On self-similar solutions of time and space fractional sub-diffusion equations

In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdelyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problem for ordinary differential equation. An essential role is played by certain properties of Erd\'elyi-Kober integral and differential operators. We have applied also successive iteration method to obtain self-similar solutions in an explicit form. The obtained self-similar solutions are represented by generalized Wright type function. We have to note that the usage of imposed conditions is important to present self-similar solutions via given data.

Spire: A Cooperative, Phase-Symmetric Solution to Distributed Consensus

All existing solutions to distributed consensus are organised around a Paxos-like structure wherein processes contend for exclusive leadership in one phase, and then either use their dominant position to propose a value in the next phase or elect an alternate leader. This approach may be characterised as adversarial and phase-asymmetric, requiring distinct message schemas and process behaviours for each phase. In over three decades of research, no algorithm has diverged from this basic model, alluding to it perhaps being the only viable solution to consensus. This paper presents a new consensus algorithm named Spire, characterised by a phase-symmetric, cooperative structure. Processes do not contend for leadership; instead, they collude to iteratively establish a dominant value and may do so concurrently without conflicting. Each successive iteration is structured identically to the previous, employing the same messages and invoking the same behaviour. By these characteristics, Spire buckles the trend in protocol design, proving that at least two disjoint cardinal solutions to consensus exist. The resulting phase symmetry halves the number of distinct messages and behaviours, offering a clear intuition and an approachable foundation for learning consensus and building practical systems.

Spire: A Cooperative, Phase-Symmetric Solution to Distributed Consensus

All existing solutions to distributed consensus are organised around a Paxos-like structure wherein processes contend for exclusive leadership in one phase, and then either use their dominant position to propose a value in the next phase or elect an alternate leader. This approach may be characterised as adversarial and phase-asymmetric, requiring distinct message schemas and process behaviours for each phase. In over three decades of research, no algorithm has diverged from this basic model, alluding to it perhaps being the only viable solution to consensus. This paper presents a new consensus algorithm named Spire, characterised by a phase-symmetric, cooperative structure. Processes do not contend for leadership; instead, they collude to iteratively establish a dominant value and may do so concurrently without conflicting. Each successive iteration is structured identically to the previous, employing the same messages and invoking the same behaviour. By these characteristics, Spire buckles the trend in protocol design, proving that at least two disjoint cardinal solutions to consensus exist. The resulting phase symmetry halves the number of distinct messages and behaviours, offering a clear intuition and an approachable foundation for learning consensus and building practical systems.

Spire: A Cooperative, Phase-Symmetric Solution to Distributed Consensus

All existing solutions to distributed consensus are organised around a Paxos-like structure wherein processes contend for exclusive leadership in one phase, and then either use their dominant position to propose a value in the next phase or elect an alternate leader. This approach may be characterised as adversarial and phase-asymmetric, requiring distinct message schemas and process behaviours for each phase. In over three decades of research, no algorithm has diverged from this basic model, alluding to it perhaps being the only viable solution to consensus. This paper presents a new consensus algorithm named Spire, characterised by a phase-symmetric, cooperative structure. Processes do not contend for leadership; instead, they collude to iteratively establish a dominant value and may do so concurrently without conflicting. Each successive iteration is structured identically to the previous, employing the same messages and invoking the same behaviour. By these characteristics, Spire buckles the trend in protocol design, proving that at least two disjoint cardinal solutions to consensus exist. The resulting phase symmetry halves the number of distinct messages and behaviours, offering a clear intuition and an approachable foundation for learning consensus and building practical systems.

Spire: A Cooperative, Phase-Symmetric Solution to Distributed Consensus

All existing solutions to distributed consensus are organised around a Paxos-like structure wherein processes contend for exclusive leadership in one phase, and then either use their dominant position to propose a value in the next phase or elect an alternate leader. This approach may be characterised as adversarial and phase-asymmetric, requiring distinct message schemas and process behaviours for each phase. In over three decades of research, no algorithm has diverged from this basic model, alluding to it perhaps being the only viable solution to consensus. This paper presents a new consensus algorithm named Spire, characterised by a phase-symmetric, cooperative structure. Processes do not contend for leadership; instead, they collude to iteratively establish a dominant value and may do so concurrently without conflicting. Each successive iteration is structured identically to the previous, employing the same messages and invoking the same behaviour. By these characteristics, Spire buckles the trend in protocol design, proving that at least two disjoint cardinal solutions to consensus exist. The resulting phase symmetry halves the number of distinct messages and behaviours, offering a clear intuition and an approachable foundation for learning consensus and building practical systems.

Existence and Uniqueness of Uncertain Fractional Backward Difference Equations of Riemann–Liouville Type

In this article, we consider the analytic solutions of the uncertain fractional backward difference equations in the sense of Riemann–Liouville fractional operators which are solved by using the Picard successive iteration method. Also, we consider the existence and uniqueness theorem of the solution to an uncertain fractional backward difference equation via the Banach contraction fixed-point theorem under the conditions of Lipschitz constant and linear combination growth. Finally, we point out some examples to confirm the validity of the existence and uniqueness of the solution.

Approximations of Fixed Points in the Hadamard Metric Space CATp(0)

In this paper, we consider the recently introduced C A T p ( 0 ) , where the comparison triangles belong to ℓ p , for p ≥ 2 . We first establish an inequality in these nonlinear metric spaces. Then, we use it to prove the existence of fixed points of asymptotically nonexpansive mappings defined in C A T p ( 0 ) . Moreover, we discuss the behavior of the successive iteration introduced by Schu for these mappings in Banach spaces. In particular, we prove that the successive iteration generates an approximate fixed point sequence.