Eigenvalues of the Fokker—Planck and BGK operators for a double-well potential

Eigenvalues of the Fokker—Planck and BGK operators for a d₂x²/2 + d₄x⁴/ double-well potential are calculated by the matrix continued-fraction method. A dependence of the eigenvalues on the friction constant or coupling strength is shown for the lowest non-zero real eigenvalue and for some higher, generally complex eigenvalues.     

Modified growth theory for high supersaturation

The classical BCF (Burton, Cabrera, Frank) growth theory assumes atomic step motion driven by the diffusion of adatoms toward the step acting as sinks. Stationary solutions of the diffusion equation have been obtained by BCF neglecting the movement of the coordinate system. By using a moving frame, as already done for a single-step flow model by Mullins and Hirth, we calculate the velocity of a periodic parallel sequence of steps, the distribution of adatoms, and the condensation coefficient as a function of the step distance for various supersaturations. This modification result in significant deviations from the original theory if a dimensionless normalized supersaturation parameterb which is proportional to the supersaturation exceeds one-half. Large values of this parameterb may occur for the high supersaturation found in Si-MBE (molecular beam epitaxy) for temperatures far below the melting point.     

Solutions of the Fokker-Planck equation for a double-well potential in terms of matrix continued fractions

Solutions of the Fokker-Planck (Kramers) equation in position-velocity space for the double-well potentiald ₂x²/2+d₄x⁴/4 in terms of matrix continued fractions are derived. It is shown that the method is also applicable to a Boltzmann equation with a BGK collision operator. Results of eigenvalues and of the Fourier transform of correlation functions are presented explicitly. The lowest nonzero eigenvalue is compared with the escape rate in the weak noise limit for various damping constants and the susceptibility is compared with the zero-friction-limit result.     

Eigenvalues and eigenfunctions of the Fokker-Planck equation for the extremely underdamped Brownian motion in a double-well potential

The eigenvalues and eigenfunctions of the Fokker-Planck equation describing the extremely underdamped Brownian motion in a symmetric double-well potential are investigated. By transforming the Fokker-Planck equation to energy and position coordinates and by performing a suitable averaging over the position coordinate, a differential equation depending only on energy is derived. For finite temperatures this equation is solved by numerical integration, whereas in the weak-noise limit an analytic result for the lowest nonzero eigenvalue is obtained. Furthermore, by using a boundary-layer theory near the critical trajectory, the correction term to the zero-friction-limit result is found.     

On dual molecules and convolution-dominated operators

We show that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the size profile of a function is fully reflected by the size profile of its sampled values. The main tool is a local holomorphic calculus for convolution-dominated operators, valid for groups with possibly non-polynomial growth. Applied to the matrix coefficients of a group representation, our methods improve on classical results on atomic decompositions and bridge a gap between abstract and concrete methods.     

Function-oriented synthesis: studies aimed at the synthesis and mode of action of 1alpha-alkyldaphnane analogues

[reaction: see text] An efficient synthetic route to the ABC tricyclic core of 1alpha-alkyldaphnanes has been developed. The conformational bias imparted by the C6-C9 oxo-bridge of BC-ring system 12 was used to elaborate the ABC-ring system precursor including the introduction of the beta-C5 hydroxyl group. A completely diastereoselective palladium-catalyzed enyne cyclization was then employed to establish the A-ring with a C1 appendage.     

Topological Properties of the Set of Functions Generated by Neural Networks of Fixed Size

We analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size. Surprisingly, this set has many undesirable properties. It is highly non-convex, except possibly for a few exotic activation functions. Moreover, the set is not closed with respect to \(L^p\)-norms, \(0< p < \infty \), for all practically used activation functions, and also not closed with respect to the \(L^\infty \)-norm for all practically used activation functions except for the ReLU and the parametric ReLU. Finally, the function that maps a family of weights to the function computed by the associated network is not inverse stable for every practically used activation function. In other words, if \(f_1, f_2\) are two functions realized by neural networks and if \(f_1, f_2\) are close in the sense that \(\Vert f_1 - f_2\Vert _{L^\infty } \le \varepsilon \) for \(\varepsilon > 0\), it is, regardless of the size of \(\varepsilon \), usually not possible to find weights \(w_1, w_2\) close together such that each \(f_i\) is realized by a neural network with weights \(w_i\). Overall, our findings identify potential causes for issues in the training procedure of deep learning such as no guaranteed convergence, explosion of parameters, and slow convergence.

     

A General Version of Price’s Theorem

Journal of Theoretical Probability - Assume that $$X_{\Sigma } \in \mathbb {R}^{n}$$ is a centered random vector following a multivariate normal distribution with positive definite covariance...