Статті в журналах з теми "Local Riesz transforms"

We characterize the weighted local Hardy spaceshρ1(ω)related to the critical radius functionρand weightsω∈A1ρ,∞(Rn)by localized Riesz transformsR^j; in addition, we give a characterization of weighted Hardy spacesHL1(ω)via Riesz transforms associated with Schrödinger operatorL, whereL=-Δ+Vis a Schrödinger operator onRn(n≥3) andVis a nonnegative function satisfying the reverse Hölder inequality.

We characterize the weighted weak local Hardy spacesWhρp(ω)related to the critical radius functionρand weightsω∈A∞ρ,∞(Rn)which locally behave as Muckenhoupt’s weights and actually include them, by the atomic decomposition. As an application, we show that localized Riesz transforms are bounded on the weighted weak local Hardy spaces.

Let [Formula: see text] be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all [Formula: see text] and [Formula: see text] satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces [Formula: see text] associated with [Formula: see text], which coincide with Goldberg’s local Hardy spaces [Formula: see text] for all [Formula: see text] when [Formula: see text] (the Laplace operator). The authors also establish a real-variable theory of [Formula: see text], which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when [Formula: see text] (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that [Formula: see text] coincides with the Hardy space [Formula: see text] associated with the operator [Formula: see text] for all [Formula: see text], where [Formula: see text] is some positive constant depending on the ellipticity and the off-diagonal estimates of [Formula: see text]. As an application, the authors establish some mapping properties for the local Riesz transforms [Formula: see text] on [Formula: see text], where [Formula: see text] and [Formula: see text].

Abstract This paper concerns a backward problem for a nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity. Such a problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2), which is usually used to model the anomalous diffusion. We show that the problem is severely ill-posed. Using the Fourier transform and a filter function, we construct a regularized solution from the data given inexactly and explicitly derive the convergence estimate in the case of the local Lipschitz reaction term. Special cases of the regularized solution are also presented. These results extend some earlier works on the space-fractional backward diffusion problem.

The Hilbert–Huang-Transform (HHT) has proven to be an appropriate multiscale analysis technique specifically for nonlinear and nonstationary time series on non-equidistant grids. It is empirically adapted to the data: first, an additive decomposition of the data (empirical mode decomposition, EMD) into certain multiscale components is computed, denoted as intrinsic mode functions. Second, to each of these components, the Hilbert transform is applied. The resulting Hilbert spectrum of the modes provides a localized time-frequency spectrum and instantaneous (time-dependent) frequencies. For the first step, the empirical decomposition of the data, a different method based on local means has been developed by Chen et al. (2006). In this paper, we extend their method to multivariate data sets in arbitrary space dimensions. We place special emphasis on deriving a method which is numerically fast also in higher dimensions. Our method works in a coarse-to-fine fashion and is based on adaptive (tensor-product) spline-wavelets. We provide some numerical comparisons to a method based on linear finite elements and one based on thin-plate-splines to demonstrate the performance of our method, both with respect to the quality of the approximation as well as the numerical efficiency. Second, for a generalization of the Hilbert transform to the multivariate case, we consider the Riesz transformation and an embedding into Clifford-algebra valued functions, from which instantaneous amplitudes, phases and orientations can be derived. We conclude with some numerical examples.

AbstractIn this paper, we study the Hardy type space {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} by means of local maximal functions associated with the heat semigroup {e^{-t\mathcal{L}}} generated by {-\mathcal{L}}, where {\mathcal{L}=-\Delta+\mu} is the generalized Schrödinger operator in {{\mathbb{R}^{n}}} ({n\geq 3}) and {\mu\not\equiv 0} is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Via the equivalence of the norms between various local maximal functions, we show that the norms {\lVert f\rVert_{H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})}^{p}} and {\lVert f\rVert_{H_{m}^{p,q}({\mathbb{R}^{n}})}^{p}} are equivalent for {\frac{n}{n+\delta^{\prime}}<p\leq 1\leq q\leq\infty} ({p\neq q}) with some {\delta^{\prime}>0}. As applications, we prove that Calderón–Zygmund operators related to the auxiliary function {m(x,\mu)} are bounded from {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into {L^{p}({\mathbb{R}^{n}})} for {\frac{n}{n+\gamma_{1}}<p\leq 1} with {\gamma_{1}>0}. In particular, we show that the Riesz transform {\nabla(-\Delta+\mu)^{-\frac{1}{2}}}, which is a special example of the above Calderón–Zygmund operators, is bounded from {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into {{H}^{p}({\mathbb{R}^{n}})} for {\frac{n}{n+\gamma_{1}}<p\leq 1} with {0<\gamma_{1}<1}.

In this paper, we propose some recent works on data analysis and synthesis based on Empirical Mode Decomposition (EMD). Firstly, a direct 2D extension of original Huang EMD algorithm with application to texture analysis, and fractional Brownian motion synthesis. Secondly, an analytical version of EMD based on PDE in 1D-space is presented. We proposed an extension in 2D-case of the so-called "sifting process" used in the original Huang's EMD. The 2D-sifting process is performed in two steps: extrema detection (by neighboring window or morphological operators) and surface interpolation by splines (thin plate splines or multigrid B-splines). We propose a multiscale segmentation approach by using the zero-crossings from each 2D-intrinsic mode function (IMF) obtained by 2D-EMD. We apply the Hilbert–Huang transform (which consists of two parts: (a) Empirical mode decomposition, and (b) the Hilbert spectral analysis) to texture analysis. We analyze each 2D-IMF obtained by 2D-EMD by studying local properties (amplitude, phase, isotropy, and orientation) extracted from the monogenic signal of each one of them. The monogenic signal proposed by Felsberg et al. is a 2D-generalization of the analytic signal, where the Riesz transform replaces the Hilbert transform. These local properties are obtained by the structure multivector such as proposed by Felsberg and Sommer. We present numerical simulations of fractional Brownian textures. Recent works published by Flandrin et al. relate that, in the case of fractional Gaussian noise (fGn), EMD acts essentially as a dyadic filter bank that can be compared to wavelet decompositions. Moreover, in the context of fGn identification, Flandrin et al. show that variance progression across IMFs is related to Hurst exponent H through a scaling law. Starting with these results, we proposed an algorithm to generate fGn, and fractional Brownian motion (fBm) of Hurst exponent H from IMFs obtained from EMD of a White noise, i.e., ordinary Gaussian noise (fGn with H = 1/2). Deléchelle et al. proposed an analytical approach (formulated as a partial differential equation (PDE)) for sifting process. This PDE-based approach is applied on signals. The analytical approach has a behavior similar to that of the EMD proposed by Huang.

Lung carcinoma, which is commonly known as lung cancer, is one of the most common cancers throughout the world. Mostly, it is not diagnosed until it has spread, and it is very difficult to treat. Hence, early diagnosis of benign and malignant pulmonary nodules can help in the risk assessment of lung cancer for patients, and with proper treatment can save their lives. In this study, a framework for the classification of pulmonary nodules from Computerized Tomography (CT) images using the machine learning-based modified gradient boosting method is proposed. Initially, the obtained CT scan images are preprocessed for better image quality. Next, a random walker method is used to segment the lung nodule boundaries based on seeds provided by the user. After that, the intensity and texture features are extracted using the Local Binary Pattern (LBP) filter and the coefficients of the Riesz wavelet transform. Finally, the proposed modified gradient boost classifier model is trained and tested using the extracted features to classify nodules as either benign or malignant. The proposed framework is verified and validated using the Lung Image Database Consortium (LIDC-IDRI) dataset. From the performance analysis, it was observed that the proposed method achieves a precision, recall, F1 score, and validation accuracy of 0.957, 0.91, 0.941, and 95.67%, respectively. The performance of the proposed method is compared with existing models and is found to be superior. It was found that the proposed classifier is able to efficiently classify pulmonary nodules as either benign or malignant.

AbstractWith $$\vec {\Delta }_j\ge 0$$ Δ → j ≥ 0 is the uniquely determined self-adjoint realization of the Laplace operator acting on j-forms on a geodesically complete Riemannian manifold M and $$\nabla $$ ∇ the Levi-Civita covariant derivative, we prove among other things a Gaussian heat kernel bound for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j , if the curvature tensor of M and its covariant derivative are bounded, an exponentially weighted $$L^p$$ L p -bound for the heat kernel of $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j , if the curvature tensor of M and its covariant derivative are bounded, that $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j is bounded in $$L^p$$ L p for all $$1\le p<\infty $$ 1 ≤ p < ∞ , if the curvature tensor of M and its covariant derivative are bounded, a second order Davies-Gaffney estimate (in terms of $$\nabla $$ ∇ and $$\vec {\Delta }_j$$ Δ → j ) for $$\mathrm {e}^{ -t\vec {\Delta }_j }$$ e - t Δ → j for small times, if the j-th degree Bochner-Lichnerowicz potential $$V_j=\vec {\Delta }_j-\nabla ^{\dagger }\nabla $$ V j = Δ → j - ∇ † ∇ of M is bounded from below (where $$V_1=\mathrm {Ric}$$ V 1 = Ric ), which is shown to fail for large time, if $$V_j$$ V j is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform $$\nabla (\vec {\Delta }_j+\kappa )^{-1/2}$$ ∇ ( Δ → j + κ ) - 1 / 2 in $$L^p$$ L p for all $$1\le p<\infty $$ 1 ≤ p < ∞ (which we prove for $$1\le p\le 2$$ 1 ≤ p ≤ 2 ), and explain its implications to geometric analysis, such as the $$L^p$$ L p -Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j .

We investigate the well-posedness in the generalized Hartree equation [Formula: see text], [Formula: see text], [Formula: see text], for low powers of nonlinearity, [Formula: see text]. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the [Formula: see text]-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

As I sit here, the radio announcer announces a feature on the forthcoming Big Brother series, another chance to engage in this collective shared experience, another opportunity to revel in your very own voyeuristic impulse, what once was private is now made public. Curiously it’s almost fifteen years ago since I released the first Scanner recordings Scanner 1 [1992] and Scanner 2 [1993] featuring the intercepted cellular phone conversations of unsuspecting talkers, which I edited into minimalist musical settings as if they were instruments, bringing into focus issues of privacy and the dichotomy between the public and the private spectrum. Sometimes the high frequency of cellular noise would pervade the atmosphere, at other junctures it would erupt into words and melt down to radio hiss. Intercepted in the data stream, transmissions would blend, blurring the voices and rupturing the light, creating audio transparencies of dreamy, cool ambience. In many ways they pre-empted our reality culture as it exists today. Having the technology to peel open virtually any zone of information and consume the contents, I used the scanner device itself – a modestly sophisticated radio receiver – to explore the relationship between the public and private spheres. Working with sound in this manner suggested a means of mapping the city, in which the scanner device provided an anonymous window into reality, cutting and pasting information to structure an alternative vernacular. It was a rare opportunity to record experience and highlight the threads of desire and interior narrative that we weave into our everyday lives. The sounds of an illicit affair, a liaison with a prostitute, a drug deal or a simple discussion of “what’s for dinner” all exist within an indiscriminate ocean of signals flying overhead, but just beyond our reach. Applying the tools in this manner, I was able to twist state-of-the-art technology in unconventional ways to intercept highly personalised and voyeuristic forms of info food: sound recordings, phone scans, modem and Net intercepts, all of which became material for my multi-layered soundscapes. Every live performance, recording or mix that has followed is still in its way a “true” representation of that moment in time and in that way relates to performance art in the temporality of its data – a “Sound Polaroid” – a way of capturing the moment in sound similar to that of a Polaroid camera, which seizes an image and immediately exposes it to permanence of interception. Is there an innate desire to remain invisible and yet hear the world, scanning it for its stories and secrets? Today for our media saturated culture, this almost fetishistic desire to know all, is expressed in the publishing of private communications, of letters, faxes or telephone conversations, giving rise to all kinds of debate on the nature of privacy and the extent to which its protection can be legislated for. Images come to mind from Wim Wender’s film Wings of Desire where the lead character is a fallen angel left on earth to try and understand the madness of mortal behaviour. At various points he is able to pass through public spaces, the library, an underground train, people’s innermost thoughts and concerns become audible to him while he remains invisible to them. It is here that these Scanner CDs mirror this fantasy of the 20th century: to know everything and to have access to all secrets without being observed. This desire continues to inform our entertainment and cultural channels and looks to continue to do so for some years yet. This listening-in and scanning of the private channels has a clear relationship to surveillance, and connects to an aesthetic explored in works such as the seminal video piece, Der Riese – The Giant (1983) where the artist collaged the contents of surveillance cameras from German supermarkets, subway platforms, traffic crossings and shopping centres, using the tools of commercial voyeurism. Without a director, nor actors nor script, this is a dehumanised exploration of a contemporary history of our post-modern times. Connecting the invisible dots between Vertov’s The Man with a Movie Camera and the Rodney King TV footage, this detached work resonates and celebrates new technology’s ability to film and map everything, scanning our landscape for future reference. We watch with a constant anticipation of resolution, of catching a moment, yet the suspension finally gives way to an exquisite boredom, the true revelation of watching others. The film closes with an alien landscape, unmarked by any human presence, moving over a simulated environment, a toy-town yet still patrolled by the power of surveillance. This corporate datasphere, revealed as a kind of digital fingerprint through its storage and distribution, has moved from security and surveillance to entertainment consumption. For me, zooming in on these spaces in between – between language and understanding, between the digital fallout of ones and zeros, between the redundant and undesired flotsam and jetsam of environmental acoustic space, led to a focus towards the cityscape. Scanning technology led towards an understanding and reading of the environment and city in a fresh manner. If an accent suggested a certain class, age or attitude, then how suggestive was the raw sound around these conversations, how influential was the location where each conversation was held? Sound is ever-present, sometimes as a constantly shifting whir, as a damp grain of footsteps, as the drone-like spangle of distant traffic, as the seemingly motionless air that ripples past our ears, or as the elegant stuttering trill of a bird overhead. How influential was this common envelope of space, the environment in which we consume sound and music? How does one define the spaces between music and sound? When we listen to a Walkman, how do we distinguish between that which is intended – the sound carrier – and that which is incidental: passing traffic, the roar of a plane, the screech of a train door, your own footsteps? Whether active (creator) or passive (listener) we set up a virtual space in which we are each free to explore the sonorous and acoustic strata of what is an intimate yet global expression of space, a simple translation of the social transformations wrought by new technologies. Projects that have followed since then have expanded upon these notions. In 1998 Liverpool became this cityscape of focus, where I produced a project, Stopstarting, which explored the acoustic debris of the city, premiering at the International Symposion on Electronic Art (ISEA) conference in September of that year. For this project I chose significant points of sound located in the city, partly based on random questions in interviewing local people, partly out of self-interest. From these I mapped out a walk that took me from one point to another, minidisc in hand, recording the acoustic data in that place, mapping out the city in sound, teasing out the language that the city speaks. I wanted to create, in a sense, a sound work similar to the opening scene in Robert Altman’s movie Short Cuts (1993), in which a helicopter hovers gently over the densely packed city landscape and the film scans into moments in the daily lives of its inhabitants. It is a motion across a city, an architectural electronic scanning of an almost invisible sound wave. Liverpool, like most cities, has its very own unique sound dialect. Historically one can recall the sound of the docks, the railway station, the Cavern Club where the Beatles played their earliest live shows, their brittle tunes floating through the air of memory. As in Der Riese, voices, traffic lights, announcement speakers, buses, building work, footsteps, telephones and cash machines became the key subjects, the lead players, and were manipulated and transformed into a composition that captured this Sound Polaroid of Liverpool at this particular point in 1998. The following year Surface Noise (1999) which explored the wow and flutter of my own city, London; taking people on a red Routemaster bus journey across the city from Big Ben to St. Paul’s Cathedral, where the sheet music of “London Bridge Is Falling Down” became the score and A-Z for both musical and geographical direction following a Cageian use of indeterminacy. Where each note fell onto the map of the city between these two points not only suggested a location at which to record but also a route that the bus would later follow with the public aboard. Performances followed this routing every night for three nights, at intervals throughout the evening, each re-assembling fragments of the city in terms of sound and image, suggesting the slight shifts in tone and shape in similar places but at very different hours, so that a busy West End street at 18:00 would transform into a ghostly emptiness at 21:00. Surface Noise became a form of alternative film soundtrack, where the film was simply the view through the dusty window of a double-decker bus. Through the brief space of a bus journey the work drew upon many of our common reserves of sonic recognition, mingling the folk memory of the nursery rhyme, the background roar of traffic and the private sounds we make, secure in the knowledge that no one else is listening. Most recently I was commissioned to create a work to celebrate Italian film director Michelangelo Antonioni’s 90th birthday. 52 Space (2002) uses sounds of the city of Rome and elements of his movie The Eclipse (1962) to create a soundtrack of an image of a city suspended in time, anonymous and surreal. The resulting work is a distilled narrative of seductive conversation, musical fragments and scanned city soundscapes. Selecting a series of 52 framed images from the closing moments of the film slowed down to a kind of mnemonic slide show and accompanied by audio culled from the movie, processed with twinkling elements from the soundtrack’s original melody, the live performance conveys a complex and mysterious chronicle, offering up a space for contemplation and reflection as the soundtrack weaves an imaginary narrative. It’s almost as if you are gently floating through the city, experiencing this dream-like state. All of my works have explored the hidden resonances and meanings within memory and, in particular, the subtle traces that people and their actions leave behind. My role has often been to discover and reveal these layers of history, scanning across the mediums, so the works are part urban guide, part urban geography and part detective fiction, raising questions about public and private space. Engaging with the tools of surveillance and scanning technology has given rise to an understanding of communication that was otherwise hidden. Revelation followed from a discovery of the possibilities of these devices. Recording and redirecting these moments back into the public stream has enabled me to construct an archaeology of loss, pathos and missed connections, a momentary forgotten past in our digital future, radioactive fossils of sound, image and the imagination. Citation reference for this article MLA Style Rimbaud, Robin. "Scan and Deliver." M/C Journal 8.4 (2005). echo date('d M. Y'); ?> <http://journal.media-culture.org.au/0508/05-rimbaud.php>. APA Style Rimbaud, R. (Aug. 2005) "Scan and Deliver," M/C Journal, 8(4). Retrieved echo date('d M. Y'); ?> from <http://journal.media-culture.org.au/0508/05-rimbaud.php>.