Functions In L2 Research Articles

MR image reconstruction using deep learning: evaluation of network structure and loss functions.

To review and evaluate approaches to convolutional neural network (CNN) reconstruction for accelerated cardiac MR imaging in the real clinical context. Two CNN architectures, Unet and residual network (Resnet) were evaluated using quantitative and qualitative assessment by radiologist. Four different loss functions were also considered: pixel-wise (L1 and L2), patch-wise structural dissimilarity (Dssim) and feature-wise (perceptual loss). The networks were evaluated using retrospectively and prospectively under-sampled cardiac MR data. Based on our assessments, we find that Resnet and Unet achieve similar image quality but that former requires only 100,000 parameters compared to 1.3 million parameters for the latter. The perceptual loss function performed significantly better than L1, L2 or Dssim loss functions as determined by the radiologist scores. CNN image reconstruction using Resnet yields comparable image quality to Unet with 10X the number of parameters. This has implications for training with significantly lower data requirements. Network training using the perceptual loss function was found to better agree with radiologist scoring compared to L1, L2 or Dssim loss functions.

The complex-time Segal–Bargmann transform

We introduce a new form of the Segal–Bargmann transform for a connected Lie group K of compact type. We show that the heat kernel (ρt(x))t>0,x∈K has a space-time analytic continuation to a holomorphic function(ρC(τ,z))Reτ>0,z∈KC, where KC is the complexification of K. The new transform is defined by the integral(Bτf)(z)=∫KρC(τ,zk−1)f(k)dk,z∈KC. If s>0 and τ∈D(s,s) (the disk of radius s centered at s), this integral defines a holomorphic function on KC for each f∈L2(K,ρs). We construct a heat kernel density μs,τ on KC such that, for all s,τ as above, Bs,τ:=Bτ|L2(K,ρs) is an isometric isomorphism from L2(K,ρs) onto the space of holomorphic functions in L2(KC,μs,τ). When τ=t=s, the transform Bt,t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When τ=t∈(0,2s), the transform Bs,t coincides with the one introduced by the first two authors.

Functions of constant geodesic x-ray transform

We show that the existence of a function in L1 with constant geodesic x-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions with constant geodesic x-ray transform always exist on manifolds with rotational symmetry.

Generalized Characteristics of Smoothness and Some Extreme Problems of the Approximation Theory of Functions in the Space L2(ℝ). II

In the second part of the paper, we establish the exact Jackson-type inequalities for the characteristic of smoothness Λw on the classes of functions $$ {L}_2^{\alpha}\left(\iota \right) $$ defined by the fractional derivatives of order v 2 (0,1) in the space L2(ℝ). The exact values of the mean L2(ℝ) -widths for the classes of functions defined by the generalized characteristics of smoothness ew and Λw are also computed.

A multiscale theory for image registration and nonlinear inverse problems

In an influential paper, Tadmor et al. (2004) [42] introduced a hierarchical decomposition of an image as a sum of constituents of different scales. Here we construct analogous hierarchical expansions for diffeomorphisms, in the context of image registration, with the sum replaced by composition of maps. We treat this as a special case of a general framework for multiscale decompositions, applicable to a wide range of imaging and nonlinear inverse problems. As a paradigmatic example of the latter, we consider the Calderón inverse conductivity problem. We prove that we can simultaneously perform a numerical reconstruction and a multiscale decomposition of the unknown conductivity, driven by the inverse problem itself. We provide novel convergence proofs which work in the general abstract settings, yet are sharp enough to prove that the hierarchical decomposition of Tadmor, Nezzar and Vese converges for arbitrary functions in L2, a problem left open in their paper. We also give counterexamples that show the optimality of our general results.

Instantaneous regularization of distributions from [formula omitted] in the one-dimensional parabolic Keller–Segel system

The Neumann problem for the Keller–Segel system ut=uxx−(uvx)x,vt=vxx−v+u(⋆)is considered in open bounded intervals Ω⊂R, with a particular focus on the question to which extent supposedly present singularities can be regularized despite the destabilizing cross-diffusive interaction in (⋆).The main results in this direction indicate that in the considered one-dimensional framework, even immediate regularization into spaces of smooth functions occurs for arbitrary distributions no more singular than Radon measures in their first and L2 functions in their second component. In particular, it is shown that given any nonnegative μ0∈(C0(Ω¯))⋆ and v0∈L2(Ω), the corresponding initial–boundary value problem for (⋆) admits at least one global weak solution which is smooth in Ω¯×(0,∞) and satisfies u(⋅,t)⇀⋆μ0 in (C0(Ω¯))⋆ and v(⋅,t)→v0 in L2(Ω) as t↘0.This apparently goes somewhat beyond precedent constructions of solutions to (⋆) for rough initial data, which inter alia seem to exclusively require the initial signal distribution to satisfy at least v0∈W1,q(Ω) for some q>1.

Generalized Characteristics of Smoothness and Some Extreme Problems of the Approximation Theory of Functions in the Space L2(ℝ). I

Generalized Characteristics of Smoothness and Some Extreme Problems of the Approximation Theory of Functions in the Space L2(ℝ). I

Influence of L2 English phonotactics in L1 Brazilian Portuguese illusory vowel perception

This study examines potential changes to L1 (Brazilian Portuguese, BP) perception of phonotactic structure as a function of L2 (English) experience. Syllables with a coda stop violate syllable structure constraints in BP, but are licit in English. As a result, BP monolinguals perceive an illusory /i/ after an illicit coda (e.g., ob/i/ter ‘to obtain’). To understand the effects on L1 perception when the L2 phonology allows a syllabic structure that is illicit in the L1, we tested 15 L1 speakers of BP in an L2 English immersion setting. In separate BP and English sessions, participants completed an explicit metalinguistic vowel identification task and two ABX tasks, with one ABX task designed to tap lower-level processing and the other, higher-level phonological processing. Our analysis reveals that, while not fully target-like in L2 English, the bilinguals’ perception data in both languages contrast sharply with existing monolingual BP data. This is the case whether participants employ what we assume here to be metalinguistic strategies, lower-level encoding strategies, or higher-level encoding strategies in perception. Based on this pattern of results, we conclude that the data provide evidence of L2 phonotactic influence in L1 perception.

INTEGRAL TRANSFORMATION IN A PARTIALLY BOUNDED REGION WITH A RADIAL THERMAL FLOW

A mathematical theory is developed for constructing integral transformations in a partially bounded region with a radial heat flow - a massive body bounded from the inside by a cylindrical cavity. Constructed: an integral transformation, the image of the operator on the right side of the equation of unsteady heat conduction, the inversion formula for the image of the desired function. The proposed approach favorably differs from the classical theory of differential equations of mathematical physics for the construction of generalized integral transformations based on the eigenfunctions of the corresponding singular Sturm-Liouville problems. The developed method is based on the operational solution of the initial boundary problems of unsteady heat conduction with an initial function of a general form L2(r0,∞) belonging to the r > r0 region and homogeneous boundary conditions and is associated with the calculation of the Riemann-Mellin contour integrals from images containing various combinations of modified Bessel functions. At the same time, for the above-mentioned region, the method of Green's functions was developed by constructing integral representations of analytical solutions of the first, second and third boundary value problems through inhomogeneities in the initial formulation of the problem (boundary conditions, source function in the initial equation). Mathematical models for finding the corresponding Green's functions are formulated, and functional relations of all three Green functions included in the presented integral formula are written out with the help of the developed theory of integral transformations. The functional relations constructed in the article can be used when considering numerous special cases of practical thermal physics. The specific possible applications of the presented results in many areas of science and technology are given.

Smart bacteria‐foraging algorithm‐based customized kernel support vector regression and enhanced probabilistic neural network for compaction quality assessment and control of earth‐rock dam

AbstractCompaction quality assessment and control for an earth‐rock dam is the key measure to ensure dam safety. However, to date, the compaction quality assessment model has not been accurate enough, and no effective feedback control measures have been developed. Hybrid data mining algorithms have great potential for solving this problem. In this study, smart bacteria‐foraging algorithm‐based customized kernel support vector regression (SBFA‐CKSVR) is proposed for compaction quality assessment, whereas an enhanced probabilistic neural network (EPNN) is adopted for compaction quality control. SBFA integrates a bacteria‐foraging algorithm, chaos mapping, and adaptive and quantum computing to solve the high‐dimensional complex problem effectively. CKSVR is proposed to approximate a function in quadratic continuous integral space L2(R) where its hyperparameters are optimized by SBFA. Finally, SBFA‐CKSVR is used to establish a high‐precision compaction quality assessment model whereas the EPNN is adopted to realize the compaction quality feedback control. A three‐dimensional real‐time monitoring system for the earth‐rock dam is also developed based on SBFA‐CKSVR and EPNN. A large‐scale hydraulic engineering application proves the effectiveness and superiority of this research compared with the previous work.

The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets

In this paper, we consider the planar self-affine measures μM,D generated by an expanding matrix M∈M2(Z) and an integer digit set D={(00),(α1α2),(β1β2)} with α1β2−α2β1≠0. We show that if det⁡(M)∉3Z, then the mutually orthogonal exponential functions in L2(μM,D) is finite, and the exact maximal cardinality is given.

Continuum versus discrete networks, graph Laplacians, and reproducing kernel Hilbert spaces

Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space X. We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of L2 functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example).

Optimal Control and Additive Perturbations Help in Estimating Ill-Posed and Uncertain Dynamical Systems

ABSTRACTOrdinary differential equations (ODE) are routinely calibrated on real data for estimating unknown parameters or for reverse-engineering. Nevertheless, standard statistical techniques can give disappointing results because of the complex relationship between parameters and states, which makes the corresponding estimation problem ill-posed. Moreover, ODE are mechanistic models that are prone to modeling errors, whose influences on inference are often neglected during statistical analysis. We propose a regularized estimation framework, called Tracking, which consists in adding a perturbation (L2 function) to the original ODE. This perturbation facilitates data fitting and represents also possible model misspecifications, so that parameter estimation is done by solving a trade-off between data fidelity and model fidelity. We show that the underlying optimization problem is an optimal control problem that can be solved by the Pontryagin maximum principle for general nonlinear and partially observed ODE. The same methodology can be used for the joint estimation of finite and time-varying parameters. We show, in the case of a well-specified parametric model that our estimator is consistent and reaches the root-n rate. In addition, numerical experiments considering various sources of model misspecifications shows that Tracking still furnishes accurate estimates. Finally, we consider semiparametric estimation on both simulated data and on a real data example. Supplementary materials for this article are available online.

Best Polynomial Approximations and Widths of Classes of Functions in the Space L2

Best Polynomial Approximations and Widths of Classes of Functions in the Space L2

How interactional needs shape information structure: An analysis of the discourse functions of topicalization in three L2 varieties of English

AbstractTopicalization refers to the sentence-initial placement of constituents other than the subject (see, e.g., Lambrecht 1994; Birner & Ward 2009) and has recently gained more attention in particular with regard to its frequency, forms, and functions in L2 and learner varieties of English (cf. Lange 2012; Winkle 2015; Leuckert 2017). Specific interactional needs, such as the intention of establishing topic continuity in the discourse, may motivate speakers to deviate from the canonical SVX pattern of English sentences. Furthermore, speakers may choose to emphasize or contrast information which has not previously been mentioned in the discourse. Therefore, this paper argues that cognitive, pragmatic, emotiveaffective, and social factors (cf. Schmid 2016) all play an important role for topicalization, but certain communicative goals may lead to speakers ignoring the cognitive status of information by topicalizing brand-new or unused information. Furthermore, it is shown in this paper that topicalized constituents which serve to create topic continuity (and are, therefore, highly salient in the discourse) tend to persist longer in the subsequent discourse than topicalized constituents which serve other discourse functions.

On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities {In}, for n∈N, valid for functions in C0n((0,∞))⊂L2((0,∞)). In particular, I1 is the classical (integral) Hardy inequality and I2 is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space Hn([0,∞)) of functions defined on [0,∞). Moreover, f∈Hn([0,∞)) implies f′∈Hn−1([0,∞)); as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite b>0, these inequalities hold on the standard Sobolev space H0n((0,b)). Furthermore, in all cases, the Birman constants [(2n−1)!!]2/22n in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in L2((0,∞)) (resp., L2((0,b))). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.

Host cell transcriptome modification upon exogenous HPV16 L2 protein expression.

Human papillomavirus type 16 minor capsid protein L2 has been shown to assist in the initial entry and intracellular trafficking events leading to nuclear translocation of the viral genome. During our investigations of L2 function, we observed that expression of L2 in a keratinocyte cell line (HaCaT) resulted in phenotypic changes. In this manuscript, we present data that expression of the L2 protein in this cellular model system HaCaTs resulted in a shift from G0/G1 phase to mitotic S phase, as well as a reduced amount of retinoblastoma protein (Rb) and an increase in Cdc2 phosphorylation. We performed genome-wide host cell mRNA sequencing and identified 2586 differentially expressed genes upon HPV16 L2 expression. Via machine learning and protein network analysis, genes involved in cellular differentiation and proliferation were highlighted as impacted by L2. Our results have implications for the role of L2 at the viral production stages when the virus needs to prevent cellular differentiation while maintaining the cells ability to replicate DNA. Our study suggests a potential novel function of the L2 protein, as a regulator of cellular gene transcription.

Hardy and Carleson Measure Spaces Associated with Operators on Spaces of Homogeneous Type

Let (X, d, μ) be a metric measure space with doubling property. The Hardy spaces associated with operators L were introduced and studied by many authors. All these spaces, however, were first defined by L 2(X) functions and finally the Hardy spaces were formally defined by the closure of these subspaces of L 2(X) with respect to Hardy spaces norms. A natural and interesting question in this context is to characterize the closure. The purpose of this paper is to answer this question. More precisely, we will introduce ${CMO}_{L}^{p}(X)$ , the Carleson measure spaces associated with operators L, and characterize the Hardy spaces associated with operators L via $({CMO}_{L}^{p}(X))'$ , the distributions of ${CMO}_{L}^{p}(X)$ .

Linear dependency of translations and square-integrable representations

Let G be a locally compact group. We examine the problem of determining when nonzero functions in L2(G) have linearly independent left translations. In particular, we establish some results for the case when G has an irreducible, square-integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group, and the Weyl–Heisenberg group. We also investigate the case when G has an abelian, closed subgroup of finite index.

Gender constraint in L1 and L2 reflexive pronoun resolution by Chinese-English bilinguals

Abstract Previous behavioral research indicates that native speakers of Chinese are not very sensitive to lexical gender information, and one question that remains is how the rule of gender constraint functions in L1 and L2 reflexive pronoun resolution by Chinese-English bilinguals. To answer the question, the present study conducted two ERP experiments in which participants' ERP responses to L1 and L2 reflexive pronouns that were either congruent or incongruent with their antecedents were recorded and analyzed. Gender incongruity effect was observed in the L1 in the form of left anterior negativity effect (LAN) within 350–450 ms, but in the L2 in the form of midline enhanced positivity within 300–500 ms. Then, within a later time window of 500–800 ms, the P600 component was observed for gender incongruity in both the L1 and the L2. However, large variance was observed in the participants' behavioral performance in the non-native language, and the amplitude of the P600 effect decreased linearly with the participants’ inability to detect gender incongruity. The above findings demonstrate that the rule of gender constraint functions similarly in L1 and L2 reflexive pronoun resolution by Chinese-English bilinguals, but nevertheless the exact mechanism employed for the L1 and the L2 was not the same, with an LAN-P600 pattern for the L1 but an Earlier Positivity-P600 pattern for the L2. Moreover, the large variance in L2 processing should be taken into consideration in future research on L2 reflexive pronoun resolution.