Wavelets as activation functions in Neural Networks

Traditionally, a few activation functions have been considered in neural networks, including bounded functions such as threshold, sigmoidal and hyperbolic-tangent, as well as unbounded ReLU, GELU, and Soft-plus, among other functions for deep learning, but the search for new activation functions still being an open research area. In this paper, wavelets are reconsidered as activation functions in neural networks and the performance of Gaussian family wavelets (first, second and third derivatives) are studied together with other functions available in Keras-Tensorflow. Experimental results show how the combination of these activation functions can improve the performance and supports the idea of extending the list of activation functions to wavelets which can be available in high performance platforms.

Engineering spectral properties of non-interacting lattice Hamiltonians

We investigate the spectral properties of one-dimensional lattices with position-dependent hopping amplitudes and on-site potentials that are smooth bounded functions of the position. We find an exact integral form for the density of states (DOS) in the limit of an infinite number of sites, which we derive using a mixed Bloch-Wannier basis consisting of piecewise Wannier functions. Next, we provide an exact solution for the inverse problem of constructing the position-dependence of hopping in a lattice model yielding a given DOS. We confirm analytic results by comparing them to numerics obtained by exact diagonalization for various incarnations of position-dependent hoppings and on-site potentials. Finally, we generalize the DOS integral form to multi-orbital tight-binding models with longer-range hoppings and in higher dimensions.

For which functions are 𝑓(𝑋_{𝑡})-𝔼𝕗(𝕏_{𝕥}) and 𝕘(𝕏_{𝕥})/𝔼𝕘(𝕏_{𝕥}) martingales?

Let X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} be a one-dimensional Lévy process such that each X t X_t has a C b 1 C^1_b -density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions f : R → R f\colon \mathbb {R}\to \mathbb {R} , and exponentially bounded functions g : R → ( 0 , ∞ ) g\colon \mathbb {R}\to (0,\infty ) , such that f ( X t ) − E f ( X t ) f(X_t)-\mathbb {E} f(X_t) , resp. g ( X t ) / E g ( X t ) g(X_t)/\mathbb {E} g(X_t) , are martingales.

Noise Management in Homomorphic Encryption Scheme: Analyzing of Algorithm

Homomorphic encryption is the Swiss army of cryptography. It allows to perform computations on encrypted data. this conjecture of processing on encrypted data was stated by [RAD78]. Until Gentry breakthrough in 2009, only partial solutions were offered. They deal with encrypted data with bounded functions in operations.Gentry raised the bound of number of operations by introducing a new method called the bootstrapping. This method allows to reduce the noise in the ciphertext and to perform more computing on it. Since Gentry's breakthrough, several improvements and several alternatives to the bootstrapping technique have been proposed to improve execution time and reduce resource consumption. This article studies the growth of noise and the noise management strategy in homomorphic encryption. It also presents the trend of hoping strategy from 2009 to 2016. Through the DGHV, it shows the management of noise on a one-bit encrypted message.

On the existence and uniqueness of the solution of the coefficient inverse problem for a semilinear parabolic equation

In this paper, we consider the problem of determining the source function and the coefficient by the derivative with respect to time in a semilinear parabolic equation with overdetermination conditions defined on two different hyperplanes. The existence and uniqueness theorems of the classical solution of the posed coefficient inverse problem in the class of smooth bounded functions were proved. An example of input data satisfying the conditions of the proved theorems is given.

Products of Toeplitz Operators on the 2-Analytic Bergman Space

Let f and g be bounded functions, and let T f and T g be Toeplitz operators on A 2 2 D . We show that if the product T f T g equals zero and one of f and g is a radial function satisfying a Mellin transform condition, then the other function must be zero.

Optimal minimax rates against non-smooth alternatives

This study investigates optimal minimax rates for specification testing when the alternative hypothesis is built on a set of non-smooth functions. The set consists of bounded functions that are not necessarily differentiable with no smoothness constraints imposed on their derivatives. In the instrumental variable regression set up with an unknown error variance structure, we find that the optimal minimax rate is n−1/4, where n is the sample size. The rate is achieved by a simple test based on the difference between non-parametric and parametric variance estimators. Simulation studies illustrate that the test has reasonable power against various non-smooth alternatives. The empirical application to Engel curves specification emphasizes the good applicability of the test.

On the conditional plurisubharmonic envelopes of bounded functions

In this paper, we extend some recent results of Guedj et al. [V. Guedj, C. H. Lu and A. Zeriahi, Plurisubharmonic envelopes and supersolutions. J. Differ. Geom. 113(2) (2019) 273–313] about psh envelopes of bounded functions on bounded domains in [Formula: see text]. We also present a result on the regularity of psh envelopes.

Generalization of Quantum Ostrowski-Type Integral Inequalities

In this paper, we prove some new Ostrowski-type integral inequalities for q-differentiable bounded functions. It is also shown that the results presented in this paper are a generalization of know results in the literarure. Applications to special means are also discussed.