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Showing 1 to 20 of 56 for “"Inner product"”.
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Arithmetic inner product formula for unitary groups
… an arithmetic analogue of the classical Rallis' inner product formula, which we call arithmetic inner product formula, and confirm it for unitary groups of two variables. In particular, we calculate the Néron-Tate height of special points on Shimura curves attached to certain unitary groups of …
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F-virtual Abelian Varieties and Rallis Inner Product Formula
… of Langlands L-function. This new case of Rallis inner product formula enables us to relate nonvanishing of L-value to the nonvanishing of theta lifting.
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Convergence Rates of Spectral Distribution of Random Inner Product Kernel Matrices
… two parts. In the first part, we focus on random inner product kernel matrices. Under various assumptions, many authors have proved that the limiting empirical spectral distribution (ESD) of such matrices A converges to the Marchenko- Pastur distribution. Here, we establish the corresponding rate …
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Completion of hinge loss has an implicit bias
… iterates for two-layer linear networks on the inner product loss function is derived. Using the derived dynamical system, a precise explicit algorithm is developed which when implemented reproduces the gradient descent iterates of two-layer ReLU nets on the inner product exactly. This result is …
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Communication complexity of permutation-invariant functions
… the relatively lesser-known upper bounds for GAP-INNER-PRODUCT (from the sketching literature) and SPARSE-GAP-INNER-PRODUCT (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of communication complexity with imperfectly shared randomness where we show …
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Thoughts on harmonic analysis on the sphere: spherical wavelet frames and kernels
… At the end of this dissertation I give an inner product formula with respect to a recently emerged surface-value dependent inner product space on a triangular mesh and prove its equivalence to the combinatoric inner product.
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Products in Lambda-Rings
… we derive a method for the computation of the inner product extending that of Lascoux. We also introduce a method that reduces the analysis of the plethysm, at least for special cases, to solving a system of linear Diophantine equations in a distinguished region of space.
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Orthogonalität und beste Approximation
… compute orthogonal projections by means of the inner product known from analytic geometry, this insight can be generalized and put to good use far beyond the boundaries of concrete geometry. While perpendicular lines are relevant only in two or three dimensional space, good approximations are in …
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Affine quantum algebras, Weyl groups and constructible functions
… A, giving a geometric description of its natural inner product, and studying the theory of cells attached to the canonical basis. In the second part we study a realization of the group algebra of the Weyl group in a convolution algebra of constructible functions on the Steinberg variety, and …
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Using information theoretic measures to evaluate support vector machine kernels
… in some input space is equivalent to an inner product on some Hilbert space. Estimators of Renyi's generalized form of information theoretic measurements reduce to a form that gives an elegant characterization of the geometric properties of the kernel in the feature space. It is shown how …
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Exponential Integrators for the Incompressible Navier-Stokes Equations
… to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit …
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Hermite-Hadamard inequality in the geometry of banach spaces
… contributions including the characterization of inner product spaces via orthogonality; the extension of means of positive numbers to a vector space setting; and the developments of some important inequalities, namely the Hermite-Hadamard inequality, Ostrowski inequality and Gruss inequality in …
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Notes on generalized Fourier series with application to gravitational field determination
… where (f, φ<sub>n</sub>) is the inner product of the functions f an φ<sub>n</sub>. The e existence of a complete orthonormal system in L² is proven. Conditions for convergence of the generalized Fourier series are presented. A discussion of orthogonal polynomials with special …
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Applied left-definite theory : the Jacobi polynomials, their Sobolev orthogonality, and self-adjoint operators.
… they are orthogonal with respect to a Sobolev inner product. In this work, we first consider the special case where α = β = –1. We shall discuss the Sobolev orthogonality of the Jacobi polynomials and construct a self-adjoint operator in a certain Hilbert-Sobolev space having the entire …
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Objective functions for nonlinear structural analysis
… presented. The objective function, which is the inner product of the equations of equilibrium (algorithm 2), is compared with another objective function which, for a static problem, is the total potential energy (algorithm 3). The method of formulating the mathematical model for algorithm 2 is …
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Iterative Techniques for Radial Basis Function Interpolation
… We establish the important role a certain semi-inner product plays in the convergence analysis of the original algorithm, and the first proof of convergence is given. This leads to a new technique using line searches described later. Then it is shown that the original algorithm is equivalent to …
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Identities between Hecke Eigenforms
… Maeda's conjecture, we prove that the Petersson inner product $\langle f^2,g\rangle$ is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for $SL_2(\mathhbb{Z})$ of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp …
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Incremental proper orthogonal decomposition for PDE simulation data: Algorithms and analysis
… SVD algorithm with respect to a weighted inner product to compute the proper orthogonal decomposition (POD). The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. We also modify the algorithm to …
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Signal approximation using the bilinear transform
… The second application is computing the inner product of two continuous-time signals for a binary detection problem. In this case, we compare the bilinear representation with Nyquist sampling.
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Galerkin Approximations of General Delay Differential Equations with Multiple Discrete or Distributed Delays
… which are orthogonal polynomials under an inner product with a point mass. A main advantage of using such polynomials is that they live in the domain of the underlying linear operator, which arguably simplifies the related numerical treatments. The obtained results generalize a previous …
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