Academic literature on the topic 'Kernel decomposition formula'

AbstractAn integral formula is derived for the spherical functions on the symmetric space G/K = SO0(p, q)/ SO(p) × SO(q). This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra a of the abelian part in the decomposition G = KAK. The corresponding result is then obtained for the heat kernel of the symmetric space SO0(p, q)/ SO(p) × SO(q) using the Plancherel formula.In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.

A sequence of special functions in Hardy space [Formula: see text] are constructed from Cauchy kernel on unit disk 𝔻. Applying projection operator of the sequence of functions leads to an analytic sampling approximation to f, any given function in [Formula: see text]. That is, f can be approximated by its analytic samples in 𝔻s. Under a mild condition, f is approximated exponentially by its analytic samples. By the analytic sampling approximation, a signal in [Formula: see text] can be approximately decomposed into components of positive instantaneous frequency. Using circular Hilbert transform, we apply the approximation scheme in [Formula: see text] to Ls(𝕋2) such that a signal in Ls(𝕋2) can be approximated by its analytic samples on ℂs. A numerical experiment is carried out to illustrate our results.

Calderón–Zygmund operators are playing an important role in real analysis. The continuity of Calderón–Zygmund operators T on L2 was studied in [2–4] and, here, we are investigating the continuity of T on the Besov spaces [Formula: see text] where 1 ≤ p, q ≤ ∞ and on the Triebel–Lizorkin spaces [Formula: see text] where 1 ≤ p < ∞, 1 ≤ q ≤ ∞. The exponents measuring the regularity of the distributional kernel K(x, y) of T away from the diagonal are playing a key role in our results. They are smaller than the ones which were assumed by other authors. Moreover, our results are sharp in the case of the Besov spaces [Formula: see text] and of the Triebel–Lizorkin spaces [Formula: see text] when 1 ≤ q ≤ ∞. The proof uses a pseudo-annular decomposition of Calderón–Zygmund operators.

We initiate a detailed study of two-parameter Besov spaces on the unit ball of [Formula: see text] consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem.

AbstractIn a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

Dans ce travail, nous nous intéressons au plongement [Formula: see text] des T-unités d'un corps de nombres K dans une partie de ses complétés p-adiques construite sur l'ensemble S. Nous montrons que l'injectivité de [Formula: see text] permet d'obtenir des informations sur la structure du groupe de Galois de certaines extensions de K où la ramification est liée à S et la décomposition à T. Nous étudions également le comportement asymptotique du noyau de [Formula: see text] le long d'une extension p-adique analytique sans p-torsion. In this article, we are interested in the embedding [Formula: see text] of the T-units of a number field K in some part of its p-adic completions at S. We show that the injectivity of [Formula: see text] allows us to obtain some information on the structure of the Galois group of some extensions of K where the ramification is attached at S and the decomposition at T. Moreover, we study the asymptotic behavior of the kernel [Formula: see text] along a p-adic analytic extension.

The focusing inversion of gravity and magnetic potential-field data using the randomized singular value decomposition (RSVD) method is considered. This approach facilitates tackling the computational challenge that arises in the solution of the inversion problem that uses the standard and accurate approximation of the integral equation kernel. We have developed a comprehensive comparison of the developed methodology for the inversion of magnetic and gravity data. The results verify that there is an important difference between the application of the methodology for gravity and magnetic inversion problems. Specifically, RSVD is dependent on the generation of a rank [Formula: see text] approximation to the underlying model matrix, and the results demonstrate that [Formula: see text] needs to be larger, for equivalent problem sizes, for the magnetic problem compared to the gravity problem. Without a relatively large [Formula: see text], the dominant singular values of the magnetic model matrix are not well approximated. We determine that this is due to the spectral properties of the matrix. The comparison also shows us how the use of the power iteration embedded within the randomized algorithm improves the quality of the resulting dominant subspace approximation, especially in magnetic inversion, yielding acceptable approximations for smaller choices of [Formula: see text]. Further, we evaluate how the differences in spectral properties of the magnetic and gravity input matrices also affect the values that are automatically estimated for the regularization parameter. The algorithm is applied and verified for the inversion of magnetic data obtained over a portion of the Wuskwatim Lake region in Manitoba, Canada.

Purpose – The purpose of this paper is to propose a Chebyshev collocation method (CCM) for Hallén’s equation of thin wire antennas. Design/methodology/approach – Since the current induced on the thin wire antennas behaves like the square root of the distance from the end, a smoothed current is used to annihilate this end effect. Then the CCM adopts Chebyshev polynomials to approximate the smoothed current from which the actual current can be quickly recovered. To handle the difficulty of the kernel singularity and to realize fast computation, a decomposition is adopted by separating the singularity from the exact kernel. The integrals including the singularity in the linear system can be given in an explicit formula while the others can be evaluated efficiently by the fast cosine transform or the fast Fourier transform. Findings – The CCM convergence rate is fast and this method is more efficient than the other existing methods. Specially, it can attain less than 1 percent relative errors by using 32 basis functions when a/h is bigger than 2×10−5 where h is the half length of wire antenna and a is the radius of antenna. Besides, a new efficient scheme to evaluate the exact kernel has been proposed by comparing with most of the literature methods. Originality/value – Since the kernel evaluation is vital to the solution of Hallén’s and Pocklington’s equations, the proposed scheme to evaluate the exact kernel may be helpful in improving the efficiency of existing methods in the study of wire antennas. Due to the good convergence and efficiency, the CCM may be a competitive method in the analysis of radiation properties of thin wire antennas. Several numerical experiments are presented to validate the proposed method.

The aim of this article is to model an ordinal response variable in terms of vector-valued functional data included on a vector-valued reproducing kernel Hilbert space (RKHS). In particular, we focus on the vector-valued RKHS obtained when a geometrical object (body) is characterized by a current and on the ordinal regression model. A common way to solve this problem in functional data analysis is to express the data in the orthonormal basis given by decomposition of the covariance operator. But our data present very important differences with respect to the usual functional data setting. On the one hand, they are vector-valued functions, and on the other, they are functions in an RKHS with a previously defined norm. We propose to use three different bases: the orthonormal basis given by the kernel that defines the RKHS, a basis obtained from decomposition of the integral operator defined using the covariance function and a third basis that combines the previous two. The three approaches are compared and applied to an interesting problem: building a model to predict the fit of children's garment sizes, based on a 3D database of the Spanish child population. Our proposal has been compared with alternative methods that explore the performance of other classifiers (Support Vector Machine and [Formula: see text]-NN), and with the result of applying the classification method proposed in this work, from different characterizations of the objects (landmarks and multivariate anthropometric measurements instead of currents), obtaining in all these cases worst results.

Exponential increase and global access to read/write memory states in quantum computing (QC) simulation limit both the number of qubits and quantum transformations which can be currently simulated. Although QC simulation is parallel by nature, spatial and temporal complexity are major performance hazards, making this a nontrivial application for high performance computing. A new methodology employing reduction and decomposition optimizations has shown relevant results, but its GPU implementation could be further improved. In this work, we develop a new kernel for in situ GPU simulation that better explores its resources without requiring further hardware. Shor’s and Grover’s algorithms are simulated up to 25 and 21 qubits respectively and compared to our previous version, to [Formula: see text] simulator and to ProjectQ framework, showing better results with relative speedups up to 4.38×, 3357.76× and 333× respectively.

Dans cette thèse, nous étudions les solutions de l'équation de Boltzmann. Nous nous intéressons au cadre homogène en espace où la solution f(t; x; v) dépend uniquement du temps t et de la vitesse v. Nous considérons des sections efficaces singulières (cas dit non cutoff) dans le cas Maxwellien. Pour l'étude du problème de Cauchy, nous considérons une fluctuation de la solution autour de la distribution Maxwellienne puis une décomposition de cette fluctuation dans la base spectrale associée à l'oscillateur harmonique quantique. Dans un premier temps, nous résolvons numériquement les solutions en utilisant des méthodes de calcul symbolique et la décomposition spectrale des fonctions de Hermite. Nous considérons des conditions initiales régulières et des conditions initiales de type distribution. Ensuite, nous prouvons qu'il n'y a plus de solution globale en temps pour une condition initiale grande et qui change de signe (ce qui ne contredit pas l'existence globale d'une solution faible pour une condition initiale positive - voir par exemple Villani Arch. Rational Mech. Anal 1998)
In this thesis, we study the solutions of the Boltzmann equation. We are interested in the homogeneous framework in which the solution f(t; x; v) depends only on the time t and the velocity v. We consider singular crosssections (non cuto_ case) in the Maxwellian case. For the study of the Cauchy problem, we consider a uctuation of the solution around the Maxwellian distribution then a decomposition of this uctuation in the spectral base associated to the quantum harmonic oscillator At first, we solve numerically the solutions using symbolic computation methods and spectral decomposition of Hermite functions. We consider regular initial data and initial conditions of distribution type. Next, we prove that there is no longer a global solution in time for a large initial condition that changes sign (which does not contradict the global existence of a weak solution for a positive initial condition - see for example Villani Arch. Rational Mech. Anal 1998)

Let $\Omega \subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(\Omega)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring $\mathbb C[\underline{z}]\subseteq \mathcal H$ is dense and the point-wise multiplication induced by $p\in \mathbb C[\underline{z}]$ is bounded on $\mathcal H$. We fix an ideal $\mathcal I \subseteq \mathbb C[\underline{z}]$ generated by $p_1,\ldots,p_t$ and let $[\mathcal I]$ denote the completion of $\mathcal I$ in $\mathcal H$. Let $X:[\mathcal I] \to \mathcal H$ be the inclusion map. Thus we have a short exact sequence of Hilbert modules \begin{tikzcd} 0 \arrow{r} &\mbox{[} \mathcal I \mbox{]} \arrow{r}{X} & {\mathcal H} \arrow{r}{\pi} & \mathcal Q \arrow{r}& 0 , \end{tikzcd} where the module multiplication in the quotient $\mathcal Q:=[\mathcal I]^\perp$ is given by the formula $m_p f = P_{[\mathcal I]^\perp} (p f),$ $p\in \mathbb C[\underline{z}],\,f\in \mathcal Q$. The analytic Hilbert module $\mathcal H$ defines a subsheaf $\mathcal S^\mathcal H$ of the sheaf $\mathcal O(\Omega)$ of holomorphic functions defined on $\Omega$. For any open $U \subset \Omega$, it is obtained by setting $$\mathcal S^\mathcal H(U) := \Big \{\, \sum_{i=1}^n ({f_i|}_U) h_i : f_i \in \mathcal H, h_i \in \mathcal O(U), n\in\mathbb N\,\Big \}.$$ This is locally free and naturally gives rise to a holomorphic line bundle on $\Omega$. However, in general, the sheaf corresponding to the sub-module $[\mathcal I]$ is not locally free but only coherent. Building on the earlier work of S. Biswas, a decomposition theorem is obtained for the kernel $K_{[\mathcal I]}$ along the zero set $V_{[\mathcal I]}:=\big\{z\in \mathbb C^m: f(z) = 0, f\in [\mathcal I]\big\}$ which is assumed to be a submanifold of codimension $t$: There exists anti-holomorphic maps $F_1, \ldots, F_t: V_{[\mathcal I]}\to [\mathcal I]$ such that $$ K_{[\mathcal I]}(\cdot, u) = \overline{p_1(u)} F^1_w(u) + \cdots \overline{p_t(u)} F_w^t(u),\, u\in \Omega_w,$$ in some neighbourhood $\Omega_w$ of each fixed but arbitrary $w\in V_{[\mathcal I]}$ for some anti-holomorphic maps $F_w^1, \ldots, F^t_w: \Omega_w \to [\mathcal I]$ extending $F_1, \ldots,F_t$. The anti-holomorphic maps $F_1, \ldots,F_t$ are linearly independent on $V_{[\mathcal I]}$, defining a rank $t$ anti-holomorphic Hermitian vector bundle on it. This gives rise to complex geometric invariants for the pair $([\mathcal I], \mathcal H)$. Next, using a decomposition formula obtained from an earlier work of Douglas, Misra and Varughese, the maps $F_1, \ldots, F_t: V_{[\mathcal I]}\to [\mathcal I]$ are explicitly determined with the additional assumption that $p_{i},p_{j}$ are relatively prime for $i\neq j$. Using this, a line bundle on $V_{[\mathcal I]}\times\mathbb{P}^{t-1}$ is constructed via the monoidal transformation around $V_{[\mathcal I]}$ which provides useful invariants for $([\mathcal I], \mathcal H)$. Localising the modules $[\mathcal I]$ and $\mathcal H$ at $w\in \Omega$, we obtain the localization $X(w)$ of the module map $X$. The localizations are nothing but the quotient modules $[\mathcal I]/{[\mathcal I]_w}$ and $\mathcal H/{\mathcal H_w}$, where $[\mathcal I]_w$ and $\mathcal H_w$ are the maximal sub-modules of functions vanishing at $w$. These clearly define anti-holomorphic line bundles $E_{[\mathcal I]}$ and $E_\mathcal H$, respectively, on $\Omega\setminus V_{[\mathcal I]}$. However, there is a third line bundle, namely, ${\rm Hom}(E_\mathcal H, E_{[\mathcal I]})$ defined by the anti-holomorphic map $X(w)^*$. The curvature of a holomorphic line bundle $\mathcal L$ on $\Omega$, computed with respect to a holomorphic frame $\gamma$ is given by the formula $$\mathcal K_\mathcal L(z) = \sum_{i,j=1}^{m}\tfrac{\partial^2}{\partial z_i \partial \bar{z}_j}\log\|\gamma(z)\|^2 dz_i \wedge d\bar{z}_j.$$ It is a complete invariant for the line bundle $\mathcal L$. The alternating sum $$ \mathcal A_{[\mathcal I], \mathcal H}(w):=\mathcal K_X(w) - \mathcal K_{[\mathcal{I}]}(w) + \mathcal K_{\mathcal{H}}(w) = 0,\,\, w\in \Omega \setminus V_{[\mathcal I]}, $$ where $\mathcal K_X$, $\mathcal K_{[\mathcal{I}]}$ and $\mathcal K_{\mathcal{H}}$ denote the curvature $(1,1)$ form of the line bundles $E_X$, $E_{[\mathcal{I}]}$ and $E_{\mathcal{H}}$, respectively. Thus it is an invariant for the pair $([\mathcal I], \mathcal H)$. However, when $\mathcal I$ is principal, by taking distributional derivatives, $\mathcal A_{[\mathcal I], \mathcal H}(w)$ extends to all of $\Omega$ as a $(1,1)$ current. Consider the following diagram of short exact sequences of Hilbert modules: $$(1)\,\,\,\,\,\,\,\,\,\,\, \begin{tikzcd} 0\arrow{r} &\mbox{[}\mathcal I\mbox{]} \arrow{d} \arrow{r} {X} & {\mathcal H}\arrow{d}{L} \arrow{r}{\pi} & \mathcal Q\arrow{d} \arrow{r}& 0\\ 0\arrow{r} &\mbox{[}\widetilde{\mathcal I}\mbox{]} \arrow{r}{\widetilde{X}} &\widetilde{\mathcal H} \arrow{r}{\tilde{\pi}}& \widetilde{\mathcal Q} \arrow{r}& 0, \end{tikzcd} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (2)\,\,\,\,\,\,\,\,\,\,\, \begin{tikzcd} \mbox{[}\mathcal I\mbox{]} \arrow{d} \arrow{r} {X} & {\mathcal H}\arrow{d}{L}\\ \mbox{[}\widetilde{\mathcal I}\mbox{]} \arrow{r}{\widetilde{X}} &\widetilde{\mathcal H} \end{tikzcd}$$ It is shown that if $\mathcal A_{[\mathcal I], \mathcal H}(w)=\mathcal A_{[\widetilde{\mathcal I}], \widetilde{\mathcal H}}(w)$, then $L|_{[\mathcal I]}$ makes the second diagram commute. Hence, if $L$ is bijective, then $[\mathcal I]$ and $[\widetilde{\mathcal I]}$ are equivalent as Hilbert modules. It follows that the alternating sum is an invariant for the ``rigidity'' phenomenon.

This chapter describes the decomposition of the geometric kernel. It considers the assumptions on the Schwartz function and decomposes the height series into local heights using arithmetic models. The intersections with the Hodge bundles are zero, and a decomposition to a sum of local heights by standard results in Arakelov theory is achieved. The chapter proceeds by reviewing the definition of the Néeron–Tate height and shows how to compute it by the arithmetic Hodge index theorem. When there is no horizontal self-intersection, the height pairing automatically decomposes to a summation of local pairings. The chapter proves that the contribution of the Hodge bundles in the height series is zero. It also compares two kernel functions and states the computational result. It concludes by deducing the kernel identity.