Have a deep view into what people are curious about.

🔗 Mathematics 🔗 Statistics

Benford's law, also called the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.

The graph to the right shows Benford's law for base 10. There is a generalization of the law to numbers expressed in other bases (for example, base 16), and also a generalization from leading 1 digit to leading n digits.

It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, physical and mathematical constants. Like other general principles about natural data—for example the fact that many data sets are well approximated by a normal distribution—there are illustrative examples and explanations that cover many of the cases where Benford's law applies, though there are many other cases where Benford's law applies that resist a simple explanation. It tends to be most accurate when values are distributed across multiple orders of magnitude, especially if the process generating the numbers is described by a power law (which are common in nature).

It is named after physicist Frank Benford, who stated it in 1938 in a paper titled "The Law of Anomalous Numbers", although it had been previously stated by Simon Newcomb in 1881.

🔗 Mathematics

In mathematics, 0.999... (also written as 0.9, among other ways) denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...). This number is equal to 1. In other words, "0.999..." and "1" represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. (In other systems, 0.999... can have the same meaning, a different definition, or be undefined.)

More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all base representations. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, or disciplines—some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.

🔗 Computer graphics

Pixel-art scaling algorithms are graphical filters that are often used in video game console emulators to enhance hand-drawn 2D pixel art graphics. The re-scaling of pixel art is a specialist sub-field of image rescaling.

As pixel-art graphics are usually in very low resolutions, they rely on careful placing of individual pixels, often with a limited palette of colors. This results in graphics that rely on a high amount of stylized visual cues to define complex shapes with very little resolution, down to individual pixels. This makes image scaling of pixel art a particularly difficult problem.

A number of specialized algorithms have been developed to handle pixel-art graphics, as the traditional scaling algorithms do not take such perceptual cues into account.

Since a typical application of this technology is improving the appearance of fourth-generation and earlier video games on arcade and console emulators, many are designed to run in real time for sufficiently small input images at 60 frames per second. This places constraints on the type of programming techniques that can be used for this sort of real-time processing. Many work only on specific scale factors: 2× is the most common, with 3×, 4×, 5× and 6× also present.

🔗 Cryptography 🔗 Cryptography/Computer science

In cryptography, the dining cryptographers problem studies how to perform a secure multi-party computation of the boolean-OR function. David Chaum first proposed this problem in the early 1980s and used it as an illustrative example to show that it was possible to send anonymous messages with unconditional sender and recipient untraceability. Anonymous communication networks based on this problem are often referred to as DC-nets (where DC stands for "dining cryptographers").

Despite the word dining, the dining cryptographers problem is unrelated to the dining philosophers problem.

🔗 Companies 🔗 Marketing & Advertising 🔗 Media 🔗 Philadelphia 🔗 Pennsylvania

N. W. Ayer & Son was a Philadelphia advertising agency founded in 1869. It called itself the oldest advertising agency in the United States. Named after Francis Ayer's father N. W. Ayer, it ventured into advertising in 1884. It created a number of memorable slogans for firms such as De Beers, AT&T and the U.S. Army. The company started to decline in the 1960s and, after a series of mergers, was closed in 2002 with its assets sold to the Publicis Groupe.

🔗 Aviation

A cyclorotor, cycloidal rotor, cycloidal propeller or cyclogiro, is a fluid propulsion device that converts shaft power into the acceleration of a fluid using a rotating axis perpendicular to the direction of fluid motion. It uses several blades with a spanwise axis parallel to the axis of rotation and perpendicular to the direction of fluid motion. These blades are cyclically pitched twice per revolution to produce force (thrust or lift) in any direction normal to the axis of rotation. Cyclorotors are used for propulsion, lift, and control on air and water vehicles. An aircraft using cyclorotors as the primary source of lift, propulsion, and control is known as a cyclogyro or cyclocopter. A unique aspect is that it can change the magnitude and direction of thrust without the need of tilting any aircraft structures. The patented application, used on ships with particular actuation mechanisms both mechanical or hydraulic, is named after German company Voith Turbo.

🔗 Graphic design

Corporate Memphis is a term used (often disparagingly) to describe a flat, geometric art style, widely used in Big Tech illustrations in the late 2010s and early 2020s. It is often criticized as seeming uninspired and dystopian.

🔗 Mathematics

In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so that men and women alternate and nobody sits next to his or her partner. This problem was formulated in 1891 by Édouard Lucas and independently, a few years earlier, by Peter Guthrie Tait in connection with knot theory. For a number of couples equal to 3, 4, 5, ... the number of seating arrangements is

12, 96, 3120, 115200, 5836320, 382072320, 31488549120, ... (sequence A059375 in the OEIS).

Mathematicians have developed formulas and recurrence equations for computing these numbers and related sequences of numbers. Along with their applications to etiquette and knot theory, these numbers also have a graph theoretic interpretation: they count the numbers of matchings and Hamiltonian cycles in certain families of graphs.

🔗 Ships

The Voith Schneider propeller (VSP), also known as a cycloidal drive is a specialized marine propulsion system (MPS). It is highly maneuverable, being able to change the direction of its thrust almost instantaneously. It is widely used on tugs and ferries.

🔗 Mathematics 🔗 History of Science

Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, yet many mathematical problems, both major and minor, still remain unsolved. These unsolved problems occur in multiple domains, including physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and more. Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention.