Category: Bookmark

🔖 The Erdős discrepancy problem | Polymath1Wiki

Bookmarked The Erdős discrepancy problem (Polymath1Wiki)

🔖 The Entropy Decrement Method and the Erdos Discrepancy Problem | Simons Institute for the Theory of Computing

Bookmarked The Entropy Decrement Method and the Erdos Discrepancy Problem (Simons Institute for the Theory of Computing)

Tuesday, April 11th, 2017 9:30 am – 10:30 am
Structure vs. Randomness
Speaker: Terry Tao, UCLA

We discuss a variant of the density and energy increment arguments that we call an "entropy decrement method", which can be used to locate a scale in which two relevant random variables share very little mutual information, and thus behave somewhat like independent random variables.  We were able to use this method to obtain a new correlation estimate for multiplicative functions, which in turn was used to establish the Erdos discrepancy conjecture that any sequence taking values in {-1,+1} had unbounded sums on homogeneous arithmetic progressions.

🔖 [1509.05363] The Erdos discrepancy problem by Terence Tao | arXiv

Bookmarked [1509.05363] The Erdos discrepancy problem by Terence TaoTerence Tao (arxiv.org)

We show that for any sequence f:N→{−1,+1} taking values in {−1,+1}, the discrepancy
supn,d∈N∣∣∣∣∑j=1nf(jd)∣∣∣∣
of f is infinite. This answers a question of Erdős. In fact the argument also applies to sequences f taking values in the unit sphere of a real or complex Hilbert space. The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when f is replaced by a (stochastic) completely multiplicative function g. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when g usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.

🔖 Sign patterns of the Mobius and Liouville functions | Terence Tao

Bookmarked Sign patterns of the Mobius and Liouville functions by Terence Tao (What's new)
Kaisa Matomäki, Maksym Radziwiłł, and I have just uploaded to the arXiv our paper “Sign patterns of the Liouville and Möbius functions”. This paper is somewhat similar to our previous p…

🔖 [1501.04585] Multiplicative functions in short intervals | arXiv

Bookmarked [1501.04585] Multiplicative functions in short intervals by Kaisa Matomäki, Maksym Radziwiłł (arxiv.org)
We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations in the sum of μ(n) in almost all intervals of the form [x,x+ψ(x)] with ψ(x)→∞ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of xϵ-smooth numbers in intervals of the form [x,x+c(ε)x−−√], recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of λ(n)λ(n+1), with λ(n) Liouville's function, is non-trivially bounded in absolute value by 1−δ for some δ>0. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function f has a positive proportion of sign changes if and only if f is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the M\"obius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

🔖 Takachizu

Bookmarked Takachizu (takachizu.org)
Takachizu is a community archive that identifies and reflects on that which is most valuable about Little Tokyo.
hat tip:

🔖 Categorical informatics

Bookmarked Categorical informatics by David Spivak (math.mit.edu)

"Category theory is a universal modeling language."

Background.

Success is founded on information. A tight connection between success (in anything) and information. It follows that we should (if we want to be more successful) study what information is.

Grant proposals. These are several grant proposals, some funded, some in the pipeline, others not funded, that explain various facets of my research project.

Introductory talk (video, slides).

Blog post, on John Baez's blog Azimuth, about my motivations for studying this subject. (Here's a .pdf version.)

🔖 A New Factor in Evolution by James Mark Baldwin | The American Naturalist: Vol 30, No 354

Bookmarked A New Factor in Evolution by James Mark Baldwin (The American Naturalist: Vol 30, No 354 )

In several recent publications I have developed, from different points of view, some considerations which tend to bring out a certain influence at work in organic evolution which I venture to call "a new factor". I give below the list of references [1] to these publications and shall refer to them by number as this paper proceeds. The object of the present paper is to gather into one sketch an outline of the view of the process of development which these different publications have hinged upon.

The problems involved in a theory of organic development may be gathered up under three great heads: Ontogeny, Phylogeny, Heredity. The general consideration, the " factor " which I propose to bring out, is operative in the first instance, in the field of Ontogeny; I shall consequently speak first of the problem of Ontogeny, then of that of Phylogeny, in so far as the topic dealt with makes it necessary, then of that of Heredity, under the same limitation, and finally, give some definitions and conclusions.

👓 Aviation Cocktail Recipe | Kitchen Swagger

Bookmarked Aviation Cocktail Recipe (Kitchen Swagger)
The Aviation cocktail is a Prohibition-era cocktail, consisting of gin, maraschino liqueur (cherry), crème de violette, and fresh lemon juice.
I can’t remember where I heard about this in the last week (perhaps a reference on a television show?), but it sounded interesting. Sadly, it’s got one exotic and infrequently used ingredient, so I’m debating about making some…

🔖 Pulling the Goalie: Hockey and Investment Implications by Clifford S. Asness, Aaron Brown | SSRN

Bookmarked Pulling the Goalie: Hockey and Investment Implications by Clifford S. Asness, Aaron Brown (SSRN)
We build a simple, but powerful and intuitive, model for when a hockey coach should pull the goalie when trailing. When the model reports that the coaches aren’t doing it nearly early enough, we then ask why, and take away some key lessons for portfolio and risk management, and business in general.

🔖 PlumX Metrics | Plum Analytics

Bookmarked PlumX Metrics (plumanalytics.com)

PlumX Metrics provide insights into the ways people interact with individual pieces of research output (articles, conference proceedings, book chapters, and many more) in the online environment. Examples include, when research is mentioned in the news or is tweeted about. Collectively known as PlumX Metrics, these metrics are divided into five categories to help make sense of the huge amounts of data involved and to enable analysis by comparing like with like.

PlumX gathers and brings together appropriate research metrics for all types of scholarly research output.

We categorize metrics into 5 separate categories: Usage, Captures, Mentions, Social Media, and Citations.

🔖 [1703.04184v2] A Theory for Gender Differences in Variability by Theodore P. Hill and Sergei Tabachnikov

Bookmarked [1703.04184v2] A Theory for Gender Differences in Variability by Theodore P. Hill, Sergei Tabachnikov (arxiv.org)
A selectivity theory is proposed to help explain how one gender of a species might tend to evolve with greater variability than the other gender. Briefly, the theory says that if one sex is relatively selective, then more variable subpopulations of the opposite sex will tend to prevail over those with lesser variability; and conversely, if one sex is relatively non-selective, then less variable subpopulations of the opposite sex will tend to prevail over those with greater variability. This theory makes no assumptions about differences in means between the sexes, nor does it presume that one sex is selective and the other non-selective. Two mathematical models are presented: a statistical analysis using normally distributed fitness values, and a deterministic analysis using a standard system of coupled ordinary differential equations with exponentially distributed fitness levels. The theory is applied to the classical greater male variability hypothesis.