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History is littered with hundreds of conflicts over the future of a community, group, location or business that were "resolved" when one of the parties stepped ahead and destroyed what was there. With the original point of contention destroyed, the debates would fall to the wayside. Archive Team believes that by duplicated condemned data, the conversation and debate can continue, as well as the richness and insight gained by keeping the materials. Our projects have ranged in size from a single volunteer downloading the data to a small-but-critical site, to over 100 volunteers stepping forward to acquire terabytes of user-created data to save for future generations.

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Bertrand Wong

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Papers

Riemann Hypothesis

This paper discusses the distribution of the non-trivial zeros of the Riemann zeta function ζ. It... more This paper discusses the distribution of the non-trivial zeros of the Riemann zeta function ζ. It looks into the question of whether any non-trivial zeros would ever possibly be found off the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1, e.g., at Re(s) = 1/4, 1/3, 3/4, 4/5, etc., and why all the non-trivial zeros are always found at the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1 and not anywhere else on this critical strip, with the first 10 13 non-trivial zeros having been found only at the critical line Re(s) = 1/2. It should be noted that a conjecture, or, hypothesis could possibly be proved by comparing it with a theorem that has been proven, which is one of the several deductions utilized in this paper. Through these several deductions presented, the paper shows how the Riemann hypothesis may be approached to arrive at a solution. In the paper, instead of merely using estimates of integrals and sums (which are imprecise and may therefore be of little or no reliability) in the support of arguments, where feasible actual computations and precise numerical facts are used to support arguments, for precision, for more sharpness in the arguments, and for "checkability" or ascertaining of the conclusions. This paper is the revised and expanded version of a paper [5] published in 2022.

Why the economy is hard to manage and how this could possibly be dealt with

This paper raises some points about the economy and economic policies, and presents some possible... more This paper raises some points about the economy and economic policies, and presents some possible economic solution, providing the stimulus for economic thought and importantly action to forestall economic problems. The practicable economic policies suggested in the paper would to some extent alleviate the serious economic problems of inflation, deflation, recessions and unemployment, though it is also hoped that the suggested economic policies could permanently eliminate these serious economic issues. The successful implementation of these economic policies would certainly lead to a better society, possibly with inclusive and sustainable economic growth, employment and decent work for all. [This paper published in an international economics journal is the sequel to the paper “We should Control the Economy and Not Let the Economy Control Us: How this May Be Achieved through a Macro-Economical Way” (which was published in 2021, https://doi.org/10.20935/AL3001 ).]

On Non-Trivial Zeros and Riemann Zeta Function

This paper examines the mysterious non-trivial zeros of the Riemann zeta function ζ and explains ... more This paper examines the mysterious non-trivial zeros of the Riemann zeta function ζ and explains their role, e.g., in the computation of the error term in Riemann’s J function for estimating the quantity of primes less than a given number. The paper also explains the close connection between the Riemann zeta function ζ and the prime numbers. [Published in international mathematics journal.]

COVID-19: Lesson learnt and using technology to possibly mitigate the pandemic and other viral infections

COVID-19 has made its unexpected entrance into the world sometime in late 2019 causing much fear,... more COVID-19 has made its unexpected entrance into the world sometime in late 2019 causing much fear, economic disasters, suffering, and fatalities throughout the world. This paper examines the pandemic and introduces an aggressive intervention and strategy, for humanity should not let themselves be sitting ducks waiting for the virus to attack, and some possible technological methods for mitigating COVID-19 and other viral infections. [This paper on saving lives is published in international public health journal.]

Non-Trivial Zeros of Riemann Zeta Function and Riemann Hypothesis

This paper, which is published in an international mathematics journal and endorsed by a mathemat... more This paper, which is published in an international mathematics journal and endorsed by a mathematics society, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis.

Primes and Their Link to the Goldbach Conjecture

This informative paper, which is published in an international mathematics journal, presents insi... more This informative paper, which is published in an international mathematics journal, presents insights and many important points on the prime numbers, which are the building-blocks or “atoms” of the integers, and the Goldbach conjecture formulated by Christian Goldbach (1690 - 1764) which are the result of years of research (the author having published two papers on the Goldbach conjecture in an international mathematics journal in 2012), all of which would be of interest to researchers working on the prime numbers and the Goldbach conjecture itself. The Goldbach conjecture, viz., every even number after 2 is the sum of 2 primes, is actually related to the distribution or “behavior” of the prime numbers. Therefore, when the distribution or “behavior” of the prime numbers is firmly understood the conjecture could be more easily solved. The paper has much to share about the distribution or “behavior” of the prime numbers, providing much numerical evidence to support the conjecture, besides suggesting ways or arguments for resolving the conjecture.

Logical Reasoning including Mathematical Reasoning: Practical Problems and Possible Solution

Logical reasoning in any form is an important aspect of life; it is persuading or convincing othe... more Logical reasoning in any form is an important aspect of life; it is persuading or convincing others with logic through writing or speech, for example, scientists, politicians, businessmen, financiers, solicitors and many others do this. This paper points out the frequent inefficacy of logical presentations, arguments and debates per se in bringing about the correct and wonted outcomes. It describes the scenario of people frequently involved in fruitless arguments and debates, and shows why the application of logic, for example, in logical argument or debate, could not often achieve the desired outcomes, much of the time ending up with frustration, unhappiness, bad feelings and poor relationships. Scenarios from mathematics, which probably represents the most rigorous form of logical reasoning, and science are described as well. The paper also delves into the problems encountered in logical reasoning as well as some modes of reasoning. It would be difficult and might be impossible to reason with and convince someone with a closed mind-set, someone who has made up the mind not to be convinced, or even someone who is not intelligent enough to be convinced. The paper, which is published in an international mathematics journal, presents a resolution to this serious problem, which is important, as that would be conducive to peace and harmony.

The Covid-19 Pandemic

Covid-19 has made its unexpected entrance into the world some time in late 2019 causing much fear... more Covid-19 has made its unexpected entrance into the world some time in late 2019 causing much fear, economic disasters, suffering and fatalities throughout the world. This paper suggests a more aggressive intervention and strategy, for humanity should not let themselves be sitting ducks waiting for the virus to attack, and some possible methods for stopping Covid-19 in its path. (Expanded & Updated)

Interpretation of Gravity

Gravity was discovered accidentally by Newton after he observed an apple falling from a tree. But... more Gravity was discovered accidentally by Newton after he observed an apple falling from a tree. But what actually is gravity and what is its role in nature? However, gravity remains much of a mystery. We experience gravity and are affected by it all the time, and yet we do not know what it really is except for interpreting it as a force or field. This paper, which is published in an international physics journal, takes on the challenging task of examining gravity and suggesting the possible actual role of gravity in nature, from our macro-world to the micro quantum world, and hence the possible unification of gravity with all the other forces of nature, through two modes of reasoning, namely reasoning by reductio ad absurdum and by analogy. The paper contains preliminary and explanatory materials which lead to the author’s hypotheses in Sections 5 and 6; Section 6 also provides some anecdotal evidence of what gravity is.

On the twin primes

This paper brings up a few possible approaches to solving the twin primes conjecture. [Published ... more

Logic in general and mathematical logic in particular

This paper brings up some important points about logic, e.g., mathematical logic, and also an inc... more This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [Published in international mathematics journal. Acknowledgments: The author expresses his gratitude to the referees and the Editor-in-Chief for their valuable comments in strengthening the contents of this paper.]

We should Control the Economy and Not Let the Economy Control Us: Is this Achievable and How is this Achievable?

Economic upturns and downturns appear to happen in cycles of say three to five years, and everyon... more Economic upturns and downturns appear to happen in cycles of say three to five years, and everyone looks forward to the impending economic downturns in trepidation and helplessness. The government also apparently couldn’t normally avert the economic downturn and the best it could normally do is to warn the people that the economic downturn is expected and they should be prepared for retrenchments and have to tighten their belts, and so on, becoming a sort of doomsday prophet. This is the situation of the economy controlling the people, who have become helpless victims of the economic downturn. This article raises some important points about the economy and economic policies, and proposes some important solutions. Economic problems that plague us seem intractable and seem to require some radical counter-measures. One of the root causes of a recession is that people are not spending sufficiently. To overcome this problem a radical change in our monetary system could be effected. This article is the revised and expanded version of an article which had been published in a journal of a well-established professional body for economists, whose editor remarked that the ideas in the article are against conventional wisdom.

We should Control the Economy and Not Let the Economy Control Us: How this May Be Achieved through a Macro-Economical Way

This letter proposes some solutions to our modern seemingly intractable economic problems. Keynes... more This letter proposes some solutions to our modern seemingly intractable economic problems. Keynesian theories, the bugbear of modern economics, seem to hold little water nowadays. Modern economies appear to be in need of more radical solutions. [Published in an international journal. The sequel to this letter "Why the economy is hard to manage and how this could possibly be dealt with" is published in an international economics journal, http://dx.doi.org/10.1590/0101-31572023-3509 .]

One may wonder what reality, which is abstract and intangible, is. This paper examines the variou... more

Some Inconsistence in Logic

This paper brings up some important points about logic, e.g., mathematical logic, and also an inc... more This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Godel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have pride of place. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, et al. Perhaps, by Godel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Godel's incompleteness theorems will have ramifications in other areas involving logic. The paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics.

The Twin Primes

The primes, including the twin primes and the other prime pairs, are the building-blocks of the i... more The primes, including the twin primes and the other prime pairs, are the building-blocks of the integers. Euclid's proof of the infinitude of the primes is generally regarded as elegant. It is a proof by contradiction, or, reductio ad absurdum, and it relies on an algorithm which always brings in larger and larger primes, an infinite number of them. However, the proof is also subtle and is misinterpreted by some, with one well-known mathematician even remarking that the algorithm might not work for extremely large numbers. A long unsettled related problem, the twin primes conjecture, has also aroused the interest of many researchers. The author has been conducting research on the twin primes for a long time and had published a paper on them (see, B. Wong, Possible solutions for the "twin" primes conjecture - The infinity of the twin primes, International Mathematical Journal, 3(8), 2003, 873-886). This informative paper presents some important facts on the twin primes which would be of interest to prime number researchers, with some remarks/reasons that point to the infinitude of the twin primes, including a reasoning which is somewhat similar to Euclid's proof of the infinity of the primes; very importantly, two algorithms are developed in Section 5 for sieving out the twin primes from the infinite list of the integers, which would be of interest to cryptographers and even to computer programmers.

[Published in an international mathematics journal. Thoroughly edited & re-arranged by the Editor-in-Chief of the journal.]

The Navier-Stokes Equations And Turbulence Or Chaos

The motion of fluids which are incompressible could be described by the Navier-Stokes differentia... more The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. Although they are relatively simple-looking, the three-dimensional Navier-Stokes equations misbehave very badly. Even with nice, smooth, reasonably harmless initial conditions, the solutions could wind up being extremely unstable. The field of fluid mechanics would be dramatically altered through a mathematical understanding of the outrageous behaviour of these equations. An explanation why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence or chaos (which is a three-dimensional phenomenon), would be provided. [Published in an international journal.]

On Quantum Entanglement

Quantum entanglement, a term coined by Erwin Schrodinger in 1935, is a mechanical phenomenon at t... more Quantum entanglement, a term coined by Erwin Schrodinger in 1935, is a mechanical phenomenon at the quantum level wherein the quantum states of two (or more) particles have to be described with reference to each other though these particles may be spatially separated. This phenomenon leads to paradox and has puzzled us for a long time. The behaviour of entangled particles is apparently inexplicable, incomprehensible and like magic at work. Locality has been a reliable and fruitful principle which has guided us to the triumphs of twentieth century physics. But the consequences of the local laws in quantum theory could seem "spooky" and nonlocal, with some theorists questioning locality itself. Could two subatomic particles on opposite sides of the universe be really instantaneously connected? Is any theory which predicts such a connection essentially flawed or incomplete? Are the results of experiments which demonstrate such a connection being misinterpreted? These questions challenge our most basic concepts of spatial distance and time. Modern physics is in the process of dismantling the space all around us and the universe will never be the same. Quantum entanglement involves the utilisation of cutting edge technology and will bring great benefits to society. This paper traces the development of quantum entanglement and presents some possible explanations for the strange behaviour of entangled particles. This paper is published in an international journal.

Modeling Turbulence With The Navier-Stokes Equations

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. ... more The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is a three-dimensional phenomenon. [The paper is published in an international journal.]

A Survey Of The Riemann Zeta Function With Its Applications

This paper explicates the Riemann hypothesis and proves its validity. [The paper is published in ... more

Riemann Hypothesis

This paper discusses the distribution of the non-trivial zeros of the Riemann zeta function ζ. It... more This paper discusses the distribution of the non-trivial zeros of the Riemann zeta function ζ. It looks into the question of whether any non-trivial zeros would ever possibly be found off the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1, e.g., at Re(s) = 1/4, 1/3, 3/4, 4/5, etc., and why all the non-trivial zeros are always found at the critical line Re(s) = 1/2 on the critical strip between Re(s) = 0 and Re(s) = 1 and not anywhere else on this critical strip, with the first 10 13 non-trivial zeros having been found only at the critical line Re(s) = 1/2. It should be noted that a conjecture, or, hypothesis could possibly be proved by comparing it with a theorem that has been proven, which is one of the several deductions utilized in this paper. Through these several deductions presented, the paper shows how the Riemann hypothesis may be approached to arrive at a solution. In the paper, instead of merely using estimates of integrals and sums (which are imprecise and may therefore be of little or no reliability) in the support of arguments, where feasible actual computations and precise numerical facts are used to support arguments, for precision, for more sharpness in the arguments, and for "checkability" or ascertaining of the conclusions. This paper is the revised and expanded version of a paper [5] published in 2022.

Why the economy is hard to manage and how this could possibly be dealt with

This paper raises some points about the economy and economic policies, and presents some possible... more This paper raises some points about the economy and economic policies, and presents some possible economic solution, providing the stimulus for economic thought and importantly action to forestall economic problems. The practicable economic policies suggested in the paper would to some extent alleviate the serious economic problems of inflation, deflation, recessions and unemployment, though it is also hoped that the suggested economic policies could permanently eliminate these serious economic issues. The successful implementation of these economic policies would certainly lead to a better society, possibly with inclusive and sustainable economic growth, employment and decent work for all. [This paper published in an international economics journal is the sequel to the paper “We should Control the Economy and Not Let the Economy Control Us: How this May Be Achieved through a Macro-Economical Way” (which was published in 2021, https://doi.org/10.20935/AL3001 ).]

On Non-Trivial Zeros and Riemann Zeta Function

This paper examines the mysterious non-trivial zeros of the Riemann zeta function ζ and explains ... more This paper examines the mysterious non-trivial zeros of the Riemann zeta function ζ and explains their role, e.g., in the computation of the error term in Riemann’s J function for estimating the quantity of primes less than a given number. The paper also explains the close connection between the Riemann zeta function ζ and the prime numbers. [Published in international mathematics journal.]

COVID-19: Lesson learnt and using technology to possibly mitigate the pandemic and other viral infections

COVID-19 has made its unexpected entrance into the world sometime in late 2019 causing much fear,... more COVID-19 has made its unexpected entrance into the world sometime in late 2019 causing much fear, economic disasters, suffering, and fatalities throughout the world. This paper examines the pandemic and introduces an aggressive intervention and strategy, for humanity should not let themselves be sitting ducks waiting for the virus to attack, and some possible technological methods for mitigating COVID-19 and other viral infections. [This paper on saving lives is published in international public health journal.]

Non-Trivial Zeros of Riemann Zeta Function and Riemann Hypothesis

This paper, which is published in an international mathematics journal and endorsed by a mathemat... more This paper, which is published in an international mathematics journal and endorsed by a mathematics society, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis.

Primes and Their Link to the Goldbach Conjecture

This informative paper, which is published in an international mathematics journal, presents insi... more This informative paper, which is published in an international mathematics journal, presents insights and many important points on the prime numbers, which are the building-blocks or “atoms” of the integers, and the Goldbach conjecture formulated by Christian Goldbach (1690 - 1764) which are the result of years of research (the author having published two papers on the Goldbach conjecture in an international mathematics journal in 2012), all of which would be of interest to researchers working on the prime numbers and the Goldbach conjecture itself. The Goldbach conjecture, viz., every even number after 2 is the sum of 2 primes, is actually related to the distribution or “behavior” of the prime numbers. Therefore, when the distribution or “behavior” of the prime numbers is firmly understood the conjecture could be more easily solved. The paper has much to share about the distribution or “behavior” of the prime numbers, providing much numerical evidence to support the conjecture, besides suggesting ways or arguments for resolving the conjecture.

Logical Reasoning including Mathematical Reasoning: Practical Problems and Possible Solution

Logical reasoning in any form is an important aspect of life; it is persuading or convincing othe... more Logical reasoning in any form is an important aspect of life; it is persuading or convincing others with logic through writing or speech, for example, scientists, politicians, businessmen, financiers, solicitors and many others do this. This paper points out the frequent inefficacy of logical presentations, arguments and debates per se in bringing about the correct and wonted outcomes. It describes the scenario of people frequently involved in fruitless arguments and debates, and shows why the application of logic, for example, in logical argument or debate, could not often achieve the desired outcomes, much of the time ending up with frustration, unhappiness, bad feelings and poor relationships. Scenarios from mathematics, which probably represents the most rigorous form of logical reasoning, and science are described as well. The paper also delves into the problems encountered in logical reasoning as well as some modes of reasoning. It would be difficult and might be impossible to reason with and convince someone with a closed mind-set, someone who has made up the mind not to be convinced, or even someone who is not intelligent enough to be convinced. The paper, which is published in an international mathematics journal, presents a resolution to this serious problem, which is important, as that would be conducive to peace and harmony.

The Covid-19 Pandemic

Covid-19 has made its unexpected entrance into the world some time in late 2019 causing much fear... more Covid-19 has made its unexpected entrance into the world some time in late 2019 causing much fear, economic disasters, suffering and fatalities throughout the world. This paper suggests a more aggressive intervention and strategy, for humanity should not let themselves be sitting ducks waiting for the virus to attack, and some possible methods for stopping Covid-19 in its path. (Expanded & Updated)

Interpretation of Gravity

Gravity was discovered accidentally by Newton after he observed an apple falling from a tree. But... more Gravity was discovered accidentally by Newton after he observed an apple falling from a tree. But what actually is gravity and what is its role in nature? However, gravity remains much of a mystery. We experience gravity and are affected by it all the time, and yet we do not know what it really is except for interpreting it as a force or field. This paper, which is published in an international physics journal, takes on the challenging task of examining gravity and suggesting the possible actual role of gravity in nature, from our macro-world to the micro quantum world, and hence the possible unification of gravity with all the other forces of nature, through two modes of reasoning, namely reasoning by reductio ad absurdum and by analogy. The paper contains preliminary and explanatory materials which lead to the author’s hypotheses in Sections 5 and 6; Section 6 also provides some anecdotal evidence of what gravity is.

On the twin primes

This paper brings up a few possible approaches to solving the twin primes conjecture. [Published ... more

Logic in general and mathematical logic in particular

This paper brings up some important points about logic, e.g., mathematical logic, and also an inc... more This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [Published in international mathematics journal. Acknowledgments: The author expresses his gratitude to the referees and the Editor-in-Chief for their valuable comments in strengthening the contents of this paper.]

We should Control the Economy and Not Let the Economy Control Us: Is this Achievable and How is this Achievable?

Economic upturns and downturns appear to happen in cycles of say three to five years, and everyon... more Economic upturns and downturns appear to happen in cycles of say three to five years, and everyone looks forward to the impending economic downturns in trepidation and helplessness. The government also apparently couldn’t normally avert the economic downturn and the best it could normally do is to warn the people that the economic downturn is expected and they should be prepared for retrenchments and have to tighten their belts, and so on, becoming a sort of doomsday prophet. This is the situation of the economy controlling the people, who have become helpless victims of the economic downturn. This article raises some important points about the economy and economic policies, and proposes some important solutions. Economic problems that plague us seem intractable and seem to require some radical counter-measures. One of the root causes of a recession is that people are not spending sufficiently. To overcome this problem a radical change in our monetary system could be effected. This article is the revised and expanded version of an article which had been published in a journal of a well-established professional body for economists, whose editor remarked that the ideas in the article are against conventional wisdom.

We should Control the Economy and Not Let the Economy Control Us: How this May Be Achieved through a Macro-Economical Way

This letter proposes some solutions to our modern seemingly intractable economic problems. Keynes... more This letter proposes some solutions to our modern seemingly intractable economic problems. Keynesian theories, the bugbear of modern economics, seem to hold little water nowadays. Modern economies appear to be in need of more radical solutions. [Published in an international journal. The sequel to this letter "Why the economy is hard to manage and how this could possibly be dealt with" is published in an international economics journal, http://dx.doi.org/10.1590/0101-31572023-3509 .]

One may wonder what reality, which is abstract and intangible, is. This paper examines the variou... more

Some Inconsistence in Logic

This paper brings up some important points about logic, e.g., mathematical logic, and also an inc... more This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Godel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have pride of place. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, et al. Perhaps, by Godel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Godel's incompleteness theorems will have ramifications in other areas involving logic. The paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics.

The Twin Primes

The primes, including the twin primes and the other prime pairs, are the building-blocks of the i... more The primes, including the twin primes and the other prime pairs, are the building-blocks of the integers. Euclid's proof of the infinitude of the primes is generally regarded as elegant. It is a proof by contradiction, or, reductio ad absurdum, and it relies on an algorithm which always brings in larger and larger primes, an infinite number of them. However, the proof is also subtle and is misinterpreted by some, with one well-known mathematician even remarking that the algorithm might not work for extremely large numbers. A long unsettled related problem, the twin primes conjecture, has also aroused the interest of many researchers. The author has been conducting research on the twin primes for a long time and had published a paper on them (see, B. Wong, Possible solutions for the "twin" primes conjecture - The infinity of the twin primes, International Mathematical Journal, 3(8), 2003, 873-886). This informative paper presents some important facts on the twin primes which would be of interest to prime number researchers, with some remarks/reasons that point to the infinitude of the twin primes, including a reasoning which is somewhat similar to Euclid's proof of the infinity of the primes; very importantly, two algorithms are developed in Section 5 for sieving out the twin primes from the infinite list of the integers, which would be of interest to cryptographers and even to computer programmers.

[Published in an international mathematics journal. Thoroughly edited & re-arranged by the Editor-in-Chief of the journal.]

The Navier-Stokes Equations And Turbulence Or Chaos

The motion of fluids which are incompressible could be described by the Navier-Stokes differentia... more The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. Although they are relatively simple-looking, the three-dimensional Navier-Stokes equations misbehave very badly. Even with nice, smooth, reasonably harmless initial conditions, the solutions could wind up being extremely unstable. The field of fluid mechanics would be dramatically altered through a mathematical understanding of the outrageous behaviour of these equations. An explanation why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence or chaos (which is a three-dimensional phenomenon), would be provided. [Published in an international journal.]

On Quantum Entanglement

Quantum entanglement, a term coined by Erwin Schrodinger in 1935, is a mechanical phenomenon at t... more Quantum entanglement, a term coined by Erwin Schrodinger in 1935, is a mechanical phenomenon at the quantum level wherein the quantum states of two (or more) particles have to be described with reference to each other though these particles may be spatially separated. This phenomenon leads to paradox and has puzzled us for a long time. The behaviour of entangled particles is apparently inexplicable, incomprehensible and like magic at work. Locality has been a reliable and fruitful principle which has guided us to the triumphs of twentieth century physics. But the consequences of the local laws in quantum theory could seem "spooky" and nonlocal, with some theorists questioning locality itself. Could two subatomic particles on opposite sides of the universe be really instantaneously connected? Is any theory which predicts such a connection essentially flawed or incomplete? Are the results of experiments which demonstrate such a connection being misinterpreted? These questions challenge our most basic concepts of spatial distance and time. Modern physics is in the process of dismantling the space all around us and the universe will never be the same. Quantum entanglement involves the utilisation of cutting edge technology and will bring great benefits to society. This paper traces the development of quantum entanglement and presents some possible explanations for the strange behaviour of entangled particles. This paper is published in an international journal.

Modeling Turbulence With The Navier-Stokes Equations

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. ... more The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is a three-dimensional phenomenon. [The paper is published in an international journal.]

A Survey Of The Riemann Zeta Function With Its Applications

This paper explicates the Riemann hypothesis and proves its validity. [The paper is published in ... more

Goldbach Conjecture

This paper is a revision and expansion of two papers on the Goldbach conjecture which the author ... more This paper is a revision and expansion of two papers on the Goldbach conjecture which the author had published in an international mathematics journal in 2012. It presents insights and many important points on the conjecture and the prime numbers which are the result of years of research, all of which would be of interest to researchers working on the prime numbers and the Goldbach itself. The Goldbach conjecture, viz., every even number after 2 is the sum of 2 primes, is actually related to the distribution or "behavior" of the prime numbers. Therefore, when the distribution or "behavior" of the prime numbers is firmly understood the conjecture could be more easily resolved. This paper, which has been refereed and accepted for publication, has much to share about the distribution or "behavior" of the prime numbers, besides resolving the conjecture.

On Logical Reasoning in Everyday Affairs

Logical reasoning in any form is an important aspect of life; it is persuading or convincing othe... more Logical reasoning in any form is an important aspect of life; it is persuading or convincing others with logic through writing or speech, for example, scientists, politicians, businessmen, financiers, solicitors and many others do this. This paper points out the frequent inefficacy of logical presentations, arguments and debates per se in bringing about the correct and wonted outcomes. It describes the scenario of people frequently involved in fruitless arguments and debates, and shows why the application of logic, for example, in logical argument or debate, could not often achieve the desired outcomes, much of the time ending up with frustration, unhappiness, bad feelings and poor relationships. Scenarios from mathematics, which probably represents the most rigorous form of logical reasoning, and science are described as well. The paper also delves into the problems encountered in logical reasoning as well as some modes of reasoning. It would be difficult and might be impossible to reason with and convince someone with a closed mind-set, someone who has made up the mind not to be convinced, or even someone who is not intelligent enough to be convinced. The paper presents a resolution to this serious problem, which is important, as that would be conducive to peace and harmony.

Logic and Intuition

This paper aims to inspire thinking on the capabilities and potential of the human brain. The bra... more This paper aims to inspire thinking on the capabilities and potential of the human brain. The brain apparently has great potential for development and great untapped capabilities. Practically everyone is keen on improving his mental capacity, especially the capability of logical reasoning, that is, the ability in utilising logic to achieve the desired outcomes. It appears that logic is equated with intelligence and is regarded as the most important aspect of thinking by many (though emotional intelligence is now the new kid in the block which seems to be gaining traction). The author here looks at reasoning or logic, as well as intuition, from a different and perhaps unique perspective.

Curbing the Spread of Covid-19

Covid-19 has made its unexpected entrance into the world some time in late 2019 causing much fear... more Covid-19 has made its unexpected entrance into the world some time in late 2019 causing much fear, economic disasters, suffering and fatalities throughout the world. This article suggests a more aggressive intervention and strategy, for humanity should not let themselves be sitting ducks waiting for the virus to attack, and some possible methods for stopping Covid-19 in its path. (A useful summary is included just below.)