Characterizations of weighted dynamic Hardy-type inequalities with higher-order derivatives

In this paper, we establish some necessary and sufficient conditions for the validity of a generalized dynamic Hardy-type inequality with higher-order derivatives with two different weighted functions on time scales. The corresponding continuous and discrete cases are captured when $\mathbb{T=R}$ T = R and $\mathbb{T=N}$ T = N , respectively. Finally, some applications to our main result are added to conclude some continuous results known in the literature and some other discrete results which are essentially new.

Toeplitz-Superposition Operators on Analytic Bloch Spaces

The important purpose of this current work is to study a new class of operators, the so-called Toeplitz-superposition operators as an expansion of the weighted known composition operators, induced by such continuous entire functions mapping on bounded specific sets. Minutely, we have deeply discussed the conditions for boundedness of this new type of operators between certain types of some holomorphic Bloch classes with some specific values of the weighted functions.

New characterizations of weights on dynamic inequalities involving a Hardy operator

In this paper, we establish some new characterizations of weighted functions of dynamic inequalities containing a Hardy operator on time scales. These inequalities contain the characterization of Ariňo and Muckenhoupt when $\mathbb{T}=\mathbb{R}$ T = R , whereas they contain the characterizations of Bennett–Erdmann and Gao when $\mathbb{T}=\mathbb{N}$ T = N .

A GENERALIZED SOLUTION OF THE DIRICHLET PROBLEM FOR A MODEL ANISOTROPIC WEIGHTED EQUATION

The paper deals with the Dirichlet problem for a model nonlinear degenerate anisotropic elliptic second-order equation. Anisotropy and degeneration (with respect to the independent variables) is characterized by the presence of different exponents q1 , q2 and weighted functions |x|^q1 та |x|^q2 in the left side of the equation. The main result of the paper is theorem on the existence of the generalized solution of the Dirichlet problem under consideration.

Creating a Novel Consensus Algorithm for Distributed Computing use Cases

There are many consensus algorithms that exist in parallel computing that involve multiple computing units like virtual machines which make use of available resources and arrive at a single agreeable state for the combined system. This is done on the basis of voting which itself branches into several arrangements like voting, functions of central tendencies, weighted functions of central tendencies etc. Some applications that consensus algorithms try to cover are: deciding on transaction operations (read, write, commit); deciding on node leaders of a system; maintaining replicas in the state of a machine (also called a state machine) and creating consistency between them. Some common algorithms of this type are Proof of Work algorithm (PoW), the practical Byzantine fault tolerance algorithm (PBFT), the proof-of-stake algorithm (PoS) and the delegated proof-of-stake algorithm (DPoS), Paxos algorithm and the Raft consensus algorithm.

On Fibonomial sums identities with special sign functions: analytically q-calculus approach

Recently Marques and Trojovsky [On some new identities for the Fibonomial coefficients, Math. Slovaca 64 (2014), 809–818] presented interesting two sum identities including the Fibonomial coefficients and Fibonacci numbers. These sums are unusual as they include a rare sign function and their upper bounds are odd. In this paper, we give generalizations of these sums including the Gaussian q-binomial coefficients. We also derive analogue q-binomial sums whose upper bounds are even. Finally we give q-binomial sums formulæ whose weighted functions are different from the earlier ones. To prove the claimed results, we analytically use q-calculus.