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🔗 Mathematics

A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles and tiles as well as wallpaper.

The simplest wallpaper group, Group p1, applies when there is no symmetry other than the fact that a pattern repeats over regular intervals in two dimensions, as shown in the section on p1 below.

Consider the following examples of patterns with more forms of symmetry:

Examples A and B have the same wallpaper group; it is called p4m in the IUC notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4g or 4*2 . The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.

The number of symmetry groups depends on the number of dimensions in the patterns. Wallpaper groups apply to the two-dimensional case, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.

A proof that there were only 17 distinct groups of such planar symmeries was first carried out by Evgraf Fedorov in 1891 and then derived independently by George Pólya in 1924. The proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in § The seventeen groups.

🔗 Mathematics 🔗 Statistics

The secretary problem is a problem that demonstrates a scenario involving optimal stopping theory. The problem has been studied extensively in the fields of applied probability, statistics, and decision theory. It is also known as the marriage problem, the sultan's dowry problem, the fussy suitor problem, the googol game, and the best choice problem.

The basic form of the problem is the following: imagine an administrator who wants to hire the best secretary out of ${\displaystyle n}$ rankable applicants for a position. The applicants are interviewed one by one in random order. A decision about each particular applicant is to be made immediately after the interview. Once rejected, an applicant cannot be recalled. During the interview, the administrator gains information sufficient to rank the applicant among all applicants interviewed so far, but is unaware of the quality of yet unseen applicants. The question is about the optimal strategy (stopping rule) to maximize the probability of selecting the best applicant. If the decision can be deferred to the end, this can be solved by the simple maximum selection algorithm of tracking the running maximum (and who achieved it), and selecting the overall maximum at the end. The difficulty is that the decision must be made immediately.

The shortest rigorous proof known so far is provided by the odds algorithm (Bruss 2000). It implies that the optimal win probability is always at least ${\displaystyle 1/e}$ (where e is the base of the natural logarithm), and that the latter holds even in a much greater generality (2003). The optimal stopping rule prescribes always rejecting the first ${\displaystyle \sim n/e}$ applicants that are interviewed and then stopping at the first applicant who is better than every applicant interviewed so far (or continuing to the last applicant if this never occurs). Sometimes this strategy is called the ${\displaystyle 1/e}$ stopping rule, because the probability of stopping at the best applicant with this strategy is about ${\displaystyle 1/e}$ already for moderate values of ${\displaystyle n}$. One reason why the secretary problem has received so much attention is that the optimal policy for the problem (the stopping rule) is simple and selects the single best candidate about 37% of the time, irrespective of whether there are 100 or 100 million applicants.

🔗 Military history

🔗 Computing

In computer science, the ski rental problem is a name given to a class of problems in which there is a choice between continuing to pay a repeating cost or paying a one-time cost which eliminates or reduces the repeating cost.

Congratulations, you have discovered Wikipedia, one of the most popular sites on the Internet and probably the leading informational resource in the world today.

Unfortunately some people are evil and have to be banned. In order to save time, we have compiled this easy questionnaire. It's multiple choice, one answer only.

Q: You have discovered a website which has enormous reach and is used by millions of people every day. You think:

1. Wow, this is awesome! How can I help?
2. Wow, this is awesome! How can I use this to my advantage?

---

Scoring:

1. 1 point
2. 0 points

Marking: 0 points: Fail. Please die in a fire. 1 point: Pass. Welcome!

---

It seems like the authors of this 'test' are evil, willing people to die and in a most horrible manner.

🔗 Astronomy 🔗 Astronomy/Astronomical objects

PSR J1748−2446ad is the fastest-spinning pulsar known, at 716 Hz, or 716 times per second. This pulsar was discovered by Jason W. T. Hessels of McGill University on November 10, 2004 and confirmed on January 8, 2005.

If the neutron star is assumed to contain less than two times the mass of the Sun, within the typical range of neutron stars, its radius is constrained to be less than 16 km. At its equator it is spinning at approximately 24% of the speed of light, or over 70,000 km per second.

The pulsar is located in a globular cluster of stars called Terzan 5, located approximately 18,000 light-years from Earth in the constellation Sagittarius. It is part of a binary system and undergoes regular eclipses with an eclipse magnitude of about 40%. Its orbit is highly circular with a 26-hour period. The other object is at least 0.14 solar masses, with a radius of 5–6 solar radii. Hessels et al. state that the companion may be a "bloated main-sequence star, possibly still filling its Roche Lobe". Hessels et al. go on to speculate that gravitational radiation from the pulsar might be detectable by LIGO.

🔗 Statistics

In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at tables in a Chinese restaurant. Imagine a Chinese restaurant with an infinite number of circular tables, each with infinite capacity. Customer 1 sits at the first table. The next customer either sits at the same table as customer 1, or the next table. This continues, with each customer choosing to either sit at an occupied table with a probability proportional to the number of customers already there (i.e., they are more likely to sit at a table with many customers than few), or an unoccupied table. At time n, the n customers have been partitioned among m ≤ n tables (or blocks of the partition). The results of this process are exchangeable, meaning the order in which the customers sit does not affect the probability of the final distribution. This property greatly simplifies a number of problems in population genetics, linguistic analysis, and image recognition.

David J. Aldous attributes the restaurant analogy to Jim Pitman and Lester Dubins in his 1983 book.

🔗 Computer science

In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space ${\displaystyle \Omega }$ on which a sensible kernel function (measuring similarity between elements of ${\displaystyle \Omega }$) may be defined. For example, various kernels have been proposed for learning from data which are: vectors in ${\displaystyle \mathbb {R} ^{d}}$, discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song , Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in.

The analysis of distributions is fundamental in machine learning and statistics, and many algorithms in these fields rely on information theoretic approaches such as entropy, mutual information, or Kullback–Leibler divergence. However, to estimate these quantities, one must first either perform density estimation, or employ sophisticated space-partitioning/bias-correction strategies which are typically infeasible for high-dimensional data. Commonly, methods for modeling complex distributions rely on parametric assumptions that may be unfounded or computationally challenging (e.g. Gaussian mixture models), while nonparametric methods like kernel density estimation (Note: the smoothing kernels in this context have a different interpretation than the kernels discussed here) or characteristic function representation (via the Fourier transform of the distribution) break down in high-dimensional settings.

Methods based on the kernel embedding of distributions sidestep these problems and also possess the following advantages:

1. Data may be modeled without restrictive assumptions about the form of the distributions and relationships between variables
2. Intermediate density estimation is not needed
3. Practitioners may specify the properties of a distribution most relevant for their problem (incorporating prior knowledge via choice of the kernel)
4. If a characteristic kernel is used, then the embedding can uniquely preserve all information about a distribution, while thanks to the kernel trick, computations on the potentially infinite-dimensional RKHS can be implemented in practice as simple Gram matrix operations
5. Dimensionality-independent rates of convergence for the empirical kernel mean (estimated using samples from the distribution) to the kernel embedding of the true underlying distribution can be proven.
6. Learning algorithms based on this framework exhibit good generalization ability and finite sample convergence, while often being simpler and more effective than information theoretic methods

Thus, learning via the kernel embedding of distributions offers a principled drop-in replacement for information theoretic approaches and is a framework which not only subsumes many popular methods in machine learning and statistics as special cases, but also can lead to entirely new learning algorithms.

🔗 Computing 🔗 Systems 🔗 Business 🔗 Engineering 🔗 Systems/Systems engineering

Parkinson's law of triviality is C. Northcote Parkinson's 1957 argument that members of an organization give disproportionate weight to trivial issues. Parkinson provides the example of a fictional committee whose job was to approve the plans for a nuclear power plant spending the majority of its time on discussions about relatively minor but easy-to-grasp issues, such as what materials to use for the staff bike shed, while neglecting the proposed design of the plant itself, which is far more important and a far more difficult and complex task.

The law has been applied to software development and other activities. The terms bicycle-shed effect, bike-shed effect, and bike-shedding were coined as metaphors to illuminate the law of triviality; it was popularised in the Berkeley Software Distribution community by the Danish software developer Poul-Henning Kamp in 1999 and has spread from there to the whole software industry.