practices. The results show that feedback generated from multiple sources has the greatest motivational function in L2 writing, followed by single-source feedback, including peer feedback, teacher feedback, and automated feedback. Moderator analysis indicates that feedback type is a statistically significant variable moderating the effectiveness of feedback. In light of the findings, implications for L2 writing instruction and future L2 writing research are discussed.

Second order regularity for the [formula omitted]-Laplace equations with [formula omitted] data on the right-hand side

This work considers p(x)-Laplace equations involving the L2 function on the right-hand side. In particular, we extend classical W1,2 regularity estimates for the gradient of a solution to linear equations to nonlinear equations with variable exponents under minimal regularity assumptions on the boundary We focus on the global W1,2 estimate, which is motivated by the L2-coercivity results for quasilinear elliptic equations with Orlicz growth, as demonstrated in Cianchi and Maz’ya (2018).

Rationally sampled Gabor frames on the half real line

Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line R+=(0,∞) is an LCA group under multiplication and the usual topology, with the Haar measure dμ=dxx. This paper addresses rationally sampled Gabor frames for L2(R+,dμ). Given a function in L2(R+,dμ), we introduce a new Zak transform matrix associated with it, which is different from the conventional Zibulski-Zeevi matrix. It allows us to define a function by designing its Zak transform matrix. Using our Zak transform matrix method, we characterize and express complete Gabor systems, Bessel sequences, Gabor frames, Riesz bases and Gabor duals of an arbitrarily given Gabor frame for L2(R+,dμ), and prove the minimality of the canonical dual frames in some sense. Some examples are also provided to illustrate the generality of our theory.

REGULARIZING EFFECT IN SINGULAR SEMILINEAR PROBLEMS

We analyze how different relations in the lower order terms lead to the same regularizing effect on singular problems whose model is in Ω, u = 0 on ∂Ω, where Ω is a bounded open set of is a nonnegative function in L1(Ω) and g(x,s) is a Carathéodory function. In a framework where no solution is expected, we prove its existence (regularizing effect) whenever the datum f interacts conveniently either with the boundary of the domain or with the lower order term.

X-ray properties and obscured fraction of AGN in the J1030 Chandra field

The 500ks Chandra ACIS-I observation of the field around the z = 6.31 quasar SDSS J1030+0524 is currently the fifth deepest extragalactic X-ray survey. The rich multi-band coverage of the field allowed an effective identification and redshift determination of the X-ray source counterparts; to date, a catalog of 243 extragalactic X-ray sources with either a spectroscopic or photometric redshift estimate in the range z ≈ 0 − 6 is available over an area of 355 arcmin2. Given its depth and the multi-band information, this catalog is an excellent resource to investigate X-ray spectral properties of distant active galactic nuclei (AGN) and derive the redshift evolution of their obscuration. We performed a thorough X-ray spectral analysis for each object in the sample, and measured its nuclear column density NH and intrinsic (de-absorbed) 2–10 keV rest-frame luminosity, L2 − 10. Whenever possible, we also used the presence of the Fe Kα emission line to improve the photometric redshift estimates. We measured the fractions of AGN hidden by column densities in excess of 1022 and 1023 cm−2 (f22 and f23, respectively) as a function of L2 − 10 and redshift, and corrected for selection effects to recover the intrinsic obscured fractions. At z ∼ 1.2, we found f22 ∼ 0.7 − 0.8 and f23 ∼ 0.5 − 0.6, respectively, in broad agreement with the results from other X-ray surveys. No significant variations in X-ray luminosity were found within the limited luminosity range probed by our sample (log L2 − 10 ∼ 42.8 − 44.3). When focusing on luminous AGN with log L2 − 10 ∼ 44 to maximize the sample completeness up to large cosmological distances, we did not observe any significant change in f22 or f23 over the redshift range z ∼ 0.8 − 3. Nonetheless, the obscured fractions we measure are significantly higher than is seen in the local Universe for objects of comparable intrinsic luminosity, pointing toward an increase in the average AGN obscuration toward early cosmic epochs, as also observed in other X-ray surveys. We finally compared our results with recent analytic models that ascribe the greater obscuration observed in AGN at high redshifts to the dense interstellar medium (ISM) of their hosts. When combined with literature measurements, our results favor a scenario in which the total column density of the ISM and the characteristic surface density of its individual clouds both increase toward early cosmic epochs as NH, ISM∝(1 + z)δ, with δ ∼ 3.3 − 4 and Σc, * ∝ (1 + z)2, respectively.

Spherical random sampling of localized functions on 𝕊ⁿ⁻¹

In this paper, we show that the set of random points uniformly distributed over S n − 1 \mathbb {S}^{n-1} is a stable set of sampling for a class of localized functions in L 2 ( S n − 1 ) L^2(\mathbb {S}^{n-1}) with overwhelming probability. The probability bound of the random sampling inequality for the localized function is obtained from the matrix Bernstein inequality on the independent random matrices.

Ellipsoidal and hyperbolic Radon transforms; microlocal properties and injectivity

We present novel microlocal and injectivity analyses of ellipsoid and hyperboloid Radon transforms. We introduce a new Radon transform, R, which defines the integrals of a compactly supported L2 function, f, over ellipsoids and hyperboloids with centers on a smooth connected surface, S. Our transform is shown to be a Fourier Integral Operator (FIO) and in our main theorem we prove that R satisfies the Bolker condition if the support of f is contained in a connected open set that is not intersected by any plane tangent to S. Under certain conditions, this is an equivalence. We give examples where our theory can be applied. Focusing specifically on a cylindrical geometry of interest in Ultrasound Reflection Tomography (URT), we prove injectivity results and investigate the visible singularities. In addition, we present example reconstructions of image phantoms in two-dimensions and validate our microlocal theory.

Language Moments in Second Language Interaction: Relevance of Understanding to Learning

Interactional modification is important in SLA research because it involves correcting problematic L2 use. However, not all modifications will lead to pedagogical changes. Participants in conversational interactions are not always oriented to linguistic forms or functions. One way to address this dilemma is to examine the process by which participants come to terms with problematic L2 use in interactional exchanges. “Language moments” refer to cases in which L2 forms and functions are objects of interactive exchanges in L2 interactions. Through conversation analysis, the present study uncovered four different types in which participants in L2 interaction discovered and acted on language moments in terms of the degree of explicitness in recognizing and addressing problematic L2 use. This study used data from ESL classroom interactions that featured native teachers of English and L2 learners in an US context. This descriptive account of interactional processes might complement prior research studies that have focused on effects of interactional modification.

Fusion Objective Function on Progressive Super-Resolution Network

Recent advancements in Single-Image Super-Resolution (SISR) have explored the network architecture of deep-learning models to achieve a better perceptual quality of super-resolved images. However, the effect of the objective function, which contributes to improving the performance and perceptual quality of super-resolved images, has not gained much attention. This paper proposes a novel super-resolution architecture called Progressive Multi-Residual Fusion Network (PMRF), which fuses the learning objective functions of L2 and Multi-Scale SSIM in a progressively upsampling framework structure. Specifically, we propose a Residual-in-Residual Dense Blocks (RRDB) architecture on a progressively upsampling platform that reconstructs the high-resolution image during intermediate steps in our super-resolution network. Additionally, the Depth-Wise Bottleneck Projection allows high-frequency information of early network layers to be bypassed through the upsampling modules of the network. Quantitative and qualitative evaluation of benchmark datasets demonstrate that the proposed PMRF super-resolution algorithm with novel fusion objective function (L2 and MS-SSIM) improves our model’s perceptual quality and accuracy compared to other state-of-the-art models. Moreover, this model demonstrates robustness against noise degradation and achieves an acceptable trade-off between network efficiency and accuracy.

Widths of the Classes of Functions in the Weight Space L2,γ(ℝ), γ = exp(−X2)

Widths of the Classes of Functions in the Weight Space L2,γ(ℝ), γ = exp(−X2)

On estimates of diameter values of classes of functions in the weight space L2,γ (ℝ2), γ = exp(–x2 – y2)

For a function f ∈ L2,γ (ℝ2), the q-averaging of the generalized m-order modulus of continuity Ωm,γ (f) with the weight function φ, i.e., the value of 𝔐q (Ωm,γ (f), φ; t) (t ∈ (0, 1); m ∈ ℕ; q ∈ (0,∞)) has been considered. With its help and with the participation of the majorant Ψ, classes of two-variable functions \( {\overset{\sim }{W}}_2\left({\Omega}_{m,\gamma}^q,\varphi; \Psi \right) \) = \( \left\{f\in {L}_{2,\gamma}\left({\mathbb{R}}^2\right):{\mathfrak{M}}_q\left({\Omega}_{m,\gamma }(f),\varphi; t\right)\leqslant \Psi (t)\forall t\in \left(0,1\right)\right\} \) were introduced, for which upper and lower estimates of various diameters in the space L2,γ (ℝ2) were found. A certain condition on the majorant Ψ was found, under which exact diameter values can be calculated. A similar problem has been considered for the classes \( {\overset{\sim }{W}}_2^{r,0}\left({\Omega}_{m,\gamma}^q,\varphi, \Psi \right) \) = \( {L}_{2,\gamma}^{r,0}\left(D,{\mathbb{R}}^2\right)\cap {\overset{\sim }{W}}_2^r\left({\Omega}_{m,\gamma}^q,\varphi, \Psi \right) \) \( \left(r,m\in \mathbb{N};q\in \left(0,\infty \right);D=-\frac{\partial^2}{{\partial x}^2}-\frac{\partial^2}{{\partial y}^2}+2x\right.\frac{\partial }{\partial x}+2y\frac{\partial }{\partial y} \) is the differential operator) consisting of functions f of class the \( {L}_{2,\gamma}^{r,0}\left(D,{\mathbb{R}}^2\right)=\left\{f\in {L}_{2,\gamma}\left({\mathbb{R}}^2\right):{c}_{i0}(f)={c}_{0j}(f)={c}_{00}(f)=0\forall i,j\in \mathbb{N}\right\} \) (here cij (f) are Fourier–Hermitian coefficients) for which the r-th iterations of Drf = D (Dr–1f) ∈ L2,γ (ℝ2) satisfy the condition \( {\mathfrak{M}}_q\left({\Omega}_{m,\gamma}\left({D}^rf\right),\varphi; t\right)\leqslant \Psi (t)\forall t\in \left(0,1\right) \). A number of results related to the specification of the exact values of various diameters are given.