Congruence relationIn abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. The prototypical example of a congruence relation is congruence modulo on the set of integers.
Borel equivalence relationIn mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borel subset of X × X (in the product topology). Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducible to F, in symbols E ≤B F, if and only if there is a Borel function Θ : X → Y such that for all x,x' ∈ X, one has x E x' ⇔ Θ(x) F Θ(x').
Equivalence relationIn mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.
MD5The MD5 message-digest algorithm is a widely used hash function producing a 128-bit hash value. MD5 was designed by Ronald Rivest in 1991 to replace an earlier hash function MD4, and was specified in 1992 as RFC 1321. MD5 can be used as a checksum to verify data integrity against unintentional corruption. Historically it was widely used as a cryptographic hash function; however it has been found to suffer from extensive vulnerabilities.
MD4The MD4 Message-Digest Algorithm is a cryptographic hash function developed by Ronald Rivest in 1990. The digest length is 128 bits. The algorithm has influenced later designs, such as the MD5, SHA-1 and RIPEMD algorithms. The initialism "MD" stands for "Message Digest". The security of MD4 has been severely compromised. The first full collision attack against MD4 was published in 1995, and several newer attacks have been published since then. As of 2007, an attack can generate collisions in less than 2 MD4 hash operations.
SandSand is a granular material composed of finely divided mineral particles. Sand has various compositions but is defined by its grain size. Sand grains are smaller than gravel and coarser than silt. Sand can also refer to a textural class of soil or soil type; i.e., a soil containing more than 85 percent sand-sized particles by mass. The composition of sand varies, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal settings is silica (silicon dioxide, or SiO2), usually in the form of quartz.
Ab initioAb initio (ˌæb_ᵻˈnɪʃioʊ ) is a Latin term meaning "from the beginning" and is derived from the Latin ab ("from") + initio, ablative singular of initium ("beginning"). Circa 1600, from Latin, literally "from the beginning", from ablative case of initium "entrance", "beginning", related to verb inire "to go into", "enter upon", "begin". Ab initio (abbreviation: ab init.) is used in several contexts, including the following: In law, ab initio refers to something being the case from the start or from the instant of the act rather than from when the court declared it so.
ExerciseExercise is a body activity that enhances or maintains physical fitness and overall health and wellness. It is performed for various reasons, including weight loss or maintenance, to aid growth and improve strength, develop muscles and the cardiovascular system, hone athletic skills, improve health, or simply for enjoyment. Many individuals choose to exercise outdoors where they can congregate in groups, socialize, and improve well-being as well as mental health.
Sand castingSand casting, also known as sand molded casting, is a metal casting process characterized by using sand known as casting sand as the mold material. The term "sand casting" can also refer to an object produced via the sand casting process. Sand castings are produced in specialized factories called foundries. In 2003, over 60% of all metal castings were produced via sand casting. Molds made of sand are relatively cheap, and sufficiently refractory even for steel foundry use.
European College of NeuropsychopharmacologyThe European College of Neuropsychopharmacology (ECNP) is a pan-European, non-profit scientific association that serves as a platform to exchange and promote research in the field of neuropsychopharmacology. The ECNP “is committed to ensuring that advances in the understanding of brain function and human behaviour are translated into better treatments and enhanced public health”. The ECNP organises a number of activities to achieve this aim, such as a yearly congress, workshops, seminars, New Frontiers Meetings, publications, awards, supported talks and much more.
Set theorySet theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.
A priori and a posterioriA priori ("from the earlier") and a posteriori ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on experience. A priori knowledge is independent from any experience. Examples include mathematics, tautologies, and deduction from pure reason. A posteriori knowledge depends on empirical evidence. Examples include most fields of science and aspects of personal knowledge. The terms originate from the analytic methods found in Organon, a collection of works by Aristotle.
