$$\ell _2$$ Similarity Metrics for Diffusion Multi-Compartment Model Images Registration

Multicompartmental modeling outperforms conventional diffusion-weighted imaging (DWI) in the assessment of prostate cancer. Optimized multicompartmental models could further improve the detection and characterization of prostate cancer. To optimize multicompartmental signal models and apply them to study diffusion in normal and cancerous prostate tissue in vivo. Retrospective. Forty-six patients who underwent MRI examination for suspected prostate cancer; 23 had prostate cancer and 23 had no detectable cancer. 3T multishell diffusion-weighted sequence. Multicompartmental models with 2-5 tissue compartments were fit to DWI data from the prostate to determine optimal compartmental apparent diffusion coefficients (ADCs). These ADCs were used to compute signal contributions from the different compartments. The Bayesian Information Criterion (BIC) and model-fitting residuals were calculated to quantify model complexity and goodness-of-fit. Tumor contrast-to-noise ratio (CNR) and tumor-to-background signal intensity ratio (SIR) were computed for conventional DWI and multicompartmental signal-contribution maps. Analysis of variance (ANOVA) and two-sample t-tests (α = 0.05) were used to compare fitting residuals between prostate regions and between multicompartmental models. T-tests (α = 0.05) were also used to assess differences in compartmental signal-fraction between tissue types and CNR/SIR between conventional DWI and multicompartmental models. The lowest BIC was observed from the 4-compartment model, with optimal ADCs of 5.2e-4, 1.9e-3, 3.0e-3, and >3.0e-2 mm2 /sec. Fitting residuals from multicompartmental models were significantly lower than from conventional ADC mapping (P < 0.05). Residuals were lowest in the peripheral zone and highest in tumors. Tumor tissue showed the largest reduction in fitting residual by increasing model order. Tumors had a greater proportion of signal from compartment 1 than normal tissue (P < 0.05). Tumor CNR and SIR were greater on compartment-1 signal maps than conventional DWI (P < 0.05) and increased with model order. The 4-compartment signal model best described diffusion in the prostate. Compartmental signal contributions revealed by this model may improve assessment of prostate cancer. Level of Evidence 3 Technical Efficacy Stage 3 J. MAGN. RESON. IMAGING 2021;53:628-639.

We consider the problem of determining the Fourier integral in the Hilbert space of square integrable func- tions. Fourier integral is the scalar product of two functions belonging to the Hilbert space of square integrable functions and the Hilbert space of almost periodic functions. Scalar product for different Hilbert spaces defined at the intersection of these spaces, which contains only one zero element. Therefore, the Fourier integral is not defined in the Hilbert space of square integrable functions with nonzero norm.

The Kac-Baker model describes a 1-dim. classical lattice spin system with exponentially fast decaying two body interaction. The model was introduced by M. Kac and G. Baker to investigate the phenomenon of phase transition in systems with weak long-range interactions like van der Waals gas. Ruelle's dynamical zeta function for this model can be expressed in terms of Fredholm determinants of two transfer operators and hence is a meromorphic function. One of the two operators, found by M. Kac, is an integral operator with symmetric kernel acting in the Hilbert space of square integrable functions on the line. The other one is Ruelle's transfer operator acting in some Banach space of holomorphic observables of the system. In this paper we show how the Kac operator can be explicitly related basically through the Segal-Bargmann transform to the Ruelle operator restricted to a certain Segal-Bargmann space of entire functions in the complex plane. This allows us to show that Ruelle's zeta function for the Kac-Baker model has infinitely many non-trivial zeros on the real axis. In a special case we can show the existence of also infinitely many trivial zeros on the line Re s = ln 2 in the complex s-plane. Hence some kind of Riemann hypothesis seems to hold for this dynamical zeta function.

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.

Harmonic analysis on a locally compact group is the study of certain spaces and algebras of functions and measures defined on the group. The function spaces in question are defined by measurability and integrability conditions; manipulation of the relevant measures requires delicate techniques from measure and integration theory. Also, the fundamental theorems of the theory of representations of locally compact groups [see §22] depend wholly upon the study of a certain space of square integrable functions. Thus the theories of measure and integration play a vital role in the present book.

ABSTRACTWe study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by homogenized solution, the Green’s function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance.

We construct a family of generalized coherent states attached to Landau levels of a charged particle moving in the Poincaré disc under a perpendicular uniform magnetic field. The corresponding coherent state transforms enable us to connect, by an integral transform, spaces of bound states of the particle with the space of square integrable functions on the real line. The established connection provides us with a new way to obtain hyperbolic Landau states.

The three-body Schrodinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length \({\kappa + 1, \kappa = 0, 1, \ldots}\) As a result, the solutions to the three-body Schrodinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve—in these subalgebras—the radial Schrodinger equation in three dimensions with the inverse power potential of the form \({r^{-{\kappa}-1}}\) . As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with \({\kappa = 0}\) , are reduced to the well-known eigenvalues of bound states at threshold.

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

The functional linear model with scalar response is a regression model where the predictor is a random function defined on some compact set of ℝ and the response is scalar. The response is modelled as Y=Ψ(X)+ɛ, where Ψ is some linear continuous operator defined on the space of square integrable functions and valued in ℝ. The random input X is independent from the noise ɛ. In this paper, we are interested in testing the null hypothesis of no effect, that is, the nullity of Ψ restricted to the Hilbert space generated by the random variable X. We introduce two test statistics based on the norm of the empirical cross‐covariance operator of (X,Y). The first test statistic relies on a χ2 approximation and we show the asymptotic normality of the second one under appropriate conditions on the covariance operator of X. The test procedures can be applied to check a given relationship between X and Y. The method is illustrated through a simulation study.

A class of irreducible unitary representations belonging to the continuous series of SUp,q is explicitly determined in a space composed of the Kronecker product of three spaces of square integrable functions. The continuous series corresponds to a Cartan subgroup whose vector part has maximal dimension. These representations are distinguished by a parameter r = 1, 2, …, p for p ≤ q in SUp,q. For r = 0, one obtains the representations in the discrete series as in [5], and all representations in the continuous series, for r ≠ 0, are obtained explicitly.

In the first section we saw that the space of C∞ regular, localized functions with all moments vanishing is characterized by a rapid decay of the modulus of the wayelet coefficients at small and large scales and at large positions so that the wavelet transform is localized over the position-scale half-plane. By duality the space of tempered distributions was characterized by an at most polynomial growth of the wavelet coefficients at small and large scales and large positions. A third space, . the Hilbert space of square integrable functions, was c aracterized by a square integrable distribution of the energy over the position-scale half-plane. In all three cases this characterization was independent of the analysing wavelet. It therefore seems natural to introduce spaces that are characterized by their distribution of the modulus of the wavelet coefficients over the position-scale half-plane. This will be done in this chapter. For supplementary information we refer the reader to the literature. In particular in the case of the wavelet transform the books by Meyer (1985) and Coifman and Meyer are certainly standard references. On the other hand, many ideas that are now unified under the wavelet approach can be found in great detail in the book by Triebel (1992). Again, all results can be generalized in the language of group theory, as was shown by Feichtinger and Grochenig (1989).

Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform