Leibniz on the Empty Term ‘Nothing’

This paper discusses Leibniz’s treatment of the term ‘nihil’ that appears in some logical papers about the notion of Real Addition. First, the paper argues that the term should be understood as an empty (singular) term and that sentences with empty terms can be true (§2). Second, it sketches a positive free logic to describe the logical behaviour of empty terms (§3). After explaining how this approach avoids a contradiction that threatens the introduction of the term ‘nihil’ in the Real Addition calculus (§4), and how this approach should be understood within Leibniz’s philosophy (§5), the paper assesses the prospects of such an approach with regard to two fundamental issues in Leibniz’s thought: the fictional nature of infinitesimals (§6), and the occurrence of the term ‘nothing’ in the proof of the existence of God that we find in the New Essays (§7).

Systematic Literature Review of Challenges in Blockchain Scalability

Blockchain technology is fast becoming the most transformative technology of recent times and has created hype and optimism, gaining much attention from the public and private sectors. It has been widely deployed in decentralized crypto currencies such as Bitcoin and Ethereum. Bitcoin is the success story of a public blockchain application that propelled intense research and development into blockchain technology. However, scalability remains a crucial challenge. Both Bitcoin and Ethereum are encountering low-efficiency issues with low throughput, high transaction latency, and huge energy consumption. The scalability issue in public Blockchains is hindering the provision of optimal solutions to businesses and industries. This paper presents a systematic literature review (SLR) on the public blockchain scalability issue and challenges. The scope of this SLR includes an in-depth investigation into the scalability problem of public blockchain, associated fundamental factors, and state-of-art solutions. This project managed to extract 121 primary papers from major scientific databases such as Scopus, IEEE explores, Science Direct, and Web of Science. The synthesis of these 121 articles revealed that scalability in public blockchain is not a singular term. A variety of factors are allied to it, with transaction throughput being the most discussed factor. In addition, other interdependent vita factors include storages, block size, number of nodes, energy consumption, latency, and cost. Generally, each term is somehow directly or indirectly reliant on the consensus model embraced by the blockchain nodes. It is also noticed that the contemporary available consensus models are not efficient in scalability and thus often fail to provide good QoS (throughput and latency) for practical industrial applications. Our findings exemplify that the Internet of Things (IoT) would be the leading application of blockchain in industries such as energy, finance, resource management, healthcare, education, and agriculture. These applications are, however, yet to achieve much-desired outcomes due to scalability issues. Moreover, Onchain and offchain are the two major categories of scalability solutions. Sagwit, block size expansion, sharding, and consensus mechanisms are examples of onchain solutions. Offchain, on the other hand, is a lighting network.

Discontinuous perturbations of nonhomogeneous strongly-singular Kirchhoff problems

In this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz–Sobolev space. The presence of both strongly-singular and non-continuous terms brings up difficulties in associating a differentiable functional to the problem with finite energy in the whole space $$W_0^{1,\Phi }(\Omega )$$ W 0 1 , Φ ( Ω ) . To overcome this obstacle, we establish an optimal condition for the existence of $$W_0^{1,\Phi }(\Omega )$$ W 0 1 , Φ ( Ω ) -solutions to a strongly-singular problem, which allows us to constrain the energy functional to a subset of $$W_0^{1,\Phi }(\Omega )$$ W 0 1 , Φ ( Ω ) in order to apply techniques of convex analysis and generalized gradient in the sense of Clarke.

A weak solution for a $(p(x),q(x))$-Laplacian elliptic problem with a singular term

Here, we consider the following elliptic problem with variable components: $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , with Dirichlet boundary condition in a bounded domain in $\mathbb{R}^{N}$ R N with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.

Existence and Nonexistence of Positive Solutions for Singular (p, q)-Equations with Superdiffusive Perturbation

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is parametric and exhibits the combined effects of a singular term and of a superdiffusive one. We prove an existence and nonexistence result for positive solutions depending on the value of the parameter $$\lambda \in \overset{\circ }{{\mathbb {R}}}_+=(0,+\infty )$$ λ ∈ R ∘ + = ( 0 , + ∞ ) .

A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

We consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

Positive Solutions for Singular Anisotropic (p, q)-Equations

In this paper, we consider a Dirichlet problem driven by an anisotropic (p, q)-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.