Spectral shift function for a discretized continuum

A spectral shift function (SSF) is an important object in the scattering theory which is related both to the spectral density and to the scattering matrix. In the paper, it is shown how to employ the SSF formalism to solve scattering problems when the continuum is discretized, e.g. when solving a scattering problem in a finite volume or in the representation of some finite square-integrable basis. A new algorithm is proposed for reconstructing integrated densities of states and the SSF using a union of discretized spectra corresponding to a set of Gaussian bases with the shifted scale parameters. The examples given show that knowledge of the discretized spectra of the total and asymptotic Hamiltonians is sufficient to find the scattering partial phase shifts at any required energy, as well as the resonances parameters.

Spectral asymptotics for the Schrödinger operator with a non-decaying potential

We study Schrödinger operators H ( h ) = − h 2 Δ + V ( x ) acting in L 2 ( R n ) for non-decaying potentials V. We give a full asymptotic expansion of the spectral shift function for a pair of such operators in the high energy limit. In particular for asymptotically homogeneous potentials W at infinity of degree zero, we also study the semiclassical asymptotics to give a Weyl formula of the spectral shift function above the threshold max W and Mourre estimates in the range of W except at its critical values.

Vendor-neutral sequences (VENUS) and fully transparent workflows improve inter-vendor reproducibility of quantitative MRI

Purpose: We developed a transparent end-to-end qMRI workflow that starts with a vendor-neutral acquisition and tested the hypothesis that vendor-neutral sequences (VENUS) decrease inter-vendor variability of T1, MTR and MTsat measurements. Methods: We developed and deployed a vendor-neutral 3D spoiled gradient-echo (SPGR) sequence on three clinical scanners by two MRI vendors and acquired T1 maps on the NIST phantom, as well as T1, MTR and MTsat maps in three healthy participants. We performed hierarchical shift function analysis in vivo to characterize the differences between scanners when VENUS is used instead of commercial vendor implementations. Inter-vendor deviations were compared for statistical significance to test the hypothesis. Results: In the NIST phantom, VENUS reduced inter-vendor differences from 8 - 19.4% to 0.2 - 5% with an overall accuracy improvement, reducing ground truth T1 deviation from 7 - 11% to 0.2 - 4%. In vivo we found that the variability between vendors is significantly reduced (p = 0.015) for all maps (T1, MTR and MTsat) using VENUS. Conclusion: We conclude that vendor-neutral workflows are feasible and compatible with clinical MRI scanners. The significant reduction of inter-vendor variability using VENUS has important implications for qMRI research and for the reliability of multicenter clinical trials.

Trace singularities in obstacle scattering and the Poisson relation for the relative trace

We consider the case of scattering by several obstacles in $${\mathbb {R}}^d$$ R d , $$d \ge 2$$ d ≥ 2 for the Laplace operator $$\Delta $$ Δ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $$\Delta _1$$ Δ 1 and $$\Delta _2$$ Δ 2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator $$g(\Delta ) - g(\Delta _1) - g(\Delta _2) + g(\Delta _0)$$ g ( Δ ) - g ( Δ 1 ) - g ( Δ 2 ) + g ( Δ 0 ) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function $$\xi _\mathrm {rel}(\lambda ) = -\frac{1}{\pi } {\text {Im}}(\Xi (\lambda ))$$ ξ rel ( λ ) = - 1 π Im ( Ξ ( λ ) ) , where $$\Xi (\lambda )$$ Ξ ( λ ) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of $$\xi _\mathrm {rel}$$ ξ rel . In particular we prove that $${\hat{\xi }}_\mathrm {rel}$$ ξ ^ rel is real-analytic near zero and we relate the decay of $$\Xi (\lambda )$$ Ξ ( λ ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function $$\Xi (\lambda )$$ Ξ ( λ ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.

Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing

The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.

TERITORIALITAS RUANG RUMAH ADAT CIKONDANG PADA TRADISI WUKU TAUN

The current activities in the Cikondang traditional house which has become a cultural heritage is currently changing in a shift function, but it remains the center of the activities for continuation of ritual events, named Wuku Taun. This celebration must be held in a traditional house as a form of respect for the traditional ancestors. Therefore ,the role of the space in traditional houses is very significant. The activities that take place in the spaces of the Cikondang traditional house have formed an interaction between the user and the territory. Starting from this background, it has several problems; how is the territorial phenomenon that occurs in the rooms of traditional houses during Wuku Taun celebration and what are the elements that affect the territoriality in the spaces, especially at the Wuku Taun celebration. This study aims to determine the elements that form territorial relationships. This research is a descriptive analysis with a phenomenological method that uses the territorial theory approach proposed by Brower. The results showed that the territoriality in the traditional house is seen in the territorial setting where the users of the space still place their respective positions because of the restrictions shown as a defensive mechanism, the indigenous people still follow to the rules of customs that apply as cultural and ritual controls, giving rise to regular territoriality and customs rules which are cultural factors are able to produce their own territorial attitude from the visible activities. In addition, strong social ties between the local and the outside peoples, a high sense of ownership within the indigenous community so that this cultural ritual continues every year in the spaces of the Cikondang traditional house. Kegiatan saat ini di rumah adat Cikondang yang telah menjadi cagar budaya saat ini mengalami pergeseran fungsi, namun tetap menjadi pusat kegiatan keberlangsungan acara ritual adat yaitu Wuku Taun. Perayaan tersebut wajib dilaksanakan dalam rumah adat sebagai bentuk penghormatan terhadap para leluhur adat. Oleh karena itu, peran ruang dalam rumah adat menjadi suatu hal yang sangat signifikan. Kegiatan yang berlangsung di dalam ruang-ruang rumah adat Cikondang telah membentuk suatu interaksi antara pengguna dengan wilayah. Bertolak dari latar belakang tersebut yang kemudian menimbulkan permasalahan yaitu, bagaimana fenomena teritorialitas yang terjadi dalam ruang-ruang rumah adat pada saat perayaan Wuku Taun dan apa saja unsur-unsur yang berpengaruh pada teritorialitas dalam ruang-ruang tersebut khususnya pada perayaan Wuku Taun. Penelitian ini bertujuan untuk mengetahui terjadinya fenomena teritorialitas terutama saat kegiatan Wuku Taun dan untuk mengetahui unsur-unsur yang membentuk hubungan teritorialitas. Penelitian ini merupakan penelitian deskriptif analisis dengan metode fenomenologi yang menggunakan pendekatan teori teritorialitas yang dikemukakan oleh Brower. Hasil penelitian menunjukkan bahwa teritorialitas di dalam rumah adat terlihat pada setting teritori di mana pengguna ruang tetap menempatkan posisinya masing-masing karena pembatasan yang ditunjukkan sebagai mekanisme defensif, masyarakat adat tetap mematuhi aturan adat istiadat yang berlaku sebagai kontrol budaya dan ritual sehingga menimbulkan teritorialitas yang teratur dan aturan adat istiadat yang merupakan faktor budaya mampu menghasilkan sikap teritorialitas tersendiri dari aktivitas yang terlihat. Selain itu, keterikatan sosial yang kuat antar masyarakat adat setempat maupun luar, rasa kepemilikan yang tinggi dalam diri masyarakat adat sehingga ritual budaya ini terus berlangsung ada setiap tahunnya di dalam ruang-ruang rumah adat Cikondang.

Bouncing Cosmology in f G , T Gravity with Logarithmic Trace Term

This study is devoted to explore bouncing cosmology in the context of f G , T theory of gravity. For this purpose, a Gauss–Bonnet cosmological model with logarithmic trace term is considered. In particular, the possibility of obtaining bouncing solutions by considering two equations of state parameters is investigated. A graphical analysis is provided for analyzing the obtained bouncing solutions. The energy conditions are discussed in detail. It is interesting to notice that null and strong energy conditions are violated near the neighborhood of bouncing points justifying the accelerating universe in the light of the recent observational data. The behavior of the scale factor, red shift function, deceleration parameter, and Hubble parameter is also debated. An important feature of the current study is the discussion of conservation equation in f G , T gravity. The possibility of some suitable constraint equations which recover the standard conservation equation is discussed, and all the free parameters are assumed accordingly. All the results in this study suggest that the proposed f G , T gravity model provides good bouncing solutions with the chosen EoS parameters.

Seismic image registration using multiscale convolutional neural networks

Aligning seismic images is important in many areas of seismic processing such as time-lapse studies, tomography, and registration of compressional and shear-wave images. This problem is especially difficult when the misalignment is large and varies rapidly and when the images are not shifted versions of each other because they are either contaminated by noise or have different phase or frequency content. In addition, the images may be related by multidimensional vector-valued shift functions. We have developed a fast, scalable, and end-to-end trainable convolutional neural network (CNN) for seismic image registration. The concept of optical flow is widely applied to the problem of image registration using variational methods. Recent developments in the field of computer vision have shown that optical flow estimation can be formulated as a supervised machine learning task and can be successfully solved using CNNs. We train our CNN, SeisFlowNet, on images warped with known shifts and corrupted with noise, frequency, and phase perturbations. We evaluate the promising performance of the trained SeisFlowNet with synthetic data sets where the shift function is known and the images are contaminated with noise and other perturbations. The accuracy of the results obtained with SeisFlowNet is favorably compared with two other popular methods for seismic registration: windowed crosscorrelation and dynamic image warping. Further, we highlight the principles adopted to create training data sets and the advantages and disadvantages of the method.