154,937 results
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2. On the Controversy over the Logical Correctness of Einstein’s First Paper on Mass-Energy Equivalence
- Author
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Lixian Yan, Patrick Moylan, and Michael Gironda
- Subjects
symbols.namesake ,Correctness ,Circular reasoning ,General Engineering ,symbols ,Criticism ,Mass–energy equivalence ,Energy–momentum relation ,Logic error ,Einstein ,Mathematical economics ,Mathematics ,Simple (philosophy) - Abstract
It is well-known that Einstein’s first attempt to explain E = mc2 which was published in Annalen der Physik in 1905, has been criticized as problematic. In particular, it has been shown by Ives and reiterated by Jammer that it suffers from the error of circular reasoning. Attempts have been made in the scientific literature to discount the circular reasoning objection of Ives, Jammer, Arzelies and others. Fritz Rohrlich in 1990 gave a remarkably simple and concise derivation of E = mc2 along lines similar to Einstein’s but based on both momentum and energy conservation, in contrast to Einstein’s which uses only energy considerations. Rohrlich’s approach using momentum conservation is an alternative to Einstein’s, which is free from objection in logical error, and we make it quite clear on the importance of the implicit assumption of momentum conservation in any attempt to refute the circular reasoning error in Einstein’s paper. It is our contention that this point is overlooked or altogether avoided by those who have attempted to uproot the circular reasoning criticism of Einstein’s paper.
- Published
- 2021
3. Remarks on a recent paper titled: 'On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces'
- Author
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Charles E. Chidume
- Subjects
Pure mathematics ,Smoothness (probability theory) ,Applied Mathematics ,lcsh:Mathematics ,Banach space ,Hilbert space ,Regular polygon ,010103 numerical & computational mathematics ,Fixed point ,lcsh:QA1-939 ,01 natural sciences ,Opial property ,010101 applied mathematics ,symbols.namesake ,Accretive ,Uniformly smooth ,Common fixed point ,symbols ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition $0 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is $E=l_{p}$ E = l p , $2\leq p 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition $0 0 < κ ≤ 1 2 , then E is necessarily $l_{2}$ l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.
- Published
- 2021
4. Remarks on a paper by El-Guindy and Papanikolas
- Author
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Takehiro Hasegawa
- Subjects
Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Continuation ,symbols.namesake ,Number theory ,010201 computation theory & mathematics ,Fourier analysis ,symbols ,0101 mathematics ,Abelian group ,Mathematics - Abstract
There are similarities between Drinfeld modules and abelian varieties. The purpose of this paper is to investigate these similarities in terms of supersingularity. More specifically, we provide explicit formulas of Hasse invariants (or, equivalently, of supersingular polynomials) for Drinfeld modules, which is a continuation of the 2013 paper by Ahmad El-Guindy and Matthew A. Papanikolas. We present several supersingular Drinfeld modules as an application.
- Published
- 2020
5. Properties of a novel stochastic rock–paper–scissors dynamics
- Author
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Hailing Wang, Zuxiong Li, Zhusong Chu, and Jun Cheng
- Subjects
Lyapunov function ,education.field_of_study ,Stochastic modelling ,Applied Mathematics ,Dynamics (mechanics) ,Population ,01 natural sciences ,010305 fluids & plasmas ,Computational Mathematics ,symbols.namesake ,Bounded function ,0103 physical sciences ,Theory of computation ,symbols ,Applied mathematics ,010306 general physics ,education ,Mathematics - Abstract
This paper is concerned with some stochastic properties of a novel rock–paper–scissors model. Firstly, the global existence of an unique positive solution of the stochastic model is obtained. Then we demonstrate the positive solution of the model is stochastically bounded. Besides, some sufficient conditions for population to be stochastically permanent and extinct are derived with the use of some appropriate Lyapunov functions. At last, some numerical simulations are carried out to illustrate our theoretical analysis results.
- Published
- 2020
6. Folding a Paper Strip to Minimize Thickness
- Author
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Hiro Ito, Erik D. Demaine, Anna Lubiw, David Eppstein, Ryuhei Uehara, Yushi Uno, Adam Hesterberg, Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science, and Massachusetts Institute of Technology. Department of Mathematics
- Subjects
Computational Geometry (cs.CG) ,FOS: Computer and information sciences ,paper folding ,0102 computer and information sciences ,02 engineering and technology ,Topology ,01 natural sciences ,Theoretical Computer Science ,NP completeness ,symbols.namesake ,Exponential growth ,crease width ,Computer Science - Data Structures and Algorithms ,0202 electrical engineering, electronic engineering, information engineering ,Discrete Mathematics and Combinatorics ,Data Structures and Algorithms (cs.DS) ,Mathematics ,NP-complete ,computational origami ,Planar graph ,rigid origami ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,symbols ,Computer Science - Computational Geometry ,020201 artificial intelligence & image processing ,linkage ,optimization problem - Abstract
In this paper, we study how to fold a specified origami crease pattern in order to minimize the impact of paper thickness. Specifically, origami designs are often expressed by a mountain-valley pattern (plane graph of creases with relative fold orientations), but in general this specification is consistent with exponentially many possible folded states. We analyze the complexity of finding the best consistent folded state according to twometrics:minimizing the total number of layers in the folded state (so that a “flat folding” is indeed close toflat), andminimizing the total amount of paper required to execute the folding (where “thicker” creases consume more paper). We prove both problems strongly NP-complete even for 1D folding. On the other hand, we prove the first problem fixed-parameter tractable in 1D with respect to the number of layers., National Science Foundation (U.S.). Origami Design for Integration of Self-assembling Systems for Engineering Innovation (grant EFRI- 124038), National Science Foundation (U.S.). Expedition grant (CCF-1138967), United States. Department of Defense. National Defense Science and Engineering Graduate (NDSEG) Fellowship (32 CFR 168a)
- Published
- 2015
7. On the Dispersions of the Gel’fand–Pinsker Channel and Dirty Paper Coding
- Author
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Jonathan Scarlett
- Subjects
Independent and identically distributed random variables ,Gaussian ,Variable-length code ,Data_CODINGANDINFORMATIONTHEORY ,Library and Information Sciences ,Computer Science Applications ,Gel’fand-Pinsker channel ,Combinatorics ,symbols.namesake ,Second-order coding rate ,Shannon–Fano coding ,Dirty paper coding ,symbols ,Channel dispersion ,Channels with state ,Algorithm ,Encoder ,Decoding methods ,Information Systems ,Mathematics ,Coding (social sciences) - Abstract
This paper studies the second-order coding rates for memoryless channels with a state sequence known non-causally at the encoder. In the case of finite alphabets, an achievability result is obtained using constant-composition random coding, and by using a small fraction of the block to transmit the empirical distribution of the state sequence. For error probabilities less than 0.5, it is shown that the second-order rate improves on an existing one based on independent and identically distributed random coding. In the Gaussian case (dirty paper coding) with an almost-sure power constraint, an achievability result is obtained using random coding over the surface of a sphere, and using a small fraction of the block to transmit a quantized description of the state power. It is shown that the second-order asymptotics are identical to the single-user Gaussian channel of the same input power without a state.
- Published
- 2015
8. A remark on a paper of P. B. Djakov and M. S. Ramanujan
- Author
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Murat Yurdakul and Elif Uyanik
- Subjects
Unbounded operator ,Combinatorics ,symbols.namesake ,Monotone polygon ,Basis (linear algebra) ,General Mathematics ,Bounded function ,Operator (physics) ,symbols ,Sequence space ,Continuous linear operator ,Ramanujan's sum ,Mathematics - Abstract
Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\"{o}the matrices when every continuous linear operator between l-K\"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.
- Published
- 2019
9. Annotated Translations of Three of the Euler’s Papers on Celestial Mechanics
- Author
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Sylvio R. Bistafa
- Subjects
General Engineering ,Motion (geometry) ,Lagrangian point ,Three-body problem ,Celestial mechanics ,symbols.namesake ,Orbit ,Classical mechanics ,Physics::Space Physics ,Euler's formula ,symbols ,Astrophysics::Earth and Planetary Astrophysics ,Center of mass ,Mathematics ,Syzygy (astronomy) - Abstract
Annotated translations from Latin of three of the Euler’s papers on celestial mechanics are presented, which fall into the category of three-body problems. The first translation deals with an exact solution of three bodies that move around the common center of mass and always line up. This is considered the first work from which the three collinear Lagrange points could be obtained. The second translation deals with motions of Sun, Earth and Moon in syzygy and Moon libration as well, where, for the first time, Euler introduces an archaic form of a Fourier sine series expansion to describe the Moon’s wagging motion. The last translation relates to a paper that was written with the goal of alleviating astronomical computations of the perturbed motion of the Moon around the Earth by the Sun, ending up with eight coupled differential equations for resolving the perturbed motion of this celestial body. Despite showing great analytical skills, Euler gave no indications on how this system of equations could be solved, which renders his efforts practically useless in the determination of the variations of the nodal line and inclination of the Moon’s orbit.
- Published
- 2019
10. Evolutionary dynamics of rock-paper-scissors game in the patchy network with mutations
- Author
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Tina Verma and Arvind Kumar Gupta
- Subjects
Hopf bifurcation ,education.field_of_study ,General Mathematics ,Applied Mathematics ,Population ,Evolutionary game theory ,Biodiversity ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Metapopulation ,symbols.namesake ,Transcritical bifurcation ,Evolutionary biology ,Mutation (genetic algorithm) ,symbols ,education ,Evolutionary dynamics ,Mathematics - Abstract
Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.
- Published
- 2021
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