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mathematics, a random compact set is essentially a compact set-valued random variable. Random compact sets are useful in the study of attractors for random dynamical
Random_compact_set
compact subsets of n-dimensional Euclidean space Rn to random compact sets. Let X be a random compact set in Rn; that is, a Borel–measurable function from some
Vitale's random Brunn–Minkowski inequality
Vitale's_random_Brunn–Minkowski_inequality
is a purely atomic measure. A random set is a set-valued random element. One specific example is a random compact set. Let ( M , d ) {\displaystyle (M
Random_element
Variable representing a random phenomenon
random variables Event (probability theory) Multivariate random variable Pairwise independent random variables Observable variable Random compact set
Random_variable
Type of random variable
particularly in probability theory and stochastic geometry, a random closed set is a random variable whose values are closed subsets of a given topological
Random_closed_set
Overview of and topical guide to probability
Probability space Sample space Standard probability space Random element Random compact set Dynkin system Probability axioms Event (probability theory)
Outline_of_probability
Probability space Sample space Standard probability space Random element Random compact set Dynkin system Probability axioms Normalizing constant Event
List_of_probability_topics
θ : R × Ω → Ω. A random compact set K : Ω → 2X is said to be absorbing if, for all d-bounded deterministic sets B ⊆ X, there exists a (finite) random time τB : Ω → 0
Absorbing set (random dynamical systems)
Absorbing_set_(random_dynamical_systems)
Function from sets to numbers
each set in the powerset of some given set. See possibility theory. A random set is a set-valued random variable. See the article random compact set. The
Set_function
Type of topological space in mathematics
X is called locally compact if every point x of X has a compact neighbourhood, i.e., there exists an open set U and a compact set K, such that x ∈ U ⊆
Locally_compact_space
Inputs for which a function's value is non-zero
Bounded function – Mathematical function whose set of values is bounded Bump function – Smooth and compactly supported function Support of a module Titchmarsh
Support_(mathematics)
minimal compact invariant set that attracts all bounded deterministic sets. If a random dynamical system has a compact random absorbing set K {\displaystyle
Pullback_attractor
Class of mathematical sets
Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many
Borel_set
index Random assignment Random compact set Random data – see randomness Random effects estimation – see Random effects model Random effects model Random element
List_of_statistics_articles
Data structure for approximate set membership
to random ( log n + log ϵ − 1 ) {\displaystyle (\log n+\log \epsilon ^{-1})} -bit fingerprints, and then storing those fingerprints in a compact hash
Bloom_filter
Functional analysis theorem
theorem is a result related to the set of compact and convex subsets of a normed vector space. It states that such sets can be isometrically embededded into
Rådström's_embedding_theorem
Form of computer data storage
Random-access memory (RAM; /ræm/) is a form of electronic computer memory that can be read and changed in any order, typically used to store working data
Random-access_memory
Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the smallest σ-algebra such that all compactly supported
Baire_set
moments of functions of random variables / (1:R) Bertrand's paradox / (1:M) Pitman–Yor process / (1:G) Random compact set / (1:G) Random element / (1:G) Coupling /
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Random set of points on a space with random number and random position
probability theory, a point process or point field is a set of a random number of mathematical points randomly located on a mathematical space such as the real
Point_process
Sums of sets of vectors are nearly convex
Zvi; Vitale, Richard A. (1975). "A strong law of large numbers for random compact sets". The Annals of Probability. 3 (5): 879–882. doi:10.1214/aop/1176996275
Shapley–Folkman_lemma
Diagram that shows all possible logical relations between a collection of sets
between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships
Venn_diagram
Concept in measure theory
or a Radon random variable. Another equivalent criterion of the tightness of a collection M {\displaystyle M} is sequential weak compactness. We say the
Tightness_of_measures
Process of finding the optimal set of variables for a machine learning algorithm
networks offer a memory efficient version of this approach by choosing a compact representation for the hypernetwork. More recently, Δ-STN has improved
Hyperparameter_optimization
High density mixed use transit oriented planning
The compact city or city of short distances is an urban planning and urban design concept, which promotes relatively high residential density with mixed
Compact_city
Covering by shapes without overlaps or gaps
square is {4}. The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons
Tessellation
Cryptographic device
hardware random number generator (HRNG), true random number generator (TRNG), non-deterministic random bit generator (NRBG), or physical random number generator
Hardware random number generator
Hardware_random_number_generator
probability Multivalued function Differential inclusion Observational error Random compact set Reliability (statistics) Confidence interval Best, worst and average
Interval_finite_element
Probability distribution modeling a coin toss which need not be fair
mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p {\displaystyle p} and
Bernoulli_distribution
Study of random spatial patterns
random object processes, allowing construction of elaborate random spatial patterns. The simplest version, the Boolean model, places a random compact
Stochastic_geometry
Type of random mathematical object
Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located
Poisson_point_process
Type of computer memory
Magnetoresistive random-access memory (MRAM) is a type of non-volatile random-access memory which stores data in magnetic domains. Developed in the mid-1980s
Magnetoresistive_RAM
Stochastic way of assigning quantities across a space
probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they
Random_measure
Left-invariant (or right-invariant) measure on locally compact topological group
the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on
Haar_measure
Type of computer memory introduced 2014
Double Data Rate 4 Synchronous Dynamic Random-Access Memory (DDR4 SDRAM) is a type of synchronous dynamic random-access memory with a high-bandwidth ("double
DDR4_SDRAM
Probability distribution
distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f ( x
Normal_distribution
Type of plane partition
space and the distance to each site is attained (e.g., when a site is a compact set or a closed ball), then each Voronoi cell can be represented as a union
Voronoi_diagram
Electronic device that plays audio compact discs
A CD player is an electronic device that plays audio compact discs, which are a digital optical disc data storage format. CD players were first sold to
CD_player
Data format used for audio compact discs
Compact Disc Digital Audio (CDDA or CD-DA), also known as Digital Audio Compact Disc or simply as Audio CD, is the standard format for audio compact discs
Compact_Disc_Digital_Audio
Data storage device
of random access, which means that it takes the same amount of time to access any memory location, so data can be efficiently accessed in any random order
Semiconductor_memory
Phenomenon in statistics
alignments of random points in a plane seeks to discover subsets of points that occupy an approximately straight line within a larger set of points that
Alignments_of_random_points
Shape containing unit line segments in all directions
k-dimensional subspaces. Define an (n, k)-Besicovitch set K to be a compact set in Rn containing a translate of every k-dimensional unit disk which has
Kakeya_set
Probability distribution
by the fact that convergent power series converge uniformly on compact subsets of the set of points where they converge. The mean of the geometric distribution
Geometric_distribution
Classical problem in combinatorics
the maximum number of sets that any element belongs to. Randomized rounding is an approximation technique for the weighted set cover problem that uses
Set_cover_problem
Typeface style used in mathematics
mathematical texts. The style is most commonly used to represent the number sets N {\displaystyle \mathbb {N} } (natural numbers), Z {\displaystyle \mathbb
Blackboard_bold
mutually non-isomorphic models) are obtained by probabilistic means (random compact sets and Brownian motion). One part of this theory (so-called type III
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
Infinite graph containing all countable graphs
non-randomly, as a graph whose vertices are hereditarily finite sets (finite sets whose elements are hereditarily finite) with edges representing set membership
Rado_graph
Plane fractal built from squares
carpet is a universal plane curve. That is: the Sierpiński carpet is a compact subset of the plane with Lebesgue covering dimension 1, and every subset
Sierpiński_carpet
Type of computer memory
Dynamic random-access memory (dynamic RAM or DRAM) is a type of random-access semiconductor memory that stores each bit of data in a memory cell. A DRAM
Dynamic_random-access_memory
American progressive rock band
managed to trim the run time down to 80 minutes so as to fit on a single compact disc. Act II reached No. 39 on the Billboard Heatseekers chart. The band
The_Dear_Hunter
Family of RISC-based computer architectures
processor executes the Thumb instruction set, a compact 16-bit encoding for a subset of the ARM instruction set. Most of the Thumb instructions are directly
Arm_architecture_family
Array data structure that compactly stores bits
bit map, bit set, bit string, or bit vector) is an array data structure that compactly stores bits. It can be used to implement a simple set data structure
Bit_array
Collection of statistical models
relations among variables. A dog show provides an example. A dog show is not a random sampling of the breed: it is typically limited to dogs that are adult, pure-bred
Analysis_of_variance
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
quaternionic inner product space. In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent k-frames in R n , C n ,
Stiefel_manifold
Binary tree selected at random
In computer science and probability theory, a random binary tree is a binary tree selected at random from some probability distribution on binary trees
Random_binary_tree
Vector quantization algorithm minimizing the sum of squared deviations
Partition places all of them close to the center of the data set. According to Hamerly et al., the Random Partition method is generally preferable for algorithms
K-means_clustering
Branch of mathematics
function.) Namely, there is always a compact subset of C P 1 {\displaystyle \mathbf {CP} ^{1}} , the Julia set, on which the dynamics of f is chaotic
Complex_dynamics
Measure on group representations
Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group G {\displaystyle G} , that describes
Plancherel_measure
2019 video game
minerals. These minerals can be found on asteroids or planets, plundered from randomly spawned ships, or recovered from unknown signals. Raw materials are extracted
Space_Engineers
Probability distribution
random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable
Distribution of the product of two random variables
Distribution_of_the_product_of_two_random_variables
Statistical model
stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate
Gaussian_process
Method for the construction of fractals
any nonempty compact set A in X. (For contractive IFS this convergence takes place even for any nonempty closed bounded set A). Random elements arbitrarily
Iterated_function_system
Type of computer memory
Non-volatile random-access memory (NVRAM) is random-access memory that retains data without applied power. This is in contrast to dynamic random-access memory
Non-volatile random-access memory
Non-volatile_random-access_memory
Type of computer memory
Double data rate synchronous dynamic random-access memory (DDR SDRAM) is a type of synchronous dynamic random-access memory (SDRAM) widely used in computers
DDR_SDRAM
Smallest convex set containing a given set
the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme
Convex_hull
Free and open-source software portal Colt is a set of open-source libraries for high-performance scientific and technical computing written in Java and
Colt_(libraries)
Measure of algorithmic complexity
theory (or Kolmogorov complexity). Kolmogorov randomness defines a string (usually of bits) as being random if the shortest computer program that can produce
Kolmogorov_complexity
Branch of mathematics that studies dynamical systems
T-invariant mod 0 set that is a union of q intervals of length a, hence it has measure qa strictly between 0 and 1. Let G be a compact abelian group, μ
Ergodic_theory
Eighteenth letter of the Greek alphabet
probability theory and statistics, Σ denotes the covariance matrix of a set of random variables, sometimes in the form | Σ {\displaystyle \;|\!\!\!\Sigma
Sigma
1977 studio album by Jay Ferguson
The album was first issued on compact disc by the Collector's Choice label. The album was reissued as part of a two disc set including Ferguson's first and
Thunder_Island_(album)
Property of a sequence or series
(local, uniform, absolute convergence) and compact normal convergence (absolute convergence on compact sets) can be defined. Normal convergence implies
Modes_of_convergence
probability distribution on compact sets; for each point ξ {\displaystyle \xi } of the Poisson point process we pick a set C ξ {\displaystyle C_{\xi }}
Boolean model (probability theory)
Boolean_model_(probability_theory)
Camera that captures photographs or video in digital format
main imager. Some compact digital cameras use a hybrid autofocus system similar to what is commonly available on DSLRs. Typically, compact digital cameras
Digital_camera
Fourier transform of the probability density function
characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Office, and the HP 110 font set. New fonts include: Albertus (Light, Roman, Italic) Antique Olive (Roman, Italic, Bold, Compact) Apple Chancery Arial (Regular
PostScript_fonts
Marked objects for finding random numbers
sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board
Dice
Subset of artificial intelligence
of its points. This process condenses extensive datasets into a more compact set of representative points. Particularly beneficial in image and signal
Machine_learning
Type of computer memory
Synchronous dynamic random-access memory (synchronous dynamic RAM or SDRAM) is any DRAM where the operation of its external pin interface is coordinated
Synchronous dynamic random-access memory
Synchronous_dynamic_random-access_memory
Intersection of Set Theory and General Topology
one new set, and allowing for fine control over the properties of the expanded universe. See the main articles for applications such as random reals. M
Set-theoretic_topology
Averages of repeated trials converge to the expected value
the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law
Law_of_large_numbers
Proving validity without revealing other data
required, except for trivial proofs of BPP problems. In the common random string and random oracle models, non-interactive zero-knowledge proofs exist. The
Zero-knowledge_proof
Chess set used for competitive play
bases: The resulting pieces were compact, well balanced, and weighted to provide an understandable, practical playing set. It may have been a combination
Staunton_chess_set
Fractal composed of triangles
but any other compact set would suffice. If one takes a point and applies each of the transformations dA, dB, and dC to it randomly, the resulting points
Sierpiński_triangle
Numerical integration process
problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences) to achieve variance reduction. This is in contrast
Quasi-Monte_Carlo_method
Property of uniformly space-filling movement
visit all parts of the space in which the system moves, in a uniform and random sense. This implies that the average behavior of the system can be deduced
Ergodicity
Problem in computer science
a computation", in Mathematics Today, Lynn Arthur Steen, Vintage Books (Random House), 1980. A wonderful little paper, perhaps the best ever written about
Halting_problem
Mathematical concept
measure, and K {\displaystyle {\mathcal {K}}} is the collection of all compact subsets of X {\displaystyle X} , the notion of K {\displaystyle {\mathcal
Uniform_integrability
System with self-optimizing transfer function
undesired signal (typically random noise) not correlated with g, g', u, u' or v', v' = an undesired signal (typically random noise) not correlated with
Adaptive_filter
Concept in measure theory
Locally compact groups which are σ-compact are σ-finite under the Haar measure. For example, all connected, locally compact groups G are σ-compact. To see
Σ-finite_measure
Concept in mathematics
curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations
Integral_geometry
Generalization of a measure
not. First restrict the content to compact sets. This gives a function λ {\displaystyle \lambda } of compact sets C {\displaystyle C} with the following
Content_(measure_theory)
Volume rendering technique
compared to Neural Radiance Field (NeRF) based solutions, though still more compact than previous point-based approaches. May require hyperparameter tuning
Gaussian_splatting
Stochastic point process in mathematics
{\displaystyle K(x,x')} . Let Λ {\displaystyle \Lambda } be a locally compact Polish space and μ {\displaystyle \mu } be a Radon measure on Λ {\displaystyle
Determinantal_point_process
Mathematical space with a notion of distance
defined for metric spaces. Other notions, such as continuity, compactness, and open and closed sets can be defined for metric spaces, but also in the even more
Metric_space
as a single product. Compact Disc-Recordable (CD-R) A variation of the optical compact disc which can be written to once. Compact Disc-ReWritable (CD-RW)
Glossary of computer hardware terms
Glossary_of_computer_hardware_terms
Mathematical concept
of continuous functions f {\displaystyle f} each vanishing outside a compact set. C 0 ( X ) {\displaystyle C_{0}(X)} the class of continuous functions
Convergence_of_measures
Novel type of computer memory
Resistive random-access memory (ReRAM or RRAM) is a type of non-volatile (NV) random-access (RAM) computer memory that works by changing the resistance
Resistive random-access memory
Resistive_random-access_memory
Divide and conquer sorting algorithm
sorting. Overall, it is slightly faster than merge sort and heapsort for randomized data, particularly on larger distributions. Quicksort is a divide-and-conquer
Quicksort
Relation among continuous functions
usually appears in the context of topological vector spaces. When X is compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every
Equicontinuity
Machine for shuffling playing cards
A shuffling machine is a machine for randomly shuffling packs of playing cards. Because standard shuffling techniques are seen as weak, and in order to
Shuffling_machine
RANDOM COMPACT-SET
RANDOM COMPACT-SET
Surname or Lastname
English
English : unexplained; perhaps a variant of Francom.
Surname or Lastname
English
English : variant of Rand 1, from the Old French oblique case.
Surname or Lastname
English
English : variant of Ransom.
Male
Norwegian
 Norwegian form of Old Norse Arnþórr, ANDOR means "eagle of Thor." Compare with another form of Andor.
Male
Scandinavian
 Scandinavian form of Old Norse Randolfr, RANDOLF means "shield-wolf." Compare with another form of Randolf.
Surname or Lastname
English
English : probably a variant of Crandon, a habitational name from Crandon in Somerset or Crandean in Falmer, Sussex. Compare Grandin.
Surname or Lastname
English
English : patronymic from Rand 1.
Surname or Lastname
English
English : variant spelling of Randall.Americanized spelling of Randel.
Female
English
Variant spelling of English Randy, RANDI means "worthy of admiration."
Male
English
Medieval form of English Randolf, RANDAL means "shield-wolf."
Female
English
Short form of English Miranda, RANDA means "worthy of admiration."Â
Surname or Lastname
English or Scottish
English or Scottish : unexplained. Possibly, as Black suggests, a reduced form of Langdon.French : from the old Germanic personal name element Lando (see Land), via the oblique case, Landonis.
Boy/Male
English American
Son of Rand.
Male
English
 Variant spelling of Middle English Randulf, RANDOLF means "shield-wolf." Compare with other forms of Randolf.
Surname or Lastname
English
English : variant of Brandon.
Male
English
Pet form of English Randall and Randolph, both RANDY means "shield-wolf." Compare with feminine Randy.
Boy/Male
English
Son of Rand.
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : patronymic from the Middle English personal name Rand(e) (see Rand 1).
Male
Hungarian
 Variant spelling of Hungarian András, ANDOR means "man; warrior." Compare with another form of Andor.
Female
English
Pet form of English Miranda, RANDY means "worthy of admiration."Â Compare with masculine Randy.Â
RANDOM COMPACT-SET
RANDOM COMPACT-SET
Girl/Female
Tamil
Rich or from hadria, Dissolved
Boy/Male
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Sindhi, Tamil, Telugu, Traditional
Name of Lord Shiva; Light; Lightning
Surname or Lastname
English
English : unexplained. Possibly a variant of Stelling.
Boy/Male
Hindu, Indian
King Name in South India
Boy/Male
English
Diminutives of any masculine or feminine name begining with Christ-, for example Christahel,...
Boy/Male
Indian, Sanskrit
High
Girl/Female
American, Australian, British, Chinese, Christian, Dutch, English, French, Greek, Latin
Laurel Tree; Crowned with Laurel; Bay Plant
Boy/Male
African, Arabic, Hindu, Indian, Kannada, Malaysian, Marathi, Muslim, Punjabi, Sikh, Sindhi, Telugu
Divine; Seeker; Another Name for God; One who Longs or Wishes; Sender (of Truth); Student; Lover; The Seeker of Truth
Girl/Female
Latin
Mild.
Girl/Female
Muslim
Singer
RANDOM COMPACT-SET
RANDOM COMPACT-SET
RANDOM COMPACT-SET
RANDOM COMPACT-SET
RANDOM COMPACT-SET
n.
The release of a captive, or of captured property, by payment of a consideration; redemption; as, prisoners hopeless of ransom.
n.
Distance to which a missile is cast; range; reach; as, the random of a rifle ball.
v. i.
To go or stray at random.
adv.
In a compact manner; with close union of parts; densely; tersely.
n.
To redeem from captivity, servitude, punishment, or forfeit, by paying a price; to buy out of servitude or penalty; to rescue; to deliver; as, to ransom prisoners from an enemy.
n.
A roving motion; course without definite direction; want of direction, rule, or method; hazard; chance; -- commonly used in the phrase at random, that is, without a settled point of direction; at hazard.
n.
Contact or impression by touch; collision; forcible contact; force communicated.
a.
Going at random or by chance; done or made at hazard, or without settled direction, aim, or purpose; hazarded without previous calculation; left to chance; haphazard; as, a random guess.
imp. & p. p.
of Compact
adv.
In a random manner.
n.
One who makes a compact.
p. p. & a
Brief; close; pithy; not diffuse; not verbose; as, a compact discourse.
a.
Compact; pressed close; concentrated; firmly united.
n.
Random.
v. t.
To manure with compost.
v. t.
To compact or join anew.
n.
Extent; reach; sweep; capacity; sphere; as, the compass of his eye; the compass of imagination.
n.
To exact a ransom for, or a payment on.