A note on Dedekind zeta values at 1/2

For a number field [Formula: see text], let [Formula: see text] be the Dedekind zeta function associated to [Formula: see text]. In this paper, we study non-vanishing and transcendence of [Formula: see text] as well as its derivative [Formula: see text] at [Formula: see text]. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of [Formula: see text] and non-vanishing of [Formula: see text]-series at [Formula: see text], J. Number Theory 161 (2016) 444–456].

IPGM: Inertial Proximal Gradient Method for Convolutional Dictionary Learning

Inspired by the recent success of the proximal gradient method (PGM) and recent efforts to develop an inertial algorithm, we propose an inertial PGM (IPGM) for convolutional dictionary learning (CDL) by jointly optimizing both an ℓ2-norm data fidelity term and a sparsity term that enforces an ℓ1 penalty. Contrary to other CDL methods, in the proposed approach, the dictionary and needles are updated with an inertial force by the PGM. We obtain a novel derivative formula for the needles and dictionary with respect to the data fidelity term. At the same time, a gradient descent step is designed to add an inertial term. The proximal operation uses the thresholding operation for needles and projects the dictionary to a unit-norm sphere. We prove the convergence property of the proposed IPGM algorithm in a backtracking case. Simulation results show that the proposed IPGM achieves better performance than the PGM and slice-based methods that possess the same structure and are optimized using the alternating-direction method of multipliers (ADMM).

Intersection curve of two parametric surfaces in Euclidean n-space

The aim of this paper is to study the differential geometric properties of the intersection curve of two parametric surfaces in Euclidean n-space. For this aim, we first present the mth order derivative formula of a curve lying on a parametric surface. Then, we obtain curvatures and Frenet vectors of the transversal intersection curve of two parametric surfaces in Euclidean n-space. We also provide computer code produced in MATLAB to simplify determining the coefficients relative to Frenet frame of higher order derivatives of a curve.

A fractional Hadamard formula and applications

We derive a shape derivative formula for the family of principal Dirichlet eigenvalues $$\lambda _s(\Omega )$$ λ s ( Ω ) of the fractional Laplacian $$(-\Delta )^s$$ ( - Δ ) s associated with bounded open sets $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N of class $$C^{1,1}$$ C 1 , 1 . This extends, with a help of a new approach, a result in Dalibard and Gérard-Varet (Calc. Var. 19(4):976–1013, 2013) which was restricted to the case $$s=\frac{1}{2}$$ s = 1 2 . As an application, we consider the maximization problem for $$\lambda _s(\Omega )$$ λ s ( Ω ) among annular-shaped domains of fixed volume of the type $$B\setminus \overline{B}'$$ B \ B ¯ ′ , where B is a fixed ball and $$B'$$ B ′ is ball whose position is varied within B. We prove that $$\lambda _s(B\setminus \overline{B}')$$ λ s ( B \ B ¯ ′ ) is maximal when the two balls are concentric. Our approach also allows to derive similar results for the fractional torsional rigidity. More generally, we will characterize one-sided shape derivatives for best constants of a family of subcritical fractional Sobolev embeddings.

Novel Criteria of Stability for Delayed Memristive Quaternionic Neural Networks: Directly Quaternionic Method

In this paper, we fixate on the stability of varying-time delayed memristive quaternionic neural networks (MQNNs). With the help of the closure of the convex hull of a set the theory of differential inclusion, MQNN are transformed into variable coefficient continuous quaternionic neural networks (QNNs). The existence and uniqueness of the equilibrium solution (ES) for MQNN are concluded by exploiting the fixed-point theorem. Then a derivative formula of the quaternionic function’s norm is received. By utilizing the formula, the M-matrix theory, and the inequality techniques, some algebraic standards are gained to affirm the global exponential stability (GES) of the ES for the MQNN. Notably, compared to the existing work on QNN, our direct quaternionic method operates QNN as a whole and markedly reduces computing complexity and the gained results are more apt to be verified. The two numerical simulation instances are provided to evidence the merits of the theoretical results.

ON THE GENERATING FUNCTION FOR BERNSTEIN POLYNOMIALS OF TRIPLE SEQUENCES

The aim of this paper is to give main properties of the generating function of the Bernstein polynomials of triple sequence spaces. It was proved the recurrence relations and derivative formula for Bernstein polynomials of triple sequences. Further more, some new results are obtained by using this generating function of these polynomials.

Further Results on the p , k − Analogue of Hypergeometric Functions Associated with Fractional Calculus Operators

In a previous article, first and last researchers introduced an extension of the hypergeometric functions which is called “ p , k -extended hypergeometric functions.” Motivated by this work, here, we derive several novel properties for these functions, including integral representations, derivative formula, k-Beta transform, Laplace and inverse Laplace transforms, and operators of fractional calculus. Relevant connections of some of the discussed results here with those presented in earlier references are outlined.

A new generalization of Mittag-Leffler function via q-calculus

The present paper deals with a new different generalization of the Mittag-Leffler function through q-calculus. We then investigate its remarkable properties like convergence, recurrence relation, integral representation, q-derivative formula, q-Laplace transformation, and image formula under q-derivative operator. In addition to this, we consider some specific cases to give the utilization of our main results.