Mathematical Proof and Its Discontents

The chapter appraises science as an intellectual activity that is appropriately carried out on its own terms. Consequently, it is not appropriate to introduce references to God as a component part of a mathematical proof, nor of a system of forces in the natural world, nor of a sequence of impersonal processes in the biosphere. This does not mean that it is inappropriate to be thankful to God and to celebrate all these aspects of the world as gifts. They can be employed as opportunities to express appreciation through studying and understanding them better in their own right. Nevertheless, there may be processes, such as those which shape a person’s self-identity, in which it is appropriate to recognize God’s more direct role. Good practice concerning acknowledgements sections in scientific publications such as doctoral theses and journal articles is then discussed.

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Nonlinear stability of two-layer shallow water flows with a free surface

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2019.0594 ◽

2020 ◽

Vol 476 (2236) ◽

pp. 20190594

Author(s):

Francisco de Melo Viríssimo ◽

Paul A. Milewski

Keyword(s):

Free Surface ◽

Initial Data ◽

Nonlinear Stability ◽

Mathematical Proof ◽

Passive Layer ◽

Physical Parameters ◽

Immiscible Fluid ◽

Dimensional System ◽

The Cauchy Problem ◽

The Difference

The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two ‘baroclinic’ parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability.

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A Relationship Between the Factor Indivisibility and the Output Elasticity of the Indivisible Factor

Studies in Microeconomics ◽

10.1177/2321022220985160 ◽

2021 ◽

pp. 232102222098516

Author(s):

Dipankar Das

Keyword(s):

Production Function ◽

Limit Point ◽

Mathematical Proof ◽

Wage Equation ◽

Marginal Value ◽

Output Elasticity ◽

New Type ◽

First Time ◽

Jel Classifications

The paper puts forth a notion and derives a special type of production function where labour is an indivisible factor and is in the integer space. Thus, Newtonian calculus is not an appropriate method of deriving the marginal value because limit point does not exist. This shows that indivisibility determines the output elasticity. In the first part, the paper propounds a notion regarding how indivisibility determines curvature of the production function. In the second part, the paper incorporates the findings within a production function and derives a new type accordingly. Moreover, it formally derives the standard wage equation considering all the entitlements of labour, namely (a) normal wages, (b) interest and (c) rent of ability. So far, no such mathematical proof is there to support this wage composition. This paper, for the first time, derives this wage equation considering indivisibility of labour. JEL Classifications: J23, J24, J31, D24, C61, E24, L8

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Mathematical Proof of the Synthetic Running Correlation Coefficient and Its Ability to Reflect Temporal Variations in Correlation

Journal of Ocean University of China ◽

10.1007/s11802-021-4826-9 ◽

2021 ◽

Vol 20 (3) ◽

pp. 562-572

Author(s):

Jinping Zhao ◽

Yong Cao ◽

Yanyue Shi ◽

Xin Wang

Keyword(s):

Correlation Coefficient ◽

Mathematical Proof ◽

Temporal Variations

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The Vogt-Russell Theorem, and New Results on an Old Problem

Symposium - International Astronomical Union ◽

10.1017/s007418090014570x ◽

1978 ◽

Vol 80 ◽

pp. 303-311

Author(s):

Helmuth Kähler

Keyword(s):

Existence And Uniqueness ◽

Thermal Equilibrium ◽

Equilibrium Configuration ◽

Mathematical Proof ◽

Stellar Structure

Half a centry ago Henry Norris Russell and Heinrich Vogt independently made a conjecture concerning the structure of spherical stars which are in hydrostatic and thermal equilibrium (Russell, 1927; Vogt, 1926). This conjecture has later come to be known as the Vogt-Russell theorem and is usually formulated as follows: The structure of a star is uniquely determined by the mass and the composition. In other words, the statement claims the existence and uniqueness of a stellar equilibrium configuration for given parameters mass and composition, and you may find what is called a mathematical proof in many textbooks on stellar structure.

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Block Design-Based Asynchronous Neighbor Discovery Protocol for Wireless Sensor Networks

Journal of Sensors ◽

10.1155/2015/951652 ◽

2015 ◽

Vol 2015 ◽

pp. 1-12 ◽

Cited By ~ 5

Author(s):

Sangil Choi ◽

Wooksik Lee ◽

Teukseob Song ◽

Jong-Hoon Youn

Keyword(s):

Wireless Sensor Networks ◽

Sensor Networks ◽

Simulation Study ◽

Mathematical Proof ◽

Block Design ◽

Wireless Sensor ◽

Neighbor Discovery ◽

Block Designs ◽

Worst Case ◽

Neighbor Discovery Protocol

Neighbor discovery is a significant research topic in wireless sensor networks. After wireless sensor devices are deployed in specific areas, they attempt to determine neighbors within their communication range. This paper proposes a new Block design-based Asynchronous Neighbor Discovery protocol for sensor networks calledBAND. We borrow the concept of combinatorial block designs for neighbor discovery. First, we summarize a practical challenge and difficulty of using the original block designs. To address this challenge, we create a new block generation technique for neighbor discovery schedules and provide a mathematical proof of the proposed concept. A key aspect of the proposed protocol is that it combines two block designs in order to construct a new block for neighbor discovery. We analyze the worst-case neighbor discovery latency numerically between our protocol and some well-known protocols in the literature. Our protocol reveals that the worst-case latency is much lower than others. Finally, we evaluate the performance ofBANDand existing representative protocols through the simulation study. The results of our simulation study show that the average and maximum latency ofBANDis about 40% lower than that of existing protocols. Furthermore,BANDspends approximately 30% less energy than others during the neighbor discovery process.

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