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https://en.wikipedia.org/wiki/Cantor%20space | In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space.
Examples
The Cantor set itself is a Cantor space. But the canonical example of a Cantor space is the countably infinite topological product of the discrete 2-point space {0, 1}. This is usually written as or 2ω (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2ω is an infinite binary sequence, that is a sequence that assumes only the values 0 or 1. Given such a sequence a0, a1, a2,..., one can map it to the real number
This mapping gives a homeomorphism from 2ω onto the Cantor set, demonstrating that 2ω is indeed a Cantor space.
Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, complete metric space. (To see this, note that in such a space, any non-empty perfect set contains two disjoint non-empty perfect subsets of arbitrarily small diameter, and so one can imitate the construction
of the usual Cantor set.) Also, every uncountable, separable, completely metrizable space contains
Cantor spaces as subspaces. This includes most of the common spaces in real analysis.
Characterization
A topological characterization of Cantor spaces is given by Brouwer's theorem:
The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as:
This theorem is also equivalent (via Stone's representation theorem for Boolean algebras) to the fact that any two countable atomless Boolean algebras are isomorphic.
Properties
As can be expected from Brouwer's theorem, Cantor spaces appear in several forms. But many properties of Cantor spaces can be established using 2ω, because its construction as a product makes it amenable to analysis.
Cantor s |
https://en.wikipedia.org/wiki/PF%20%28firewall%29 | PF (Packet Filter, also written pf) is a BSD licensed stateful packet filter, a central piece of software for firewalling. It is comparable to netfilter (iptables), ipfw, and ipfilter.
PF was developed for OpenBSD, but has been ported to many other operating systems.
History
PF was originally designed as replacement for Darren Reed's IPFilter, from which it derives much of its rule syntax. IPFilter was removed from OpenBSD's CVS tree on 30 May 2001 due to OpenBSD developers' concerns with its license.
The initial version of PF was written by Daniel Hartmeier. It appeared in OpenBSD 3.0, which was released on 1 December 2001.
It was later extensively redesigned by Henning Brauer and Ryan McBride with most of the code written by Henning Brauer. Henning Brauer is currently the main developer of PF.
Features
The filtering syntax is similar to IPFilter, with some modifications to make it clearer. Network address translation (NAT) and quality of service (QoS) have been integrated into PF. Features such as pfsync and CARP for failover and redundancy, authpf for session authentication, and ftp-proxy to ease firewalling the difficult FTP protocol, have also extended PF.
Also PF supports SMP (Symmetric multiprocessing) & STO (Stateful Tracking Options).
One of the many innovative features is PF's logging. PF's logging is configurable per rule within the pf.conf and logs are provided from PF by a pseudo-network interface called pflog, which is the only way to lift data from kernel-level mode for user-level programs. Logs may be monitored using standard utilities such as tcpdump, which in OpenBSD has been extended especially for the purpose, or saved to disk in the tcpdump/pcap binary format using the pflogd daemon.
Ports
Apart from running on its home platform OpenBSD, PF has been ported to many other operating systems, however there are major differences in capabilities. Some ports date back many years. OpenBSD always has the latest version with the most features.
|
https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey%20FFT%20algorithm | The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size in terms of N1 smaller DFTs of sizes N2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below.
Because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT. For example, Rader's or Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley–Tukey, or the prime-factor algorithm can be exploited for greater efficiency in separating out relatively prime factors.
The algorithm, along with its recursive application, was invented by Carl Friedrich Gauss. Cooley and Tukey independently rediscovered and popularized it 160 years later.
History
This algorithm, including its recursive application, was invented around 1805 by Carl Friedrich Gauss, who used it to interpolate the trajectories of the asteroids Pallas and Juno, but his work was not widely recognized (being published only posthumously and in Neo-Latin). Gauss did not analyze the asymptotic computational time, however. Various limited forms were also rediscovered several times throughout the 19th and early 20th centuries. FFTs became popular after James Cooley of IBM and John Tukey of Princeton published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer.
Tukey reportedly came up with the idea during a meeting of President Kennedy's Science Advisory Committee discussing ways to detect nuclear-weapon tests in the Soviet Union by employing seismometers located outside the country. These sensors would generate seismological time series. However, analysis |
https://en.wikipedia.org/wiki/Feistel%20cipher | In cryptography, a Feistel cipher (also known as Luby–Rackoff block cipher) is a symmetric structure used in the construction of block ciphers, named after the German-born physicist and cryptographer Horst Feistel, who did pioneering research while working for IBM; it is also commonly known as a Feistel network. A large proportion of block ciphers use the scheme, including the US Data Encryption Standard, the Soviet/Russian GOST and the more recent Blowfish and Twofish ciphers. In a Feistel cipher, encryption and decryption are very similar operations, and both consist of iteratively running a function called a "round function" a fixed number of times.
History
Many modern symmetric block ciphers are based on Feistel networks. Feistel networks were first seen commercially in IBM's Lucifer cipher, designed by Horst Feistel and Don Coppersmith in 1973. Feistel networks gained respectability when the U.S. Federal Government adopted the DES (a cipher based on Lucifer, with changes made by the NSA) in 1976. Like other components of the DES, the iterative nature of the Feistel construction makes implementing the cryptosystem in hardware easier (particularly on the hardware available at the time of DES's design).
Design
A Feistel network uses a round function, a function which takes two inputs a data block and a subkey and returns one output of the same size as the data block. In each round, the round function is run on half of the data to be encrypted, and its output is XORed with the other half of the data. This is repeated a fixed number of times, and the final output is the encrypted data. An important advantage of Feistel networks compared to other cipher designs such as substitution–permutation networks is that the entire operation is guaranteed to be invertible (that is, encrypted data can be decrypted), even if the round function is not itself invertible. The round function can be made arbitrarily complicated, since it does not need to be designed to be invertible |
https://en.wikipedia.org/wiki/Germ%20%28mathematics%29 | In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets. In specific implementations of this idea, the functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed (the functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word local has some meaning.
Name
The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain.
Formal definition
Basic definition
Given a point x of a topological space X, and two maps (where Y is any set), then and define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal; meaning that for all u in U.
Similarly, if S and T are any two subsets of X, then they define the same germ at x if there is again a neighbourhood U of x such that
It is straightforward to see that defining the same germ at x is an equivalence relation (be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written
Given a map f on X, then its germ at x is usually denoted [f ]x. Similarly, the germ at x of a set S is written [S]x. Thus,
A map germ at x in X that maps the point x in X to the point y in Y is denoted as
When using this notation, f is then intended as an entire equivalence class of maps, using the same letter f for any representative map.
Notice that two sets are germ-equivalent at x if and only if their characteristic functions are germ-equivalent at x:
More generally
Maps need not be defined on all of X, and in particular they don't need to |
https://en.wikipedia.org/wiki/Phylogenesis | Phylogenesis (from Greek φῦλον phylon "tribe" + γένεσις genesis "origin") is the biological process by which a taxon (of any rank) appears. The science that studies these processes is called phylogenetics.
These terms may be confused with the term phylogenetics, the application of molecular - analytical methods (i.e. molecular biology and genomics), in the explanation of phylogeny and its research.
Phylogenetic relationships are discovered through phylogenetic inference methods that evaluate observed heritable traits, such as DNA sequences or overall morpho-anatomical, ethological, and other characteristics.
Phylogeny
The result of these analyses is a phylogeny (also known as a phylogenetic tree) – a diagrammatic hypothesis about the history of the evolutionary relationships of a group of organisms. Phylogenetic analyses have become central to understanding biodiversity, evolution, ecological genetics and genomes.
Cladistics
Cladistics (Greek , klados, i.e. "branch") is an approach to biological classification in which organisms are categorized based on shared, derived characteristics that can be traced to a group's most recent common ancestor and are not present in more distant ancestors. Therefore, members of a group are assumed to share a common history and are considered to be closely related.
The cladistic method interprets each character state transformation implied by the distribution of shared character states among taxa (or other terminals) as a potential piece of evidence for grouping. The outcome of a cladistic analysis is a cladogram – a tree-shaped diagram (dendrogram) that is interpreted to represent the best hypothesis of phylogenetic relationships.
Although traditionally such cladograms were generated largely on the basis of morphological characteristics calculated by hand, genetic sequencing data and computational phylogenetics are now commonly used and the parsimony criterion has been abandoned by many phylogeneticists in favor of more "sop |
https://en.wikipedia.org/wiki/Delay%20slot | In computer architecture, a delay slot is an instruction slot being executed without the effects of a preceding instruction. The most common form is a single arbitrary instruction located immediately after a branch instruction on a RISC or DSP architecture; this instruction will execute even if the preceding branch is taken. Thus, by design, the instructions appear to execute in an illogical or incorrect order. It is typical for assemblers to automatically reorder instructions by default, hiding the awkwardness from assembly developers and compilers.
Branch delay slots
When a branch instruction is involved, the location of the following delay slot instruction in the pipeline may be called a branch delay slot. Branch delay slots are found mainly in DSP architectures and older RISC architectures. MIPS, PA-RISC, ETRAX CRIS, SuperH, and SPARC are RISC architectures that each have a single branch delay slot; PowerPC, ARM, Alpha, and RISC-V do not have any. DSP architectures that each have a single branch delay slot include the VS DSP, μPD77230 and TMS320C3x. The SHARC DSP and MIPS-X use a double branch delay slot; such a processor will execute a pair of instructions following a branch instruction before the branch takes effect. The TMS320C4x uses a triple branch delay slot.
The following example shows delayed branches in assembly language for the SHARC DSP including a pair after the RTS instruction. Registers R0 through R9 are cleared to zero in order by number (the register cleared after R6 is R7, not R9). No instruction executes more than once.
R0 = 0;
CALL fn (DB); /* call a function, below at label "fn" */
R1 = 0; /* first delay slot */
R2 = 0; /* second delay slot */
/***** discontinuity here (the CALL takes effect) *****/
R6 = 0; /* the CALL/RTS comes back here, not at "R1 = 0" */
JUMP end (DB);
R7 = 0; /* first delay slot */
R8 = 0; /* second delay slot */
|
https://en.wikipedia.org/wiki/Spin%E2%80%93statistics%20theorem | In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ħ, all particles that move in 3 dimensions have either integer spin or half-integer spin.
Background
Quantum states and indistinguishable particles
In a quantum system, a physical state is described by a state vector. A pair of distinct state vectors are physically equivalent if they differ only by an overall phase factor, ignoring other interactions. A pair of indistinguishable particles such as this have only one state. This means that if the positions of the particles are exchanged (i.e., they undergo a permutation), this does not identify a new physical state, but rather one matching the original physical state. In fact, one cannot tell which particle is in which position.
While the physical state does not change under the exchange of the particles' positions, it is possible for the state vector to change sign as a result of an exchange. Since this sign change is just an overall phase, this does not affect the physical state.
The essential ingredient in proving the spin-statistics relation is relativity, that the physical laws do not change under Lorentz transformations. The field operators transform under Lorentz transformations according to the spin of the particle that they create, by definition.
Additionally, the assumption (known as microcausality) that spacelike-separated fields either commute or anticommute can be made only for relativistic theories with a time direction. Otherwise, the notion of being spacelike is meaningless. However, the proof involves looking at a Euclidean version of spacetime, in which the time direction is treated as a spatial one, as will be now explained.
Lorentz transformations include 3-dimensional rotations and boosts. A boost transfers to a frame of reference with a different velocity and is mathematically |
https://en.wikipedia.org/wiki/List%20of%20general%20topology%20topics | This is a list of general topology topics.
Basic concepts
Topological space
Topological property
Open set, closed set
Clopen set
Closure (topology)
Boundary (topology)
Dense (topology)
G-delta set, F-sigma set
closeness (mathematics)
neighbourhood (mathematics)
Continuity (topology)
Homeomorphism
Local homeomorphism
Open and closed maps
Germ (mathematics)
Base (topology), subbase
Open cover
Covering space
Atlas (topology)
Limits
Limit point
Net (topology)
Filter (topology)
Ultrafilter
Topological properties
Baire category theorem
Nowhere dense
Baire space
Banach–Mazur game
Meagre set
Comeagre set
Compactness and countability
Compact space
Relatively compact subspace
Heine–Borel theorem
Tychonoff's theorem
Finite intersection property
Compactification
Measure of non-compactness
Paracompact space
Locally compact space
Compactly generated space
Axiom of countability
Sequential space
First-countable space
Second-countable space
Separable space
Lindelöf space
Sigma-compact space
Connectedness
Connected space
Separation axioms
T0 space
T1 space
Hausdorff space
Completely Hausdorff space
Regular space
Tychonoff space
Normal space
Urysohn's lemma
Tietze extension theorem
Paracompact
Separated sets
Topological constructions
Direct sum and the dual construction product
Subspace and the dual construction quotient
Topological tensor product
Examples
Discrete space
Locally constant function
Trivial topology
Cofinite topology
Finer topology
Product topology
Restricted product
Quotient space
Unit interval
Continuum (topology)
Extended real number line
Long line (topology)
Sierpinski space
Cantor set, Cantor space, Cantor cube
Space-filling curve
Topologist's sine curve
Uniform norm
Weak topology
Strong topology
Hilbert cube
Lower limit topology
Sorgenfrey plane
Real tree
Compact-open topology
Zariski topology
Kuratowski closure axioms
Unicoherent
Solenoid (mathematics)
Uniform spaces
Uniform continuity
Lipschitz continuity
Uniform isomorphism
Uniform property
Uni |
https://en.wikipedia.org/wiki/Magic%20number%20%28programming%29 | In computer programming, a magic number is any of the following:
A unique value with unexplained meaning or multiple occurrences which could (preferably) be replaced with a named constant
A constant numerical or text value used to identify a file format or protocol; for files, see List of file signatures
A distinctive unique value that is unlikely to be mistaken for other meanings (e.g., Globally Unique Identifiers)
Unnamed numerical constants
The term magic number or magic constant refers to the anti-pattern of using numbers directly in source code. This has been referred to as breaking one of the oldest rules of programming, dating back to the COBOL, FORTRAN and PL/1 manuals of the 1960s. The use of unnamed magic numbers in code obscures the developers' intent in choosing that number, increases opportunities for subtle errors (e.g. is every digit correct in 3.14159265358979323846 and is this equal to 3.14159?) and makes it more difficult for the program to be adapted and extended in the future. Replacing all significant magic numbers with named constants (also called explanatory variables) makes programs easier to read, understand and maintain.
Names chosen to be meaningful in the context of the program can result in code that is more easily understood by a maintainer who is not the original author (or even by the original author after a period of time). An example of an uninformatively named constant is int SIXTEEN = 16, while int NUMBER_OF_BITS = 16 is more descriptive.
The problems associated with magic 'numbers' described above are not limited to numerical types and the term is also applied to other data types where declaring a named constant would be more flexible and communicative. Thus, declaring const string testUserName = "John" is better than several occurrences of the 'magic value' "John" in a test suite.
For example, if it is required to randomly shuffle the values in an array representing a standard pack of playing cards, this pseudocode does |
https://en.wikipedia.org/wiki/Moore%20machine | In the theory of computation, a Moore machine is a finite-state machine whose current output values are determined only by its current state. This is in contrast to a Mealy machine, whose output values are determined both by its current state and by the values of its inputs. Like other finite state machines, in Moore machines, the input typically influences the next state. Thus the input may indirectly influence subsequent outputs, but not the current or immediate output. The Moore machine is named after Edward F. Moore, who presented the concept in a 1956 paper, “Gedanken-experiments on Sequential Machines.”
Formal definition
A Moore machine can be defined as a 6-tuple consisting of the following:
A finite set of states
A start state (also called initial state) which is an element of
A finite set called the input alphabet
A finite set called the output alphabet
A transition function mapping a state and the input alphabet to the next state
An output function mapping each state to the output alphabet
A Moore machine can be regarded as a restricted type of finite-state transducer.
Visual representation
Table
A state transition table is a table listing all the triples in the transition relation .
Diagram
The state diagram for a Moore machine, or Moore diagram, is a diagram state diagram that associates an output value with each state.
Relationship with Mealy machines
As Moore and Mealy machines are both types of finite-state machines, they are equally expressive: either type can be used to parse a regular language.
The difference between Moore machines and Mealy machines is that in the latter, the output of a transition is determined by the combination of current state and current input ( as the domain of ), as opposed to just the current state ( as the domain of ). When represented as a state diagram,
for a Moore machine, each node (state) is labeled with an output value;
for a Mealy machine, each arc (transition) is labeled with an output v |
https://en.wikipedia.org/wiki/Homeomorphism%20%28graph%20theory%29 | In graph theory, two graphs and are homeomorphic if there is a graph isomorphism from some subdivision of to some subdivision of . If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the topological sense.
Subdivision and smoothing
In general, a subdivision of a graph G (sometimes known as an expansion) is a graph resulting from the subdivision of edges in G. The subdivision of some edge e with endpoints {u,v } yields a graph containing one new vertex w, and with an edge set replacing e by two new edges, {u,w } and {w,v }.
For example, the edge e, with endpoints {u,v }:
can be subdivided into two edges, e1 and e2, connecting to a new vertex w:
The reverse operation, smoothing out or smoothing a vertex w with regards to the pair of edges (e1, e2) incident on w, removes both edges containing w and replaces (e1, e2) with a new edge that connects the other endpoints of the pair. Here, it is emphasized that only degree-2 (i.e., 2-valent) vertices can be smoothed.
For example, the simple connected graph with two edges, e1 {u,w } and e2 {w,v }:
has a vertex (namely w) that can be smoothed away, resulting in:
Determining whether for graphs G and H, H is homeomorphic to a subgraph of G, is an NP-complete problem.
Barycentric subdivisions
The barycentric subdivision subdivides each edge of the graph. This is a special subdivision, as it always results in a bipartite graph. This procedure can be repeated, so that the nth barycentric subdivision is the barycentric subdivision of the n−1st barycentric subdivision of the graph. The second such subdivision is always a simple graph.
Embedding on a surface
It is evident that subdividing a graph preserves planarity. Kuratowski's theorem states that
a finite graph is planar if and o |
https://en.wikipedia.org/wiki/CW%20complex | A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.
Definition
CW complex
A CW complex is constructed by taking the union of a sequence of topological spaces such that each is obtained from by gluing copies of k-cells , each homeomorphic to , to by continuous gluing maps . The maps are also called attaching maps.
Each is called the k-skeleton of the complex.
The topology of is weak topology: a subset is open iff is open for each cell .
In the language of category theory, the topology on is the direct limit of the diagram The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:
This partition of X is also called a cellulation.
The construction, in words
The CW complex construction is a straightforward generalization of the following process:
A 0-dimensional CW complex is just a set of zero or more discrete points (with the discrete topology).
A 1-dimensional CW complex is constructed by taking the disjoint union of a 0-dimensional CW complex with one or more copies of the unit interval. For each copy, there is a map that "glues" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points). The topology of the CW complex is the topology of the quotient space defined by these gluing maps.
In general, an n-dimensional CW complex is constructed by taking the disjoint union of a k-dimensional CW complex (for some ) with one or more copies of the n-dimensional ball. For each copy, there is a map that "glues" its boundary (the -dimensiona |
https://en.wikipedia.org/wiki/EROS%20%28microkernel%29 | Extremely Reliable Operating System (EROS) is an operating system developed starting in 1991 at the University of Pennsylvania, and then Johns Hopkins University, and The EROS Group, LLC. Features include automatic data and process persistence, some preliminary real-time support, and capability-based security. EROS is purely a research operating system, and was never deployed in real world use. , development stopped in favor of a successor system, CapROS.
Key concepts
The overriding goal of the EROS system (and its relatives) is to provide strong support at the operating system level for the efficient restructuring of critical applications into small communicating components. Each component can communicate with the others only through protected interfaces, and is isolated from the rest of the system. A protected interface, in this context, is one that is enforced by the lowest level part of the operating system, the kernel. That is the only part of the system that can move information from one process to another. It also has complete control of the machine and (if properly constructed) cannot be bypassed. In EROS, the kernel-provided mechanism by which one component names and invokes the services of another is a capability, using inter-process communication (IPC). By enforcing capability-protected interfaces, the kernel ensures that all communications to a process arrive via an intentionally exported interface. It also ensures that no invocation is possible unless the invoking component holds a valid capability to the invoked component. Protection in capability systems is achieved by restricting the propagation of capabilities from one component to another, often through a security policy termed confinement.
Capability systems naturally promote component-based software structure. This organizational approach is similar to the programming language concept of object-oriented programming, but occurs at larger granularity and does not include the concept of inheritanc |
https://en.wikipedia.org/wiki/Malacology | Malacology is the branch of invertebrate zoology that deals with the study of the Mollusca (mollusks or molluscs), the second-largest phylum of animals in terms of described species after the arthropods. Mollusks include snails and slugs, clams, and cephalopods, along with numerous other kinds, many of which have shells. Malacology derives .
Fields within malacological research include taxonomy, ecology and evolution. Several subdivisions of malacology exist, including conchology, devoted to the study of mollusk shells, and teuthology, the study of cephalopods such as octopus, squid, and cuttlefish. Applied malacology studies medical, veterinary, and agricultural applications, for example the study of mollusks as vectors of schistosomiasis and other diseases.
Archaeology employs malacology to understand the evolution of the climate, the biota of the area, and the usage of the site.
Zoological methods are used in malacological research. Malacological field methods and laboratory methods (such as collecting, documenting and archiving, and molecular techniques) were summarized by Sturm et al. (2006).
History
In 1681, Filippo Bonanni wrote the first book ever published that was solely about seashells, the shells of marine mollusks. The book was entitled: In 1868, the German Malacological Society was founded.
Malacologists
Those who study malacology are known as malacologists. Those who study primarily or exclusively the shells of mollusks are known as conchologists, while those who study mollusks of the class Cephalopoda are teuthologists.
Societies
American Malacological Society
Association of Polish Malacologists ()
Belgian Malacological Society () – French speaking
– Dutch speaking
Brazilian Malacological Society ()
Conchological Society of Great Britain and Ireland
Conchologists of America
Dutch Malacological Society
Estonian Malacological Society
European Quaternary Malacologists
Freshwater Mollusk Conservation Society
German Malacological S |
https://en.wikipedia.org/wiki/Interlisp | Interlisp (also seen with a variety of capitalizations) is a programming environment built around a version of the programming language Lisp. Interlisp development began in 1966 at Bolt, Beranek and Newman (renamed BBN Technologies) in Cambridge, Massachusetts with Lisp implemented for the Digital Equipment Corporation (DEC) PDP-1 computer by Danny Bobrow and D. L. Murphy. In 1970, Alice K. Hartley implemented BBN LISP, which ran on PDP-10 machines running the operating system TENEX (renamed TOPS-20). In 1973, when Danny Bobrow, Warren Teitelman and Ronald Kaplan moved from BBN to the Xerox Palo Alto Research Center (PARC), it was renamed Interlisp. Interlisp became a popular Lisp development tool for artificial intelligence (AI) researchers at Stanford University and elsewhere in the community of the Defense Advanced Research Projects Agency (DARPA). Interlisp was notable for integrating interactive development tools into an integrated development environment (IDE), such as a debugger, an automatic correction tool for simple errors (via do what I mean (DWIM) software design), and analysis tools.
Adaptations
At Xerox PARC, an early attempt was made to define a virtual machine to facilitate porting, termed the Interlisp virtual machine. However, it was not useful as a basis for porting.
Peter Deutsch defined a byte-coded instruction set for Interlisp, and implemented it as a microcode emulator for the Xerox Alto. This was then ported to a series of workstation designs produced by Xerox for internal use and for commercial exploitation, including on the Xerox 1100 (Dolphin), 1108 (Dandelion), 1109 (the floating-point enabled Dandetiger), 1186 (Daybreak), and 1132 (Dorado). Interlisp implementations for these were known collectively as Interlisp-D. Commercially, these were sold as Lisp machines and branded as Xerox AI Workstations when Larry Masinter was the chief scientist of that group. The same designs, but with different software, were also sold under different na |
https://en.wikipedia.org/wiki/Uuencoding | uuencoding is a form of binary-to-text encoding that originated in the Unix programs uuencode and uudecode written by Mary Ann Horton at the University of California, Berkeley in 1980, for encoding binary data for transmission in email systems.
The name "uuencoding" is derived from Unix-to-Unix Copy, i.e. "Unix-to-Unix encoding" is a safe encoding for the transfer of arbitrary files from one Unix system to another Unix system but without guarantee that the intervening links would all be Unix systems. Since an email message might be forwarded through or to computers with different character sets or through transports which are not 8-bit clean, or handled by programs that are not 8-bit clean, forwarding a binary file via email might cause it to be corrupted. By encoding such data into a character subset common to most character sets, the encoded form of such data files was unlikely to be "translated" or corrupted, and would thus arrive intact and unchanged at the destination. The program uudecode reverses the effect of uuencode, recreating the original binary file exactly. uuencode/decode became popular for sending binary (and especially compressed) files by email and posting to Usenet newsgroups, etc.
It has now been largely replaced by MIME and yEnc. With MIME, files that might have been uuencoded are instead transferred with Base64 encoding.
Encoded format
A uuencoded file starts with a header line of the form:
begin <mode> <file><newline>
<mode> is the file's Unix file permissions as three octal digits (e.g. 644, 744). This is typically only significant to Unix-like operating systems.
<file> is the file name to be used when recreating the binary data.
<newline> signifies a newline character, used to terminate each line.
Each data line uses the format:
<length character><formatted characters><newline>
<length character> is a character indicating the number of data bytes which have been encoded on that line. This is an ASCII character determined by adding |
https://en.wikipedia.org/wiki/List%20of%20computability%20and%20complexity%20topics | This is a list of computability and complexity topics, by Wikipedia page.
Computability theory is the part of the theory of computation that deals with what can be computed, in principle. Computational complexity theory deals with how hard computations are, in quantitative terms, both with upper bounds (algorithms whose complexity in the worst cases, as use of computing resources, can be estimated), and from below (proofs that no procedure to carry out some task can be very fast).
For more abstract foundational matters, see the list of mathematical logic topics. See also list of algorithms, list of algorithm general topics.
Calculation
Lookup table
Mathematical table
Multiplication table
Generating trigonometric tables
History of computers
Multiplication algorithm
Peasant multiplication
Division by two
Exponentiating by squaring
Addition chain
Scholz conjecture
Presburger arithmetic
Computability theory: models of computation
Arithmetic circuits
Algorithm
Procedure, recursion
Finite state automaton
Mealy machine
Minsky register machine
Moore machine
State diagram
State transition system
Deterministic finite automaton
Nondeterministic finite automaton
Generalized nondeterministic finite automaton
Regular language
Pumping lemma
Myhill-Nerode theorem
Regular expression
Regular grammar
Prefix grammar
Tree automaton
Pushdown automaton
Context-free grammar
Büchi automaton
Chomsky hierarchy
Context-sensitive language, context-sensitive grammar
Recursively enumerable language
Register machine
Stack machine
Petri net
Post machine
Rewriting
Markov algorithm
Term rewriting
String rewriting system
L-system
Knuth–Bendix completion algorithm
Star height
Star height problem
Generalized star height problem
Cellular automaton
Rule 110 cellular automaton
Conway's Game of Life
Langton's ant
Edge of chaos
Turing machine
Deterministic Turing machine
Non-deterministic Turing machine
Alternating automaton
Alternating Turing machine
Turing-complete
Turing tarpit
Oracle machine
Lambda |
https://en.wikipedia.org/wiki/Outline%20of%20linear%20algebra | <noinclude>This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices.
Linear equations
Linear equation
System of linear equations
Determinant
Minor
Cauchy–Binet formula
Cramer's rule
Gaussian elimination
Gauss–Jordan elimination
Overcompleteness
Strassen algorithm
Matrices
Matrix
Matrix addition
Matrix multiplication
Basis transformation matrix
Characteristic polynomial
Trace
Eigenvalue, eigenvector and eigenspace
Cayley–Hamilton theorem
Spread of a matrix
Jordan normal form
Weyr canonical form
Rank
Matrix inversion, invertible matrix
Pseudoinverse
Adjugate
Transpose
Dot product
Symmetric matrix
Orthogonal matrix
Skew-symmetric matrix
Conjugate transpose
Unitary matrix
Hermitian matrix, Antihermitian matrix
Positive-definite, positive-semidefinite matrix
Pfaffian
Projection
Spectral theorem
Perron–Frobenius theorem
List of matrices
Diagonal matrix, main diagonal
Diagonalizable matrix
Triangular matrix
Tridiagonal matrix
Block matrix
Sparse matrix
Hessenberg matrix
Hessian matrix
Vandermonde matrix
Stochastic matrix
Toeplitz matrix
Circulant matrix
Hankel matrix
(0,1)-matrix
Matrix decompositions
Matrix decomposition
Cholesky decomposition
LU decomposition
QR decomposition
Polar decomposition
Reducing subspace
Spectral theorem
Singular value decomposition
Higher-order singular value decomposition
Schur decomposition
Schur complement
Haynsworth inertia additivity formula
Relations
Matrix equivalence
Matrix congruence
Matrix similarity
Matrix consimilarity
Row equivalence
Computations
Elementary row operations
Householder transformation
Least squares, linear least squares
Gram–Schmidt process
Woodbury matrix identity
Vector spaces
Vector space
Linear combination
Linear span
Linear independence
Scalar multiplication
Basis
Change of basis
Hamel basis
Cyclic decomposition theorem
Dimension theorem for vector spaces
Hamel dimension
Examp |
https://en.wikipedia.org/wiki/Confidentiality | Confidentiality involves a set of rules or a promise usually executed through confidentiality agreements that limits the access or places restrictions on certain types of information.
Legal confidentiality
By law, lawyers are often required to keep confidential anything pertaining to the representation of a client. The duty of confidentiality is much broader than the attorney–client evidentiary privilege, which only covers communications between the attorney and the client.
Both the privilege and the duty serve the purpose of encouraging clients to speak frankly about their cases. This way, lawyers can carry out their duty to provide clients with zealous representation. Otherwise, the opposing side may be able to surprise the lawyer in court with something he did not know about his client, which may weaken the client's position. Also, a distrustful client might hide a relevant fact he thinks is incriminating, but that a skilled lawyer could turn to the client's advantage (for example, by raising affirmative defenses like self-defense) However, most jurisdictions have exceptions for situations where the lawyer has reason to believe that the client may kill or seriously injure someone, may cause substantial injury to the financial interest or property of another, or is using (or seeking to use) the lawyer's services to perpetrate a crime or fraud. In such situations the lawyer has the discretion, but not the obligation, to disclose information designed to prevent the planned action. Most states have a version of this discretionary disclosure rule under Rules of Professional Conduct, Rule 1.6 (or its equivalent). A few jurisdictions have made this traditionally discretionary duty mandatory. For example, see the New Jersey and Virginia Rules of Professional Conduct, Rule 1.6.
In some jurisdictions, the lawyer must try to convince the client to conform his or her conduct to the boundaries of the law before disclosing any otherwise confidential information. These exce |
https://en.wikipedia.org/wiki/Universality%20class | In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite scales, their behavior will become increasingly similar as the limit scale is approached. In particular, asymptotic phenomena such as critical exponents will be the same for all models in the class.
Some well-studied universality classes are the ones containing the Ising model or the percolation theory at their respective phase transition points; these are both families of classes, one for each lattice dimension. Typically, a family of universality classes will have a lower and upper critical dimension: below the lower critical dimension, the universality class becomes degenerate (this dimension is 2d for the Ising model, or for directed percolation, but 1d for undirected percolation), and above the upper critical dimension the critical exponents stabilize and can be calculated by an analog of mean-field theory (this dimension is 4d for Ising or for directed percolation, and 6d for undirected percolation).
List of critical exponents
Critical exponents are defined in terms of the variation of certain physical properties of the system near its phase transition point. These physical properties will include its reduced temperature , its order parameter measuring how much of the system is in the "ordered" phase, the specific heat, and so on.
The exponent is the exponent relating the specific heat C to the reduced temperature: we have . The specific heat will usually be singular at the critical point, but the minus sign in the definition of allows it to remain positive.
The exponent relates the order parameter to the temperature. Unlike most critical exponents it is assumed positive, since the order parameter will usually be zero at the critical point. So we have .
The exponent relates the temperature with the system's respo |
https://en.wikipedia.org/wiki/Full-time%20equivalent | Full-time equivalent (FTE), or whole time equivalent (WTE), is a unit of measurement that indicates the workload of an employed person (or student) in a way that makes workloads or class loads comparable across various contexts. FTE is often used to measure a worker's or student's involvement in a project, or to track cost reductions in an organization. An FTE of 1.0 is equivalent to a full-time worker or student, while an FTE of 0.5 signals half of a full work or school load.
In government
United States
According to the Federal government of the United States, FTE is defined by the Government Accountability Office (GAO) as the number of total hours worked divided by the maximum number of compensable hours in a full-time schedule as defined by law. For example, if the normal schedule for a quarter is defined as 411.25 hours ([35 hours per week × (52 weeks per year – 5 weeks' regulatory vacation)] / 4), then someone working 100 hours during that quarter represents 100/411.25 = 0.24 FTE. Two employees working in total 400 hours during that same quarterly period represent 0.97 FTE. Ex.
The U.S. Office of Management and Budget, or OMB, the President's budget office, will often place upper limits on the total number of FTE that a given agency may utilize each year. In the past, if agencies were given a ceiling on the actual number of employed workers, which was reported on a given day of the year, the agency could employ more than this number for much of the year. Then, as the reporting deadline approached, employees could be let go to reduce the total number to the authorized ceiling on the reporting date. Providing agencies with an FTE ceiling, which is calculated based on the total number of hours worked by all employees throughout the year, irrespective of the total numbers employed at any point in time, prevents agencies from using such a strategy.
Although the generally accepted human-resources meaning for the "E" in FTE is "equivalent", the term is often overl |
https://en.wikipedia.org/wiki/Suspended%20animation | Suspended animation is the temporary (short- or long-term) slowing or stopping of biological function so that physiological capabilities are preserved. It may be either hypometabolic or ametabolic in nature. It may be induced by either endogenous, natural or artificial biological, chemical or physical means. In its natural form, it may be spontaneously reversible as in the case of species demonstrating hypometabolic states of hibernation. When applied with therapeutic intent, as in deep hypothermic circulatory arrest (DHCA), usually technologically mediated revival is required.
Basic principles
Suspended animation is understood as the pausing of life processes by exogenous or endogenous means without terminating life itself. Breathing, heartbeat and other involuntary functions may still occur, but they can only be detected by artificial means. For this reason, this procedure has been associated with a lethargic state in nature when animals or plants appear, over a period, to be dead but then can wake up or prevail without suffering any harm. This has been termed in different contexts hibernation, dormancy or anabiosis (the latter in some aquatic invertebrates and plants in scarcity conditions).
In July 2020, marine biologists reported that aerobic microorganisms (mainly), in "quasi-suspended animation", were found in organically-poor sediments, up to 101.5 million years old, below the seafloor in the South Pacific Gyre (SPG) ("the deadest spot in the ocean"), and could be the longest-living life forms ever found.
This condition of apparent death or interruption of vital signs may be similar to a medical interpretation of suspended animation. It is only possible to recover signs of life if the brain and other vital organs suffer no cell deterioration, necrosis or molecular death principally caused by oxygen deprivation or excess temperature (especially high temperature).
Some examples of people that have returned from this apparent interruption of life lasting |
https://en.wikipedia.org/wiki/Barry%20Boehm | Barry William Boehm (May 16, 1935 – August 20, 2022) was an American software engineer, distinguished professor of computer science, industrial and systems engineering; the TRW Professor of Software Engineering; and founding director of the Center for Systems and Software Engineering at the University of Southern California. He was known for his many contributions to the area of software engineering.
In 1996, Boehm was elected as a member into the National Academy of Engineering for contributions to computer and software architectures and to models of cost, quality, and risk for aerospace systems.
Biography
Boehm was born on May 16, 1935. He received a BA in mathematics from Harvard University in 1957, and an MS in 1961, and PhD from UCLA in 1964, both in mathematics as well. He also received honorary Sc.D. in Computer Science from the U. of Massachusetts in 2000 and in Software Engineering from the Chinese Academy of Sciences in 2011.
In 1955 he started working as a programmer-analyst at General Dynamics. In 1959 he switched to the RAND Corporation, where he was head of the Information Sciences Department until 1973. From 1973 to 1989 he was chief scientist of the Defense Systems Group at TRW Inc. From 1989 to 1992 he served within the U.S. Department of Defense (DoD) as director of the DARPA Information Science and Technology Office, and as director of the DDR&E Software and Computer Technology Office. From 1992 he was TRW Professor of Software Engineering, Computer Science Department, and director, USC Center for Systems and Software Engineering, formerly Center for Software Engineering.
He served on the board of several scientific journals, including the IEEE Transactions on Software Engineering, Computer, IEEE Software, ACM Computing Reviews, Automated Software Engineering, Software Process, and Information and Software Technology.
Awards
Later awards for Boehm included the Office of the Secretary of Defense Award for Excellence in 1992, the ASQC Lifetim |
https://en.wikipedia.org/wiki/List%20of%20rules%20of%20inference | This is a list of rules of inference, logical laws that relate to mathematical formulae.
Introduction
Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.
Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation
indicates such a subderivation from the temporary assumption to .
Rules for propositional calculus
Rules for negations
Reductio ad absurdum (or Negation Introduction)
Reductio ad absurdum (related to the law of excluded middle)
Ex contradictione quodlibet
Rules for conditionals
Deduction theorem (or Conditional Introduction)
Modus ponens (or Conditional Elimination)
Modus tollens
Rules for conjunctions
Adjunction (or Conjunction Introduction)
Simplification (or Conjunction Elimination)
Rules for disjunctions
Addition (or Disjunction Introduction)
Case analysis (or Proof by Cases or Argument by Cases or Disjunction elimination)
Disjunctive syllogism
Constructive dilemma
Rules for biconditionals
Biconditional introduction
Biconditional elimination
Rules of classical predicate calculus
In the following rules, is exactly like except for having the term wherever has the free variable .
Universal Generalization (or Universal Introduction)
Restriction 1: is a variable which does not occur in .
Restriction 2: is not mentioned in any hypothesis or undischarged assumptions.
Universal Instantiation (or Universal Elimination)
Restriction: No free occurrence of in falls within the scope of a quantifier quantifying a variable occurring in .
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https://en.wikipedia.org/wiki/Outline%20of%20category%20theory | The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.
Essence of category theory
Category
Functor
Natural transformation
Branches of category theory
Homological algebra
Diagram chasing
Topos theory
Enriched category theory
Higher category theory
Categorical logic
Specific categories
Category of sets
Concrete category
Category of vector spaces
Category of graded vector spaces
Category of chain complexes
Category of finite dimensional Hilbert spaces
Category of sets and relations
Category of topological spaces
Category of metric spaces
Category of preordered sets
Category of groups
Category of abelian groups
Category of rings
Category of magmas
Category of medial magmas
Objects
Initial object
Terminal object
Zero object
Subobject
Group object
Magma object
Natural number object
Exponential object
Morphisms
Epimorphism
Monomorphism
Zero morphism
Normal morphism
Dual (category theory)
Groupoid
Image (category theory)
Coimage
Commutative diagram
Cartesian morphism
Slice category
Functors
Isomorphism of categories
Natural transformation
Equivalence of categories
Subcategory
Faithful functor
Full functor
Forgetful functor
Yoneda lemma
Representable functor
Functor category
Adjoint functors
Galois connection
Pontryagin duality
Affine scheme
Monad (category theory)
Comonad
Combinatorial species
E |
https://en.wikipedia.org/wiki/KERNAL | KERNAL is Commodore's name for the ROM-resident operating system core in its 8-bit home computers; from the original PET of 1977, followed by the extended but related versions used in its successors: the VIC-20, Commodore 64, Plus/4, Commodore 16, and Commodore 128.
Description
The Commodore 8-bit machines' KERNAL consists of the low-level, close-to-the-hardware OS routines roughly equivalent to the BIOS in IBM PC compatibles (in contrast to the BASIC interpreter routines, also located in ROM) as well as higher-level, device-independent I/O functionality, and is user-callable via a jump table in RAM whose central (oldest) part, for reasons of backwards compatibility, remains largely identical throughout the whole 8-bit series. The KERNAL ROM occupies the last 8 KB of the 8-bit CPU's 64 KB address space ($E000–$FFFF).
The jump table can be modified to point to user-written routines, for example to integrate a fast loader so that its fast replacement routines are used system-wide, or replacing the system text output routine with one that works in bitmapped mode rather than character mode. This use of a jump table was new to small computers at the time.
The Adventure International games published for the VIC-20 on cartridge are an example of software that uses the KERNAL. Because they only use the jump table, the games can be memory dumped to disk, loaded into a Commodore 64, and run without modification.
The KERNAL was initially written for the Commodore PET by John Feagans, who introduced the idea of separating the BASIC routines from the operating system. It was further developed by several people, notably Robert Russell, who added many of the features for the VIC-20 and the C64.
Example
A simple, yet characteristic, example of using the KERNAL is given by the following 6502 assembly language subroutine (written in ca65 assembler format/syntax):
CHROUT = $ffd2 ; CHROUT is the address of the character output routine
CR = $0d |
https://en.wikipedia.org/wiki/Bank%20switching | Bank switching is a technique used in computer design to increase the amount of usable memory beyond the amount directly addressable by the processor instructions. It can be used to configure a system differently at different times; for example, a ROM required to start a system from diskette could be switched out when no longer needed. In video game systems, bank switching allowed larger games to be developed for play on existing consoles.
Bank switching originated in minicomputer systems. Many modern microcontrollers and microprocessors use bank switching to manage random-access memory, non-volatile memory, input-output devices and system management registers in small embedded systems. The technique was common in 8-bit microcomputer systems. Bank-switching may also be used to work around limitations in address bus width, where some hardware constraint prevents straightforward addition of more address lines, and to work around limitations in the ISA, where the addresses generated are narrower than the address bus width. Some control-oriented microprocessors use a bank-switching technique to access internal I/O and control registers, which limits the number of register address bits that must be used in every instruction.
Unlike memory management by paging, data is not exchanged with a mass storage device like disk storage. Data remains in quiescent storage in a memory area that is not currently accessible to the processor (although it may be accessible to the video display, DMA controller, or other subsystems of the computer) without the use of special prefix instructions.
Technique
Bank switching can be considered as a way of extending the address space of processor instructions with some register. Examples:
The follow-on system to a processor with a 12 bit address has a 15 bit address bus, but there is no way to directly specify the high three bits on the address bus. Internal bank registers can be used to provide those bits.
The follow-on system to a processor |
https://en.wikipedia.org/wiki/Gen%20Digital | Gen Digital Inc. (formerly Symantec Corporation and NortonLifeLock) is a multinational software company co-headquartered in Tempe, Arizona and Prague, Czech Republic. The company provides cybersecurity software and services. Gen is a Fortune 500 company and a member of the S&P 500 stock-market index. The company also has development centers in Pune, Chennai and Bangalore. Its portfolio includes Norton, Avast, LifeLock, Avira, AVG, ReputationDefender, and CCleaner.
On October 9, 2014, Symantec declared it would split into two independent publicly traded companies by the end of 2015. One company would focus on security, the other on information management. On January 29, 2016, Symantec sold its information-management subsidiary, named Veritas Technologies, and which Symantec had acquired in 2004, to The Carlyle Group. On August 9, 2019, Broadcom Inc. announced they would be acquiring the Enterprise Security software division of Symantec for $10.7 billion, and the company became known as NortonLifeLock. After completing its merger with Avast in September 2022, the company adopted the name Gen Digital Inc.
History
1982 to 1989
Founded in 1982 by Gary Hendrix with a National Science Foundation grant, Symantec was originally focused on artificial intelligence-related projects, including a database program. Hendrix hired several Stanford University natural language processing researchers as the company's first employees.
In 1984, it became clear that the advanced natural language and database system that Symantec had developed could not be ported from DEC minicomputers to the PC. This left Symantec without a product, but with expertise in natural language database query systems and technology. As a result, later in 1984, Symantec was acquired by another, smaller software startup company, C&E Software, founded by Denis Coleman and Gordon Eubanks and headed by Eubanks. C&E Software developed a combined file management and word processing program called Q&A. Barry Greens |
https://en.wikipedia.org/wiki/Affine%20representation | In mathematics, an affine representation of a topological Lie group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A.
An example is the action of the Euclidean group E(n) on the Euclidean space En.
Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space A. If it does, we may take that as origin and regard A as a vector space; in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general.
See also
Group action
Projective representation
References
.
Homological algebra
Representation theory of Lie algebras
Representation theory of Lie groups |
https://en.wikipedia.org/wiki/CAcert.org | CAcert.org is a community-driven certificate authority that issues free X.509 public key certificates. CAcert.org relies heavily on automation and therefore issues only Domain-validated certificates (and not Extended validation or Organization Validation certificates).
These certificates can be used to digitally sign and encrypt email; encrypt code and documents; and to authenticate and authorize user connections to websites via TLS/SSL.
CAcert Inc. Association
On 24 July 2003, Duane Groth incorporated CAcert Inc. as a non-profit association registered in New South Wales, Australia. CAcert Inc runs CAcert.org—a community-driven certificate authority.
In 2004, the Dutch Internet pioneer Teus Hagen became involved. He served as board member and, in 2008, as president.
Certificate Trust status
A disadvantage of CAcert.org is that its root certificates are not included in the most widely deployed certificate stores and it has to be added by its customers. As of 2021, most browsers, email clients, and operating systems do not automatically trust certificates issued by CAcert. Thus, users receive an "untrusted certificate" warning upon trying to view a website providing X.509 certificate issued by CAcert, or view emails authenticated with CAcert certificates in Microsoft Outlook, Mozilla Thunderbird, etc. CAcert uses its own certificate on its website.
Web browsers
Discussion for inclusion of CAcert root certificate in Mozilla Application Suite and Mozilla Firefox started in 2004. Mozilla had no CA certificate policy at the time. Eventually, Mozilla developed a policy which required CAcert to improve their management system and conduct audits. In April 2007, CAcert formally withdrew its application for inclusion in the Mozilla root program. At the same time, the CA/Browser Forum was established to facilitate communication among browser vendors and Certificate Authorities. Mozilla's advice was incorporated into "Baseline Requirements" used by most major browser ve |
https://en.wikipedia.org/wiki/Quiver%20%28mathematics%29 | In mathematics, especially representation theory, a quiver is another name for a multidigraph; that is, a directed graph where loops and multiple arrows between two vertices are allowed. Quivers are commonly used in representation theory: a representation of a quiver assigns a vector space to each vertex of the quiver and a linear map to each arrow .
In category theory, a quiver can be understood to be the underlying structure of a category, but without composition or a designation of identity morphisms. That is, there is a forgetful functor from (the category of categories) to (the category of multidigraphs). Its left adjoint is a free functor which, from a quiver, makes the corresponding free category.
Definition
A quiver consists of:
The set of vertices of
The set of edges of
Two functions: giving the start or source of the edge, and another function, giving the target of the edge.
This definition is identical to that of a multidigraph.
A morphism of quivers is defined as follows. If and are two quivers, then a morphism of quivers consists of two functions and such that the following diagrams commute:
That is,
and
Category-theoretic definition
The above definition is based in set theory; the category-theoretic definition generalizes this into a functor from the free quiver to the category of sets.
The free quiver (also called the walking quiver, Kronecker quiver, 2-Kronecker quiver or Kronecker category) is a category with two objects, and four morphisms: The objects are and . The four morphisms are and the identity morphisms and That is, the free quiver is
A quiver is then a functor
More generally, a quiver in a category is a functor The category of quivers in is the functor category where:
objects are functors
morphisms are natural transformations between functors.
Note that is the category of presheaves on the opposite category .
Path algebra
If is a quiver, then a path in is a sequence of arrows
s |
https://en.wikipedia.org/wiki/Monad%20%28category%20theory%29 | In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors of some fixed category. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also useful in the theory of datatypes, the denotational semantics of imperative programming languages, and in functional programming languages, allowing languages with non-mutable states to do things such as simulate for-loops; see Monad (functional programming).
Introduction and definition
A monad is a certain type of endofunctor. For example, if and are a pair of adjoint functors, with left adjoint to , then the composition is a monad. If and are inverse functors, the corresponding monad is the identity functor. In general, adjunctions are not equivalences—they relate categories of different natures. The monad theory matters as part of the effort to capture what it is that adjunctions 'preserve'. The other half of the theory, of what can be learned likewise from consideration of , is discussed under the dual theory of comonads.
Formal definition
Throughout this article denotes a category. A monad on consists of an endofunctor together with two natural transformations: (where denotes the identity functor on ) and (where is the functor from to ). These are required to fulfill the following conditions (sometimes called coherence conditions):
(as natural transformations ); here and are formed by "horizontal composition"
(as natural transformations ; here denotes the identity transformation from to ).
We can rewrite these conditions using the following commutative diagrams:
See the article on natural transfor |
https://en.wikipedia.org/wiki/Solomon%20Feferman | Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, recursion theory, and set theory, he was known for his contributions to the history of logic (for instance, via biographical writings on figures such as Kurt Gödel, Alfred Tarski, and Jean van Heijenoort) and as a vocal proponent of the philosophy of mathematics known as predicativism, notably from an anti-platonist stance.
Life
Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to the United States after World War I and had met and married in New York. Neither parent had any advanced education. The family moved to Los Angeles, where Feferman graduated from high school at age 16.
He received his B.S. from the California Institute of Technology in 1948, and in 1957 his Ph.D. in mathematics from the University of California, Berkeley, under Alfred Tarski, after having been drafted and having served in the U.S. Army from 1953 to 1955. In 1956 he was appointed to the Departments of Mathematics and Philosophy at Stanford University, where he later became the Patrick Suppes Professor of Humanities and Sciences. While the majority of his career was spent at Stanford, he also spent time as a post-doctoral fellow at the Institute for Advanced Study in Princeton, a visiting professor at MIT, and a visiting fellow at the University of Oxford (Wolfson College and All Souls College).
Feferman died on 26 July 2016 at his home in Stanford, following an illness that lasted three months and a stroke. At his death, he had been a member of the MAA for 37 years.
Contributions
Feferman was editor-in-chief of the five-volume Collected Works of Kurt Gödel, published by Oxford University Press between 2001 and 2013.
In 2004, together with his wife Anita Burdman Feferman, he published a biography of Alfred Tarski: Alfred Tarski: Life and Logic.
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https://en.wikipedia.org/wiki/Integration%20testing | Integration testing (sometimes called integration and testing, abbreviated I&T) is the phase in software testing in which the whole software module is tested or if it consists of multiple software modules they are combined and then tested as a group. Integration testing is conducted to evaluate the compliance of a system or component with specified functional requirements. It occurs after unit testing and before system testing. Integration testing takes as its input modules that have been unit tested, groups them in larger aggregates, applies tests defined in an integration test plan to those aggregates, and delivers as its output the integrated system ready for system testing.
Approach
Some different types of integration testing are big-bang, mixed (sandwich), risky-hardest, top-down, and bottom-up. Other Integration Patterns are: collaboration integration, backbone integration, layer integration, client-server integration, distributed services integration and high-frequency integration.
In big-bang testing, most of the developed modules are coupled together to form a complete software system or major part of the system and then used for integration testing. This method is very effective for saving time in the integration testing process. However, if the test cases and their results are not recorded properly, the entire integration process will be more complicated and may prevent the testing team from achieving the goal of integration testing.
In bottom-up testing, the lowest level components are tested first, and are then used to facilitate the testing of higher level components. The process is repeated until the component at the top of the hierarchy is tested. All the bottom or low-level modules, procedures or functions are integrated and then tested. After the integration testing of lower level integrated modules, the next level of modules will be formed and can be used for integration testing. This approach is helpful only when all or most of the modules of |
https://en.wikipedia.org/wiki/Call%20signs%20in%20North%20America | Call signs are frequently still used by North American broadcast stations, in addition to amateur radio and other international radio stations that continue to identify by call signs worldwide. Each country has a different set of patterns for its own call signs. Call signs are allocated to ham radio stations in Barbados, Canada, Mexico and the United States.
Many countries have specific conventions for classifying call signs by transmitter characteristics and location. The call sign format for radio and television call signs follows a number of conventions. All call signs begin with a prefix assigned by the International Telecommunication Union. For example, the United States has been assigned the following prefixes: AAA–ALZ, K, N, W. For a complete list, see international call sign allocations.
Bermuda, Bahamas, and the Caribbean
Pertaining to their status as former or current colonies, all of the British West Indies islands shared the VS, ZB–ZJ, and ZN–ZO prefixes. The current, largely post-independence, allocation list is as follows:
Anguilla (in amateur radio VP2E prefix)
Antigua and Barbuda (uses V2 prefix)
Bahamas (has the C6 prefix)
Barbados (uses 8P)
Bermuda (also uses VS, in amateur radio normally VP9)
British Virgin Islands (for amateur radio uses VP2V)
Cayman Islands (ZF for amateur operation, ZF1 for Grand Cayman, ZF8 for Little Cayman and ZF9 for Cayman Brac islands. Visiting reciprocal for all islands is ZF2)
Dominica (Commonwealth of Dominica, uses J7)
Grenada (uses J3)
Jamaica (uses 6Y)
Montserrat (for amateur operation VP2M prefix)
St. Kitts and Nevis (uses V4)
St. Lucia (uses J6)
St. Vincent and the Grenadines (uses J8)
Turks and Caicos Islands (typically uses VP5)
Cuba
Cuba uses the prefixes CL–CM, CO, and T4, with district numbers from 0 to 9 for amateur operations.
Dominican Republic
The Dominican Republic uses the prefixes HI–HJ.
French West Indies
All of the French possessions share the prefix F. Further divisions that are |
https://en.wikipedia.org/wiki/Aviator%20call%20sign | An aviator call sign or aviator callsign is a call sign given to a military pilot, flight officer, and even some enlisted aviators. The call sign is a specialized form of nickname that is used as a substitute for the aviator's given name. It is used on flight suit and flight jacket name tags, painted/displayed beneath the officer's or enlisted aircrewman's name on aircraft fuselages or canopy rails, and in radio conversations. They are most commonly used in tactical jet aircraft communities (i.e., fighter, bomber, attack) than in other aircraft communities (i.e., airlift, mobility, maritime patrol), but their use is not totally exclusive to the former. Many NASA Astronauts with military aviator backgrounds are referred to during spaceflights by their call signs rather than their first names.
The origins of aviator call signs are varied. Most call signs play on or reference on variants of the aviator's firstname or surname. Other inspirations for call signs may include personality traits, middle name, references to historical figures, or past exploits during the pilot's career. Aviator call signs nearly always must come from a member or members of the aviator's squadron, training class, or other cohort.
It is considered bad form to try to give oneself a call sign and it is also common for aviators to be given a fairly derogatory call sign, and the more they complain about it, the more likely it is to stick.
Some aviators use the same call sign throughout their careers; in other cases an aviator might have a series of call signs. For example, U.S. Navy Lieutenant Kara Hultgreen was originally given the call sign "Hulk" because of her ability to bench-press 200 pounds. Later, after a television appearance in which she wore noticeable makeup, she received the call sign "Revlon", and a 1998 biography was entitled Call Sign Revlon.
In fiction
Film
The 1986 film Top Gun, set at the United States Navy Fighter Weapons School, featured several aviators with call signs, |
https://en.wikipedia.org/wiki/Outline%20of%20discrete%20mathematics | Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and analysis.
Included below are many of the standard terms used routinely in university-level courses and in research papers. This is not, however, intended as a complete list of mathematical terms; just a selection of typical terms of art that may be encountered.
Subjects in discrete mathematics
Logic – a study of reasoning
Modal Logic: A type of logic for the study of necessity and probability
Set theory – a study of collections of elements
Number theory – study of integers and integer-valued functions
Combinatorics – a study of Counting
Finite mathematics – a course title
Graph theory – a study of graphs
Digital geometry and digital topology
Algorithmics – a study of methods of calculation
Information theory – a mathematical representation of the conditions and parameters affecting the transmission and processing of information
Computability and complexity theories – deal with theoretical and practical limitations of algorithms
Elementary probability theory and Markov chains
Linear algebra – a study of related linear equations
Functions – an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable)
Partially ordered set –
Probability – concerns with numerical descriptions of the chances of occurrence of an event
Proofs –
Relation – a collection of ordered pairs containing one object from each set
Discrete mathematical disciplines
For further reading in discrete mathematics, beyond a basic level, see thes |
https://en.wikipedia.org/wiki/Homochronous | In telecommunication, the term homochronous describes the relationship between two signals
such that their corresponding significant instants are displaced by a constant interval of time.
Synchronization |
https://en.wikipedia.org/wiki/UPGMA | UPGMA (unweighted pair group method with arithmetic mean) is a simple agglomerative (bottom-up) hierarchical clustering method. It also has a weighted variant, WPGMA, and they are generally attributed to Sokal and Michener.
Note that the unweighted term indicates that all distances contribute equally to each average that is computed and does not refer to the math by which it is achieved. Thus the simple averaging in WPGMA produces a weighted result and the proportional averaging in UPGMA produces an unweighted result (see the working example).
Algorithm
The UPGMA algorithm constructs a rooted tree (dendrogram) that reflects the structure present in a pairwise similarity matrix (or a dissimilarity matrix).
At each step, the nearest two clusters are combined into a higher-level cluster. The distance between any two clusters and , each of size (i.e., cardinality) and , is taken to be the average of all distances between pairs of objects in and in , that is, the mean distance between elements of each cluster:
In other words, at each clustering step, the updated distance between the joined clusters and a new cluster is given by the proportional averaging of the and distances:
The UPGMA algorithm produces rooted dendrograms and requires a constant-rate assumption - that is, it assumes an ultrametric tree in which the distances from the root to every branch tip are equal. When the tips are molecular data (i.e., DNA, RNA and protein) sampled at the same time, the ultrametricity assumption becomes equivalent to assuming a molecular clock.
Working example
This working example is based on a JC69 genetic distance matrix computed from the 5S ribosomal RNA sequence alignment of five bacteria: Bacillus subtilis (), Bacillus stearothermophilus (), Lactobacillus viridescens (), Acholeplasma modicum (), and Micrococcus luteus ().
First step
First clustering
Let us assume that we have five elements and the following matrix of pairwise distances between them :
In |
https://en.wikipedia.org/wiki/Annulus%20%28mathematics%29 | In mathematics, an annulus (: annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word anulus or annulus meaning 'little ring'. The adjectival form is annular (as in annular eclipse).
The open annulus is topologically equivalent to both the open cylinder and the punctured plane.
Area
The area of an annulus is the difference in the areas of the larger circle of radius and the smaller one of radius :
The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord tangent to the inner circle, in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so and are sides of a right-angled triangle with hypotenuse , and the area of the annulus is given by
The area can also be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width and area and then integrating from to :
The area of an annulus sector of angle , with measured in radians, is given by
Complex structure
In complex analysis an annulus in the complex plane is an open region defined as
If is , the region is known as the punctured disk (a disk with a point hole in the center) of radius around the point .
As a subset of the complex plane, an annulus can be considered as a Riemann surface. The complex structure of an annulus depends only on the ratio . Each annulus can be holomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map
The inner radius is then .
The Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus.
The Joukowsky transform conformally maps an annulus onto an ellipse with a slit cut between foci.
See also
References
External links
Annulus d |
https://en.wikipedia.org/wiki/Muller%27s%20ratchet | In evolutionary genetics, Muller's ratchet (named after Hermann Joseph Muller, by analogy with a ratchet effect) is a process which, in the absence of recombination (especially in an asexual population), results in an accumulation of irreversible deleterious mutations. This happens because in the absence of recombination, and assuming reverse mutations are rare, offspring bear at least as much mutational load as their parents. Muller proposed this mechanism as one reason why sexual reproduction may be favored over asexual reproduction, as sexual organisms benefit from recombination and consequent elimination of deleterious mutations. The negative effect of accumulating irreversible deleterious mutations may not be prevalent in organisms which, while they reproduce asexually, also undergo other forms of recombination. This effect has also been observed in those regions of the genomes of sexual organisms that do not undergo recombination.
Etymology
Although Muller discussed the advantages of sexual reproduction in his 1932 talk, it does not contain the word "ratchet". Muller first introduced the term "ratchet" in his 1964 paper, and the phrase "Muller's ratchet" was coined by Joe Felsenstein in his 1974 paper, "The Evolutionary Advantage of Recombination".
Explanation
Asexual reproduction compels genomes to be inherited as indivisible blocks so that once the least mutated genomes in an asexual population begin to carry at least one deleterious mutation, no genomes with fewer such mutations can be expected to be found in future generations (except as a result of back mutation). This results in an eventual accumulation of mutations known as genetic load. In theory, the genetic load carried by asexual populations eventually becomes so great that the population goes extinct. Also, laboratory experiments have confirmed the existence of the ratchet and the consequent extinction of populations in many organisms (under intense drift and when recombinations are not allowed) |
https://en.wikipedia.org/wiki/Gene%20regulatory%20network | A gene (or genetic) regulatory network (GRN) is a collection of molecular regulators that interact with each other and with other substances in the cell to govern the gene expression levels of mRNA and proteins which, in turn, determine the function of the cell. GRN also play a central role in morphogenesis, the creation of body structures, which in turn is central to evolutionary developmental biology (evo-devo).
The regulator can be DNA, RNA, protein or any combination of two or more of these three that form a complex, such as a specific sequence of DNA and a transcription factor to activate that sequence. The interaction can be direct or indirect (through transcribed RNA or translated protein). In general, each mRNA molecule goes on to make a specific protein (or set of proteins). In some cases this protein will be structural, and will accumulate at the cell membrane or within the cell to give it particular structural properties. In other cases the protein will be an enzyme, i.e., a micro-machine that catalyses a certain reaction, such as the breakdown of a food source or toxin. Some proteins though serve only to activate other genes, and these are the transcription factors that are the main players in regulatory networks or cascades. By binding to the promoter region at the start of other genes they turn them on, initiating the production of another protein, and so on. Some transcription factors are inhibitory.
In single-celled organisms, regulatory networks respond to the external environment, optimising the cell at a given time for survival in this environment. Thus a yeast cell, finding itself in a sugar solution, will turn on genes to make enzymes that process the sugar to alcohol. This process, which we associate with wine-making, is how the yeast cell makes its living, gaining energy to multiply, which under normal circumstances would enhance its survival prospects.
In multicellular animals the same principle has been put in the service of gene cascades |
https://en.wikipedia.org/wiki/Computer-aided%20maintenance | Computer-aided maintenance (not to be confused with CAM which usually stands for Computer Aided Manufacturing) refers to systems that utilize software to organize planning, scheduling, and support of maintenance and repair. A common application of such systems is the maintenance of computers, either hardware or software, themselves. It can also apply to the maintenance of other complex systems that require periodic maintenance, such as reminding operators that preventive maintenance is due or even predicting when such maintenance should be performed based on recorded past experience.
Computer aided configuration
The first computer-aided maintenance software came from DEC in the 1980s to configure VAX computers. The software was built using the techniques of artificial intelligence expert systems, because the problem of configuring a VAX required expert knowledge. During the research, the software was called R1 and was renamed XCON when placed in service. Fundamentally, XCON was a rule-based configuration database written as an expert system using forward chaining rules. As one of the first expert systems to be pressed into commercial service it created high expectations, which did not materialize, as DEC lost commercial pre-eminence.
Help Desk software
Help desks frequently use help desk software that captures symptoms of a bug and relates them to fixes, in a fix database. One of the problems with this approach is that the understanding of the problem is embodied in a non-human way, so that solutions are not unified.
Strategies for finding fixes
The bubble-up strategy simply records pairs of symptoms and fixes. The most frequent set of pairs is then presented as a tentative solution, which is then attempted. If the fix works, that fact is further recorded, along with the configuration of the presenting system, into a solutions database.
Oddly enough, shutting down and booting up again manages to 'fix,' or at least 'mask,' a bug in many computer-based systems; |
https://en.wikipedia.org/wiki/Lookup%20table | In computer science, a lookup table (LUT) is an array that replaces runtime computation with a simpler array indexing operation, in a process termed as direct addressing. The savings in processing time can be significant, because retrieving a value from memory is often faster than carrying out an "expensive" computation or input/output operation. The tables may be precalculated and stored in static program storage, calculated (or "pre-fetched") as part of a program's initialization phase (memoization), or even stored in hardware in application-specific platforms. Lookup tables are also used extensively to validate input values by matching against a list of valid (or invalid) items in an array and, in some programming languages, may include pointer functions (or offsets to labels) to process the matching input. FPGAs also make extensive use of reconfigurable, hardware-implemented, lookup tables to provide programmable hardware functionality.
LUTs differ from hash tables in a way that, to retrieve a value with key , a hash table would store the value in the slot where is a hash function i.e. is used to compute the slot, while in the case of LUT, the value is stored in slot , thus directly addressable.
History
Before the advent of computers, lookup tables of values were used to speed up hand calculations of complex functions, such as in trigonometry, logarithms, and statistical density functions.
In ancient (499 AD) India, Aryabhata created one of the first sine tables, which he encoded in a Sanskrit-letter-based number system. In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" Modern school children are often taught to memorize "times tables" to avoid calculations of t |
https://en.wikipedia.org/wiki/ATX | ATX (Advanced Technology Extended) is a motherboard and power supply configuration specification, patented by David Dent in 1995 at Intel, to improve on previous de facto standards like the AT design. It was the first major change in desktop computer enclosure, motherboard and power supply design in many years, improving standardization and interchangeability of parts. The specification defines the dimensions; the mounting points; the I/O panel; and the power and connector interfaces among a computer case, a motherboard, and a power supply.
Overview
ATX is the most common motherboard design. Other standards for smaller boards (including microATX, FlexATX, nano-ITX, and mini-ITX) usually keep the basic rear layout but reduce the size of the board and the number of expansion slots. Dimensions of a full-size ATX board are , which allows many ATX chassis to accept microATX boards. The ATX specifications were released by Intel in 1995 and have been revised numerous times since. The most recent ATX motherboard specification is version 2.2. The most recent ATX12V power supply unit specification is ATX 3.0 released in February 2022.
EATX (Extended ATX) is a bigger version of the ATX motherboard with dimensions. While some dual CPU socket motherboards have been implemented in ATX, the extra size of EATX makes it the typical form factor for dual socket systems, and with sockets that support four or eight memory channels, for single socket systems with a large number of memory slots.
In 2004, Intel announced the BTX (Balanced Technology eXtended) standard, intended as a replacement for ATX. While some manufacturers adopted the new standard, Intel discontinued any future development of BTX in 2006. , the ATX design still remains the de facto standard for personal computers.
Connectors
On the back of the computer case, some major changes were made to the AT standard. Originally AT style cases had only a keyboard connector and expansion slots for add-on card backplates |
https://en.wikipedia.org/wiki/Stelzer%20engine | The Stelzer engine is a two-stroke opposing-piston free-piston engine design proposed by Frank Stelzer. It uses conjoined pistons in a push-pull arrangement which allows for fewer moving parts and simplified manufacturing. An engine of the same design appeared on the cover of the February 1969 issue of Mechanix Illustrated magazine.
Operation
There are two combustion chambers and a central precompression chamber. Control of the air flow between the precompression chamber and the combustion chambers is made by stepped piston rods.
Applications
Applications envisaged for the engine include driving:
An air compressor
A hydraulic pump
A linear generator
Prototypes
A prototype engine was demonstrated in Frankfurt in 1983 and Opel was reported to be interested in it. In 1982, the Government of Ireland agreed to pay half the cost of a factory at Shannon Airport to manufacture the engines. A prototype car with a Stelzer engine and electric transmission was shown at a German motor show in 1983.
See also
Linear alternator
References
External links
-- Two-Stroke Internal Combustion Engine 1983
Diagrams of Stelzer engine and linear alternator
Proposed engines
Free-piston engines |
https://en.wikipedia.org/wiki/Wizardry%20II%3A%20The%20Knight%20of%20Diamonds | Wizardry II: The Knight of Diamonds (originally known as Wizardry: Knight of Diamonds - The Second Scenario) is the second game in the Wizardry series of role-playing video games. It was published in 1982 by Sir-Tech.
Gameplay
The game begins with the city of Llylgamyn under siege. Llylgamyn's rulers have been killed, and the city's only hope is for the recovery of the staff of Gnilda, only obtainable from trading the mystic "Knight of Diamonds" armor from the legendary Knight of Diamonds to fend off the invaders.
The game functions virtually identically to the first scenario, Wizardry: Proving Grounds of the Mad Overlord, with the player guiding a party of up to six adventurers into a six-level dungeon. The original version required players to import characters from the first game, whilst later versions include a pre-generated party and the ability to create new characters. As the game is intended to be played by those who have successfully completed the first game, the difficulty level is intended for characters of at least level 13, and no training area means that lower level characters will go through a "baptism by fire". Mechanical differences include the ability to save the game in the dungeon rather than forcing the characters to exit the dungeon and return to the training grounds, and some of the spells increasing in power (as noted by a message in the dungeon).
Unlike the first scenario, where half of the levels had no purpose plot-wise and could be skipped if the player wished, exploring every level in Knight of Diamonds is necessary to complete the game. Each of the six levels has a piece of the Knight's armor somewhere in the level, and all of the pieces must be collected in order to finish the game. Furthermore, unlike in the first scenario, there are no elevators that can be used to skip levels.
Reception
Softline in 1982 praised Knight of Diamonds variety of monsters and liked that each level of the dungeon had quests. The magazine concluded, "One |
https://en.wikipedia.org/wiki/Xcode | Xcode is Apple's integrated development environment (IDE) for macOS, used to develop software for macOS, iOS, iPadOS, watchOS, tvOS, and visionOS. It was initially released in late 2003; the latest stable release is version 15, released on September 18, 2023, and is available free of charge via the Mac App Store and the Apple Developer website. Registered developers can also download preview releases and prior versions of the suite through the Apple Developer website. Xcode includes command-line tools which enable UNIX-style development via the Terminal app in macOS. They can also be downloaded and installed without the GUI.
Before Xcode, Apple offered developers Project Builder and Interface Builder to develop Mac OS X applications.
Major features
Xcode supports source code for the programming languages: C, C++, Objective-C, Objective-C++, Java, AppleScript, Python, Ruby, ResEdit (Rez), and Swift, with a variety of programming models, including but not limited to Cocoa, Carbon, and Java. Third parties have added support for GNU Pascal, Free Pascal, Ada, C#, Go, Perl, and D.
Xcode can build fat binary (universal binary) files containing code for multiple architectures with the Mach-O executable format. These helped ease the transitions from 32-bit PowerPC to 64-bit PowerPC, from PowerPC to Intel x86, from 32-bit to 64-bit Intel, and most recently from Intel x86 to Apple silicon by allowing developers to distribute a single application to users and letting the operating system automatically choose the appropriate architecture at runtime. Using the iOS SDK, tvOS SDK, and watchOS SDK, Xcode can also be used to compile and debug applications for iOS, iPadOS, tvOS, and watchOS.
Xcode includes the GUI tool Instruments, which runs atop a dynamic tracing framework, DTrace, created by Sun Microsystems and released as part of OpenSolaris.
Xcode also integrates built-in support for source code management using the Git version control system and protocol, allowing the use |
https://en.wikipedia.org/wiki/Automated%20analyser | An automated analyser is a medical laboratory instrument designed to measure various substances and other characteristics in a number of biological samples quickly, with minimal human assistance. These measured properties of blood and other fluids may be useful in the diagnosis of disease.
Photometry is the most common method for testing the amount of a specific analyte in a sample. In this technique, the sample undergoes a reaction to produce a color change. Then, a photometer measures the absorbance of the sample to indirectly measure the concentration of analyte present in the sample. The use of an ion-selective electrode (ISE) is another common analytical method that specifically measures ion concentrations. This typically measures the concentrations of sodium, calcium or potassium present in the sample.
There are various methods of introducing samples into the analyser. Test tubes of samples are often loaded into racks. These racks can be inserted directly into some analysers or, in larger labs, moved along an automated track. More manual methods include inserting tubes directly into circular carousels that rotate to make the sample available. Some analysers require samples to be transferred to sample cups. However, the need to protect the health and safety of laboratory staff has prompted many manufacturers to develop analysers that feature closed tube sampling, preventing workers from direct exposure to samples. Samples can be processed singly, in batches, or continuously.
The automation of laboratory testing does not remove the need for human expertise (results must still be evaluated by medical technologists and other qualified clinical laboratory professionals), but it does ease concerns about error reduction, staffing concerns, and safety.
Routine biochemistry analysers
These are machines that process a large portion of the samples going into a hospital or private medical laboratory. Automation of the testing process has reduced testing time for many |
https://en.wikipedia.org/wiki/Lake%20Powell | Lake Powell is an artificial reservoir on the Colorado River in Utah and Arizona, United States. It is a major vacation destination visited by approximately two million people every year. It is the second largest artificial reservoir by maximum water capacity in the United States behind Lake Mead, storing of water when full. However, Lake Mead has fallen below Lake Powell in size several times during the 21st century in terms of volume of water, depth and surface area.
Lake Powell was created by the flooding of Glen Canyon by the Glen Canyon Dam, which also led to the 1972 creation of Glen Canyon National Recreation Area, a popular summer destination of public land managed by the National Park Service. The reservoir is named for John Wesley Powell, a civil war veteran who explored the river via three wooden boats in 1869. It primarily lies in parts of Garfield, Kane, and San Juan counties in southern Utah, with a small portion in Coconino County in northern Arizona. The northern limits of the lake extend at least as far as the Hite Crossing Bridge.
Lake Powell is a water storage facility for the Upper Basin states of the Colorado River Compact (Colorado, Utah, Wyoming and New Mexico). The Compact specifies that the Upper Basin states are to provide a minimum annual flow of to the Lower Basin states (Arizona, Nevada, and California).
According to US Geological Survey and the Bureau of Reclamation report - In addition to water loss, Lake Powell faced an average annual loss in storage capacity of about 33,270 acre-feet, or 11 billion gallons, per year between 1963 and 2018 because of sediments flowing in from the Colorado and San Juan rivers, according to the report. Those sediments settle at the bottom of the reservoir and decrease the total amount of water the reservoir can hold.
History
Planning
In the 1940s and early 1950s, the United States Bureau of Reclamation planned to construct a series of Colorado River dams in the rugged Colorado Plateau province of |
https://en.wikipedia.org/wiki/Blind%20carbon%20copy | Blind carbon copy (abbreviated Bcc) allows the sender of a message to conceal the person entered in the Bcc field from the other recipients. This concept originally applied to paper correspondence and now also applies to email.
In some circumstances, the typist creating a paper correspondence must ensure that multiple recipients of such a document do not see the names of other recipients. To achieve this, the typist can:
Add the names in a second step to each copy, without carbon paper;
Set the ribbon not to strike the paper, which leaves names off the top copy (but may leave letter impressions on the paper).
With email, recipients of a message are specified using addresses in any of these three fields:
To: Primary recipients
Cc: Carbon copy to secondary recipients—other interested parties
Bcc: Blind carbon copy to tertiary recipients who receive the message. The primary and secondary recipients cannot see the tertiary recipients. Depending on email software, the tertiary recipients may only see their own email address in Bcc, or they may see the email addresses of all primary and secondary recipients but will not see other tertiary recipients.
It is common practice to use the Bcc: field when addressing a very long list of recipients, or a list of recipients who should not (necessarily) know each other, e.g. in mailing lists.
Benefits
There are a number of reasons for using this feature:
Bcc is often used to prevent an accidental "Reply All" from sending a reply intended for only the originator of the message to the entire recipient list. Using Bcc can prevent an email storm from happening.
To send a copy of one's correspondence to a third party (for example, a colleague) when one does not want to let the recipient know that this is being done (or when one does not want the recipient to know the third party's e-mail address, assuming the other recipient is in the To: or Cc: fields).
To send a message to multiple parties with none of them knowing the o |
https://en.wikipedia.org/wiki/Bus%20snooping | Bus snooping or bus sniffing is a scheme by which a coherency controller (snooper) in a cache (a snoopy cache) monitors or snoops the bus transactions, and its goal is to maintain a cache coherency in distributed shared memory systems. This scheme was introduced by Ravishankar and Goodman in 1983, under the name "write-once" cache coherency. A cache containing a coherency controller (snooper) is called a snoopy cache.
How it works
When specific data is shared by several caches and a processor modifies the value of the shared data, the change must be propagated to all the other caches which have a copy of the data. This change propagation prevents the system from violating cache coherency. The notification of data change can be done by bus snooping. All the snoopers monitor every transaction on a bus. If a transaction modifying a shared cache block appears on a bus, all the snoopers check whether their caches have the same copy of the shared block. If a cache has a copy of the shared block, the corresponding snooper performs an action to ensure cache coherency. The action can be a flush or an invalidation of the cache block. It also involves a change of cache block state depending on the cache coherence protocol.
Types of snooping protocols
There are two kinds of snooping protocols depending on the way to manage a local copy of a write operation:
Write-invalidate
When a processor writes on a shared cache block, all the shared copies in the other caches are invalidated through bus snooping. This method ensures that only one copy of a datum can be exclusively read and written by a processor. All the other copies in other caches are invalidated. This is the most commonly used snooping protocol. MSI, MESI, MOSI, MOESI, and MESIF protocols belong to this category.
Write-update
When a processor writes on a shared cache block, all the shared copies of the other caches are updated through bus snooping. This method broadcasts a write data to all caches throughout a |
https://en.wikipedia.org/wiki/Gr%C3%B6bner%20basis | In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps.
Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and
Gaussian elimination for linear systems.
Gröbner bases were introduced by Bruno Buchberger in his 1965 Ph.D. thesis, which also included an algorithm to compute them (Buchberger's algorithm). He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work.
However, the Russian mathematician Nikolai Günther had introduced a similar notion in 1913, published in various Russian mathematical journals. These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for multivariate power series was developed independently by Heisuke Hironaka in 1964, who named them standard bases. This term has been used by some authors to also denote Gröbner bases.
The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras, like Ore algebras.
Tools
Polynomial ring
Gröbner bases are primarily defined for ideals in a polynomial ring over |
https://en.wikipedia.org/wiki/Correctness%20%28computer%20science%29 | In theoretical computer science, an algorithm is correct with respect to a specification if it behaves as specified. Best explored is functional correctness, which refers to the input-output behavior of the algorithm (i.e., for each input it produces an output satisfying the specification).
Within the latter notion, partial correctness, requiring that if an answer is returned it will be correct, is distinguished from total correctness, which additionally requires that an answer is eventually returned, i.e. the algorithm terminates. Correspondingly, to prove a program's total correctness, it is sufficient to prove its partial correctness, and its termination. The latter kind of proof (termination proof) can never be fully automated, since the halting problem is undecidable.
For example, successively searching through integers 1, 2, 3, … to see if we can find an example of some phenomenon—say an odd perfect number—it is quite easy to write a partially correct program (see box). But to say this program is totally correct would be to assert something currently not known in number theory.
A proof would have to be a mathematical proof, assuming both the algorithm and specification are given formally. In particular it is not expected to be a correctness assertion for a given program implementing the algorithm on a given machine. That would involve such considerations as limitations on computer memory.
A deep result in proof theory, the Curry–Howard correspondence, states that a proof of functional correctness in constructive logic corresponds to a certain program in the lambda calculus. Converting a proof in this way is called program extraction.
Hoare logic is a specific formal system for reasoning rigorously about the correctness of computer programs. It uses axiomatic techniques to define programming language semantics and argue about the correctness of programs through assertions known as Hoare triples.
Software testing is any activity aimed at evaluating an att |
https://en.wikipedia.org/wiki/Dickson%27s%20lemma | In mathematics, Dickson's lemma states that every set of -tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it to prove a result in number theory about perfect numbers. However, the lemma was certainly known earlier, for example to Paul Gordan in his research on invariant theory.
Example
Let be a fixed number, and let be the set of pairs of numbers whose product is at least . When defined over the positive real numbers, has infinitely many minimal elements of the form , one for each positive number ; this set of points forms one of the branches of a hyperbola. The pairs on this hyperbola are minimal, because it is not possible for a different pair that belongs to to be less than or equal to in both of its coordinates. However, Dickson's lemma concerns only tuples of natural numbers, and over the natural numbers there are only finitely many minimal pairs. Every minimal pair of natural numbers has and , for if x were greater than K then (x − 1, y) would also belong to S, contradicting the minimality of (x, y), and symmetrically if y were greater than K then (x, y − 1) would also belong to S. Therefore, over the natural numbers, has at most minimal elements, a finite number.
Formal statement
Let be the set of non-negative integers (natural numbers), let n be any fixed constant, and let be the set of -tuples of natural numbers. These tuples may be given a pointwise partial order, the product order, in which if and only if for every .
The set of tuples that are greater than or equal to some particular tuple forms a positive orthant with its apex at the given tuple.
With this notation, Dickson's lemma may be stated in several equivalent forms:
In every non-empty subset of there is at least one but no more than a finite number of elements that are minimal elements of for the pointwise partial order.
For every infinite sequence of -t |
https://en.wikipedia.org/wiki/Monomial | In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, is a monomial. The constant is a monomial, being equal to the empty product and to for any variable . If only a single variable is considered, this means that a monomial is either or a power of , with a positive integer. If several variables are considered, say, then each can be given an exponent, so that any monomial is of the form with non-negative integers (taking note that any exponent makes the corresponding factor equal to ).
A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is . For example, in this interpretation and are monomials (in the second example, the variables are and the coefficient is a complex number).
In the context of Laurent polynomials and Laurent series, the exponents of a monomial may be negative, and in the context of Puiseux series, the exponents may be rational numbers.
Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the prefix "bi-" (two in Latin), a monomial should theoretically be called a "mononomial". "Monomial" is a syncope by haplology of "mononomial".
Comparison of the two definitions
With either definition, the set of monomials is a subset of all polynomials that is closed under multiplication.
Both uses of this notion can be found, and in many cases the distinction is simply ignored, see for instance examples for the first and second meaning. In informal discussions the distinction is seldom important, and tendency is towards the broad |
https://en.wikipedia.org/wiki/Induced%20representation | In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of , the induced representation is, in a sense, the "most general" representation of that extends the given one. Since it is often easier to find representations of the smaller group than of , the operation of forming induced representations is an important tool to construct new representations.
Induced representations were initially defined by Frobenius, for linear representations of finite groups. The idea is by no means limited to the case of finite groups, but the theory in that case is particularly well-behaved.
Constructions
Algebraic
Let be a finite group and any subgroup of . Furthermore let be a representation of . Let be the index of in and let be a full set of representatives in of the left cosets in . The induced representation can be thought of as acting on the following space:
Here each is an isomorphic copy of the vector space V whose elements are written as with . For each g in and each gi there is an hi in and j(i) in {1, ..., n} such that . (This is just another way of saying that is a full set of representatives.) Via the induced representation acts on as follows:
where for each i.
Alternatively, one can construct induced representations by extension of scalars: any K-linear representation of the group H can be viewed as a module V over the group ring K[H]. We can then define
This latter formula can also be used to define for any group and subgroup , without requiring any finiteness.
Examples
For any group, the induced representation of the trivial representation of the trivial subgroup is the right regular representation. More generally the induced representation of the trivial representation of any subgroup is the permutation representation on the cosets of that subgroup.
An induced representation of a one dimensional representation is called a |
https://en.wikipedia.org/wiki/Potassium%20sodium%20tartrate | Potassium sodium tartrate tetrahydrate, also known as Rochelle salt, is a double salt of tartaric acid first prepared (in about 1675) by an apothecary, Pierre Seignette, of La Rochelle, France. Potassium sodium tartrate and monopotassium phosphate were the first materials discovered to exhibit piezoelectricity. This property led to its extensive use in "crystal" gramophone (phono) pick-ups, microphones and earpieces during the post-World War II consumer electronics boom of the mid-20th century. Such transducers had an exceptionally high output with typical pick-up cartridge outputs as much as 2 volts or more. Rochelle salt is deliquescent so any transducers based on the material deteriorated if stored in damp conditions.
It has been used medicinally as a laxative. It has also been used in the process of silvering mirrors. It is an ingredient of Fehling's solution (reagent for reducing sugars). It is used in electroplating, in electronics and piezoelectricity, and as a combustion accelerator in cigarette paper (similar to an oxidizer in pyrotechnics).
In organic synthesis, it is used in aqueous workups to break up emulsions, particularly for reactions in which an aluminium-based hydride reagent was used. Sodium Potassium tartrate is also important in the food industry.
It is a common precipitant in protein crystallography and is also an ingredient in the Biuret reagent which is used to measure protein concentration. This ingredient maintains cupric ions in solution at an alkaline pH.
Preparation
The starting material is tartar with a minimum tartaric acid content 68 %. This is first dissolved in water or in the mother liquor of a previous batch. It is then basified with hot saturated sodium hydroxide solution to pH 8, decolorized with activated charcoal, and chemically purified before being filtered. The filtrate is evaporated to 42 °Bé at 100 °C, and passed to granulators in which Seignette's salt crystallizes on slow cooling. The salt is separated from the mo |
https://en.wikipedia.org/wiki/Outline%20of%20software%20engineering | The following outline is provided as an overview of and topical guide to software engineering:
Software engineering – application of a systematic, disciplined, quantifiable approach to the development, operation, and maintenance of software; that is the application of engineering to software.
The ACM Computing Classification system is a poly-hierarchical ontology that organizes the topics of the field and can be used in semantic web applications and as a de facto standard classification system for the field. The major section "Software and its Engineering" provides an outline and ontology for software engineering.
Software applications
Software engineers build software (applications, operating systems, system software) that people use.
Applications influence software engineering by pressuring developers to solve problems in new ways. For example, consumer software emphasizes low cost, medical software emphasizes high quality, and Internet commerce software emphasizes rapid development.
Business software
Accounting software
Analytics
Data mining closely related to database
Decision support systems
Airline reservations
Banking
Automated teller machines
Cheque processing
Credit cards
Commerce
Trade
Auctions (e.g. eBay)
Reverse auctions (procurement)
Bar code scanners
Compilers
Parsers
Compiler optimization
Interpreters
Linkers
Loaders
Communication
E-mail
Instant messengers
VOIP
Calendars — scheduling and coordinating
Contact managers
Computer graphics
Animation
Special effects for video and film
Editing
Post-processing
Cryptography
Databases, support almost every field
Embedded systems Both software engineers and traditional engineers write software control systems for embedded products.
Automotive software
Avionics software
Heating ventilating and air conditioning (HVAC) software
Medical device software
Telephony
Telemetry
Engineering All traditional engineering branches use software extensively. Engineers use spreadsheets, more than they ever used calculators |
https://en.wikipedia.org/wiki/Promiscuous%20mode | In computer networking, promiscuous mode is a mode for a wired network interface controller (NIC) or wireless network interface controller (WNIC) that causes the controller to pass all traffic it receives to the central processing unit (CPU) rather than passing only the frames that the controller is specifically programmed to receive. This mode is normally used for packet sniffing that takes place on a router or on a computer connected to a wired network or one being part of a wireless LAN. Interfaces are placed into promiscuous mode by software bridges often used with hardware virtualization.
In IEEE 802 networks such as Ethernet or IEEE 802.11, each frame includes a destination MAC address. In non-promiscuous mode, when a NIC receives a frame, it drops it unless the frame is addressed to that NIC's MAC address or is a broadcast or multicast addressed frame. In promiscuous mode, however, the NIC allows all frames through, thus allowing the computer to read frames intended for other machines or network devices.
Many operating systems require superuser privileges to enable promiscuous mode. A non-routing node in promiscuous mode can generally only monitor traffic to and from other nodes within the same broadcast domain (for Ethernet and IEEE 802.11) or ring (for Token Ring). Computers attached to the same Ethernet hub satisfy this requirement, which is why network switches are used to combat malicious use of promiscuous mode. A router may monitor all traffic that it routes.
Promiscuous mode is often used to diagnose network connectivity issues. There are programs that make use of this feature to show the user all the data being transferred over the network. Some protocols like FTP and Telnet transfer data and passwords in clear text, without encryption, and network scanners can see this data. Therefore, computer users are encouraged to stay away from insecure protocols like telnet and use more secure ones such as SSH.
Detection
As promiscuous mode can be used |
https://en.wikipedia.org/wiki/Test-driven%20development | Test-driven development (TDD) is a software development process relying on software requirements being converted to test cases before software is fully developed, and tracking all software development by repeatedly testing the software against all test cases. This is as opposed to software being developed first and test cases created later.
Software engineer Kent Beck, who is credited with having developed or "rediscovered" the technique, stated in 2003 that TDD encourages simple designs and inspires confidence.
Test-driven development is related to the test-first programming concepts of extreme programming, begun in 1999, but more recently has created more general interest in its own right.
Programmers also apply the concept to improving and debugging legacy code developed with older techniques.
Test-driven development cycle
The following sequence is based on the book Test-Driven Development by Example:
1. Add a test
The adding of a new feature begins by writing a test that passes iff the feature's specifications are met. The developer can discover these specifications by asking about use cases and user stories. A key benefit of test-driven development is that it makes the developer focus on requirements before writing code. This is in contrast with the usual practice, where unit tests are only written after code.
2. Run all tests. The new test should fail for expected reasons
This shows that new code is actually needed for the desired feature. It validates that the test harness is working correctly. It rules out the possibility that the new test is flawed and will always pass.
3. Write the simplest code that passes the new test
Inelegant or hard code is acceptable, as long as it passes the test. The code will be honed anyway in Step 5. No code should be added beyond the tested functionality.
4. All tests should now pass
If any fail, the new code must be revised until they pass. This ensures the new code meets the test requirements and does not break exis |
https://en.wikipedia.org/wiki/Proof%20by%20infinite%20descent | In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions.
Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction, the original premise—that any solution exists—is incorrect: its correctness produces a contradiction.
An alternative way to express this is to assume one or more solutions or examples exists, from which a smallest solution or example—a minimal counterexample—can then be inferred. Once there, one would try to prove that if a smallest solution exists, then it must imply the existence of a smaller solution (in some sense), which again proves that the existence of any solution would lead to a contradiction.
The earliest uses of the method of infinite descent appear in Euclid's Elements. A typical example is Proposition 31 of Book 7, in which Euclid proves that every composite integer is divided (in Euclid's terminology "measured") by some prime number.
The method was much later developed by Fermat, who coined the term and often used it for Diophantine equations. Two typical examples are showing the non-solvability of the Diophantine equation and prov |
https://en.wikipedia.org/wiki/Alien%20Syndrome | is a run and gun video game developed by Sega and released in arcades in 1987, and later ported to the Master System in 1988. The game utilizes a side-scrolling feature that allows the player to take control of either a male (Ricky) or female (Mary) soldier whilst hunting aliens and saving hostages before they run out of time.
Alien Syndrome released with a strong opening, becoming one of Sega's best-selling video games for the Master System. It received generally positive reviews, with praise towards the gameplay, character designs, sounds, and horror-themed gameplay. It was included as a bonus game in the Genesis compilation game, Sonic's Ultimate Genesis Collection (2009). A sequel of the same name was released in 2007 to a strong negative response from critics.
Gameplay
Two players control two soldiers, named Ricky and Mary, who fight their way through large eight-way scrolling levels while rescuing their comrades who are being held by aliens. After they have rescued a certain number of hostages, the exit opens and they can pass through it in order to fight the end-of-level guardian. If this monstrosity is defeated, they are then able to move onto the next stage.
Alien Syndrome features two player simultaneous gameplay and pickups which assist the player, including better weapons and maps of the current level.
Ports
In 1988, the game was ported to the Master System, MSX, Amiga, Atari ST, Amstrad CPC, Commodore 64, and Famicom/Nintendo Entertainment System (published by Tengen without a Nintendo license). Later, the game was ported to the ZX Spectrum (1989), Game Gear (1992), and X68000 (1992).
Reception
In Japan, Game Machine listed Alien Syndrome on their May 15, 1987 issue as being the third most-successful table arcade unit of the month. The original arcade version of the game was reviewed in the July 1987 issue of Computer and Video Games, where Clare Edgeley described it as "one of the most gripping games" she "played in months", praising the Aliens-l |
https://en.wikipedia.org/wiki/Optimal%20solutions%20for%20the%20Rubik%27s%20Cube | Optimal solutions for the Rubik's Cube are solutions that are the shortest in some sense. There are two common ways to measure the length of a solution. The first is to count the number of quarter turns. The second is to count the number of outer-layer twists, called "face turns". A move to turn an outer layer two quarter (90°) turns in the same direction would be counted as two moves in the quarter turn metric (QTM), but as one turn in the face metric (FTM, or HTM "Half Turn Metric", or OBTM "Outer Block Turn Metric").
The maximal number of face turns needed to solve any instance of the Rubik's Cube is 20, and the maximal number of quarter turns is 26. These numbers are also the diameters of the corresponding Cayley graphs of the Rubik's Cube group. In STM (slice turn metric), the minimal number of turns is unknown.
There are many algorithms to solve scrambled Rubik's Cubes. An algorithm that solves a cube in the minimum number of moves is known as God's algorithm.
Move notation
To denote a sequence of moves on the 3×3×3 Rubik's Cube, this article uses "Singmaster notation", which was developed by David Singmaster.
The followings are standard moves, which do not move centre cubies of any face to another location:
The letters L, R, F, B, U, and D indicate a clockwise quarter turn of the left, right, front, back, up, and down face respectively. Half turns are indicated by appending a 2. A counterclockwise quarter turn is indicated by appending a prime symbol ( ′ ).
However, because these notations are human-oriented, we use clockwise as positive, and not mathematically-oriented, which is counterclockwise as positive.
The following are non-standard moves, which move centre cubies of faces to other locations:
The letters M, S and E are used to denote the turning of a middle layer. M represents turning the layer between the R and L faces 1 quarter turn top to bottom. S represents turning the layer between the F and B faces 1 quarter turn clockwise, as seen |
https://en.wikipedia.org/wiki/Cayley%20graph | In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing expander graphs.
Definition
Let be a group and be a generating set of . The Cayley graph is an edge-colored directed graph constructed as follows:
Each element of is assigned a vertex: the vertex set of is identified with
Each element of is assigned a color .
For every and , there is a directed edge of color from the vertex corresponding to to the one corresponding to .
Not every convention requires that generate the group. If is not a generating set for , then is disconnected and each connected component represents a coset of the subgroup generated by .
If an element of is its own inverse, then it is typically represented by an undirected edge.
The set is often assumed to be finite, especially in geometric group theory, which corresponds to being locally finite and being finitely generated.
The set is sometimes assumed to be symmetric () and not containing the group identity element. In this case, the uncolored Cayley graph can be represented as a simple undirected graph.
Examples
Suppose that is the infinite cyclic group and the set consists of the standard generator 1 and its inverse (−1 in the additive notation); then the Cayley graph is an infinite path.
Similarly, if is the finite cyclic group of order and the set consists of two elements, the standard generator of and its inverse, then the Cayley graph is the cycle . More generally, the Cayley graphs of finite cyclic groups are exactly the circulant graphs.
The Cayley graph of the direct product of grou |
https://en.wikipedia.org/wiki/Random%20graph | In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.
Models
A random graph is obtained by starting with a set of n isolated vertices and adding successive edges between them at random. The aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise. Different random graph models produce different probability distributions on graphs. Most commonly studied is the one proposed by Edgar Gilbert, denoted G(n,p), in which every possible edge occurs independently with probability 0 < p < 1. The probability of obtaining any one particular random graph with m edges is with the notation .
A closely related model, the Erdős–Rényi model denoted G(n,M), assigns equal probability to all graphs with exactly M edges. With 0 ≤ M ≤ N, G(n,M) has elements and every element occurs with probability . The latter model can be viewed as a snapshot at a particular time (M) of the random graph process , which is a stochastic process that starts with n vertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges.
If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probabi |
https://en.wikipedia.org/wiki/Desargues%27s%20theorem | In projective geometry, Desargues's theorem, named after Girard Desargues, states:
Two triangles are in perspective axially if and only if they are in perspective centrally.
Denote the three vertices of one triangle by and , and those of the other by and . Axial perspectivity means that lines and meet in a point, lines and meet in a second point, and lines and meet in a third point, and that these three points all lie on a common line called the axis of perspectivity. Central perspectivity means that the three lines and are concurrent, at a point called the center of perspectivity.
This intersection theorem is true in the usual Euclidean plane but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by adding points at infinity, following Jean-Victor Poncelet. This results in a projective plane.
Desargues's theorem is true for the real projective plane and for any projective space defined arithmetically from a field or division ring; that includes any projective space of dimension greater than two or in which Pappus's theorem holds. However, there are many "non-Desarguesian planes", in which Desargues's theorem is false.
History
Desargues never published this theorem, but it appeared in an appendix entitled Universal Method of M. Desargues for Using Perspective (Manière universelle de M. Desargues pour practiquer la perspective) to a practical book on the use of perspective published in 1648. by his friend and pupil Abraham Bosse (1602–1676).
Coordinatization
The importance of Desargues's theorem in abstract projective geometry is due especially to the fact that a projective space satisfies that theorem if and only if it is isomorphic to a projective space defined over a field or division ring.
Projective versus affine spaces
In an affine space such as the Euclidean |
https://en.wikipedia.org/wiki/Method%20of%20complements | In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (or mechanism) for addition throughout the whole range. For a given number of places half of the possible representations of numbers encode the positive numbers, the other half represents their respective additive inverses. The pairs of mutually additive inverse numbers are called complements. Thus subtraction of any number is implemented by adding its complement. Changing the sign of any number is encoded by generating its complement, which can be done by a very simple and efficient algorithm. This method was commonly used in mechanical calculators and is still used in modern computers. The generalized concept of the radix complement (as described below) is also valuable in number theory, such as in Midy's theorem.
The nines' complement of a number given in decimal representation is formed by replacing each digit with nine minus that digit. To subtract a decimal number y (the subtrahend) from another number x (the minuend) two methods may be used:
In the first method the nines' complement of x is added to y. Then the nines' complement of the result obtained is formed to produce the desired result.
In the second method the nines' complement of y is added to x and one is added to the sum. The leftmost digit '1' of the result is then discarded. Discarding the leftmost '1' is especially convenient on calculators or computers that use a fixed number of digits: there is nowhere for it to go so it is simply lost during the calculation. The nines' complement plus one is known as the ten's complement.
The method of complements can be extended to other number bases (radices); in particular, it is used on most digital computers to perform subtraction, represent negative numbers in base 2 or binary arithmetic and test underflow and overflow in calculation.
Numeric complements
The radix complem |
https://en.wikipedia.org/wiki/Log-structured%20file%20system | A log-structured filesystem is a file system in which data and metadata are written sequentially to a circular buffer, called a log. The design was first proposed in 1988 by John K. Ousterhout and Fred Douglis and first implemented in 1992 by Ousterhout and Mendel Rosenblum for the Unix-like Sprite distributed operating system.
Rationale
Conventional file systems tend to lay out files with great care for spatial locality and make in-place changes to their data structures in order to perform well on optical and magnetic disks, which tend to seek relatively slowly.
The design of log-structured file systems is based on the hypothesis that this will no longer be effective because ever-increasing memory sizes on modern computers would lead to I/O becoming write-heavy because reads would be almost always satisfied from memory cache. A log-structured file system thus treats its storage as a circular log and writes sequentially to the head of the log.
This has several important side effects:
Write throughput on optical and magnetic disks is improved because they can be batched into large sequential runs and costly seeks are kept to a minimum.
The structure is naturally suited to media with append-only zones or pages such as flash storages and shingled magnetic recording HDDs
Writes create multiple, chronologically-advancing versions of both file data and meta-data. Some implementations make these old file versions nameable and accessible, a feature sometimes called time-travel or snapshotting. This is very similar to a versioning file system.
Recovery from crashes is simpler. Upon its next mount, the file system does not need to walk all its data structures to fix any inconsistencies, but can reconstruct its state from the last consistent point in the log.
Log-structured file systems, however, must reclaim free space from the tail of the log to prevent the file system from becoming full when the head of the log wraps around to meet it. The tail can release spa |
https://en.wikipedia.org/wiki/Serre%20duality | In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an n-dimensional variety, the theorem says that a cohomology group is the dual space of another one, . Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf.
The Serre duality theorem is also true in complex geometry more generally, for compact complex manifolds that are not necessarily projective complex algebraic varieties. In this setting, the Serre duality theorem is an application of Hodge theory for Dolbeault cohomology, and may be seen as a result in the theory of elliptic operators.
These two different interpretations of Serre duality coincide for non-singular projective complex algebraic varieties, by an application of Dolbeault's theorem relating sheaf cohomology to Dolbeault cohomology.
Serre duality for vector bundles
Algebraic theorem
Let X be a smooth variety of dimension n over a field k. Define the canonical line bundle to be the bundle of n-forms on X, the top exterior power of the cotangent bundle:
Suppose in addition that X is proper (for example, projective) over k. Then Serre duality says: for an algebraic vector bundle E on X and an integer i, there is a natural isomorphism:
of finite-dimensional k-vector spaces. Here denotes the tensor product of vector bundles. It follows that the dimensions of the two cohomology groups are equal:
As in Poincaré duality, the isomorphism in Serre duality comes from the cup product in sheaf cohomology. Namely, the composition of the cup product with a natural trace map on is a perfect pairing:
The trace map is the analog for coherent sheaf cohomology of integration in de Rham |
https://en.wikipedia.org/wiki/Phase%20detector | A phase detector or phase comparator is a frequency mixer, analog multiplier or logic circuit that generates a signal which represents the difference in phase between two signal inputs.
The phase detector is an essential element of the phase-locked loop (PLL). Detecting phase difference is important in other applications, such as motor control, radar and telecommunication systems, servo mechanisms, and demodulators.
Types
Phase detectors for phase-locked loop circuits may be classified in two types. A Type I detector is designed to be driven by analog signals or square-wave digital signals and produces an output pulse at the difference frequency. The Type I detector always produces an output waveform, which must be filtered to control the phase-locked loop voltage-controlled oscillator (VCO). A type II detector is sensitive only to the relative timing of the edges of the input and reference pulses and produces a constant output proportional to phase difference when both signals are at the same frequency. This output will tend not to produce ripple in the control voltage of the VCO.
Analog phase detector
The phase detector needs to compute the phase difference of its two input signals. Let α be the phase of the first input and β be the phase of the second. The actual input signals to the phase detector, however, are not α and β, but rather sinusoids such as sin(α) and cos(β). In general, computing the phase difference would involve computing the arcsine and arccosine of each normalized input (to get an ever-increasing phase) and doing a subtraction. Such an analog calculation is difficult. Fortunately, the calculation can be simplified by using some approximations.
Assume that the phase differences will be small (much less than 1 radian, for example). The small-angle approximation for the sine function and the sine angle addition formula yield:
The expression suggests a quadrature phase detector can be made by summing the outputs of two multipliers. The q |
https://en.wikipedia.org/wiki/Abstract%20structure | An abstract structure is an abstraction that might be of the geometric spaces or a set structure, or a hypostatic abstraction that is defined by a set of mathematical theorems and laws, properties and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects. Abstract structures are studied not only in logic and mathematics but in the fields that apply them, as computer science and computer graphics, and in the studies that reflect on them, such as philosophy (especially the philosophy of mathematics). Indeed, modern mathematics has been defined in a very general sense as the study of abstract structures (by the Bourbaki group: see discussion there, at algebraic structure and also structure).
An abstract structure may be represented (perhaps with some degree of approximation) by one or more physical objects this is called an implementation or instantiation of the abstract structure. But the abstract structure itself is defined in a way that is not dependent on the properties of any particular implementation.
An abstract structure has a richer structure than a concept or an idea. An abstract structure must include precise rules of behaviour which can be used to determine whether a candidate implementation actually matches the abstract structure in question, and it must be free from contradictions. Thus we may debate how well a particular government fits the concept of democracy, but there is no room for debate over whether a given sequence of moves is or is not a valid game of chess (for example Kasparovian approaches).
Examples
A sorting algorithm is an abstract structure, but a recipe is not, because it depends on the properties and quantities of its ingredients.
A simple melody is an abstract structure, but an orchestration is not, because it depends on the properties of particular instruments.
Euclidean geometry is an abstract structure, but t |
https://en.wikipedia.org/wiki/Balance%20of%20Power%20%28video%20game%29 | Balance of Power is a strategy video game of geopolitics during the Cold War, created by Chris Crawford and published in 1985 on the Macintosh by Mindscape, followed by ports to a variety of platforms over the next two years.
In the game, the player takes the role of the President of the United States or General Secretary of the Soviet Union. The goal is to improve the player's country's standing in the world relative to the other superpower. During each yearly turn, random events occur that may have effects on the player's international prestige. The player can choose to respond to these events in various ways, which may prompt a response from the other superpower. This creates brinkmanship situations between the two nations, potentially escalating to a nuclear war, which ends the game.
Crawford was already well-known, especially for Eastern Front (1941). His 1984 announcement that he was moving to the Macintosh platform to work on a new concept generated considerable interest. It was widely reviewed after its release, including an extremely positive review in The New York Times Magazine. It was praised for its inventive non-action gameplay that was nevertheless exciting and distinct. It has been named by Computer Gaming World as one of the most innovative computer games of all time.
Balance of Power was successful on the Mac, and combined with ports it ultimately sold over a quarter million units.
Gameplay
The player may choose to be either the President of the United States or the General Secretary of the Communist Party of the Soviet Union, and must lead the chosen superpower for eight years, seeking to maximize "prestige" and avoiding a nuclear war. Each turn is one year long; at the beginning of each year, the player is presented with a set of incidents and crises in various countries around the globe and must choose a response to each one. Responses may range from no action to diplomatic notes to the other superpower, to military maneuvers. Each respons |
https://en.wikipedia.org/wiki/Paul%20R.%20Ehrlich | Paul Ralph Ehrlich (born May 29, 1932) is an American biologist known for his predictions and warnings about the consequences of population growth, including famine and resource depletion. Ehrlich is the Bing Professor Emeritus of Population Studies of the Department of Biology of Stanford University, and President of Stanford's Center for Conservation Biology.
Ehrlich became well known for the controversial 1968 book The Population Bomb, which he co-authored with his wife Anne H. Ehrlich, in which they famously stated that "[i]n the 1970s hundreds of millions of people will starve to death in spite of any crash programs embarked upon now." Among the solutions suggested in that book was population control, including "various forms of coercion" such as eliminating "tax benefits for having additional children," to be used if voluntary methods were to fail, as well as letting "hopeless" countries like India starve to death.
Scholars, journalists and public intellectuals have mixed views on Ehrlich's assertions on the dangers of expanding human populations. While Paul A. Murtaugh, associate professor of statistics at Oregon State University, says that Ehrlich was largely correct, Ehrlich has been criticized for his approach and views, both for their pessimistic outlook and, later on, for the repeated failure of his predictions to come true. For example, in response to Ehrlich's assertion that all major marine wildlife would die by 1980, Ronald Bailey termed Ehrlich an "irrepressible doomster". Ehrlich has acknowledged that "some" of what he predicted has not occurred, but nevertheless maintains that his predictions about disease and climate change were essentially correct and that human overpopulation is a major problem. Whereas American journalist Jonathan V. Last has called The Population Bomb "one of the most spectacularly foolish books ever published", journalist Fred Pearce argues that overconsumption is the real problem.
Early life, education, and academic care |
https://en.wikipedia.org/wiki/List%20of%20mathematical%20examples | This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a general theorem, could perhaps be considered an "example"). The discussion page for list of mathematical topics has some comments on this. Eventually this page may have its own discussion page. This page links to itself in order that edits to this page will be included among related changes when the user clicks on that button.
The concrete example within the article titled Rao-Blackwell theorem is perhaps one of the best ways for a probabilist ignorant of statistical inference to get a quick impression of the flavor of that subject.
Uncategorized examples, alphabetized
Alexander horned sphere
All horses are the same color
Cantor function
Cantor set
Checking if a coin is biased
Concrete illustration of the central limit theorem
Differential equations of mathematical physics
Dirichlet function
Discontinuous linear map
Efron's non-transitive dice
Example of a game without a value
Examples of contour integration
Examples of differential equations
Examples of generating functions
Examples of groups
List of the 230 crystallographic 3D space groups
Examples of Markov chains
Examples of vector spaces
Fano plane
Frieze group
Gray graph
Hall–Janko graph
Higman–Sims graph
Hilbert matrix
Illustration of a low-discrepancy sequence
Illustration of the central limit theorem
An infinitely differentiable function that is not analytic
Leech lattice
Lewy's example on PDEs
List of finite simple groups
Long line
Normally distributed and uncorrelated does not imply independent
Pairwise independence of random variables need not imply mutual independence.
Petersen graph
Sierpinski space
Simple examp |
https://en.wikipedia.org/wiki/Cumulant | In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.
The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the -th-order cumulant of their sum is equal to the sum of their -th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property.
Just as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants.
Definition
The cumulants of a random variable are defined using the cumulant-generating function , which is the natural logarithm of the moment-generating function:
The cumulants are obtained from a power series expansion of the cumulant generating function:
This expansion is a Maclaurin series, so the -th cumulant can be obtained by differentiating the above expansion times and evaluating the result at zero:
If the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later.
Alternative definition of the cumulant generating function
Some writers prefer to define the cumulant-generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function,
An advantage of —in some sense the function evaluated for purely imaginary arguments—is that is well defined for all real values of even whe |
https://en.wikipedia.org/wiki/Shoutcast | Shoutcast (formerly SHOUTcast) is a service for streaming media over the internet to media players, using its own cross-platform proprietary software. It allows digital audio content, primarily in MP3 or High-Efficiency Advanced Audio Coding format. The most common use of Shoutcast is for creating or listening to Internet audio broadcasts; however, there are also video streams. The software is available to use for free or as a paid cloud service with additional professional features.
In the early days of esports for video games, Shoutcast was used by some to stream play-by-play commentary, leading to the term "shoutcaster" as a name for esports commentators.
History
Created in 1998, Shoutcast's streaming protocol uses metadata tags and responses that all start with ICY, which stands for "I Can Yell." Nullsoft was purchased by AOL on June 1, 1999.
On January 14, 2014, AOL sold Nullsoft to Belgian online radio aggregator Radionomy Group; no financial details were publicly announced. In 2018 the software was rebranded from its original name of SHOUTcast to Shoutcast. In 2020 Radionomy shut down its own streaming service and migrated to the Shoutcast platform.
Software
The Shoutcast software uses a client–server model, with each component communicating via a network protocol that intermingles audio or video data with metadata such as song titles and the station name. It uses HTTP as a transport protocol.
Shoutcast servers and clients are available for FreeBSD, Linux, macOS, Microsoft Windows, and Solaris. There are client-only versions for Android, BlackBerry OS, iOS (iPad, iPhone), Palm OS and webOS (Radio Hibiki), PlayStation Portable, Windows Mobile, Symbian S60 and UIQ, Nintendo DS (DSOrganize), and Wii.
The output format is supported by multiple clients, including Nullsoft's own Winamp as well as Amarok, Exaile, foobar2000, iTunes, Songbird, Totem, XMMS, and Zinf. If the client does not support the Shoutcast protocol, then the Shoutcast server sends the strea |
https://en.wikipedia.org/wiki/Proof%20by%20exhaustion | Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages:
A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases.
A proof of each of the cases.
The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis of mathematical elegance. Expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the proof by exhaustion method can be used whenever the number of cases is finite. However, because most mathematical sets are infinite, this method is rarely used to derive general mathematical results.
In the Curry–Howard isomorphism, proof by exhaustion and case analysis are related to ML-style pattern matching.
Example
Proof by exhaustion can be used to prove that if an integer is a perfect cube, then it must be either a multiple of 9, 1 more than a multiple of 9, or 1 less than a multiple of 9.
Proof:
Each perfect cube is the cube of some integer n, where n is either a multiple of 3, 1 more than a multiple of 3, or 1 less than a multiple of 3. So these three cases are exhaustive:
Case 1: If n = 3p, then n3 = 27p3, which is a multiple of 9.
Case 2: If n = 3p + 1, then n3 = 27p3 + 27p2 + 9p + 1, which is 1 more than a multiple of 9. For instance, if n = 4 then n3 = 64 = 9×7 + 1.
Case 3: If n = 3p − 1, then n3 = 27p3 − 27p2 + 9p − 1, which is 1 less than a multiple of 9. For instance, if n = 5 th |
https://en.wikipedia.org/wiki/Ordnance%20Survey%20National%20Grid | The Ordnance Survey National Grid reference system (OSGB) (also known as British National Grid (BNG)) is a system of geographic grid references used in Great Britain, distinct from latitude and longitude.
The Ordnance Survey (OS) devised the national grid reference system, and it is heavily used in its survey data, and in maps based on those surveys, whether published by the Ordnance Survey or by commercial map producers. Grid references are also commonly quoted in other publications and data sources, such as guide books and government planning documents.
A number of different systems exist that can provide grid references for locations within the British Isles: this article describes the system created solely for Great Britain and its outlying islands (including the Isle of Man); the Irish grid reference system was a similar system created by the Ordnance Survey of Ireland and the Ordnance Survey of Northern Ireland for the island of Ireland. The Universal Transverse Mercator coordinate system (UTM) is used to provide grid references for worldwide locations, and this is the system commonly used for the Channel Islands and Ireland (since 2001). European-wide agencies also use UTM when mapping locations, or may use the Military Grid Reference System (MGRS), or variants of it.
Grid letters
The first letter of the British National Grid is derived from a larger set of 25 squares of size 500 km by 500 km, labelled A to Z, omitting one letter (I) (refer diagram below), previously used as a military grid. Four of these largest squares contain significant land area within Great Britain: S, T, N and H. The O square contains a tiny area of North Yorkshire, Beast Cliff at , almost all of which lies below mean high tide.
For the second letter, each 500 km square is subdivided into 25 squares of size 100 km by 100 km, each with a letter code from A to Z (again omitting I) starting with A in the north-west corner to Z in the south-east corner. These squares are outlined i |
https://en.wikipedia.org/wiki/Free%20product | In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from to K. Unless one of the groups G and H is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group (the universal group with a given set of generators).
The free product is the coproduct in the category of groups. That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group. Therefore, the free product is not the coproduct in the category of abelian groups.
The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental group of the union of two path-connected topological spaces whose intersection is also path-connected is always an amalgamated free product of the fundamental groups of the spaces. In particular, the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is, under certain conditions given in the Seifert van-Kampen theorem, the free product of the fundamental groups of the spaces.
Free products are also important in Bass–Serre theory, the study of groups acting by automorphisms on trees. Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and HNN extensions. Using the action of the modular group on a certain tessellation of the hyperbolic plane, it follows from this theor |
https://en.wikipedia.org/wiki/Constructive%20proof | In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
A constructive proof may also refer to the stronger concept of a proof that is valid in constructive mathematics.
Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice, and induces a different meaning for some terminology (for example, the term "or" has a stronger meaning in constructive mathematics than in classical).
Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues; consequently the proposition must be true (proof by contradiction). However, the principle of explosion (ex falso quodlibet) has been accepted in some varieties of constructive mathematics, including intuitionism.
Constructive proofs can be seen as defining certified mathematical algorithms: this idea is explored in the Brouwer–Heyting–Kolmogorov interpretation of constructive logic, the Curry–Howard correspondence between proofs and programs, and such logical systems as Per Martin-Löf's intuitionistic type theory, and Thierry Coquand and Gérard Huet's calculus of constructions.
A historical example
Until the end of 19th century, all mathematical proofs were essentially constructive. The first non-constructive constructions appeared with Georg Cantor’s theory of infinite sets, and the formal definition of real numbers.
The first use of non-constructive |
https://en.wikipedia.org/wiki/Function%20of%20several%20complex%20variables | The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space , that is, -tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading.
As in complex analysis of functions of one variable, which is the case , the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domain(), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex variables, this is not the case; there exist domains () that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so the domain of holomorphy is one of the themes in this field. Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and complex projective varieties () and has a different flavour to complex analytic geometry in or on Stein manifolds, these are much similar to study of algebraic varieties that is study of the algebraic geometry than complex analytic geometry.
Historical perspective
Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem. Naturally also same function of one variable that depends on |
https://en.wikipedia.org/wiki/List%20of%20sound%20chips | Sound chips come in different forms and use a variety of techniques to generate audio signals. This is a list of sound chips that were produced by a certain company or manufacturer, categorized by the sound generation of the chips.
Programmable sound generators (PSG)
Wavetable synthesis
Frequency modulation (FM) synthesis
Pulse-code modulation (PCM) sampling
See also
List of sound card standards
List of Yamaha sound chips
Sound recording and reproduction
References
External links
Sound generators of the 1980s home computers - Has a list of chips, pictures, datasheets, etc.
Video game music technology |
https://en.wikipedia.org/wiki/Tur%C3%A1n%27s%20theorem | In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that does not have a given subgraph.
An example of an -vertex graph that does not contain any -vertex clique may be formed by partitioning the set of vertices into parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the Turán graph . Turán's theorem states that the Turán graph has the largest number of edges among all -free -vertex graphs.
Turán's theorem, and the Turán graphs giving its extreme case, were first described and studied by Hungarian mathematician Pál Turán in 1941. The special case of the theorem for triangle-free graphs is known as Mantel's theorem; it was stated in 1907 by Willem Mantel, a Dutch mathematician.
Statement
Turán's theorem states that every graph with vertices that does not contain as a subgraph has at most as many edges as the Turán graph . For a fixed value of , this graph hasedges, using little-o notation. Intuitively, this means that as gets larger, the fraction of edges included in gets closer and closer to . Many of the following proofs only give the upper bound of .
Proofs
list five different proofs of Turán's theorem.
Many of the proofs involve reducing to the case where the graph is a complete multipartite graph, and showing that the number of edges is maximized when there are parts of size as close as possible to equal.
Induction
This was Turán's original proof. Take a -free graph on vertices with the maximal number of edges. Find a (which exists by maximality), and partition the vertices into the set of the vertices in the and the set |
https://en.wikipedia.org/wiki/Planisphere | In astronomy, a planisphere () is a star chart analog computing instrument in the form of two adjustable disks that rotate on a common pivot. It can be adjusted to display the visible stars for any time and date. It is an instrument to assist in learning how to recognize stars and constellations. The astrolabe, an instrument that has its origins in Hellenistic astronomy, is a predecessor of the modern planisphere.
The term planisphere contrasts with armillary sphere, where the celestial sphere is represented by a three-dimensional framework of rings.
Description
A planisphere consists of a circular star chart attached at its center to an opaque circular overlay that has a clear elliptical window or hole so that only a portion of the sky map will be visible in the window or hole area at any given time. The chart and overlay are mounted so that they are free to rotate about a common axis. The star chart contains the brightest stars, constellations and (possibly) deep-sky objects visible from a particular latitude on Earth. The night sky that one sees from the Earth depends on whether the observer is in the northern or southern hemispheres and the latitude. A planisphere window is designed for a particular latitude and will be accurate enough for a certain band either side of that. Planisphere makers will usually offer them in a number of versions for different latitudes. Planispheres only show the stars visible from the observer's latitude; stars below the horizon are not included.
A complete twenty-four-hour time cycle is marked on the rim of the overlay. A full twelve months of calendar dates are marked on the rim of the starchart. The window is marked to show the direction of the eastern and western horizons. The disk and overlay are adjusted so that the observer's local time of day on the overlay corresponds to that day's date on the star chart disc. The portion of the star chart visible in the window then represents (with a distortion because it is a flat surf |
https://en.wikipedia.org/wiki/Backdoor%20%28computing%29 | A backdoor is a typically covert method of bypassing normal authentication or encryption in a computer, product, embedded device (e.g. a home router), or its embodiment (e.g. part of a cryptosystem, algorithm, chipset, or even a "homunculus computer"—a tiny computer-within-a-computer such as that found in Intel's AMT technology). Backdoors are most often used for securing remote access to a computer, or obtaining access to plaintext in cryptosystems. From there it may be used to gain access to privileged information like passwords, corrupt or delete data on hard drives, or transfer information within autoschediastic networks.
A backdoor may take the form of a hidden part of a program, a separate program (e.g. Back Orifice may subvert the system through a rootkit), code in the firmware of the hardware, or parts of an operating system such as Windows. Trojan horses can be used to create vulnerabilities in a device. A Trojan horse may appear to be an entirely legitimate program, but when executed, it triggers an activity that may install a backdoor. Although some are secretly installed, other backdoors are deliberate and widely known. These kinds of backdoors have "legitimate" uses such as providing the manufacturer with a way to restore user passwords.
Many systems that store information within the cloud fail to create accurate security measures. If many systems are connected within the cloud, hackers can gain access to all other platforms through the most vulnerable system. Default passwords (or other default credentials) can function as backdoors if they are not changed by the user. Some debugging features can also act as backdoors if they are not removed in the release version. In 1993, the United States government attempted to deploy an encryption system, the Clipper chip, with an explicit backdoor for law enforcement and national security access. The chip was unsuccessful.
Recent proposals to counter backdoors include creating a database of backdoors' triggers |
https://en.wikipedia.org/wiki/Coercivity | Coercivity, also called the magnetic coercivity, coercive field or coercive force, is a measure of the ability of a ferromagnetic material to withstand an external magnetic field without becoming demagnetized. Coercivity is usually measured in oersted or ampere/meter units and is denoted .
An analogous property in electrical engineering and materials science, electric coercivity, is the ability of a ferroelectric material to withstand an external electric field without becoming depolarized.
Ferromagnetic materials with high coercivity are called magnetically hard, and are used to make permanent magnets. Materials with low coercivity are said to be magnetically soft. The latter are used in transformer and inductor cores, recording heads, microwave devices, and magnetic shielding.
Definitions
Coercivity in a ferromagnetic material is the intensity of the applied magnetic field (H field) required to demagnetize that material, after the magnetization of the sample has been driven to saturation by a strong field. This demagnetizing field is applied opposite to the original saturating field. There are however different definitions of coercivity, depending on what counts as 'demagnetized', thus the bare term "coercivity" may be ambiguous:
The normal coercivity, , is the H field required to reduce the magnetic flux (average B field inside the material) to zero.
The intrinsic coercivity, , is the H field required to reduce the magnetization (average M field inside the material) to zero.
The remanence coercivity, , is the H field required to reduce the remanence to zero, meaning that when the H field is finally returned to zero, then both B and M also fall to zero (the material reaches the origin in the hysteresis curve).
The distinction between the normal and intrinsic coercivity is negligible in soft magnetic materials, however it can be significant in hard magnetic materials. The strongest rare-earth magnets lose almost none of the magnetization at HCn.
Experi |
https://en.wikipedia.org/wiki/ReactOS | ReactOS is a free and open-source operating system for amd64/i686 personal computers intended to be binary-compatible with computer programs and device drivers developed for Windows Server 2003 and later versions of Microsoft Windows. ReactOS has been noted as a potential open-source drop-in replacement for Windows and for its information on undocumented Windows APIs.
ReactOS has been in development since 1996. , it is still considered feature-incomplete alpha software, and is therefore recommended by the developers only for evaluation and testing purposes. However, many Windows applications are working, such as Adobe Reader 9.3, GIMP 2.6, and LibreOffice 5.4.
ReactOS is primarily written in C, with some elements, such as ReactOS File Explorer, written in C++. The project partially implements Windows API functionality and has been ported to the AMD64 processor architecture. ReactOS, as part of the FOSS ecosystem, re-uses and collaborates with many other FOSS projects, most notably the Wine project, which presents a Windows compatibility layer for Unix-like operating systems.
History
Early development
Around 1996, a group of free and open-source software developers started a project called FreeWin95 to implement a clone of Windows 95. The project stalled in discussions of the design of the system.
While FreeWin95 had started out with high expectations, there still had not been any builds released to the public by the end of 1997. As a result, the project members, led by then coordinator Jason Filby, joined together to revive the project. The revived project sought to duplicate the functionality of Windows NT. In creating the new project, a new name, ReactOS, was chosen. The project began development in February 1998 by creating the basis for a new NT kernel and basic drivers. The name ReactOS was coined during an IRC chat. While the term "OS" stood for operating system, the term "react" referred to the group's dissatisfaction with – and reaction to – Microsoft's |
https://en.wikipedia.org/wiki/Nitrification | Nitrification is the biological oxidation of ammonia to nitrate via the intermediary nitrite. Nitrification is an important step in the nitrogen cycle in soil. The process of complete nitrification may occur through separate organisms or entirely within one organism, as in comammox bacteria. The transformation of ammonia to nitrite is usually the rate limiting step of nitrification. Nitrification is an aerobic process performed by small groups of autotrophic bacteria and archaea.
Microbiology
Ammonia oxidation
The process of nitrification begins with the first stage of ammonia oxidation, where ammonia (NH3) or ammonium (NH4+) get converted into nitrite (NO2-). This first stage is sometimes known as nitritation. It is performed by two groups of organisms, ammonia-oxidizing bacteria (AOB) and ammonia-oxidizing archaea (AOA).
Ammonia-Oxidizing Bacteria
Ammonia-Oxidizing Bacteria (AOB) are typically Gram-negative bacteria and belong to Betaproteobacteria and Gammaproteobacteria including the commonly studied genera including Nitrosomonas and Nitrococcus. They are known for their ability to utilize ammonia as an energy source and are prevalent in a wide range of environments, such as soils, aquatic systems, and wastewater treatment plants.
AOB possess enzymes called ammonia monooxygenases (AMOs), which are responsible for catalyzing the conversion of ammonia to hydroxylamine (NH2OH), a crucial intermediate in the process of nitrification. This enzymatic activity is sensitive to environmental factors, such as pH, temperature, and oxygen availability.
AOB play a vital role in soil nitrification, making them key players in nutrient cycling. They contribute to the transformation of ammonia derived from organic matter decomposition or fertilizers into nitrite, which subsequently serves as a substrate for nitrite-oxidizing bacteria (NOB).
Ammonia-Oxidizing Archaea
Prior to the discovery of archaea capable of ammonia oxidation, ammonia-oxidizing bacteria (AOB) were consi |
https://en.wikipedia.org/wiki/Bio-inspired%20computing | Bio-inspired computing, short for biologically inspired computing, is a field of study which seeks to solve computer science problems using models of biology. It relates to connectionism, social behavior, and emergence. Within computer science, bio-inspired computing relates to artificial intelligence and machine learning. Bio-inspired computing is a major subset of natural computation.
History
Early Ideas
The ideas behind biological computing trace back to 1936 and the first description of an abstract computer, which is now known as a Turing machine. Turing firstly described the abstract construct using a biological specimen. Turing imagined a mathematician that has three important attributes. He always has a pencil with an eraser, an unlimited number of papers and a working set of eyes. The eyes allow the mathematician to see and perceive any symbols written on the paper while the pencil allows him to write and erase any symbols that he wants. Lastly, the unlimited paper allows him to store anything he wants memory. Using these ideas he was able to describe an abstraction of the modern digital computer. However Turing mentioned that anything that can perform these functions can be considered such a machine and he even said that even electricity should not be required to describe digital computation and machine thinking in general.
Neural Networks
First described in 1943 by Warren McCulloch and Walter Pitts, neural networks are a prevalent example of biological systems inspiring the creation of computer algorithms. They first mathematically described that a system of simplistic neurons was able to produce simple logical operations such as logical conjunction, disjunction and negation. They further showed that a system of neural networks can be used to carry out any calculation that requires finite memory. Around 1970 the research around neural networks slowed down and many consider a 1969 book by Marvin Minsky and Seymour Papert as the main cause. Their book s |
https://en.wikipedia.org/wiki/Generalized%20hypergeometric%20function | In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.
Notation
A hypergeometric series is formally defined as a power series
in which the ratio of successive coefficients is a rational function of n. That is,
where A(n) and B(n) are polynomials in n.
For example, in the case of the series for the exponential function,
we have:
So this satisfies the definition with and .
It is customary to factor out the leading term, so β0 is assumed to be 1. The polynomials can be factored into linear factors of the form (aj + n) and (bk + n) respectively, where the aj and bk are complex numbers.
For historical reasons, it is assumed that (1 + n) is a factor of B. If this is not already the case then both A and B can be multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality.
The ratio between consecutive coefficients now has the form
,
where c and d are the leading coefficients of A and B. The series then has the form
,
or, by scaling z by the appropriate factor and rearranging,
.
This has the form of an exponential generating function. This series is usually denoted by
or
Using the rising factorial or Pochhammer symbol
this can be written
(Note that this use of the Pochhammer sym |
https://en.wikipedia.org/wiki/Hanover%20bars | Hanover bars, in one of the PAL television video formats, are an undesirable visual artifact in the reception of a television image. The name refers to the city of Hannover, in which the PAL system developer Telefunken Fernseh und Rundfunk GmbH was located.
The PAL system encodes color as YUV. The U (corresponding to B-Y) and V (corresponding to R-Y) signals carry the color information for a picture, with the phase of the V signal reversed (i.e. shifted through 180 degrees) on alternate lines (hence the name PAL, or phase alternate line). This is done to cancel minor phase errors in the reception process. However, if gross errors occur, complementary errors from the V signal carry into the U signal, and thus visible stripes occur.
Later PAL systems introduced alterations to ensure that Hanover bars do not occur, introducing a swinging burst to the color synchronization. Other PAL systems may handle this problem differently.
Suppression of Hanover bars
To suppress Hanover bars, PAL color decoders use a delay line that repeats the chroma information from each previous line and blends it with the current line. This causes phase errors to cancel out, at the cost of vertical color resolution, and in early designs, also a loss of color saturation proportional to the phase error.
References
See also
Dot crawl
PAL
PAL-S
Television technology |
https://en.wikipedia.org/wiki/Rubik%27s%20Magic | Rubik's Magic, like the Rubik's Cube, is a mechanical puzzle invented by Ernő Rubik and first manufactured by Matchbox in the mid-1980s.
The puzzle consists of eight black square tiles (changed to red squares with goldish rings in 1997) arranged in a 2 × 4 rectangle; diagonal grooves on the tiles hold wires that connect them, allowing them to be folded onto each other and unfolded again in two perpendicular directions (assuming that no other connections restrict the movement) in a manner similar to a Jacob's ladder toy. The front side of the puzzle shows, in the initial state, three separate, rainbow-coloured rings; the back side consists of a scrambled picture of three interconnected rings. The goal of the game is to fold the puzzle into a heart-like shape and unscramble the picture on the back side, thus interconnecting the rings.
Numerous ways to accomplish this exist, and experienced players can transform the puzzle from its initial into the solved state in less than 2 seconds. Other challenges for Rubik's Magic include reproducing given shapes (which are often three-dimensional), sometimes with certain tiles required to be in certain positions and/or orientations.
History
Rubik's Magic was first manufactured by Matchbox in 1986. Professor Rubik holds both a Hungarian patent (HU 1211/85, issued 19 March 1985) and a US patent (US 4,685,680, issued 11 August 1987) on the mechanism of Rubik's Magic.
In 1987, Rubik's Magic: Master Edition was published by Matchbox; it consisted of 12 silver tiles arranged in a 2 × 6 rectangle, showing 5 interlinked rings that had to be unlinked by transforming the puzzle into a shape reminiscent of a W. Around the same time, Matchbox also produced Rubik's Magic Create the Cube, a "Level Two" version of Rubik's Magic, in which the puzzle is solved when folded into a cube with a base of two tiles, and the tile colors match at the corners of the cube. It did not have as wide a release, and is rare to find.
In 1996, the original v |
https://en.wikipedia.org/wiki/Negentropy | In information theory and statistics, negentropy is used as a measure of distance to normality. The concept and phrase "negative entropy" was introduced by Erwin Schrödinger in his 1944 popular-science book What is Life? Later, French physicist Léon Brillouin shortened the phrase to néguentropie (negentropy). In 1974, Albert Szent-Györgyi proposed replacing the term negentropy with syntropy. That term may have originated in the 1940s with the Italian mathematician Luigi Fantappiè, who tried to construct a unified theory of biology and physics. Buckminster Fuller tried to popularize this usage, but negentropy remains common.
In a note to What is Life? Schrödinger explained his use of this phrase.
Information theory
In information theory and statistics, negentropy is used as a measure of distance to normality. Out of all distributions with a given mean and variance, the normal or Gaussian distribution is the one with the highest entropy. Negentropy measures the difference in entropy between a given distribution and the Gaussian distribution with the same mean and variance. Thus, negentropy is always nonnegative, is invariant by any linear invertible change of coordinates, and vanishes if and only if the signal is Gaussian.
Negentropy is defined as
where is the differential entropy of the Gaussian density with the same mean and variance as and is the differential entropy of :
Negentropy is used in statistics and signal processing. It is related to network entropy, which is used in independent component analysis.
The negentropy of a distribution is equal to the Kullback–Leibler divergence between and a Gaussian distribution with the same mean and variance as (see for a proof). In particular, it is always nonnegative.
Correlation between statistical negentropy and Gibbs' free energy
There is a physical quantity closely linked to free energy (free enthalpy), with a unit of entropy and isomorphic to negentropy known in statistics and information theory. In 1 |
https://en.wikipedia.org/wiki/Line%20bundle | In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of rank 1.
Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy type of a circle.
From the perspective of homotopy theory, a real line bundle therefore behaves much the same as a fiber bundle with a two-point fiber, that is, like a double cover. A special case of this is the orientable double cover of a differentiable manifold, where the corresponding line bundle is the determinant bundle of the tangent bundle (see below). The Möbius strip corresponds to a double cover of the circle (the θ → 2θ mapping) and by changing the fiber, can also be viewed as having a two-point fiber, the unit interval as a fiber, or the real line.
Complex line bundles are closely related to circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres.
In algebraic geometry, an invertible sheaf (i.e., locally free sheaf of rank one) is often called a line bundle.
Every line bundle arises from a divisor with the following conditions
(I) If X is reduced and irreducible sc |
https://en.wikipedia.org/wiki/Descent%20%28mathematics%29 | In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.
Descent of vector bundles
The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start.
Suppose X is a topological space covered by open sets Xi. Let Y be the disjoint union of the Xi, so that there is a natural mapping
We think of Y as 'above' X, with the Xi projection 'down' onto X. With this language, descent implies a vector bundle on Y (so, a bundle given on each Xi), and our concern is to 'glue' those bundles Vi, to make a single bundle V on X. What we mean is that V should, when restricted to Xi, give back Vi, up to a bundle isomorphism.
The data needed is then this: on each overlap
intersection of Xi and Xj, we'll require mappings
to use to identify Vi and Vj there, fiber by fiber. Further the fij must satisfy conditions based on the reflexive, symmetric and transitive properties of an equivalence relation (gluing conditions). For example, the composition
for transitivity (and choosing apt notation). The fii should be identity maps and hence symmetry becomes (so that it is fiberwise an isomorphism).
These are indeed standard conditions in fiber bundle theory (see transition map). One important application to note is change of fiber: if the fij are all you need to make a bundle, then there are many ways to make an associated bundle. That is, we can take essentially same fij, acting on various fibers.
Another major point is the relation with the chain rule: the discussion of the way there of constructing tensor fields can be summed up as 'once you learn to descend the tangent bundle, for which transitivity is the Jacobian chain rule, the rest is just 'naturality of tensor constructions'.
To move closer towards the abstract theory we need to interpret t |
https://en.wikipedia.org/wiki/System%20console | One meaning of system console, computer console, root console, operator's console, or simply console is the text entry and display device for system administration messages, particularly those from the BIOS or boot loader, the kernel, from the init system and from the system logger. It is a physical device consisting of a keyboard and a screen, and traditionally is a text terminal, but may also be a graphical terminal. System consoles are generalized to computer terminals, which are abstracted respectively by virtual consoles and terminal emulators. Today communication with system consoles is generally done abstractly, via the standard streams (stdin, stdout, and stderr), but there may be system-specific interfaces, for example those used by the system kernel.
Another, older, meaning of system console, computer console, hardware console, operator's console or simply console is a hardware component used by an operator to control the hardware, typically some combination of front panel, keyboard/printer and keyboard/display.
History
Prior to the development of alphanumeric CRT system consoles, some computers such as the IBM 1620 had console typewriters and front panels while the very first programmable computer, the Manchester Baby, used a combination of electromechanical switches and a CRT to provide console functions—the CRT displaying memory contents in binary by mirroring the machine's Williams-Kilburn tube CRT-based RAM.
Some early operating systems supported either a single keyboard/print or keyboard/display device for controlling the OS. Some also supported a single alternate console, and some supported a hardcopy console for retaining a record of commands, responses and other console messages. However, in the late 1960s it became common for operating systems to support many more consoles than 3, and operating systems began appearing in which the console was simply any terminal with a privileged user logged on.
On early minicomputers, the console was a seri |
https://en.wikipedia.org/wiki/Computer%20lab | A computer lab is a space where computer services are provided to a defined community. These are typically public libraries and academic institutions. Generally, users must follow a certain user policy to retain access to the computers. This usually consists of rules such as no illegal activity during use or attempts to circumvent any security or content-control software while using the computers.
Computer labs are often subject to time limits in order to allow more people access to use the lab. It is also common for personal login credentials to be required for access. This allows institutions to track the user's activities for any possible fraudulent use. The computers in computer labs are typically equipped with internet access, scanners, and printers and are typically arranged in rows. This is to give the workstation a similar view to facilitate lecturing or presentations, and also to facilitate small group work.
For some academic institutions, student laptops or laptop carts take place of dedicated computer labs. However, computer labs still have a place in applications requiring special software or hardware which are not easily accessible in personal computers.
Purposes
While computer labs are generally multipurpose, some labs may contain computers with hardware or software optimized for certain tasks or processes, depending on the needs of the institution operating the lab. These specialized purposes may include video editing, stock trading, 3-D computer-aided design, programming, and GIS. Increasingly, these have become the main purposes for the existence of traditional desktop-style computer labs, due to rising ownership of inexpensive personal computers making use of the lab only necessary when the expensive, specialized software and more powerful computers needed to run it are required.
Arrangements
Alternatives
In some settings, traditional desktop computer labs are impractical due to the requirement of a dedicated space. Because of this, some |
https://en.wikipedia.org/wiki/Order%20theory | Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.
Background and motivation
Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference of two numbers, which is not given by the order). Other familiar examples of orderings are the alphabetical order of words in a dictionary and the genealogical property of lineal descent within a group of people.
The notion of order is very general, extending beyond contexts that have an immediate, intuitive feel of sequence or relative quantity. In other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the subset relation, e.g., "Pediatricians are physicians," and "Circles are merely special-case ellipses."
Some orders, like "less-than" on the natural numbers and alphabetical order on words, have a special property: each element can be compared to any other element, i.e. it is smaller (earlier) than, larger (later) than, or identical to. However, many other orders do not. Consider for example the subset order on a collection of sets: though the set of birds and the set of dogs are both subsets of the set of animals, neither the |
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