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Complicated set of real numbers
In mathematics, the Markov spectrum, devised by Andrey Markov, is a complicated set of real numbers arising in Markov Diophantine equations and also in
Markov_spectrum
Solution to x*x + y*y + z*z = 3xyz
A Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equation x 2 + y 2 + z 2 = 3 x y
Markov_number
Gauss–Markov theorem Gauss–Markov process Markov blanket Markov boundary Markov chain Markov chain central limit theorem Additive Markov chain Markov additive
List of things named after Andrey Markov
List_of_things_named_after_Andrey_Markov
Stochastic processes
Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both
Gauss–Markov_process
On the approximate structure of sets whose sumset is small
mathematicians successfully formalizing contemporary mathematics. Markov spectrum Plünnecke–Ruzsa inequality Kneser's theorem (combinatorics) Freiman
Freiman's_theorem
Analytic series
that the conclusion holds for a set of denominators of density 1. Markov spectrum Rockett, Andrew M.; Szüsz, Peter (1992). Continued Fractions. World
Restricted_partial_quotients
Property of an irrational number
In number theory, specifically in Diophantine approximation theory, the Markov constant M ( α ) {\displaystyle M(\alpha )} of an irrational number α {\displaystyle
Markov_constant
American computer scientist and engineer
Igor Leonidovich Markov (born 31 March 1973) is a Ukrainian-American computer scientist and engineer. A former professor of electrical engineering and
Igor_L._Markov
Metric used to gauge ideology of SCOTUS jurists
paper by Andrew D. Martin and Kevin M. Quinn. The Martin–Quinn score uses Markov chain Monte Carlo (MCMC) methods to fit a Bayesian model of ideal points
Martin–Quinn_score
Construction in functional analysis, useful to solve differential equations
The spectrum of a linear operator T {\displaystyle T} that operates on a Banach space X {\displaystyle X} is a fundamental concept of functional analysis
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Linear algebra aspects of graph theory
{2d(d-\lambda _{2})}}.} This inequality is closely related to the Cheeger bound for Markov chains and can be seen as a discrete version of Cheeger's inequality in
Spectral_graph_theory
process Markov information source Markov kernel Markov logic network Markov model Markov network Markov process Markov property Markov random field Markov renewal
List_of_statistics_articles
Syllabus in college and university mathematics
and business, such as finite probability spaces, matrix multiplication, Markov processes, finite graphs, or mathematical models. These topics were used
Finite_mathematics
Stochastic matrix representing links between entities
matrix of links. A related matrix S corresponding to the transitions in a Markov chain of given network is constructed from A by dividing the elements of
Google_matrix
Fictional organization appearing in comics published by DC Comics
unlike the other Lantern corps – represent a space outside of the emotional spectrum, with black being the absence of light. The group is composed of deceased
Black_Lantern_Corps
Concepts from linear algebra
components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-normalized adjacency matrix; however, the adjacency
Eigenvalues_and_eigenvectors
Mental disorder involving eccentricity and social isolation in afflicted individuals
S1500. PMC 2656336. PMID 19300629. S2CID 8816485. Sheldrick AJ, Krug A, Markov V, Leube D, Michel TM, Zerres K, et al. (September 2008). "Effect of COMT
Schizotypal personality disorder
Schizotypal_personality_disorder
Sequence of data points over time
See also Markov switching multifractal (MSMF) techniques for modeling volatility evolution. A hidden Markov model (HMM) is a statistical Markov model in
Time_series
Intelligence in machines
actions and evaluate situations while being uncertain of the outcome. A Markov decision process has a transition model that describes the probability that
Artificial_intelligence
Periodicity computation method
spectral analysis (LSSA) is a class of methods for estimating a frequency spectrum by fitting sinusoids to data using a least-squares fit. Unlike Fourier
Least-squares spectral analysis
Least-squares_spectral_analysis
Russian mathematician (1934–1967)
mathematicians united around E. B. Dynkin who were developing the theory of Markov processes. His thesis introduced the concept of a strong Feller process
Igor_Girsanov
Russian mathematician (1930–1997)
problem of optimal non-linear filtering based on his theory of conditional Markov processes, which was published in his papers in 1959 and 1960. The Kalman-Bucy
Ruslan_Stratonovich
Theorem in linear algebra
theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type);
Perron–Frobenius_theorem
Machine learning software library in C++
classification problems. Shogun also offers a full implementation of Hidden Markov models. The core of Shogun is written in C++ and offers interfaces for MATLAB
Shogun_(toolbox)
Study of speech signals and the processing methods of these signals
the dominant speech processing strategy started to shift away from Hidden Markov Models towards more modern neural networks and deep learning. In 2012, Geoffrey
Speech_processing
Resource problem in machine learning
to be played. The bandit problem is formally equivalent to a one-state Markov decision process. The regret ρ {\displaystyle \rho } after T {\displaystyle
Multi-armed_bandit
Constant in Riemannian geometry
only in Riemannian geometry and global analysis, but also in the theory of Markov chains and in graph theory, where they have inspired the analogous Cheeger
Cheeger_constant
Transition rate formula
earlier times t ′ {\displaystyle t'} , i.e. it is non-Markovian. We make the Markov approximation, i.e. that it only depends on a i {\displaystyle a_{i}} at
Fermi's_golden_rule
Israeli-American academic
Hidden Markov Processes and Connections to Dynamical Systems, Cambridge University Press, July 2011. Stanford profile, Itschak Weissman IEEE Spectrum, "A
Tsachy_Weissman
Communication between a vehicle and any entity that may affect the vehicle
doi:10.1109/TMC.2020.2992045. S2CID 218931192. Gu, X.; et al. (2022). "Markov Analysis of C-V2X Resource Reservation for Vehicle Platooning". 2022 IEEE
Vehicle-to-everything
Automatic conversion of spoken language into text
recognition. During the late 1960s, Leonard Baum developed the mathematics of Markov chains at the Institute for Defense Analysis. A decade later, at CMU, Raj
Speech_recognition
Index of articles associated with the same name
graph is strongly connected.) A detailed definition is given here. Also, a Markov chain is irreducible if there is a non-zero probability of transitioning
Irreducibility_(mathematics)
Integral transform useful in probability theory, physics, and engineering
probability theory, including first passage times of stochastic processes such as Markov chains, and renewal theory. Of particular use is the ability to recover
Laplace_transform
Indian control theorist
Computational Cancer Biology: An Interaction Networks Approach 2014. Hidden Markov Processes: Theory and Applications to Biology Vidyasagar, Mathukumalli (2011)
Mathukumalli_Vidyasagar
Descriptor of computer vision
classification in computer vision. LBP is the particular case of the Texture Spectrum model proposed in 1990. LBP was first described in 1994. It has since been
Local_binary_patterns
Algorithmic production of digital images
smoothing in step 3 makes the output image look blurred. These methods, using Markov fields, non-parametric sampling, tree-structured vector quantization and
Texture_synthesis
Country in Eastern Europe and North Asia
17 (2). Modern Humanities Research Association: 494–528. JSTOR 4212688. Markov, Vladimir (1969). "Balmont: A Reappraisal". Slavic Review. 28 (2): 221–264
Russia
Projection of data onto lower-dimensional manifolds
diffusion and a random walk (Markov Chain); an analogy is drawn between the diffusion operator on a manifold and a Markov transition matrix operating on
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Concept in statistics
cosmic inflation are thought to be a GRF with a nearly scale invariant spectrum. One way of constructing a GRF is by assuming that the field is the sum
Gaussian_random_field
Army/faction in the Russian Civil War
Revolution, the arrested generals Lavr Kornilov, Anton Denikin, Sergey Markov and others were released by Commander-in-Chief Nikolay Dukhonin before his
White_Army
Diagram that shows all possible logical relations between a collection of sets
variable Bernoulli process Continuous or discrete Expected value Variance Markov chain Observed value Random walk Stochastic process Complementary event
Venn_diagram
homology. The SUPERFAMILY annotation is based on a collection of hidden Markov models (HMM), which represent structural protein domains at the SCOP superfamily
Superfamily_database
Generalized function whose value is zero everywhere except at zero
In this context, the semigroup condition is then an expression of the Markov property of Brownian motion. In higher-dimensional Euclidean space Rn, the
Dirac_delta_function
Topological space in which the closure of every open set is open
Stone–Čech compactification of a discrete space. Hartig (1983) proves the Riesz–Markov–Kakutani representation theorem by reducing it to the case of extremally
Extremally_disconnected_space
Type of machine learning model
November 2025. "AI Developers Look Beyond Chain-of-Thought Prompting". IEEE Spectrum. 8 May 2025. Retrieved 16 November 2025. Metz, Cade (20 December 2024)
Large_language_model
Biochemical method in mass spectrometry
a mass array instead of a spectrum graph. Fisher et al. proposed the NovoHMM method of de novo sequencing. A hidden Markov model (HMM) is applied as a
De_novo_peptide_sequencing
Proof that every structure with certain properties is isomorphic to another structure
v\rangle } for all v ∈ H {\displaystyle v\in H} . The more general Riesz–Markov–Kakutani representation theorem has several versions, one of them identifying
Representation_theorem
Brightest star in the constellation Orion
September 2016. Retrieved 11 February 2019. Markova, N.; Prinja, R. K.; Markov, H.; Kolka, I.; Morrison, N.; Percy, J.; Adelman, S. (2008). "Wind structure
Rigel
Probabilistic problem-solving algorithms
can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depends on the distributions of the
Mean-field_particle_methods
Card trick and probabilistic concept
count - convergent Markov chains". Archived from the original on 2023-08-20. Retrieved 2023-08-20. [...] We looked at the Markov chains, where a given
Kruskal_count
Bernays, the constructive recursive mathematics of mathematicians Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes
Mathematical_object
Proof all ranked voting rules have spoilers
elections. Under some models of voter preferences (like the left-right spectrum assumed in the median voter theorem), spoilers disappear entirely for these
Arrow's_impossibility_theorem
Italian-American engineer and businessman
speech recognition, DNA analysis, and many other applications of Hidden Markov models. On advice of a lawyer, Viterbi did not patent the algorithm. Viterbi
Andrew_Viterbi
Methods in artificial intelligence research
acquisition. Uncertainty was addressed with formal methods such as hidden Markov models, Bayesian reasoning, and statistical relational learning. Symbolic
Symbolic artificial intelligence
Symbolic_artificial_intelligence
uniqueness conjecture for Markov numbers that every Markov number is the largest number in exactly one normalized solution to the Markov Diophantine equation
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Fermented tea beverage
.409D. doi:10.1016/S0963-9969(00)00067-3. Velicanski, A; Cvetkovic, D; Markov, S; Tumbas, V; et al. (2007). "Antimicrobial and antioxidant activity of
Kombucha
Geometric algorithm
Diffusion maps exploit the relationship between heat diffusion and random walk Markov chain. The basic observation is that if we take a random walk on the data
Diffusion_map
Comics character
individual; enter a relationship with Brion Markov; and befriend Harper Row. However, Halo breaks up with Markov after he kills his uncle Baron Bedlam. In
Halo_(DC_Comics)
Deep reinforcement learning method
chip design researcher and former University of Michigan professor Igor L. Markov in Communications of the ACM, from an arXiv preprint in 2023. The critique
AlphaChip
Concept in probability and statistics
that came before it. In this way, an i.i.d. sequence is different from a Markov sequence, where the probability distribution for the nth random variable
Independent and identically distributed random variables
Independent_and_identically_distributed_random_variables
Types of numerical variables in mathematics
Continuous geometry Continuous modelling Continuous or discrete spectrum Continuous spectrum Count data Discrete-time stochastic process Discrete geometry
Continuous or discrete variable
Continuous_or_discrete_variable
calculating the theoretical power spectrum given the cosmological parameters. These codes are capable of calculating the power spectrum from the standard LCDM model
List of cosmological computation software
List_of_cosmological_computation_software
Problem in computer science
that is equivalent in its computational power to Turing machines, such as Markov algorithms, Lambda calculus, Post systems, register machines, or tag systems
Halting_problem
Overview of and topical guide to statistics
Concentration inequality Convergence of random variables Computational statistics Markov chain Monte Carlo Bootstrapping (statistics) Jackknife resampling Integrated
Outline_of_statistics
Discrete Fourier transform algorithm
Applications including efficient spherical harmonic expansion, analyzing certain Markov processes, robotics etc. Quantum FFTs Shor's fast algorithm for integer
Fast_Fourier_transform
Academy of Sciences. In 2003, he received the Institute for Nuclear Research Markov Prize for his contributions to neutrino physics. In 2006, he received the
Vadim_Kuzmin_(physicist)
Representation of a type of random process
larger time lags. This is then a low-pass filter, when applied to full spectrum light, everything except for the red light will be filtered. If φ 1 < 0
Autoregressive_model
Indirect method for finding extrasolar planets and brown dwarfs
radial-velocity measurements via observation of Doppler shifts in the spectrum of the planet's parent star. As of January 2026, over 1,100 known extrasolar
Doppler_spectroscopy
Brazilian mathematician
Lima, C. G. Moreira and S. Romaña, Classical and Dynamical Lagrange and Markov spectra ISBN 978-981-12-2528-4 Época – Os segredos das ilhas de excelência
Carlos_Matheus
Random motion of particles suspended in a fluid
be modeled by a random walk. In the general case, Brownian motion is a Markov process and described by stochastic integral equations. The French mathematician
Brownian_motion
Academic journal
models of atoms; Author: Markov, M.; Source: Doklady Akademii Nauk SSSR, Seriya A (1935), 8, 103-4. Title: Absorption spectrum of lead vapor in the Schumann
Proceedings of the USSR Academy of Sciences
Proceedings_of_the_USSR_Academy_of_Sciences
Indian-American computer scientist (born 1948)
statistical models to unsupervised segmentation based on texture analysis using Markov random field models (Cross and Jain, 1984) and Gabor filter banks (Jain
Anil K. Jain (computer scientist, born 1948)
Anil_K._Jain_(computer_scientist,_born_1948)
Natural number
(22222224), and palindromic in base 8 (252528) 10946 = Fibonacci number, Markov number 10958 = the smallest positive integer that cannot be represented
10,000
American academic
interval of values. He also showed that noise can speed up the convergence of Markov chains to equilibrium. Nonfiction Noise. Viking Press. 2006. ISBN 0-670-03495-9
Bart_Kosko
Pandemic caused by SARS-CoV-2
in those who lack a protective immune response or are immunocompromised. Markov PV, Ghafari M, Beer M, Lythgoe K, Simmonds P, Stilianakis NI, et al. (June
COVID-19_pandemic
American electrical engineer (born 1943)
of the statistical method of representing speech that is known as hidden Markov modeling (HMM). Rabiner was the first to publish the scaling algorithm for
Lawrence_Rabiner
Film genre
partly inspired by The Defiant Ones (1958) starring Pam Grier and Margaret Markov in the roles originated by Sidney Poitier and Tony Curtis. Blacula is a
Blaxploitation
Electric charge generated in certain solids due to mechanical stress
Bibcode:1979MinM...43..237A. doi:10.1180/minmag.1979.043.326.05. Radusinović, Dušan & Markov, Cvetko (1971). "Macedonite – lead titanate: a new mineral" (PDF). American
Piezoelectricity
Source of error in statistics
sample mean. For the effect of serial or temporal correlation also see Markov chain central limit theorem. The problem of inadequate specification arises
Pseudoreplication
Robot capable of locomotion
Positioning systems often use triangulation, relative position and/or Monte-Carlo/Markov localization to determine the location and orientation of the platform,
Mobile_robot
Dimensionless number that quantifies the strength of the electromagnetic interaction
method of Chand et al., discrediting those results. King et al. have used Markov chain Monte Carlo methods to investigate the algorithm used by the UNSW
Fine-structure_constant
Artificial production of human speech
synthesis method based on hidden Markov models, also called Statistical Parametric Synthesis. In this system, the frequency spectrum (vocal tract), fundamental
Speech_synthesis
Algorithm employed by process and network schedulers in computing
Round-robin scheduling does not utilize this. Higher throughput and system spectrum efficiency may be achieved by channel-dependent scheduling, for example
Round-robin_scheduling
Symbols for constants, special functions
to its diameter the prime-counting function the state distribution of a Markov chain in reinforcement learning, a policy function defining how a software
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Measure of algorithmic complexity
almost all x {\displaystyle x} . It can be shown that for the output of Markov information sources, Kolmogorov complexity is related to the entropy of
Kolmogorov_complexity
American economist (born 1948)
firm, Auctionomics, won a technical Emmy Award for their contributions to spectrum auction design. Paul Milgrom was born in Detroit, Michigan, April 20, 1948
Paul_Milgrom
Human disease
different diseases from Multiple Sclerosis, but considered by others to form a spectrum differing only in terms of chronicity, severity, and clinical course. Multiple
Inflammatory demyelinating diseases
Inflammatory_demyelinating_diseases
multicanonical ensemble (also called multicanonical sampling or flat histogram) is a Markov chain Monte Carlo sampling technique that uses the Metropolis–Hastings algorithm
Multicanonical_ensemble
Idealization of a large number of atomic-sized systems
full set of possible states. For example, a collection of walkers in a Markov chain Monte Carlo iteration is called an ensemble in some of the literature
Ensemble (mathematical physics)
Ensemble_(mathematical_physics)
Type of inter-vehicle communication network
Martin, Isabel (2018). "Transient Analysis of Idle Time in VANETs Using Markov-Reward Models". IEEE Transactions on Vehicular Technology. 67 (4): 2833–2847
Vehicular_ad_hoc_network
Correlation of a signal with a time-shifted copy of itself, as a function of shift
assumption that the error terms are uncorrelated, meaning that the Gauss Markov theorem does not apply, and that OLS estimators are no longer the Best Linear
Autocorrelation
Scientific study of digital information
Kirchhoff, M.; Parr, T.; Palacios, E.; Friston, K.; Kiverstein, J. (2018). "The Markov blankets of life: autonomy, active inference and the free energy principle"
Information_theory
System for reasoning about vagueness
direction was made by E. S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program (see Santos 1970). Successively
Fuzzy_logic
Computational model used in machine learning
learning to proceed more quickly. Formally, the environment is modeled as a Markov decision process (MDP) with states s 1 , . . . , s n ∈ S {\displaystyle
Neural network (machine learning)
Neural_network_(machine_learning)
They have been applied to the following fields: solving Markov decision processes and Markov chains for machine learning, transfer learning, value function
Diffusion_wavelets
Thesis on the nature of computability
modern notion of the computer. Other models include combinatory logic and Markov algorithms. Gurevich adds the pointer machine model of Kolmogorov and Uspensky
Church–Turing_thesis
Method of data analysis
Sham M.; Zhang, Tong (2008). A spectral algorithm for learning hidden markov models. arXiv:0811.4413. Bibcode:2008arXiv0811.4413H. Markopoulos, Panos
Principal_component_analysis
theorem Five color theorem Five lemma Fundamental theorem of arithmetic Gauss–Markov theorem (brief pointer to proof) Gödel's incompleteness theorem Gödel's
List_of_mathematical_proofs
Digital electronic system with no global clock signal
assembling large-scale heterogeneous and scalable systems. There is a large spectrum of asynchronous design styles, with tradeoffs between robustness and performance
Asynchronous_system
Integrated circuit technology
Katherine (May 23, 2017). "A Neuromorphic Chip That Makes Music". IEEE Spectrum. Retrieved October 1, 2019. Moradi, Sina; Qiao, Ning; Stefanini, Fabio;
Neuromorphic_computing
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MARKOV SPECTRUM
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