RandomnessIn common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4.
Randomness testA randomness test (or test for randomness), in data evaluation, is a test used to analyze the distribution of a set of data to see whether it can be described as random (patternless). In stochastic modeling, as in some computer simulations, the hoped-for randomness of potential input data can be verified, by a formal test for randomness, to show that the data are valid for use in simulation runs. In some cases, data reveals an obvious non-random pattern, as with so-called "runs in the data" (such as expecting random 0–9 but finding "4 3 2 1 0 4 3 2 1.
Statistical randomnessA numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll or the digits of π exhibit statistical randomness. Statistical randomness does not necessarily imply "true" randomness, i.e., objective unpredictability. Pseudorandomness is sufficient for many uses, such as statistics, hence the name statistical randomness. Global randomness and local randomness are different.
Random number generationRandom number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols that cannot be reasonably predicted better than by random chance is generated. This means that the particular outcome sequence will contain some patterns detectable in hindsight but unpredictable to foresight. True random number generators can be hardware random-number generators (HRNGs), wherein each generation is a function of the current value of a physical environment's attribute that is constantly changing in a manner that is practically impossible to model.
Algorithmically random sequenceIntuitively, an algorithmically random sequence (or random sequence) is a sequence of binary digits that appears random to any algorithm running on a (prefix-free or not) universal Turing machine. The notion can be applied analogously to sequences on any finite alphabet (e.g. decimal digits). Random sequences are key objects of study in algorithmic information theory. As different types of algorithms are sometimes considered, ranging from algorithms with specific bounds on their running time to algorithms which may ask questions of an oracle machine, there are different notions of randomness.
Randomized algorithmA randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables.
Hardware random number generatorIn computing, a hardware random number generator (HRNG), true random number generator (TRNG) or non-deterministic random bit generator (NRBG) is a device that generates random numbers from a physical process capable of producing entropy (in other words, the device always has access to a physical entropy source), rather than by means of an algorithm. Such devices are often based on microscopic phenomena that generate low-level, statistically random "noise" signals, such as thermal noise, the photoelectric effect, involving a beam splitter, and other quantum phenomena.
Jury trialA jury trial, or trial by jury, is a legal proceeding in which a jury makes a decision or findings of fact. It is distinguished from a bench trial in which a judge or panel of judges makes all decisions. Jury trials are used in a significant share of serious criminal cases in many but not all common law judicial systems. The majority of common law jurisdictions in Asia (such as Singapore, India, Pakistan and Malaysia) have abolished jury trials on the grounds that juries are susceptible to bias.
Petit juryIn common law, a petit jury (or trial jury; pronounced 'pEt@t or p@'ti:t, depending on the jurisdiction) hears the evidence in a trial as presented by both the plaintiff (petitioner) and the defendant (respondent). After hearing the evidence and often jury instructions from the judge, the group retires for deliberation, to consider a verdict. The majority required for a verdict varies. In some cases it must be unanimous, while in other jurisdictions it may be a majority or supermajority.
JuryA jury is a sworn body of people (jurors) convened to hear evidence and render an impartial verdict (a finding of fact on a question) officially submitted to them by a court, or to set a penalty or judgment. Juries developed in England during the Middle Ages and are a hallmark of the English common law system. As such, they are used by the United Kingdom, the United States, Canada, Ireland, Australia, and other countries whose legal systems were derived from the British Empire.
Group representationIn the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.
Representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
Bench trialA bench trial is a trial by judge, as opposed to a trial by jury. The term applies most appropriately to any administrative hearing in relation to a summary offense to distinguish the type of trial. Many legal systems (Roman, Islamic) use bench trials for most or all cases or for certain types of cases. While a jury renders a verdict, a judge in a bench trial does the same by making a finding. The majority of civil trials proceed without a jury and are heard by a judge sitting alone.
Irreducible representationIn mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of . Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.
Hung juryA hung jury, also called a deadlocked jury, is a judicial jury that cannot agree upon a verdict after extended deliberation and is unable to reach the required unanimity or supermajority. Hung juries usually result in the case being tried again. This situation can occur only in common law legal systems, because civil law systems either do not use juries at all or provide that the defendant is immediately acquitted if the majority or supermajority required for conviction is not reached during a single, solemn vote.
Jury nullificationJury nullification (US/UK), jury equity (UK), or a perverse verdict (UK) occurs when the jury in a criminal trial gives a not guilty verdict regardless of whether they believe a defendant has broken the law. The jury's reasons may include the belief that the law itself is unjust, that the prosecutor has misapplied the law in the defendant's case, that the punishment for breaking the law is too harsh, or general frustrations with the criminal justice system. Some juries have also refused to convict due to their own prejudices in favor of the defendant.
Deliberative democracyDeliberative democracy or discursive democracy is a form of democracy in which deliberation is central to decision-making. Deliberative democracy seeks quality over quantity by limiting decision-makers to a smaller but more representative sample of the population that is given the time and resources to focus on one issue. It often adopts elements of both consensus decision-making and majority rule. Deliberative democracy differs from traditional democratic theory in that authentic deliberation, not mere voting, is the primary source of legitimacy for the law.
Lie algebra representationIn the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
Fundamental frequencyThe fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. In some contexts, the fundamental is usually abbreviated as 0, indicating the lowest frequency counting from zero.
Online deliberationOnline deliberation is a broad term used to describe many forms of non-institutional, institutional and experimental online discussions. The term also describes the emerging field of practice and research related to the design, implementation and study of deliberative processes that rely on the use of electronic information and communications technologies (ICT). Although the Internet and social media have fostered discursive participation and deliberation online through computer-mediated communication, the academic study of online deliberation started in the early 2000s.
