#combinatorial group theory

[janmisali]

the additive group of day-2 normal partizan games

recently I've been trying to gain a deeper understanding of one very small subset of the class of normal partizan games: the games that are born on or before day 2. I've explained the general concept on main before, so I'll assume you're already familiar with the idea.

to recap, there are 22 games under the normal play convention that end after at most two moves. three moves extends that to 1474, and beyond that many moves it's an open question.

normal partizan games can be added together to form new normal partizan games. analyzing when a complicated game can be broken down into the sum of multiple other simpler games is one of the most important concepts in combinatorial game theory.

the sum of games has the properties that it's associative ((F + G) + H = F + (G + H)) and commutative (G + H = H + G), just like regular addition. so what I'm interested in is what happens if you ignore the game-ness of these games and just explore the behavior of the sums of these particular small games as an abelian group. that is, given two games G and H expressed in the form of some number of day-2 games all added together (allowing duplicates), when does G equal H? and I uh, almost understand it? but not everything.

the generative set

while these 22 games are all very simple, we don't really need all 22 of them. I'm not 100% sure how small of a generative set you can reduce them to, but I've been able to cut it down to just ten:

*2 = {0,*|0,*}

½ = {0|1}

-½ = {-1|0}

↑ = {0|*}

↓ = {*|0}

±1 = {1|-1}

±½ = {½|-½}

{½|-½*}

{1|0,*}

{0,*|-1}

the ones I've put in red are not actually day-2 games, but they can be expressed as the sums of other day-2 games, so they're fine to use. they could be replaced with actual day-2 games, but I think these particular forms make the math easier. and the last two are uh, weird. I don't know what their deal is.

anyway, all day-2 games (and by extension all sums of day-2 games) can be represented as the sum of some combination of members of this generative set. I won't explain how for all of them because it's pretty trivial a lot of the time, but here are some of the highlights:

the zero game can be reached by adding any game to its negative, or alternatively it just doesn't need to be part of the generative set because the "empty sum" comes for free

the switches {1|0} and {0|-1} can be represented as ½±½ and -½±½, respectively

the hot game {1|*} has the property that {1|*} + {1|*} = 1 + * = 1*. this suggests that it could be broken down into a "one half" component and a "star halves" component. */2 is not a unique value (there are in fact infinitely many games which when added to themselves equal star), but {½|-½*}, one half less than {1|*}, does indeed have the property that when added to itself you get * as we'd expect, and since * is its own negative, three copies of {½|-½*} can be added together to get its negative.

adding these games together

a lot about the behavior of the sums of games is already very well-documented. as I said, this is one of the key concepts in combinatorial game theory.

within this generative set, three games are their own negatives (*2, ±1, and ±½), which simplifies things a lot. they essentially don't interact with anything else in the group. "star halves" works in a similar way, as described earlier.

numbers and arrows also don't interact with each other. ↑ is infinitesimal, smaller than every positive surreal number but still positive.

if we ignore the hot fuzzy game {1|0,*} and its negative, everything else then is pretty straightforward. games in this group can be broken down into a number part, an arrow part, a "star halves" part, a *2 part, and two switch parts. the sum of two games in this group can then be handled in all of those parts individually without needing to worry about any of these parts bleeding into each other.

but what of the hot fuzzy games {1|0,*} and {0,*|-1}? do they interact with anything else?

and I don't have the answer to that. they're really weird! {1|0,*} is hot (players have an incentive to move) and fuzzy (whoever makes the first move wins), but {1|0,*} added to itself is positive! I don't think these games will ever combine with something in this group to form something that could have been expressed without using them, but I also don't know how I would go about proving that.

anyway that's just what I've been thinking about recently

[annadeedee]

why must all combinatorial group theory be terrible

[xeter-group]

Exceptional Automorphisms of S_6

The symmetric group S_6 has a special property that S_n does not have for ANY n ≠6. Really? 6, of all numbers?? How odd.

For any group G, and every g in G, the conjugation map f_g: h ↦ g h g^-1 is an automorphism of G. That is, it is an isomorphism from G -> G. The group of these particular automorphisms (under composition) is called the group of inner automorphisms of a group. Did you know that for every symmetric group S_n EXCEPT n=6, this is the entire automorphism group?

For some reason, S_6 has basically ONE (and only one) weird automorphism. What I mean by this is if we denote the inner automorphisms of S_6 by Inn(S_6), then S_6/Inn(S_6) ~= Z_2. (In other words, you can pick a single automorphism sigma such that the automorphisms of S_6 that are not inner automorphisms can be written as a composition of inner automorphisms and this one sigma.)

So we have this weird situation where |Aut(S_2)|=|Aut(C_2)| = 1, which is kinda trivial, |Aut(S_6)| = 2n!, and |Aut(S_n)| = n! otherwise. Kinda weird, huh?

What is also interesting is the proofs that there are no outer automorphisms for n≠6. Basically you can eliminate all but finitely many n, then you can pick off cases, and you are left with an n=6 shaped hole in your cases that CANNOT be filled, which feels so weird to me. If I was proving this myself I'd be going crazy having proven it for every case except 6, and having to resort to some proof which for some reason doesn't work for finitely many values.

Construction

There are so many different ways to construct the weird automorphism of S_6 - I have some links at the bottom. I particularly like the graph theory/geometric ones. Something about using factorisations of K_6 or a dodecahedron just makes the 6 feel more unique. I will admit I don't think I understand what is fundamentally special about 6 enough yet, though, on a philosophical level.

Practically, though, the constructions all basically boil down to the fact you can put a copy of S_5 inside S_6 in a way that isn't the obvious way, which you can only do for n=6. That is the special part.

After that, you have these 6 cosets of S_5 inside S_6 that S_6 acts upon. In other words, each element of S_6 permutes the 6 cosets of S_5 living inside it. But the group of permutations of 6 cosets is S_6. So we have a mapping from elements of S_6 to S_6 - an automorphism! Is this automorphism an inner automorphism? Each construction shows why they are outer differently, but a common theme is to show that the mapping S_6 to S_6 does not take transpositions to transpositions, but inner automorphisms preserve the cycle type.

Some proofs and examples

The wikipedia article. I like the construction about graph partitions. It does not, however, have much detail sadly.

Fortunately, the graph partitions thing from wikipedia is explained here. It's very short and to the point, also quite nice to look at:

Fairly elementary explanations, followed by more intense ones:

Requires only basic group theory to understand the first few explanations they provide (although it isn't trivial if you just learned group theory). Bonus points for "MyStIc PeNtAgOnS" (capitalisation mine):

Allegedly useful, but I can't grab a copy: (I think its on mathscinet under MR1240362 as per David Leep's personal website's publications section)

Combinatorial Structure of the automorphism group of S6 by T.Y. Lam and David B. Leep, Expositiones Mathematicae11 (1993), no. 4, pp. 289-903.

The comment by Matthew Towers here is also interesting:

[poutpoutlilith]

Do you study math by any chance? I studied it in undergrad! Always watching out for other math girlies lol 🤓

Believe it or not, math is my field of choice! Combinatorial and applied group theory is my current fave subfield. And it is my current research focus.

[bwhitex]

What is the paradox of Human Connection in a World of 8 Billion?

In a world bustling with over 8 billion souls, the paradox of human connection resonates through the corridors of our social existence. Each encounter carries the potential for a relationship, a shared story, a woven thread in the intricate tapestry of life. Yet, it's a universal truth a cognitive and emotional threshold that we simply cannot forge deep connections with everyone. This is not a tale of tragic scarcity but rather a narrative of selectivity and profound appreciation for the relationships that truly resonate with us.

Life's Tapestry and Human Bonds

Ever mourned the lack of human connection between you and a stranger? Before you let regret unfold, imagining the 'coulda, woulda, shoulda' moments that never came to be pause. What about the understanding that not every friend is destined to become a meaningful chapter in your life's story. You wouldn’t want them to all be, either. Not a human, even one who is capable of believing in infinite possibilities, would dare to say emotinal dynamics are just not logic, they defy our prefrontal cortex, it’s because they are governed admydala. Literally, nothing makes sense with the limited but profound experience of relational emotinal dynamics.

Good guys are never enough. I hate that I know that’s not true, but feel different about the odds of “good guys”, being enough. Total bummer cause I want to think it’s true. It almost feels tragic to think till you see this other perspective from my unique equation below and one realizes that feels sad but not tragic.

(Life's Tapestry + Human Bonds) - Superficial Ties + (Deep Resonance x Mutual Understanding) + (Acceptance / Judgement) + (Shared Moments of Harmony) - (The Need for Universal Approval)= Profound Connections that Last

This simple yet profound equation was developed to encapsulate the essence of human interpersonal relationships. Human life is enriched by weaving genuine bonds into our complex tapestry of experiences: subtracting superficiality and multiplying mutual understanding deepens the resonance of our connections. Acceptance is pivotal in the equation of relationships, outweighing judgment to foster profound and lasting bonds: shared moments of harmony add to the strength and substance of these ties. Ultimately, a meaningful social tapestry is woven not through the pursuit of universal approval, but through the cultivation of authentically connected relationships. It's natural only to have a select few relationships that deepen into something significant. Our emotional capacity, as suggested by Robin Dunbar (1992), is capped at about 150 meaningful relationships at a time—perhaps even over a lifetime.

Understanding Dunbar's Number

Dunbar's number, typically cited as around 150, is more than a statistic. It's a reflection of our cognitive limits to maintain stable and meaningful social relationships. This concept is grounded in the theory that the human brain has a finite capacity for managing the emotional complexities involved in our social lives (Dunbar, 1992).

Yet, the abstraction of this number contrasts starkly with the reality of our global population. In order physically manifest this, he is another equation to calculate the number of potential unique groups of 150 people out of 8 billion, we’d actually be able to chose from we're faced with a figure so large it defies comprehension. Here is my own equations for measuring that :

C(8,000,000,000, 150) = 8,000,000,000! / (150! X (8,000,000,000 - 150)!)

To summarize the expression of C(8,000,000,000, 150) \) represents the number of different ways a group of 150 people can be chosen from a larger population of 8 billion. This is a combinatorial problem, often referred to as a "combination," which calculates how many possible subsets can be formed from a larger set when the order of selection does not matter. The factorial notation (e.g., \( n! \)) signifies the product of all positive integers up to \( n \), which is a key part of the formula for combinations. The result of this equation is an astronomically large number, reflecting the immense variety of potential unique groups of 150 people within the global population.

While theoretically intriguing, that’s what we think cognitively, or no to be true, through logical means. But the emotional truth teaches us that the probability of forming meaningful connections with even a fraction of these combinations is virtually nil. The vastness of potential social networks is a mathematical marvel, but the intimate realities of our emotional world are far more selective.

The Probability of Connection

The mathematical probability of any specific group of 150 people coming together is infinitesimally small. Yet, the quest for meaningful connections continues. It's not the quantity of social interactions that defines our experience but the quality of the connections we do make.

We are adventurers in a grand landscape of social possibilities, yet navigators of a much smaller, more intimate map of personal relationships. The realization that we cannot possibly connect with every potential soul we encounter can inspire a sense of longinga wistful reflection on the bonds that could have shaped our lives in other circumstances.

The Practical Realities

As we travel this world, the reality is that we will only ever engage with a finite subset of the population. You have so many options, quantity but really none very much at all. Your temperament would never permit everyone being quality. Everyone too crazy to fully optimize every single potential. There are practical limitations of time, geography, and cognitive capacity ensure that the 150 people who constitute our social world at any given time are a carefully curated ensemble a reflection of who we are and what we feel close too. It may exist in the tension between the infinite potential what think exists in of our social networks and the finite nature of our emotional connections. Our crazy saves us, our tempermant, makes us too crazy for everyone and everyone too crazy for us. We would never tolerate to such ideal level of human optimization, even if we could and there is infirmity possibility to do taht, really too. Except when feelings don’t make sense are not working on the best possibilities out there. This dichotomy is hard to integrate fully into our personal realities, we want to be liked and warm with as many as possible. We think “all”, we know it, never feel it though!

What is really at the core of the essence of that probability is this: physiological and psychological sensitivities in our social existence. There is a world of boundless possibilities for connection, yet that’s mitigated by unique personal reality of meaningful and manageable emotional experiences too.

In Conclusion

The exploration of human connection in a populous world is a journey of understanding the limits of our cognitive and emotional capacities. It's a journey that teaches us the value of each relationship we choose to cultivate and the precious nature of the finite bonds that shape our existence.

As we ponder the vast array of paths not taken, let us also celebrate the connections we have made—the ones that have become part of our unique story in a world brimming with untold billions. In the landscape of human connection, it's a common feeling to worry about the depth of our relationships with others. We live in a world where the quantity of connections—friends on social media, contacts in our phones, followers on various platforms—can often seem to overshadow the quality of those connections. But it's important to remember that not every interaction will, or even should, lead to a deep connection. And that's perfectly okay. Here's a poem I invented to that captures this sentiment of this high level analysis. It summarizes what I mean perfectly too. Took me a few hours to make it, and I hope you enjoy it:

In the tapestry of life, each thread plays its part. Some bonds are fleeting, while others bind the heart. It's in the depth, not breadth, where true connections are cast,

For in the realm of human ties, some are meant to last. Not every soul we meet will walk our path till the end,

Nor every passer-by will turn to a confidant or friend. But that's the beauty of the journey, with its ebb and flow,

We cherish those who resonate, and gracefully let others go. For in the symphony of life, not every note is key,

Some harmonies are brief, yet sweet in their brevity. And though we may strive to understand each score,

Some melodies will touch us deep, some just brush the core. So fret not if crowds don't echo your inner song. For those who truly matter will hear it clear and strong. A few profound connections can outweigh scores of shallow ties and in the hearts that truly Certainly! I'll rephrase the poem to maintain its essence while aiming for clarity and coherence:

In life's woven tapestry, each thread holds its place,Some bonds might be transient, others embrace.

It's the depth, not the range, where true ties are found, Within the vast weave of human bounds, some are profound. Not all will journey with us until our story's end,

Not every stranger will transform into a friend. Yet that's the journey's beauty, with its rise and its fall,

We treasure the resonant few, and with grace, release all.

In life's grand symphony, not all notes are drawn out long, Some melodies are fleeting, yet their brief echo is strong.

While we seek to decipher each life's intricate score,

Certain tunes captivate us deep, others just skim the shore. Worry not if the multitude doesn't mirror your tune,

For those who truly matter will synchronize soon. A handful of deep connections surpass many that are slight,

In hearts that truly meld, our spirits unite.

Seek to invest attention and time with whom your soul finds peace, for they show you as you are. The rare ones who with a gesture or word, can heal or lightly scar.

It's not a fault to connect with only a few, as we navigate humanity's quest,

It's simply life's design, a cycle, not a deficiency to be suppressed.

For it's among the select bonds that we find our truest selves,

In this emotional dynamic lies our temperance on life's shelves.

It's more than a place of individual emotion—it's a shared embrace,

A haven of mutual understanding, acceptance, and grace.

Among the current billions, these gems shine amidst the gravel,

The stars that pierce the night, in a vast emotional travel.

We don't bond with all, but where we do, it's emotion that ignites the spark,

Humans seek their reflection in others, in this space where they embark.

We navigate the masses by feeling first, then contemplating our reflection,

Desiring more possibilities, we adopt a cognitive direction.

This emotional dynamic is not just ours but shared in duality's dance,

It's the truth we discover when we give different perspectives a chance.

Reference:

Dunbar, R. I. M. (1992). Neocortex size as a constraint on group size in primates. Journal of Human Evolution, 22(6), 469-493. https://doi.org/10.1016/0047-2484(92)90081-J

[coming-up-with-usernames-is-hard]

PhD Blog Week 5

Courses

CFT: So much happens in this course without any of the details filled out, makes the lectures hard to follow and then takes ages to review and fill out the details. We finally got onto actual conformal field theories, although only just. Got referred to a book on string theory for a proof, so that's how you know we're doing real maths

Lie theory: Replacement lecturer this week, started the section on Weyl's theorem, introduced Casimir operators (without ever defining the universal enveloping algebra, which seems wrong), ended with a cool proof chaining maps together, felt like doing real maths (i.e., category theory)

DIff Top: Somewhat confused lecture introducing integral curves, ended with an introduction to Lie groups which is always nice

Talks

Attended the algebra seminar, got lost after about five minutes, something about quantum groups and then the algebraic geometry started and I was confused

Example showcases, the three I watched were interesting, the first was on rep theory and constructing the Specht modules, which lead nicely into my talk (in that everyone then knew what a Young diagram is). The second was on topoi, something I've come across but never bothered to learn properly, and the third was on constructing the 5-adics, again, something I've come across but never in details. My talk went ok I think, I tried to fit in too much material and it went a bit off the rails at the end when I realised I'd changed an earlier slide and not a later one so the example didn't quite match up

Supervisor Meeting

Met with just one of my supervisors this week, introduced the 6-vertex model as a lattice model explaining some of the combinatorial aspects of the boson representation. It's a neat trick shifting perspective to view the transitioning fermions rather than the initial and final states. Finally saw how one of the results we'd been building up to (the Murnaghan-Nakayama rule) drops out neatly once you've put in all of this work up-front with definitions

Reading Groups

Complex geometry: Went to the talk on the Plucker embedding which I had been planning to give until a week ago, so I feel like I followed quite well this time

Infinity Categories: Looked at model categories this week, I think I followed most of it, mainly becuase this topic was light on the homotopy theory

Teaching

TA'd two first year tutorials, fortunately the problem sheet was more reasonable this week

Marked another first year assessment, there's already a marked improvement in their ability (and willingness) to write sentences, so they've definitely learnt from the feedback on the last one

[govindhtech]

SQBM+ Optimization from Toshiba Facilitates Success on Azure

SQBM+: An Optimization Solution Inspired by Quantum Theory

Toshiba Corporation created SQBM+, a software-based optimization tool for handling challenging combinatorial issues. It is based on the ideas of quantum computing but is easily accessible without the need for specialist gear.

SQBM+ for Azure, a version of Toshiba Digital Solutions Corporation’s SQBM+ software tailored for Microsoft Corporation’s Azure Marketplace, was released today. Toshiba Digital Solutions Corporation is a pioneer in the industry when it comes to implementing quantum-inspired solutions to challenging real-world challenges.

SQBM+ for Azure complements the version already offered on Amazon Web Services, demonstrating Toshiba Digital Solutions’ commitment to providing SQBM+ to clients operating in various IT environments.

Software for optimization swiftly selects the best selections from a vast array of possibilities. The Simulated Bifurcation Machine, a combinatorial optimization solution based on Toshiba Corporation’s quantum-inspired Simulated Bifurcation Algorithm, is the source of SQBM+. It finds high-precision, approximate (high-quality) solutions quickly to complicated, large-scale problems running on conventional machines.

Toshiba Group and its partners were able to confirm the viability of quasi-quantum computing and its application to high-frequency, high-speed stock trading, computational drug discovery, energy management, and materials research via the commercial introduction of SQBM+. Toshiba Digital Solutions wants to help solve challenging issues by using SQBM+ in a variety of domains.

Important characteristics of SQBM+ for Azure

Offered as an Azure Virtual Machine

Utilizing SQBM+ is simple and fast since an instance of the program is launched by the preconfigured Azure virtual machine. SQBM+ does not need installation.

All queries are processed rapidly by the user-only SQBM+ instance. Users have complete control over the resource and all of its features, including auto-scale, redundancy, and start/stop, since it is a unique copy.

May be integrated into the system of the client to cut down on network overhead.

SQBM+ Version 2 increases scalability, speed, and accuracy and supports 10 million variables

A quadratic unconstrained binary optimization (QUBO) solution that can handle QUBO problems with up to 10 million variables is supported by SQBM+ Version 2.

An enhanced method increases accuracy and expedites the search for an approximative answer.

In order to increase usability and overall performance, Toshiba Digital Solutions has expanded the functionalities and solvers available in SQBM+.

->A solution for cubic and quartic problems is called a PUBO solver.

->QPLIB solver: For quadratic binary programming problems with linear constraints.

SQBM+ applications:

Finance: Risk management, fraud detection, portfolio optimization.

Drug development involves two key processes: identifying potential therapeutic options and streamlining clinical trials.

Logistics: scheduling, inventory control, and route planning.

Enhancement of AI: Neural network training, machine learning algorithm improvement.

Genetic engineering: enhancing gene editing procedures, creating artificial proteins.

Gaining from the use of SQBM+

Problem-solving more quickly: Obtain ideal solutions more quickly than with conventional techniques.

Better decision-making: Acquire a deeper understanding of intricate issues and make more knowledgeable choices.

Lower expenses: Get rid of pricey gear and streamline the process of correcting problems.

Enhanced effectiveness may be attained by optimizing operations and resource allocation.

All things considered, SQBM+ proves to be an effective tool for solving difficult optimization issues in a variety of domains. Its software-based methodology, together with its accuracy, speed, and scalability, make it a desirable choice for businesses looking to maximize their productivity and decision-making.

Accessible via the Azure Marketplace

Azure Marketplace is an online marketplace designed for developers and IT professionals that offers certified, optimized solutions developed on or for Azure. The best location to get certified and Azure-optimized software for your requirements is the Azure Marketplace.

SQBM+: What is it?

Quantum-inspired: The Simulated Bifurcation Machine (SBM), a device originating from quantum computing research, serves as the model for SQBM+. It uses ideas from quantum physics to approach optimization issues in a novel way, even if it is not a real quantum solution.

Optimization solver: SQBM+ is useful for a variety of applications since it is excellent at identifying the best answers in situations when there are many options.

Software-based: SQBM+ doesn’t need specific hardware since it can operate on regular computers that include GPUs or FPGAs, unlike quantum computers. It becomes more affordable and accessible as a result.

How is SQBM+ operated?

Distinctive hardware is not needed for SQBM+, unlike real quantum computers. Rather, it functions as a cloud service or on regular PCs equipped with GPUs and FPGAs. The SB Algorithm finds the best solutions by effectively navigating through large solution spaces using the simulation of quantum events.

Is SQBM+ a replacement for quantum computing?

SQBM+ is not a straight substitute, sorry. It is a supplementary tool that uses conventional hardware to accomplish its goals while using ideas derived from quantum computing. Although it has benefits in certain domains, genuine quantum computers might eventually be able to solve problems much more effectively.

Read more on Govindhtech.com

[physics-scholars]

Who can give us a list of the different divisions of mathematics, eg. Statistics, Algebra, Geometry, Trigonometry, How many others are missing?

Some of the major divisions of mathematics:

1. Algebra

2. Analysis (which includes calculus, real analysis, complex analysis, functional analysis, etc.)

3. Geometry (which includes Euclidean geometry, topology, differential geometry, etc.)

4. Number theory

5. Combinatorics

6. Logic

7. Set theory

8. Statistics (which includes probability theory, inferential statistics, and descriptive statistics)

9. Numerical analysis

10. Optimization

11.Game theory

12. Graph theory

13. Cryptography

14. Information theory

15. Applied mathematics (which includes many of the above areas applied to specific fields like physics, engineering, economics, etc.)

16. Algebraic geometry

17. Differential equations

18. Dynamical systems

19. Mathematical physics

20. Mathematical logic

21. Mathematical modeling

22. Nonlinear analysis

23. Partial differential equations

24. Representation theory

25. Topology

26. Category theory

27. Probability theory

28. Stochastic processes

29. Number theory

30. Mathematical finance

31. Arithmetic geometry

32. Homological algebra

33. Lie theory

34. Algebraic topology

35. Functional analysis

36. Operator theory

37. Geometric analysis

38. Mathematical biology

39. Control theory

40. Differential topology

41. Computational mathematics

42. Information geometry

43. Symplectic geometry

44. Measure theory

45. Algebraic combinatorics

46. Mathematical education

47. History of mathematics

48. Mathematical philosophy

49. Quantum field theory

50. Representation theory

51. Noncommutative geometry

52. Topological data analysis

53. Group theory

54. Algebraic number theory

55. Discrete mathematics

56. Game theory

57. Information theory

58. Mathematical physics

59. Mathematical logic

60. Model theory

61. Commutative algebra

62. Complex dynamics

63. Differential geometry

64. Harmonic analysis

65. Mathematical optimization

66. Mathematical statistics

67. Category theory

68. Combinatorial game theory

69. Computational algebra

70. Financial mathematics

71. Quantum computing

72. Graph theory

73. Mathematical biology

74. Mathematical physics

75. Numerical linear algebra

76. Operator algebras

77. Partial differential equations

78. Probabilistic combinatorics

79. Quantum information theory

80. Random matrix theory

81. Representation theory

82. Topological quantum field theory

83. Algebraic coding theory

84. Arithmetic combinatorics

85. Convex geometry

86. Discrete geometry

87. Financial mathematics

88. Game theory

89.Mathematical logic

90. Mathematical optimization

91. Ergodic theory

92. Geometric group theory

93. Homotopy theory

94. Inverse problems

95. K-theory

96. Model categories

97. Nonlinear analysis

Number theory

98. Optimal control

99. PDEs on manifolds

100. Quantum field theory

101. Random walks

102. Statistical mechanics

103. Algebraic topology

104. Combinatorial optimization

105. Combinatorial representation theory

106. Computational group theory

107. Convex analysis

108. Cryptography

109. Dynamical systems

110. Fractional calculus

111. Geometric topology

112. Lie groups and Lie algebras

113. Measure theory and integration

114. Operator theory and functional analysis

115. Spectral theory

116. Topological dynamics

117. Topological graph theory

118. Algorithmic graph theory

119. Applied algebraic geometry

120. Arithmetic dynamics

121. Combinatorial geometry

122. Complex analysis

123. Differential topology

124. Discrete optimization

125. Elliptic curves

126. Geometric measure theory

127. Graph coloring

128. Homological stability

129. Information-based complexity

130. Mathematical physics

131. Model theory

132. Nonlinear waves

133. Number theory

134. Orthogonal polynomials

134. Quantum mechanics

135. Smooth dynamical systems

137. Topological combinatorics

138. Topological quantum computing

139. Algebraic K-theory

140. Analytic number theory

141. Combinatorial designs

142. Computational complexity theory

143. Dynamical systems and chaos

144. Geometric analysis

145. Homotopy type theory

146. Mathematical modeling

147. Model categories

148. Nonlinear functional analysis

149. Numerical analysis

150. Operator algebras and quantum groups

151. Random structures and algorithms

152. Topological methods in combinatorics

153. Algebraic geometry and topology

154. C*-algebras

155. Computational algebraic geometry

156. Ergodic theory and entropy

157. Formal proof

158. Lie theory and representation theory

159. Mathematical finance

These are just a small selection of the many different subfields of mathematics.

The subject of mathematics is incredibly vast and diverse, encompassing many different areas of study and application.

[thousandmaths]

Day 28: Björner and Brenti Compute Things

Today I continued reading Chapter 3 of Combinatorics of Coxeter Groups by Björner and Brenti, and I woke up early to compute examples and preparing notes for section 3.4, which I then presented.

------

Bj/Br

Section 3.4 is devoted to computational aspects of Coxeter groups, and in particular focuses on the normal form, which is the lexicographically smallest reduced word that represents a given group element. This is a lot of words but it really just means “numerical order”: if 165312 and 165132 both represent the same element, then the former is inelligible to be the normal form because it is too big.

The reason that we want to do this are many, but chief among them is that if you can find a systematic way to write group elements in normal form, then you can solve the Word Problem. And that’s cool because in many groups you simply can’t solve the Word Problem at all. It’s less cool in that the technicalities for doing so are numerous, and there are lots of fiddly details that I will not be able to completely expunge.

I’ll write $w_*$ for $w_{[n-1]}$ and $^*w$ for $^{[n-1]}w$, when $n$ is clear from context, and we write $[w]$ for the normal form of $w$.

The point of the section is to build up enough technicalities to state and prove Theorem 3.4.7.

Theorem 3.4.7. Let $(W,S)$ be a Coxeter group of rank $n$, with $v\in W$ and $s\in S$. Then there exists $j\in[n-1]$ such that

$$ [vs] = \left\{ \begin{array}{ll} {[v_*]} \, s_n \, [s_n{^*v}s] & \text{if ~} ^*vs\neq e \text{ is distinguished}, \\\ {[v_*s_j]} \, [{^*}v] & \text{if ~} ^*vs \text{ is not distinguished}, \\\ {[v_*]} & \text{if ~}^*v=s=s_n, \\\ {s} & \text{if ~} v=e. \end{array} \right. $$

As the authors remark in the text, this theorem does not quite have all the pieces to create an algorithm: missing in particular are how to check whether an arbitrary group element $ws$ is in the right-quotient $^{[n-1]}W$, and if not, the explicit computation of $j$.

It is worth noting that the first problem can be solved in a somewhat inefficient way, which are nevertheless not horrible if we could compute lengths in constant time (i.e. an oracle for $\ell$). Namely, we can use the naïve algorithm of multiplying by all possible $t\in J$ and checking if any of them decrease the length. Notice that in type A we do have such an oracle, since $\ell(\sigma)$ is the number of inversions.

The latter problem has no such recourse: the naïve algorithm is of course to try all the $j$ to see which one works, but to see whether “it works”, you have to solve the word problem in $W$. But if we could do that, then there really wouldn't be any reason for all the normal form business in the first place. In a footnote, Björner and Brenti hint at a possible way to get around that, but that will have to wait for Chapter 4 :)

I also used the Theorem to compute an actual example and trust me, you don’t want to see it. However, I’m kind of glad I did it because it brought me to the following realization: the Theorem is almost entirely unusable as a recursive solution (i.e. starting the word and peeling off $s_i$ from the right), even for type A. Instead, the right thing to do is solve the problem iteratively, starting with the identity and adding on $s_i$ successively to the right. In this way you don't have to do any work to compute $^*v$ and $v_*$ since these can be read immediately from $[v]$, which you compute in the previous step.

---

Besides the Theorem (and some technical niceties which follow it), the other major plot point is the normal form forest, which provides a very efficient (and rather cryptic) graphical representation of all the normal forms of group elements. Not much time is devoted to computational aspects here, but it seems that if you are given this forest you do not improve the efficiency of the Theorem much, and you essentially need the Theorem to compute the forest.

The way this works is that you have $n$ trees, one for each generator. In the $i^\text{th}$ tree, you draw a root vertex, and an edge to another labelled $i$. From there, you draw an edge to another vertex labelled $j$ if $ij$ is the normal form of the group element $s_is_j$. Similarly you draw an edge from the most recent vertex to another labelled $k$ if $ijk$ is the normal form of the group element $s_is_js_k$. And so on.

Here is an example: Let $W$ be the type H Coxeter group of rank $3$. We construct the normal form forest as follows:

The first two trees are pretty much always constructed this way. Beyond that things get harder. We show some false branches:

3213 is not permitted since 3213 = 3231 = 2321, which is lexicographically smaller

321232 is not permitted since 321232 = 321323 = 323123 = 232123$,

3212132121 is not permitted since 3212132121 = 3212321212 = 3231231212 = 2321231212.

Other potential false branches were much more easily eliminated by using braid moves directly: no two adjacent labels have the same number, the pattern $31$ (that is, 3 below 1) can never occur since $31=13$, similarly $323$ and $21212$ are forbidden. These seven restrictions account for all possible false branches except for the ultimate case (32121321232) which the interested reader may enjoy showing is not permitted since it is equal to 13212132123.

[thewertsearch]

Anonymous asked: Reading along with you has my old Homestuck theorybrain warmed up after years and years, and seeing Casey captcha'd had me thinking again about how with "only" a few hundred trillion codes, there can't *possibly* be a code for every item you could possibly store, the level of specificity needed to encode a living being is too high. But with how they seem closely tied in to SBURB with their "game-mechanic-y ness" and how important punchcard alchemy is to the game...

...what if they are ALSO taking advantage of the same predestination-y timey-wimeyness SBURB seems to love so much, and rather than allocating codes for every item someone 'could' possibly store (let alone alchemize!), the codes are simply and only allocated to the items someone *will* actually use? The system could even sneakily re-use codes without anyone noticing, if it knows 'ahead of time' that two groups of people are never going to communicate with each other!

So that would mean the codes would actually be allocated individually, every person having their own personal set of item->code correspondences, but nobody would ever *notice*, it would all appear completely seamless. (It's kinda like ... dynamic memory allocation in programming languages, with garbage collection clearing out slots and re-using them when it can be sure the previous occupant of a slot will never be referenced again.)

There's other weird edge cases, like if you uncaptcha an item, it picks up some superficial damage that doesn't change its fundamental identity like a scuff or something or Casey makes a new memory, and then you recaptcha it, obviously the code's gotta stay the same, right? But if you had previously shared that code with a friend and they alchemize it, does it make the *current* version of that item, or the version *at the time you shared it*?

Fuck yeah, Sylladexposting! That’s what I’m talking about! 

Yeah, let’s talk about this.

“Draw the causal arrows going backwards in time!”

The idea that the Sylladex is leveraging predestination to save on index space is absolutely fascinating, and completely in line with how Sburb has been demonstrated to work. It also solves alchemy’s combinatorial problems - you don’t have to worry about item collisions if you just predict ahead of time which items are going to be combined and fill those codes with something that makes sense. 

However, there’s a problem: timeline divergence. 

For the Sylladex to work like this, it would have to somehow account for the fact that every timeline will come with a different set of ‘destined captchalogues’ - far more than the index could hold, if timelines are as easy to create as they seem to be. 

It’s a hard one to solve. If you give John a code, can you really know for sure that in all the myriad futures, he’ll never meet someone whose index is using that code for something else? You could, if there’s some sort of ‘timeline predestination’ in effect - a second-order predestination that limits timeline divergence to a predetermined set of parameters. We’re still in the early days of Homestuck’s temporal dynamics, so I don’t want to speculate further. I’m sure we’ll see more, when more timelines start to pop up.

Anyway, I love this idea. It’s a really elegant way to solve the index’s space problems, and I can’t stop thinking about how predestination would interact with entropy, uncertainty and information theory. I think that this system, or some variation of is, is how the index works, and I’ll be coming back to it later.

Do codes equal items?

Captcha your laptop, restart it, and captcha it again. Its state has changed, but has its code? 

What about a glass of water? Presumably a full glass of water isn’t the same as an empty one. What if you take a sip, does that change the code? What about a single drop? Do two brand-new copies of Wise Guy have different codes? What about two piles of sand?

We don’t, as of yet, have much evidence either way. The kids haven’t done these kinds of experiments, and we haven’t seen the codes of two similar items to compare them. I will, however, list a few ways to resolve this, using anon’s Casey scenario as an example. 

Old Casey and New Casey get their own codes.

An item’s code changes every time it’s capchalogued. Objects are changing microscopically all the time anyway, so an argument could be made that this is the most ‘accurate’ interpretation. It would be incredibly space-inefficient, but some variation of Predestination Theory would solve this.

Old Casey’s code now yields New Casey.

An item’s code always yields the most recently captchalogued version of an object, no matter who’s using it. (This might actually enable faster-than light communication, but we’ve already got a time machine so whatever.)

Depends on what happened to Casey.

There’s some ship-of-theseus tipping point where an object is ‘no longer itself’ anymore. This one is kind of weird, because we’re talking about questions of definition and identity that have been hotly debated for millennia, but maybe Sburb has its own opinion on this. 

You decide! You get the same code, until the object is recognizably different to the player. Captchalogue the same book twice, and whether their codes match is a matter of opinion - your opinion.

[prokopetz]

What's the point of class systems in RPGs? And I don't mean the kind of system where choosing your class just means choosing the skills/feats/equipment your character starts with, which you're then able to expand in whatever direction you want, but the kind of system that's like "if you choose this class, you can have skills A and B, but never skills X and Y. if you want X and Y, you have to play this other class, but then you don't get A and B." it just always seemed needlessly restrctive to me

There are several benefits to having character classes – or something like them – in your game. The following list is by no means complete, but I think it hits most of the high points:

1. They clearly establish the player character archetypes the game expects to see in play. Such archetypes are beneficial from a worldbuilding standpoint (i.e., these are some of the sorts of people you’ll encounter in this setting), a functional standpoint (i.e., these are the things the game’s rules assume a group of player characters will be able to do), and a guiding standpoint (i.e., these are the sorts of things that player characters do). The last one is arguably the most important: many tabletop RPGs suffer from failing to provide any direction with respect to what player characters actually do. With a well-constructed set of character classes, you can simply point to any given class and say “this”.

2. The use of classes or class-like templates makes creating and modifying characters faster, easier, and more reliable. “Faster” and “easier” are obvious, since literally any set of options that’s smaller than “all of them” naturally would be, but “more reliable” is often overlooked. Given an RPG of any non-trivial complexity, it’s fairly easy for a novice player to accidentally create a character who isn’t good at anything, or who’s good at a completely different set of things than they intended them to be good at. Assuming your character classes are competently designed, they come with a guarantee that any given character will be able to contribute within that class’s defined game-mechanical niche. (But don’t undervalue “faster” and “easier”, either; not everybody who’s experienced enough to deal with the nuts and bolts of character construction actually wants to. Sometimes you just want to hit things with your sword!)

3. As an extension of points 1 and 2, a well-designed set of character classes confers strong niche protection. Rather than having to sit everybody down and have a whole discussion to make sure that nobody’s character concept too strongly overlaps with anybody else’s, and that no roles that are vital to the group’s identity and/or the game’s premise are being overlooked, you can just say “everybody pick a different character class” and things will generally work themselves out. Among other things, this means that systems with character classes are typically a better fit for groups who prefer to create characters independently, or groups with erratic or rotating participation where the attending players – and thus, the cast of characters – may vary considerably from session to session.

4. In systems with high mechanical complexity, character classes can reduce the scope of playtesting and the likelihood of unexpected failure modes by limiting the number of combinations of traits that need to be examined. In a system with à la carte character creation, every game-mechanical trait is, in theory, equally likely to be paired with every other game-mechanical trait; even a small number of player character options can lead to a combinatorial explosion of scenarios that need to be playtested. Template-based character creation is a convenient (if not necessarily easy!) way to get a handle on that: not only does it reduce the total number of possible setups you need to test, but it gives you a way to reason about which of those setups are part of the game’s core (and should thus receive lots of attention) and which are edge cases (and can thus be treated more briefly).

5. Restrictions can be a good thing. We all know people who can write for pages and pages if given a very specific prompt, but if they’re told to just write whatever they want, they’ll stare at a blank document forever. You might well be one of those people! Character classes aren’t the only way to provide a structured creative framework during character creation, but they’re one of the better-explored ones.

Of course, if you turn these benefits around it’s also easy to see where a game wouldn’t benefit from having character classes. If your system has a shallow learning curve, the range of game-mechanical traits that player characters can possess is small, character creation is treated as a group activity, and player character archetypes are established through other means, character classes don’t have a whole lot to offer.

[blasphemouscontenttrigger-bct]

Visual Essay Source dump #1

1. “Wired in? Genetic traits and entrepreneurship around the world “

This paper posits that entrepreneurship is directly related to genetics which disagrees with effectuation theory that it’s a learnable skill 

https://www.sciencedirect.com/science/article/abs/pii/S0040162521002201#!

2. “Does crowdfunding really foster innovation? Evidence from the board game industry”

Shamelessly stolen from Ricardo’s blog about how crowdsourcing allows for more novelty ideas with distinct aspects to thrive in crowdsourcing vs traditional investment. 

https://www.sciencedirect.com/science/article/pii/S0040162521001797#sec0001

3.  “ Combinatorial design “

While its mainly used as a mathematical principal I'm using the basic idea of this along with a bastardization of the 4 creative hats to create my on explanation on how new ideas can be created and differing mindsets can produce similar effects. 

https://en.wikipedia.org/wiki/Combinatorial_design#:~:text=Combinatorial%20design%20theory%20is%20the,of%20balance%20and%2For%20symmetry.&text=Combinatorial%20design%20theory%20can%20be,area%20of%20design%20of%20experiments.

4. “ Is innovation design- or technology-driven? Citation as a measure of innovation pollination “

A case study on how having innovation being tech or design based can depend on the product, using Dyson as an example

https://www.sciencedirect.com/science/article/pii/S0959652621008994

5. No! : The Power of Disagreement in a World that Wants to Get Along

“great minds think alike but fool’s rarely differ” and its hard to tell which group you fall into. This book makes the argument that genuine and open discourse is the best path for solid and thought-out ideas 

https://ebookcentral.proquest.com/lib/AUT/detail.action?docID=5164781

[justforbooks]

John Horton Conway: the world’s most charismatic mathematician known for the Game of Life whose inventions ranged from the surreal numbers to the Doomsday Rule

John Horton Conway, who has died aged 82 after contracting Covid-19, was one of the most prolific and charismatic British mathematicians of the 20th century. He was active in many branches of mathematics, including group theory, coding theory, knot theory, geometry, number theory and quadratic forms, as well as in recreational mathematics.

An iconoclastic academic, he held court for over half a century in mathematics departments worldwide, notably at Cambridge and Princeton universities. Although he thought it one of his lesser accomplishments, John is best known for his late 1960s invention of the Game of Life. It is an “organic life” simulation carried out on a square grid of cells, each of which is alive or dead according to how many living neighbours it has. Despite the simplicity of John’s defining rules, it turned out that anything that can be algorithmically computed can be done so within the zero-player Game of Life. While John’s approach was done with pen and paper, the game was a catalyst for computer programmers in the 70s and is now viewed as a watershed development in the field of cellular automata.

John, together with Elwyn Berlekamp and Richard Guy, is credited with co-founding the field of combinatorial game theory. That is the mathematical analysis of games such as noughts and crosses, draughts, chess and Go, as well as a wealth of original games that John and assorted collaborators devised over the decades, such as Phutball (short for Philosopher’s Football), a two-person board game played on a grid using one white stone (the ball) and numerous black stones (representing men), Hackenbush, and Sprouts. This led to a joint book, Winning Ways for Your Mathematical Plays (1982).

In the early 70s, John came up with the definitive refinement of what is now known as the Monster Group. Groups arise from the study of symmetry of objects in mathematics and in nature. For instance, the group of symmetries of an equilateral triangle has six things in it: three rotations and three flips. There are standard “atomic building blocks” of all finite groups of symmetries, which are well understood, and 26 strange additional ones, called sporadic groups, of which the Monster Group, predicted independently by two mathematicians, Bernd Fischer and Bob Griess, is the largest. In 1973 Griess finally constructed the Monster itself, and John then came up with a simpler construction. This group has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 symmetries in it. That is vast beyond comprehension, and about 1,000 times the current estimate for the number of atoms in the Earth.

A few years later, John went further. Along with his former PhD student Simon Norton, he came up with the Monstrous Moonshine conjecture, connecting group theory with the seemingly unrelated study of modular forms. Another of John’s former students, Richard Borcherds, got a Fields medal (one of the two highest honours in mathematics) in 1998 for his successful proof of the conjecture.

John often said that his proudest invention was the surreal numbers, a unifying number system that he conceived in the late 60s, which encompassed ordinary numbers as well as those that are infinitely small or large. He was also particularly pleased with the free will theorem in quantum mechanics from early in the current century, which he formulated with Simon Kochen. In his own words, it says that “if experimenters have free will, then so do elementary particles”.

John was a prodigious mental calculator too. He delighted in being challenged to shout out the day of the week for any randomly chosen date in history, which he could do using his own so-called Doomsday Rule; his methods were adapted and streamlined from Lewis Carroll’s 1887 algorithm. He attributed his success in so many different arenas to his habit of always working simultaneously on several unrelated problems: he might be stuck on most of them but suddenly have an idea leading to a breakthrough on another one.

While many of John’s creations were the products of his own fertile mind, he thrived on collaboration. His best known book, the Atlas of Finite Groups (1985), was co-authored by Norton, Robert Curtis, Richard Parker and Robert Wilson. Around 1976, he helped Roger Penrose refine the analysis of what we now know as Penrose tiles, and he came up with the catchy terms kite and dart for the two basic shapes from which they are made.

John left his mark on the groundbreaking book series The Art of Computer Programming. The computer scientist and author of that work, Donald Knuth, who also wrote a book about Conway’s surreal numbers, said: “Although John was a pure mathematician, he covered so many bases that I’ve cited him more than 25 times (so far) for different contributions to The Art of Computer Programming.”

The American writer Martin Gardner helped to popularise much of John’s output in his Scientific American columns, going back to the late 50s. A decade ago, John worked with the editor Peter Renz on updated versions of some of Gardner’s books of collected columns, again contributing fresh results of significance, including a new proof of Morley’s theorem about triangles. “Anyone who has sat with John has been touched, changed for ever,” said Renz. “He had a way of refining things to their purest forms.”

Born in Liverpool, John was the son of Agnes (nee Boyce) and Cyril Horton Conway. His father was a chemistry lab technician at the Liverpool Institute high school for boys, which John, and later Paul McCartney and George Harrison of the Beatles, attended. He then went to Gonville and Caius College, Cambridge, gaining his BA in 1959. A PhD (1964) followed, under Harold Davenport.

He became an assistant lecturer at Cambridge, later rising to be professor of mathematics. In 1987 he took up the position of John von Neumann professor of applied and computational mathematics at Princeton University, New Jersey.

While he was reportedly shy as a young man, over time John developed a disarming charm and an extrovert Pied Piper persona. He earned a reputation for delivering one brilliant lecture after another, and his classes at both Cambridge and Princeton were invariably oversubscribed. His infectious enthusiasm turned on generations of young people to the joy of research mathematics.

His biographer, Siobhan Roberts, dubbed him “the world’s most lovable egomaniac”, adding, “He is Archimedes, Mick Jagger, Salvador Dalí and Richard Feynman, all rolled into one.”

John was elected FRS, a fellow of the Royal Society, in 1981 – following which he told people he was now officially a Filthy Rotten Swine – and was the first recipient of the London Mathematical Society’s Pólya prize in 1987. He was awarded the Nemmers prize in mathematics (1998), the Leroy P Steele prize for mathematical exposition (2000), and the Joseph Priestley award (2001-02), and received honorary doctorates from the University of Liverpool (2001) and Jacobs University, Bremen (2015). He retired in 2013, when he became emeritus professor at Princeton.

In a 2014 video for the online channel Numberphile, John discussed getting over his feelings of inadequacy in his 20s, facing mortality in his 70s, and his lingering mathematical wish to really understand why the extraordinarily large Monster Group exists.

His first two marriages, to Eileen Howe and Larissa Queen, ended in divorce. He is survived by his third wife, Diana (nee Cutsogeorge), whom he married in 2001, and their son, Gareth; four daughters, Susie, Rosie, Ellie and Annie, from his first marriage; two sons, Alex and Oliver, from his second marriage; three grandchildren; and six great-grandchildren.

• John Horton Conway, mathematician, born 26 December 1937; died 11 April 2020

Daily inspiration. Discover more photos at http://justforbooks.tumblr.com

[lthmath]

Cheryl Praeger

Cheryl Elisabeth Praeger (born 7 September 1948, Toowoomba, Queensland) is an Australian mathematician. Praeger received BSc (1969) and MSc degrees from the University of Queensland (1974), and doctorate from the University of Oxford in 1973 under direction of Peter M. Neumann. She has published widely and has advised 27 PhD students (as of March 2018). She is currently Emeritus Professor of Mathematics at the University of Western Australia. She is best known for her works in group theory, algebraic graph theory and combinatorial designs. Praeger is a Fellow of the Australian Academy of Science, former president of the Australian Mathematical Society (1992–1994 and first female President of the Society), and was appointed as a member of the Order of Australia in 1999 for her service to mathematics in Australia, especially through research and professional associations.

Make a difference Donate to Salvation Army Australia http://bit.ly/2FkQa6V 

[poutpoutlilith]

So what is combinatorics all about?

Combinatorics is a branch of mathematics that deals with enumeration, combination, and permutation of sets of elements. Basically, a very general, abstract notion of the theory of counting.

We have theorems to generalize notions like number sequences to the ring algebra of formal power series (through ordinary, exponential, and Dirichlet series generating functions). We can count under equivalence. For example, consider the black-white colorings of the corners of a square. Some can be turned into others by rotating and flipping the square, being in some sense the same coloring. We want to know how many “same coloring” families there are. The idea is to use group theory, identifying equivalence classes with orbits. We study a lot of recurrences on number families. Stirling numbers as a polynomial change of basis. Partition numbers and Young tableaux, asymptotic equivalence. We also talk a lot about distribution problems and certain identities on finite graphs, usually to do with enumeration and counting. There are also objects called combinatorial designs, which are a good precursor to learning about partially ordered sets (POSETS).

Thanks for the ask! Hope some of that was useful :)

[thatsnotcanonpodcasts]

Robin Hood, Bohemia Interactive, ESRB & Mathematically perfect steak

Welcome back. We've been expecting you.

We have a bit of a longer episode than usual this week because we just had so many interesting people to talk about, including a liar who looks like Hugh Jackman, and one of the most prolific Astronomers to ever live.

But, first up, the Nerds discuss the rumoured Disney Robin Hood remake. In live action. With photorealistic CGI. This sounds terrifying. This is a terrible idea. This will haunt your nightmares forever.

One of Professor's favourite game studios has had a great year, and Professor wants to talk about their future. Bohemia Interactive has some great projects in the works that are well worth checking out, so we've got a summary for you.

Dev-i-Boy has brought us the ESRB's disappointing attempt to resolve the Lootbox debate. He and Professor agree that this is a poor response. Maybe one day there will be a solution, but not today.

Dev-i has also found the algorithm for creating the perfect steak. It involves dozens of factors and complicated equations. But don't pull this paper out next time you go to a barbecue, or everyone will go home before you start cooking.

As usual, we bring you the games of the week. Professor and his girlfriend are finding out why they shouldn't have kids in Think of the Children. DJ and Professor are still playing Generation Zero. Professor is better at surviving the robot apocalypse than he is at raising kids. Dev-i is playing VR chat again. We wish him luck in his quest to become an anime girl.

Live action Robin Hood movie starring animals

            -https://www.hollywoodreporter.com/heat-vision/robin-hood-remake-works-at-disney-blindspotting-director-1289702

Bohemia Interactive sales reaching 68 million USD

            -https://www.bohemia.net/blog/bohemia-interactive-sales-reaching-68-milion-usd-in-2019

ESRB’s new measures to combat loot boxes

                - https://www.theverge.com/2020/4/13/21219192/esrb-new-label-loot-boxes-gacha-game

The mathematically perfect steak

            -https://www.sciencenews.org/article/math-equations-cooking-perfect-steak-beef-meat-simulation

                - https://link.springer.com/article/10.1140%2Fepjp%2Fs13360-020-00311-0

Games Played

Professor

– Think of the Children - https://store.steampowered.com/app/573600/Think_of_the_Children/

Rating: 4.5/5

DJ

– Generation Zero - https://store.steampowered.com/app/704270/Generation_Zero/ 

Rating: 4.5/5

Dev-i-Boy

– VRChat - https://store.steampowered.com/app/438100/VRChat/ 

Rating: 4/5

Other topics discussed

Cats movie butthole cut coming soon

- https://www.polygon.com/2020/4/6/21207710/cats-release-the-butthole-cut

ARMA 3 (open-world, realism-based, military tactical shooter video game developed and published by Bohemia Interactive.)

- https://en.wikipedia.org/wiki/ARMA_3

DayZ (DayZ is a survival video game developed and published by Bohemia Interactive.)

- https://en.wikipedia.org/wiki/DayZ_(video_game)

ARMA 3 APEX : Old man

- https://arma3.com/news/arma-3-apex-old-man-is-now-available

ARMA 3 developers arrested in Greece

- https://en.wikipedia.org/wiki/ARMA_3#Espionage_arrests

ARMA 3 banned in Iran

- https://www.polygon.com/gaming/2012/9/19/3357600/arma-3-banned-in-iran

Vigor (Free-to-play online action game by Bohemia Interactive for the Xbox One.)

- https://en.wikipedia.org/wiki/Vigor_(video_game)

Minecraft Hunger Games

- https://www.yahoo.com/news/blogs/technology-blog/minecraft-hunger-games-exists-just-amazing-imagining-165117705.html

Star Wars Battlefront II (action shooter video game based on the Star Wars film franchise.)

- https://en.wikipedia.org/wiki/Star_Wars_Battlefront_II_(2017_video_game)

Heston Blumenthal's perfect steak

- https://www.sbs.com.au/food/recipes/heston-blumenthals-perfect-steak

Perfect steak journal article

- https://arxiv.org/pdf/1908.10787.pdf

Flory-Huggin’s theory (Flory–Huggins solution theory is a lattice model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing.)

- https://en.wikipedia.org/wiki/Flory%E2%80%93Huggins_solution_theory

Incredible dads save kids compilation

- https://www.youtube.com/watch?v=4RIhUUt88ZM

Oculus Quest (Oculus Quest is our first all-in-one gaming system for virtual reality.)

- https://www.oculus.com/quest/?locale=en_US

Ugandan Knuckles (Ugandan Knuckles is the nickname given to a depiction of the character Knuckles from the Sonic franchise created by YouTuber Gregzilla, which is often used as an avatar by players in the multiplayer game VRChat who repeat phrases like "do you know the way" and memes associated with the country Uganda, most notably the film Who Killed Captain Alex? and Zulul.)

- https://knowyourmeme.com/memes/ugandan-knuckles

Simp (Simp, often interpreted as an acronym for Sucker Idolizing Mediocre Pussy or a portmanteau of "sissy" and "pimp," is a slang expression used to ridicule males who are perceived as being overly invested in a woman and acting submissive to that person.)

- https://knowyourmeme.com/memes/simp

Amiga 500 (The Amiga 500, also known as the A500, is the first low-end Commodore Amiga 16/32-bitmultimedia home/personal computer.)

- https://en.wikipedia.org/wiki/Amiga_500

Conway’s Game Of Life (The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematicianJohn Horton Conway in 1970.)

- https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

 Build a working game of Tetris in Conway's Game of Life

- https://codegolf.stackexchange.com/questions/11880/build-a-working-game-of-tetris-in-conways-game-of-life

The Avengers (British espionage television programme created in 1961.)

- https://en.wikipedia.org/wiki/The_Avengers_(TV_series)

The Avengers (1998 American action spy film adaptation of the British television series of the same name directed by Jeremiah Chechik.)

- https://en.wikipedia.org/wiki/The_Avengers_(1998_film)

Brown note (a infrasonic frequency that would cause humans to lose control of their bowels due to resonance.)

- https://en.wikipedia.org/wiki/Brown_note

Ted Kaczynski (also known as the Unabomber, is an American domestic terrorist, anarchist, and former mathematics professor.)

- https://en.wikipedia.org/wiki/Ted_Kaczynski

Ken Kesey (American novelist, essayist, and countercultural figure. He considered himself a link between the Beat Generation of the 1950s and the hippies of the 1960s.)

- https://en.wikipedia.org/wiki/Ken_Kesey

That’s not COVID (TNC podcast)

- https://thatsnotcanon.com/thatsnotcovidpodcast

Shout Outs

11 April 2020 – John Conway, a renowned mathematician who created one of the first computer games passes away - https://www.cnn.com/2020/04/14/us/john-conway-death-obit-trnd/index.html

John Conway, English mathematician active in the theory of finite groups, knot theory,number theory,combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. A Google search for "Conway's Game of Life" prompts the search engine to automatically start playing the game. It is now commonly used as an introductory exercise in computing classes. Conway used his love of games to connect with children, spending time at math camps across the country. He passed away from complications from COVID-19 at the age of 82 in New Brunswick, New Jersey.

12 April 2020 – Sir Stirling Moss, F1 driver known as one of the best behind the wheel, passes away - https://www.washingtonpost.com/local/obituaries/stirling-moss-f1-driver-known-as-one-of-the-best-behind-the-wheel-dies-at-90/2020/04/12/91f03b9c-7cd3-11ea-9040-68981f488eed_story.html

Sir Stirling Craufurd Moss, a British Formula One racing driver. An inductee into the International Motorsports Hall of Fame, he won 212 of the 529 races he entered across several categories of competition and has been described as "the greatest driver never to win the World Championship". Mr. Moss was known in his sport as “Mr. Motor Racing.” Long after his retirement, he was also considered a British national treasure — a dashing gentleman racer who was chivalrous and always sportsmanlike to his competitors despite the cut and thrust of motor racing. He was knighted by Prince Charles, standing in for the queen, in 2000. Mr. Moss’s sportsmanship was perhaps most evident in 1958, when he could have won the world championship after taking the Portuguese Grand Prix in Porto in his British-made Vanwall racecar. His archrival, Mike Hawthorn, finished second, giving him a key six points, which would have clinched the world title. But Hawthorn, a fellow Englishman, was threatened with disqualification for pushing his stalled Ferrari back onto the track after a spin. His disqualification would have put Mr. Moss in the driver’s seat for the world title. But Mr. Moss told race officials that Hawthorn had pushed his Ferrari only on an off-the-track area and should not be disqualified. His intervention swayed the officials, who awarded Hawthorn second place, eventually enabling him to win the F1 world championship by a single point over Mr. Moss. He passed away from a long illness at the age of 90 in Mayfair, London.

12 April 2020 – Tim Brooke Taylor, best known for his work on The Goodies and I'm Sorry I Haven't A Clue passes away - https://www.etonline.com/tim-brooke-taylor-the-goodies-star-dies-at-79-of-coronavirus-complications-144654

Timothy Julian Brooke-Taylor, English comedian and actor. He was best known as a member of The Goodies, starring in the television series throughout the 1970s and picking up international recognition in Australia and New Zealand. He also appeared as an actor in various sitcoms, and was a panellist on I'm Sorry I Haven't a Clue for almost 50 years. In 2008, Brooke-Taylor was heard in the Doctor Who audio story The Zygon Who Fell To Earth, made by Big Finish Productions. Paul McGann played the Eighth Doctor, and Brooke-Taylor played the part of Mims, a Zygon taking the shape of a human. In 2011, Brooke-Taylor was appointed Officer of the Order of the British Empire (OBE) during Queen Elizabeth II's Birthday Honors, for his services to entertainment. He passed away from complications from COVID-19 at the age of 79 in the United Kingdom.

13 April 2020 – Rick May, who voiced Star Fox 64 and Team Fortress II passed away - https://www.dailymail.co.uk/tvshowbiz/article-8216159/Rick-voiced-Star-Fox-64-Team-Fortress-II-characters-dies-79-coronavirus.html

Rick May, American voice actor and theatrical performer, director, and teacher from Seattle, Washington. He began voice acting in video games in the late 1990s, including roles as Peppy Hare and Andross in Star Fox 64, Peppy Hare might not be one of gaming's most famous characters, but May’s line in 1997's Star Fox 64 where he played Fox McCloud’s mentor is one of the most iconic lines in gaming history - so much so that even Google got in on the beloved meme. Go ahead, Google "Do a barrel roll". His other various campaign characters, include Genghis Khan, in Age of Empires II'; and Soldier in Team Fortress 2. He passed away from complications from COVID-19 at the age of 79 in Seattle,Washington.

14 April 2020 – Pip Baker, one half of the Dr Who writing duo known as Pip and Jane Baker passes away - http://www.doctorwhonews.net/2020/04/pip-baker-died-2020.html

Pip Baker, along with his wife and writing partner Jane, was one of the best-known writers from the mid 80's era of Doctor Who, writing eleven episodes for the series. Together they created the Rani, a female Time Lord scientist who was brought to life so vividly by the late Kate O'Mara, as well a creating the companion Mel. The Bakers scripted or contributed to four serials for the programme in the 1980s: The Mark of the Rani, The Trial of a Time Lord, Parts 9–12 and 14 (also known as Terror of the Vervoids and The Ultimate Foe); and Time and the Rani. They have also written novelisations of these stories, as well as a Make Your Own Adventure With Doctor Who (Find Your Fate With Doctor Who in the United States) gamebook titled Race Against Time. Pip and Jane's audio story The Rani Reaps the Whirlwind featured the return of the Rani and was released in 2000. He passed away from complication from a fall at the age of 91 in the United Kingdom.

Remembrances

5 April 2020 – Honor Blackman - https://en.wikipedia.org/wiki/Honor_Blackman

English actress, widely known for the roles of Cathy Gale in The Avengers, Bond girlPussy Galore in Goldfinger, Julia Daggett in Shalako and Hera in Jason and the Argonauts. She is also known for her role as Laura West in the ITV sitcom The Upper Hand. At 38, she was one of the oldest actresses to play a Bond girl, and was five years older than the star Sean Connery. Albert R. Broccoli said Blackman was cast opposite Sean Connery in the James Bond films based on her success in the British television series The Avengers. He knew that most American audiences would not have seen the programme. Broccoli said, "The Brits would love her because they knew her as Mrs. Gale, the Yanks would like her because she was so good, it was a perfect combination." She died from natural causes at the age of 94 in Lewes, Sussex.

13 April 1938 – Grey Owl - https://en.wikipedia.org/wiki/Grey_Owl

Archibald Stansfeld Belaney, commonly known as Grey Owl, was a British-born conservationist, fur trapper, and writer who pretended to be a First Nations person. While he achieved fame as a conservationist during his life, after his death the revelation that he was not Indigenous, along with other autobiographical fabrications, negatively affected his reputation. Belaney rose to prominence as a notable author and lecturer, primarily on environmental issues. In working with the National Parks Branch, Grey Owl became the subject of many films, and was established as the "'caretaker of park animals' at Riding Mountain National Park in Manitoba" in 1931. Together with his numerous articles, books, films and lectures, his views on conservation reached audiences beyond the borders of Canada. His conservation views largely focused on humans' negative impact on nature through their commodification of nature's resources for profits, and a need for humans to develop a respect for the natural world. Recognition of Belaney has included biographies, a historic plaque at his birthplace, and a 1999 biopic about his life by the director Richard Attenborough. He died from pneumonia at the age of 49 in Prince Albert, Saskatchewan.

13 April 1941 – Annie Jump Cannon - https://en.wikipedia.org/wiki/Annie_Jump_Cannon

American astronomer whose cataloging work was instrumental in the development of contemporary stellar classification. With Edward C. Pickering, she is credited with the creation of the Harvard Classification Scheme, which was the first serious attempt to organize and classify stars based on their temperatures and spectral types. She was nearly deaf throughout her career. She was a suffragist and a member of the National Women's Party. Cannon manually classified more stars in a lifetime than anyone else, with a total of around 350,000 stars. She discovered 300 variable stars, five novas, and one spectroscopic binary, creating a bibliography that included about 200,000 references. She discovered her first star in 1898, though she was not able to confirm it until 1905. When she first started cataloging the stars, she was able to classify 1,000 stars in three years, but by 1913, she was able to work on 200 stars an hour. Cannon could classify three stars a minute just by looking at their spectral patterns and, if using a magnifying glass, could classify stars down to the ninth magnitude, around 16 times fainter than the human eye can see. Her work was also highly accurate. In 1925 she became the first woman to receive an honorary doctorate of science from Oxford University. In 1935, she created the Annie J. Cannon Prize for "the woman of any country, whose contributions to the science of astronomy are the most distinguished." She died from congestive heart failure at the age of 77 in Cambridge, Massachusetts.

13 April 1944 - Cécile Chaminade - https://en.wikipedia.org/wiki/C%C3%A9cile_Chaminade

French composer and pianist. In 1913, she was awarded the Légion d'Honneur, a first for a female composer. Ambroise Thomas said, "This is not a woman who composes, but a composer who is a woman." In 1908 she visited the United States, where she was accorded a hearty welcome. Her compositions were tremendous favorites with the American public, and such pieces as the Scarf Dance or the Ballet No. 1 were to be found in the music libraries of many lovers of piano music of the time. She composed a Konzertstück for piano and orchestra, the ballet music to Callirhoé and other orchestral works. Her songs, such as The Silver Ring and Ritournelle, were also great favorites. In London in November 1901, she made gramophone recordings of seven of her compositions for the Gramophone and Typewriter Company; these are among the most sought-after piano recordings by collectors, though they have been reissued on compact disk. Chaminade was relegated to obscurity for the second half of the 20th century, her piano pieces and songs mostly forgotten, with the Flute Concertino in D major, Op. 107, composed for the 1902 Paris Conservatoire Concours, her most popular piece today. Chaminade's music has been described as tuneful, highly accessible and mildly chromatic, and it may be regarded as bearing the typical characteristics of late-Romantic French music. She died at the age of 86 in Monte Carlo.

Famous Birthdays

13 April 1570 – Guy Fawkes - https://en.wikipedia.org/wiki/Guy_Fawkes

Also known as Guido Fawkes while fighting for the Spanish, was a member of a group of provincial English Catholics who planned the failed Gunpowder Plot of 1605. Fawkes converted to Catholicism and left for mainland Europe, where he fought for Catholic Spain in the Eighty Years' War against Protestant Dutch reformers in the Low Countries. He travelled to Spain to seek support for a Catholic rebellion in England without success. He later met Thomas Wintour, with whom he returned to England. Wintour introduced him to Robert Catesby, who planned to assassinate King James I and restore a Catholic monarch to the throne. The plotters leased an undercroft beneath the House of Lords; Fawkes was placed in charge of the gunpowder which they stockpiled there. The authorities were prompted by an anonymous letter to search Westminster Palace during the early hours of 5 November, and they found Fawkes guarding the explosives. He was questioned and tortured over the next few days and confessed to wanting to blow up the House of Lords. He became synonymous with the Gunpowder Plot, the failure of which has been commemorated in the UK as Guy Fawkes Night since 5 November 1605, when his effigy is traditionally burned on a bonfire, commonly accompanied by fireworks. He was born in Stonegate, York.

13 April 1892 - Robert Watson-Watt - https://en.wikipedia.org/wiki/Robert_Watson-Watt

Sir Robert Alexander Watson-Watt, Scottish pioneer of radio direction finding and radar technology. Watt began his career in radio physics with a job at the Met Office, where he began looking for accurate ways to track thunderstorms using the radio signals given off by lightning. This led to the 1920s development of a system later known as huff-duff. Huff-duff allowed operators to determine the location of an enemy radio in seconds and it became a major part of the network of systems that helped defeat the U-boat threat. It is estimated that huff-duff was used in about a quarter of all attacks on U-boats. In 1935 Watt was asked to comment on reports of a German death ray based on radio. Watt and his assistant Arnold Frederic Wilkins quickly determined it was not possible, but Wilkins suggested using radio signals to locate aircraft at long distances. This led to a February 1935 demonstration where signals from a BBC short-wave transmitter were bounced off a Handley Page Heyford aircraft. Watt led the development of a practical version of this device, which entered service in 1938 under the code name Chain Home. Watson-Watt justified his choice of a non-optimal frequency for his radar, with his often-quoted “cult of the imperfect,” which he stated as “Give them the third-best to go on with; the second-best comes too late, [and] the best never comes.” He was born in Brechin,Angus.

13 April 1899 - Alfred Mosher Butts - https://en.wikipedia.org/wiki/Alfred_Mosher_Butts

American architect, famous for inventing the board gameScrabble in 1938. In the early 1930s after working as an architect but now unemployed, Butts set out to design a board game. He studied existing games and found that games fell into three categories: number games such as dice and bingo; move games such as chess and checkers; and word games such as anagrams. Butts decided to create a game that utilized both chance and skill by combining elements of anagrams and crossword puzzles, a popular pastime of the 1920s. Players would draw seven lettered tiles from a pool and then attempt to form words from their seven letters. A key to the game was Butts' analysis of the English language. Butts studied the front page of The New York Times to calculate how frequently each letter of the alphabet was used. He then used each letter's frequency to determine how many of each letter he would include in the game. He included only four "S" tiles so that the ability to make words plural would not make the game too easy. Butts initially called the game "Lexiko", but later changed the name to "Criss Cross Words", after considering "It", and began to look for a buyer. The game makers he originally contacted rejected the idea, but Butts was tenacious. Eventually, he sold the rights to entrepreneur and game-lover James Brunot, who made a few minor adjustments to the design and renamed the game "Scrabble." To memorialize Butts's importance to the invention of the game, there is a street sign at 35th Avenue and 81st Street in Jackson Heights that is stylized using letters, with their values in Scrabble as a subscript. He was born in Poughkeepsie, New York.

Events of Interest

13 April 1953 – Project MKUltra begins - https://en.wikipedia.org/wiki/Project_MKUltra

Project MKUltra (or MK-Ultra), also called the CIA mind control program, is the code name given to a program of experiments on human subjects that were designed and undertaken by the U.S. Central Intelligence Agency, some of which were illegal. Experiments on humans were intended to identify and develop drugs and procedures to be used in interrogations in order to weaken the individual and force confessions through mind control. The project's intentionally obscure CIA cryptonym is made up of the digraph MK, meaning that the project was sponsored by the agency's Technical Services Staff, followed by the word Ultra which had previously been used to designate the most secret classification of World War II intelligence. Other related cryptonyms include Project MKNAOMI and Project MKDELTA. The project was organized through the Office of Scientific Intelligence of the CIA and coordinated with the United States Army Biological Warfare Laboratories. Code names for drug-related experiments were Project Bluebird and Project Artichoke. The program engaged in many illegal activities, including the use of U.S. and Canadian citizens as its unwitting test subjects, which led to controversy regarding its legitimacy. MKUltra used numerous methods to manipulate its subjects' mental states and brain functions. Techniques included the covert administration of high doses ofpsychoactive drugs (especially LSD) and other chemicals, electroshocks, hypnosis,sensory deprivation, isolation, verbal and sexual abuse, as well as other forms of torture. In December 2018, declassified documents included a letter to an unidentified doctor discussing work on six dogs made to run, turn and stop via remote control and brain implants.

13 April 1970 - Apollo 13 oxygen tank explodes - https://www.history.com/this-day-in-history/apollo-13-oxygen-tank-explodes

On April 13, 1970, disaster strikes 200,000 miles from Earth when oxygen tank No. 2 blows up on Apollo 13, the third manned lunar landing mission. Astronauts James A. Lovell, John L. Swigert, and Fred W. Haise had left Earth two days before for the Fra Mauro highlands of the moon but were forced to turn their attention to simply making it home alive. Mission commander Lovell reported to mission control on Earth: “Houston, we’ve had a problem here,” and it was discovered that the normal supply of oxygen, electricity, light, and water had been disrupted. The landing mission was aborted, and the astronauts and controllers on Earth scrambled to come up with emergency procedures. The crippled spacecraft continued to the moon, circled it, and began a long, cold journey back to Earth. The astronauts and mission control were faced with enormous logistical problems in stabilizing the spacecraft and its air supply and providing enough energy to the damaged fuel cells to allow successful reentry into Earth’s atmosphere. Navigation was another problem, and Apollo 13‘s course was repeatedly corrected with dramatic and untested maneuvers. On April 17, with the world anxiously watching, tragedy turned to triumph as the Apollo 13 astronauts touched down safely in the Pacific Ocean.

13 April 2017 - The US drops the largest ever non-nuclear weapon on Nangarhar Province,Afghanistan.

- https://en.wikipedia.org/wiki/GBU-43/B_MOAB

- https://en.wikipedia.org/wiki/2017_Nangarhar_airstrike

 The GBU-43/B Massive Ordnance Air Blast (commonly known as "Mother of All Bombs") is a large-yield bomb, developed for the United States military by Albert L. Weimorts, Jr. of the Air Force Research Laboratory. At the time of development, it was said to be the most powerful non-nuclear weapon in the American arsenal. The basic principle resembles that of the BLU-82 Daisy Cutter, which was used to clear heavily wooded areas in the Vietnam War. Pentagon officials suggested MOAB might be used as an anti-personnel weapon, as part of the "shock and awe" strategy integral to the 2003 invasion of Iraq. The MOAB is not a penetrator weapon and is primarily intended for soft to medium surface targets covering extended areas and targets in a contained environment such as a deep canyon or within a cave system. The MOAB was first dropped in combat in the 13 April 2017 airstrike against an Islamic State of Iraq and the Levant – Khorasan Province (ISIS) tunnel complex in Achin District, Afghanistan. Casualty figures were initially reported as 36 but increased over the following days as reconnaissance units investigated the site. On 18 April 2017, one senior Afghan security official said the bomb killed 96 Islamic State militants, among them 13 major commanders. Stars and Stripes reported that General Dawlat Waziri, spokesman for Afghanistan's Defense Ministry said that since the strike, the offensive operation in the area was resumed. An Afghan officer also said that trees 100 metres from the impact point had remained standing.

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