The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approach...The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the problem is described as much as possible in the tradition and the spirit—and as much as possible by means of the algebraic conventions—of George Boole’s Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn’s Symbolic Logic (second edition, 1894), which is squarely based on Boole’s Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is analyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productivity, the Monty Hall problem is extended and generalized in parts 3 and 4 to related cases in light of the axioms of probability theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent mathematical equations are developed and presented and illustrated by means of examples.展开更多
In Advances in Pure Mathematics (www.scirp.org/journal/apm), Vol. 1, No. 4 (July 2011), pp. 136-154, the mathematical structure of the much discussed problem of probability known as the Monty Hall problem was mapped i...In Advances in Pure Mathematics (www.scirp.org/journal/apm), Vol. 1, No. 4 (July 2011), pp. 136-154, the mathematical structure of the much discussed problem of probability known as the Monty Hall problem was mapped in detail. It is styled here as Monty Hall 1.0. The proposed analysis was then generalized to related cases involving any number of doors (d), cars (c), and opened doors (o) (Monty Hall 2.0) and 1 specific case involving more than 1 picked door (p) (Monty Hall 3.0). In cognitive terms, this analysis was interpreted in function of the presumed digital nature of rational thought and language. In the present paper, Monty Hall 1.0 and 2.0 are briefly reviewed (§§2-3). Additional generalizations of the problem are then presented in §§4-7. They concern expansions of the problem to the following items: (1) to any number of picked doors, with p denoting the number of doors initially picked and q the number of doors picked when switching doors after doors have been opened to reveal goats (Monty Hall 3.0;see §4);(3) to the precise conditions under which one’s chances increase or decrease in instances of Monty Hall 3.0 (Monty Hall 3.2;see §6);and (4) to any number of switches of doors (s) (Monty Hall 4.0;see §7). The afore-mentioned article in APM, Vol. 1, No. 4 may serve as a useful introduction to the analysis of the higher variations of the Monty Hall problem offered in the present article. The body of the article is by Leo Depuydt. An appendix by Richard D. Gill (see §8) provides additional context by building a bridge to modern probability theory in its conventional notation and by pointing to the benefits of certain interesting and relevant tools of computation now available on the Internet. The cognitive component of the earlier investigation is extended in §9 by reflections on the foundations of mathematics. It will be proposed, in the footsteps of George Boole, that the phenomenon of mathematics needs to be defined in empirical terms as something that happens to the brain or something that the brain does. It is generally assumed that mathematics is a property of nature or reality or whatever one may call it. There is not the slightest intention in this paper to falsify this assumption because it cannot be falsified, just as it cannot be empirically or positively proven. But there is no way that this assumption can be a factual observation. It can be no more than an altogether reasonable, yet fully secondary, inference derived mainly from the fact that mathematics appears to work, even if some may deem the fact of this match to constitute proof. On the deepest empirical level, mathematics can only be directly observed and therefore directly analyzed as an activity of the brain. The study of mathematics therefore becomes an essential part of the study of cognition and human intelligence. The reflections on mathematics as a phenomenon offered in the present article will serve as a prelude to planned articles on how to redefine the foundations of probability as one type of mathematics in cognitive fashion and on how exactly Boole’s theory of probability subsumes, supersedes, and completes classical probability theory. §§2-7 combined, on the one hand, and §9, on the other hand, are both self-sufficient units and can be read independently from one another. The ultimate design of the larger project of which this paper is part remains the increase of digitalization of the analysis of rational thought and language, that is, of (rational, not emotional) human intelligence. To reach out to other disciplines, an effort is made to describe the mathematics more explicitly than is usual.展开更多
We report a theoretic study on the inverse spin-Hall effect （ISHE） in a two-terminal nano-device that consists of a two-dimensional electron gas （2DEG） with Rashba spin-orbit coupling （RSOC） and two ideal leads....We report a theoretic study on the inverse spin-Hall effect （ISHE） in a two-terminal nano-device that consists of a two-dimensional electron gas （2DEG） with Rashba spin-orbit coupling （RSOC） and two ideal leads. Based on a two-site toy model and Keldysh Green＇s function method, we derive an analytic result of ISHE, which shows clearly that a nonzero transverse charge current stems from the combined effect of the RSOC, the spin bias, and its spin polarization direction in spin space. Our further numerical calculations in a larger system other than two-site lattice model demonstrate that the transverse charge current, dependent on the strength of the RSOC, the Fermi energy of the system, as well as the system size, can exhibit oscillating behavior and even reverse its sign due to Rashba spin precession. These properties may be helpful for eficient detection of the spin current （spin bias） by measuring the transverse charge current in a spin-orbital coupling system.展开更多
By adopting a complex formulation of Ohm’s law, we arrive at combined equations connecting the conductivities of conductors. The horizontal resistivity is equal to the inverse of Drude’s conductivity δo( ), and the...By adopting a complex formulation of Ohm’s law, we arrive at combined equations connecting the conductivities of conductors. The horizontal resistivity is equal to the inverse of Drude’s conductivity δo( ), and the vertical resistivity (ρy) is equal to the Hall’s conductivity ( δH). At high magnetic field, the horizontal conductivity becomes exceedingly small, whereas the vertical conductivity equals to Hall’s conductivity. The Hall’s conductivity is shown to represent the maximal conductivity of conductors. Drude’s and Hall’s conductivities are related by δo =δHωC , where ωC is the cyclotron frequency, and is the relaxation time. The quantization of Hall’s conductivity is attributed to the fact that the magnetic flux enclosed by the conductor is carried by electrons each with h/e, where h is the Planck’s constant and e is the electron’s charge. The Drude’s conductance is found to be equal to Hall's conductance provided the magnetic flux enclosed by the conductor is a multiple of h/e.展开更多
Taking into account the non separable solution for the quantum problem of the motion of a charged particle in a flat surface of lengths L<sub>x</sub> and L<sub>y</sub> with transversal static m...Taking into account the non separable solution for the quantum problem of the motion of a charged particle in a flat surface of lengths L<sub>x</sub> and L<sub>y</sub> with transversal static magnetic field B and longitudinal static electric field E, the quantum current, the transverse (Hall) and longitudinal resistivities are calculated for the state n = 0 and j = 0. We found that the transverse resistivity is proportional to an integer number, due to the quantization of the magnetic flux, and longitudinal resistivity can be zero for times t >> L<sub>x</sub>B/cE. In addition, using a modified periodicity of the solution, a modified quantization of the magnetic flux is found which allows to have IQHE and FQHE of any filling factor of the form v = k/l, with k, l ∈Z.展开更多
As the first book of“Tudor series”written by British famous historical novelist Hilary Mantel,Wolf Hall depicted the process of the blacksmith’s son Cromwell becoming a powerful minister.Meanwhile,with her distinct...As the first book of“Tudor series”written by British famous historical novelist Hilary Mantel,Wolf Hall depicted the process of the blacksmith’s son Cromwell becoming a powerful minister.Meanwhile,with her distinctive writing style,Mantel also skillfully presented women’s living predicament as“the other”and created various typical female images.This paper will focus on those images and analyze their living predicaments under male-dominated society.展开更多
In this paper, we make an initial value investigation of the unsteady flow of incompressible viscous fluid between two rigid non-conducting rotating parallel plates bounded by a porous medium under the influence of a ...In this paper, we make an initial value investigation of the unsteady flow of incompressible viscous fluid between two rigid non-conducting rotating parallel plates bounded by a porous medium under the influence of a uniform magnetic field of strength H0 inclined at an angle of inclination α with normal to the boundaries taking hall current into account. The perturbations are created by a constant pressure gradient along the plates in addition to the non-torsional oscillations of the upper plate while the lower plate is at rest. The flow in the porous medium is governed by the Brinkman’s equations. The exact solution of the velocity in the porous medium consists of steady state and transient state. The time required for the transient state to decay is evaluated in detail and the ultimate quasi-steady state solution has been derived analytically. Its behaviour is computationally discussed with reference to the various governing parameters. The shear stresses on the boundaries are also obtained analytically and their behaviour is computationally discussed.展开更多
The Tenant of Wildfell Hall,firstly published in 1848,was the second and final novel of Anne Brontë,the youngest of the Brontësisters,telling a story of the mysterious young widow Helen Huntingdon,who flees ...The Tenant of Wildfell Hall,firstly published in 1848,was the second and final novel of Anne Brontë,the youngest of the Brontësisters,telling a story of the mysterious young widow Helen Huntingdon,who flees her abusive husband and hides with her son at Wildfell Hall.Just as most contemporary critic defines it as one of the first feminist novels,the overall characterization of Helen sends a strong message of the empowerment of women.What confuses modern readers,however,is the subtle confrontation between the empowerment and its subversion.Thus,this paper conducts an in-depth analysis of the text concerning Anne's empow⁃erment of Helen and the corresponding subversion of the established empowerment before unveiling the fundamental cause of such arrangement with a view of shedding light on the comprehension of Anne Brontë's belief in universal social issues.展开更多
文摘The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the problem is described as much as possible in the tradition and the spirit—and as much as possible by means of the algebraic conventions—of George Boole’s Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn’s Symbolic Logic (second edition, 1894), which is squarely based on Boole’s Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is analyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productivity, the Monty Hall problem is extended and generalized in parts 3 and 4 to related cases in light of the axioms of probability theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent mathematical equations are developed and presented and illustrated by means of examples.
文摘In Advances in Pure Mathematics (www.scirp.org/journal/apm), Vol. 1, No. 4 (July 2011), pp. 136-154, the mathematical structure of the much discussed problem of probability known as the Monty Hall problem was mapped in detail. It is styled here as Monty Hall 1.0. The proposed analysis was then generalized to related cases involving any number of doors (d), cars (c), and opened doors (o) (Monty Hall 2.0) and 1 specific case involving more than 1 picked door (p) (Monty Hall 3.0). In cognitive terms, this analysis was interpreted in function of the presumed digital nature of rational thought and language. In the present paper, Monty Hall 1.0 and 2.0 are briefly reviewed (§§2-3). Additional generalizations of the problem are then presented in §§4-7. They concern expansions of the problem to the following items: (1) to any number of picked doors, with p denoting the number of doors initially picked and q the number of doors picked when switching doors after doors have been opened to reveal goats (Monty Hall 3.0;see §4);(3) to the precise conditions under which one’s chances increase or decrease in instances of Monty Hall 3.0 (Monty Hall 3.2;see §6);and (4) to any number of switches of doors (s) (Monty Hall 4.0;see §7). The afore-mentioned article in APM, Vol. 1, No. 4 may serve as a useful introduction to the analysis of the higher variations of the Monty Hall problem offered in the present article. The body of the article is by Leo Depuydt. An appendix by Richard D. Gill (see §8) provides additional context by building a bridge to modern probability theory in its conventional notation and by pointing to the benefits of certain interesting and relevant tools of computation now available on the Internet. The cognitive component of the earlier investigation is extended in §9 by reflections on the foundations of mathematics. It will be proposed, in the footsteps of George Boole, that the phenomenon of mathematics needs to be defined in empirical terms as something that happens to the brain or something that the brain does. It is generally assumed that mathematics is a property of nature or reality or whatever one may call it. There is not the slightest intention in this paper to falsify this assumption because it cannot be falsified, just as it cannot be empirically or positively proven. But there is no way that this assumption can be a factual observation. It can be no more than an altogether reasonable, yet fully secondary, inference derived mainly from the fact that mathematics appears to work, even if some may deem the fact of this match to constitute proof. On the deepest empirical level, mathematics can only be directly observed and therefore directly analyzed as an activity of the brain. The study of mathematics therefore becomes an essential part of the study of cognition and human intelligence. The reflections on mathematics as a phenomenon offered in the present article will serve as a prelude to planned articles on how to redefine the foundations of probability as one type of mathematics in cognitive fashion and on how exactly Boole’s theory of probability subsumes, supersedes, and completes classical probability theory. §§2-7 combined, on the one hand, and §9, on the other hand, are both self-sufficient units and can be read independently from one another. The ultimate design of the larger project of which this paper is part remains the increase of digitalization of the analysis of rational thought and language, that is, of (rational, not emotional) human intelligence. To reach out to other disciplines, an effort is made to describe the mathematics more explicitly than is usual.
基金Supported by National Natural Science Foundation of China under Grant No.10704016National Natural Science Foundation of Jiangsu Province under Grant No.BK2007100New Teacher Fund of Ministry of Education of China under Grant No.20070286036
文摘We report a theoretic study on the inverse spin-Hall effect （ISHE） in a two-terminal nano-device that consists of a two-dimensional electron gas （2DEG） with Rashba spin-orbit coupling （RSOC） and two ideal leads. Based on a two-site toy model and Keldysh Green＇s function method, we derive an analytic result of ISHE, which shows clearly that a nonzero transverse charge current stems from the combined effect of the RSOC, the spin bias, and its spin polarization direction in spin space. Our further numerical calculations in a larger system other than two-site lattice model demonstrate that the transverse charge current, dependent on the strength of the RSOC, the Fermi energy of the system, as well as the system size, can exhibit oscillating behavior and even reverse its sign due to Rashba spin precession. These properties may be helpful for eficient detection of the spin current （spin bias） by measuring the transverse charge current in a spin-orbital coupling system.
文摘By adopting a complex formulation of Ohm’s law, we arrive at combined equations connecting the conductivities of conductors. The horizontal resistivity is equal to the inverse of Drude’s conductivity δo( ), and the vertical resistivity (ρy) is equal to the Hall’s conductivity ( δH). At high magnetic field, the horizontal conductivity becomes exceedingly small, whereas the vertical conductivity equals to Hall’s conductivity. The Hall’s conductivity is shown to represent the maximal conductivity of conductors. Drude’s and Hall’s conductivities are related by δo =δHωC , where ωC is the cyclotron frequency, and is the relaxation time. The quantization of Hall’s conductivity is attributed to the fact that the magnetic flux enclosed by the conductor is carried by electrons each with h/e, where h is the Planck’s constant and e is the electron’s charge. The Drude’s conductance is found to be equal to Hall's conductance provided the magnetic flux enclosed by the conductor is a multiple of h/e.
文摘Taking into account the non separable solution for the quantum problem of the motion of a charged particle in a flat surface of lengths L<sub>x</sub> and L<sub>y</sub> with transversal static magnetic field B and longitudinal static electric field E, the quantum current, the transverse (Hall) and longitudinal resistivities are calculated for the state n = 0 and j = 0. We found that the transverse resistivity is proportional to an integer number, due to the quantization of the magnetic flux, and longitudinal resistivity can be zero for times t >> L<sub>x</sub>B/cE. In addition, using a modified periodicity of the solution, a modified quantization of the magnetic flux is found which allows to have IQHE and FQHE of any filling factor of the form v = k/l, with k, l ∈Z.
基金the initial result of“A Study on Ethics Thought of Mantel’s Novels”(Project Number:2021GH016)key humanities and social science research project among universities in Zhejiang Province.
文摘As the first book of“Tudor series”written by British famous historical novelist Hilary Mantel,Wolf Hall depicted the process of the blacksmith’s son Cromwell becoming a powerful minister.Meanwhile,with her distinctive writing style,Mantel also skillfully presented women’s living predicament as“the other”and created various typical female images.This paper will focus on those images and analyze their living predicaments under male-dominated society.
文摘In this paper, we make an initial value investigation of the unsteady flow of incompressible viscous fluid between two rigid non-conducting rotating parallel plates bounded by a porous medium under the influence of a uniform magnetic field of strength H0 inclined at an angle of inclination α with normal to the boundaries taking hall current into account. The perturbations are created by a constant pressure gradient along the plates in addition to the non-torsional oscillations of the upper plate while the lower plate is at rest. The flow in the porous medium is governed by the Brinkman’s equations. The exact solution of the velocity in the porous medium consists of steady state and transient state. The time required for the transient state to decay is evaluated in detail and the ultimate quasi-steady state solution has been derived analytically. Its behaviour is computationally discussed with reference to the various governing parameters. The shear stresses on the boundaries are also obtained analytically and their behaviour is computationally discussed.
文摘The Tenant of Wildfell Hall,firstly published in 1848,was the second and final novel of Anne Brontë,the youngest of the Brontësisters,telling a story of the mysterious young widow Helen Huntingdon,who flees her abusive husband and hides with her son at Wildfell Hall.Just as most contemporary critic defines it as one of the first feminist novels,the overall characterization of Helen sends a strong message of the empowerment of women.What confuses modern readers,however,is the subtle confrontation between the empowerment and its subversion.Thus,this paper conducts an in-depth analysis of the text concerning Anne's empow⁃erment of Helen and the corresponding subversion of the established empowerment before unveiling the fundamental cause of such arrangement with a view of shedding light on the comprehension of Anne Brontë's belief in universal social issues.
