Use of a childhood toy, the yo-yo, to present a mechanical proof of the namesake theorem of Pythagoras

2020 ◽  
Vol 9 (1) ◽  
pp. 21-28
Author(s):  
Amitava Biswas ◽  
Abhishek Bisaria
Keyword(s):  
Mechanical Proof

scholarly journals Mechanical Proof of the Maxwell-Boltzmann Speed Distribution With Analytical Integration

2021 ◽  
Vol 10 (3) ◽  
pp. 135
Author(s):  
Hejie Lin ◽  
Tsung-Wu Lin
Keyword(s):  
Integral Form ◽  
Point Of View ◽  
Speed Ratio ◽  
Mass Ratio ◽  
Exact Integration ◽  
Speed Distribution ◽  
Ideal Gases ◽  
Mechanical Proof ◽  
Theoretical Limitation ◽  
New Distribution

The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. The Maxwell-Boltzmann speed distribution of mixed particles is based on kinetic theory; however, it has never been derived from a mechanical point of view. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles based on probability analysis and Newton’s law of motion. This paper requires the probability density function (PDF) ψ^ab(u_a; v_a, v_b) of the speed u_a  of the particle with mass M_a  after the collision of two particles with mass M_a  in speed v_a  and mass M_b  in speed v_b . The PDF ψ^ab(u_a; v_a, v_b)  in integral form has been obtained before. This paper further performs the exact integration from the integral form to obtain the PDF ψ^ab(u_a; v_a, v_b)  in an evaluated form, which is used in the following equation to get new distribution P_new^a(u_a)  from old distributions P_old^a(v_a) and P_old^b(v_b). When P_old^a(v_a) and P_old^b(v_b)  are the Maxwell-Boltzmann speed distributions, the integration P_new^a(u_a)  obtained analytically is exactly the Maxwell-Boltzmann speed distribution. P_new^a(u_a)=∫_0^∞ ∫_0^∞ ψ^ab(u_a;v_a,v_b) P_old^a(v_a) P_old^b(v_b) dv_a dv_b,    a,b = 1 or 2 The mechanical proof of the Maxwell-Boltzmann speed distribution presented in this paper reveals the unsolved mechanical mystery of the Maxwell-Boltzmann speed distribution since it was proposed by Maxwell in 1860. Also, since the validation is carried out in an analytical approach, it proves that there is no theoretical limitation of mass ratio to the Maxwell-Boltzmann speed distribution. This provides a foundation and methodology for analyzing the interaction between particles with an extreme mass ratio, such as gases and neutrinos.

scholarly journals A Formalism for Primitive Logic and Mechanical Proof-Checking

1966 ◽  
Vol 26 ◽  
pp. 195-203 ◽  
Author(s):  
Katuzi Ono
Keyword(s):  
Simple Form ◽  
Predicate Logic ◽  
View Point ◽  
Proof Checking ◽  
Mechanical Proof ◽  
Classical Predicate

The universal character of the primitive logic LO in the sense that popular logics such as the lower classical predicate logic LK, the intuitionistic predicate logic LJ, Johansson’s minimal predicate logic LM, etc. can be faithfully interpreted in LO is very remarkable even from the view point of mechanical proof-checking. Since LO is very simple, deductions in LO could be mechanized in a simple form if a suitable formalism for LO is found out. Main purpose of this paper is to introduce a practical formalism for LO, practical in the sense that it is suitable at least for mechanical proof-checking business.

scholarly journals Note relating to M. Foucault’s new mechanical proof of the rotation of the Earth

1854 ◽  
Vol 6 ◽  
pp. 65-68
Keyword(s):  
The Earth ◽  
Rotation Of The Earth ◽  
Mechanical Proof

The experiment which led M. Foucault to his ingenious and interesting researches relating to the rotation of the earth, is stated by him thus:—"Having fixed on the arbor of a lathe and in the direction of the axis, a round and flexible steel rod, it was put in vibration by deflecting it from its position of equilibrium and leaving it to itself. A plane of oscillation is thus determined, which, from the persistence of the visual impressions, is clearly delineated in space; now it was remarked that, on turning by the hand the arbor which serves as a support to this vibrating rod, the plane of oscillation is not carried with it."

III. On Fessel's gyroscope

1856 ◽  
Vol 7 ◽  
pp. 43-48
Keyword(s):  
Royal Society ◽  
Rotary Motion ◽  
The Earth ◽  
Rotation Of The Earth ◽  
Mechanical Proof ◽  
Present Opportunity

Since the announcement of M. Foucault’s beautiful experiment which has afforded us a new mechanical proof of the rotation of the earth on its axis, the phenomena of rotary motion have received renewed attention, and many ingenious instruments have been contrived to exhibit and to explain them. One of the most instructive of these is the Gyroscope invented by M. Fessel of Cologne, described in its earlier form in Poggendorff’s Annalen for September 1853, and which, with some improvements by Prof. Plücker and some further modifications suggested by myself, I take the present opportunity of bringing before the Royal Society.

scholarly journals The Design and Verification of Dynamic-sized Nonblocking Data Structures

2021 ◽  
Author(s):  
◽  
Simon Doherty
Keyword(s):  
Shared Memory ◽  
Data Structures ◽  
Limited Range ◽  
Modern Computer ◽  
Multiple Processes ◽  
Efficient Data ◽  
Collective Experience ◽  
Simulation Relations ◽  
Mechanical Proof ◽  
Efficient Data Structures

<p>Modern computer systems often involve multiple processes or threads of control that communicate through shared memory. However, the implementation of correct and efficient data structures that can be shared by several processes is frequently challenging. This thesis is concerned with the design and verification of a class of shared memory algorithms known as nonblocking algorithms, which are implementations of shared data structures that provide strong progress guarantees. Nonblocking algorithms offer an appealing alternative to traditional techniques for the implementation of shared memory data structures, but they are difficult to design, and extant algorithms can often be applied in only a limited range of systems. Furthermore, because of their subtlety, it is notoriously difficult to determine whether a given nonblocking algorithm is correct. This thesis addresses these difficulties in two ways. First, we present techniques for the verification of nonblocking algorithms that dynamically allocate memory. These techniques allow the construction of formal and complete proofs of correctness, so that each proof may be checked by a mechanical proof assistant. Applying techniques first developed for the verification of distributed algorithms, we use labelled-transition systems to model algorithms and their specifications, and simulation relations to prove that an implementation meets its specification. Nonblocking algorithms often require a particular notion of simulation, called backward simulation, that is rarely necessary in other contexts. This thesis contributes to the relatively limited collective experience in the use of backward simulation. The second set of contributions addresses the limitations of many extant nonblocking algorithms. While many nonblocking algorithms allocate memory dynamically, it is difficult to determine in a nonblocking context when it is safe to free memory. We present techniques to accomplish this. Furthermore, many nonblocking algorithms depend on the availability of two powerful synchronisation primitives, known as load-linked and store-conditional, which are not normally provided by hardware. We present implementations of these primitives that work on commonly available platforms.</p>

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