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Set of isolated points in the spectrum of an operator with finite-rank Riesz projectors
In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum
Discrete spectrum (mathematics)
Discrete_spectrum_(mathematics)
Study of discrete mathematical structures
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a one-to-one
Discrete_mathematics
Concept relating to waves and signals
numbers. It is the opposite of a discrete spectrum, a set of achievable values that are discrete in the mathematical sense where there is a positive gap
Spectrum_(physical_sciences)
Set of eigenvalues of a matrix
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a
Spectrum (functional analysis)
Spectrum_(functional_analysis)
Signal representation
which is discrete and periodic results in a frequency spectrum which is also discrete and periodic; this is the usual context for a discrete Fourier transform
Frequency_domain
Construction in functional analysis, useful to solve differential equations
that the "discreteness" of the spectrum is intimately related to the corresponding states being "localized". However, a careful mathematical analysis shows
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Aspect of mathematical spectrum theory
In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of
Essential_spectrum
constructive methods of discrete geometric objects. Discrete mathematics the study of mathematical structures that are fundamentally discrete rather than continuous
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Types of numerical variables in mathematics
or discrete spectrum Continuous spectrum Count data Discrete-time stochastic process Discrete geometry Discrete mathematics Discrete measure Discrete modelling
Continuous or discrete variable
Continuous_or_discrete_variable
Function in discrete mathematics
In mathematics, the discrete Fourier transform (DFT) is a discrete version of the Fourier transform that converts a finite sequence of numbers into another
Discrete_Fourier_transform
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
Branch of mathematics
Fourier series (see Deferent and epicycle § Mathematical formalism). In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut
Fourier_analysis
Analog of the continuous Laplace operator
mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid
Discrete_Laplace_operator
combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Property that is not changed by mathematical transformations
Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations
Invariant_(mathematics)
Mathematical finite graph-associated function
to the spectrum of the adjacency matrix. The Ihara zeta function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups
Ihara_zeta_function
Linear algebra aspects of graph theory
Theory of Graph Spectra. Annals of Discrete Mathematics. ISBN 0-444-70361-6. Sunada, Toshikazu (2008), "Discrete geometric analysis", Analysis on Graphs
Spectral_graph_theory
Mathematical structures that allow quantum mechanics to be explained
) = | ⟨ a n | ψ ⟩ | 2 (Discrete, nondegenerate spectrum) P ( a n ) = ∑ i g n | ⟨ a n i | ψ ⟩ | 2 (Discrete, degenerate spectrum) d P ( α ) = | ⟨ α | ψ
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Additional mathematical object
Mathematics: Theory of Sets". Hermann, Addison-Wesley. pp. 259–346, 383–385. Foldes, Stephan (1994). Fundamental Structures of Algebra and Discrete Mathematics
Mathematical_structure
System with a countable number of states
computational theory. Because discrete systems have a countable number of states, they may be described in precise mathematical models. A computer is a finite-state
Discrete_system
Syllabus in college and university mathematics
facilitate computations, teaching and usage shifted from a broad-spectrum Finite Mathematics with paper and pen, into development and usage of software. 1957:
Finite_mathematics
Sufficiency theorem for reconstructing signals from samples
which serves as a fundamental bridge between continuous-time signals and discrete-time signals. In the case of uniformly spaced (periodic) sampling, it establishes
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Relative importance of certain frequencies in a composite signal
In signal processing, the power spectrum S x x ( f ) {\displaystyle S_{xx}(f)} of a continuous time signal x ( t ) {\displaystyle x(t)} describes the distribution
Spectral_density
Fourier analysis technique applied to sequences
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values. The DTFT
Discrete-time Fourier transform
Discrete-time_Fourier_transform
German physicist and mathematician (1824–1887)
existence of discrete spectral lines had been known since Fraunhofer discovered them in 1814. That the lines formed a discrete mathematical pattern was
Gustav_Kirchhoff
Branch of mathematics
applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of a mathematical function. Real analysis
Mathematical_analysis
Generalisation of Fourier transform to any ring
In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
Type of representation of a linear semisimple Lie group
In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp
Tempered_representation
Discrete Fourier transform algorithm
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform
Fast_Fourier_transform
Matrix representation of a graph
the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian
Laplacian_matrix
Topics referred to by the same term
numbers) to a discrete set (such as the integers). The term quantization may refer to: Quantization (signal processing), in mathematics and digital signal
Quantization
Mathematical theorem
the space L2(Γ\G) of square-integrable functions, where Γ is a cofinite discrete group. The character is given by the trace of certain functions on G. The
Selberg_trace_formula
Locally compact topological group with an invariant averaging operation
the whole space of irreducible representations. In discrete group theory, where G has the discrete topology, a simpler definition is used. In this setting
Amenable_group
Method of spectral density estimation
applied mathematics for estimating the power of a signal at different frequencies. The method is based on the concept of using periodogram spectrum estimates
Welch's_method
Mathematics award
(2021). "Isolation of the cuspidal spectrum, with applications to the Gan–Gross–Prasad conjecture". Annals of Mathematics. 194 (2): 519–584. doi:10.4007/annals
Alexanderson_Award
Theorem relating stationary processes' autocorrelations and power spectra
sequences. As mentioned earlier, the relation of this discrete sampled data to a mathematical model is often misleading, and related errors can show
Wiener–Khinchin_theorem
Limit on data transfer rate
of a communication channel, it is possible (in theory) to communicate discrete data (digital information) nearly error-free up to a computable maximum
Noisy-channel_coding_theorem
Simple polynomial map exhibiting chaotic behavior
The logistic map is a discrete dynamical system defined by the quadratic difference equation It is a recurrence relation and a polynomial mapping of degree 2
Logistic_map
Type of nonlinear wave in physics
of breathers in discrete lattices is that the breather main frequency and all its multipliers are located outside of the phonon spectrum of the lattice
Breather
Psychological questionnaire
Although most students with autism spectrum disorder have average mathematical ability and test slightly worse in mathematics than in general intelligence,
Autism-Spectrum_Quotient
Mathematics term
1007/BF01075866MR 0209390 Lubotzky, A. (1994), Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Basel: Birkhäuser Verlag
Kazhdan's_property_(T)
Partition of a polygon into triangles of equal area
"Constructing Equidissections for Certain Classes of Trapezoids" (PDF), Discrete Mathematics, 308 (23): 5672–5681, doi:10.1016/j.disc.2007.10.031, Zbl 1156.51304
Equidissection
Area of discrete mathematics
principal objects of study in discrete mathematics. Graph theory is a branch of mathematics that studies graphs, mathematical structures for modelling pairwise
Graph_theory
Mathematical concept
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations
Spectrum_of_a_C*-algebra
Topological space in which the closure of every open set is open
connected. The Stone–Čech compactification of a discrete space is extremally disconnected. The spectrum of an abelian von Neumann algebra is extremally
Extremally_disconnected_space
Russian mathematician
Millennium Problem. Dynin, A (2009). "Energy-mass spectrum of Yang-Mills bosons is infinite and discrete". arXiv:0903.4727 [math-ph]. "The many faces of
Alexander_Dynin
Reasoning for mathematical statements
Introduction to Mathematical Proofs (Third ed.). Academic Press. p. 3. ISBN 978-0-12-088509-1. Gossett, Eric (July 2009). Discrete Mathematics with Proof.
Mathematical_proof
Length in a vector space
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance
Norm_(mathematics)
Transition rate formula
as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the
Fermi's_golden_rule
to some kind of continuous spectrum, of representations involving a continuous parameter, as well as a discrete spectrum. The principal series representations
Principal series representation
Principal_series_representation
Concept in algebraic geometry
and special points, but a more complicated spectrum, since they represent general dimensions. The discrete valuation case is much like the complex unit
Generic_point
Symbol representing a mathematical object
In mathematics, a variable (from Latin variabilis 'changeable') is a symbol, typically a letter, that refers to an unspecified mathematical object. One
Variable_(mathematics)
Symbols for constants, special functions
type of small-signal model is referred to as a hybrid-pi model in discrete mathematics, a permutation Projection parallax in astronomy ϖ {\displaystyle
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Statistical model
Coloured noise – Power spectrum of a noise signal Discrete Fourier transform – Function in discrete mathematics Likelihood function – Function related to statistics
Whittle_likelihood
Interdisciplinary field of research
of Mathematical Sociology (started in 1971) has been open to papers covering a broad spectrum of topics employing a variety of types of mathematics, especially
Mathematical_sociology
Distance over which a wave's shape repeats
bands and lattice vibrations. It is mathematically equivalent to the aliasing of a signal that is sampled at discrete intervals. The concept of wavelength
Wavelength
Collection of mathematical objects
Topics in Contemporary Mathematics. Cengage. p. 47. ISBN 978-1-133-10742-2. Epp, Susanna S. (4 August 2010). Discrete Mathematics with Applications. Cengage
Set_(mathematics)
Electronic testing device
modern spectrum analyzers use analog-to-digital converters to sample spectrum amplitude after the VBW filter. Since displays have a discrete number of
Spectrum_analyzer
Graph of chess rook moves
dimension-normalized boundary in Hamming graphs", SIAM Journal on Discrete Mathematics, 17 (2): 219–236, doi:10.1137/S0895480100375053, MR 2032290. Goethals
Rook's_graph
Generalized function whose value is zero everywhere except at zero
used it as a continuum analog of the discrete Kronecker delta. However, it had been used by multiple mathematical scientists in the nineteenth century
Dirac_delta_function
Algorithm in digital signal processing
engineering, and applied mathematics. Common applications of Bartlett's method are frequency response measurements and general spectrum analysis. The method
Bartlett's_method
Branch of mathematics that studies sets
Wikibooks has a book on the topic of: Discrete mathematics/Set theory "Axiomatic set theory", Encyclopedia of Mathematics, EMS Press, retrieved 2026-05-12
Set_theory
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
Frequency of a chirp pulse
time-domain waveform, and the two versions are mathematically related by the Fourier transform. The spectrum is of particular interest when pulses are subject
Chirp_spectrum
Collection of mathematical theories
the spectrum lies on the real line and (in general) is a spectral combination of a point spectrum of discrete eigenvalues and a continuous spectrum. In
Spectral_theory
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly
Philosophy_of_mathematics
Interdisciplinary theory behind quantum computing
(January 1999). "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer". SIAM Review. 41 (2): 303–332. Bibcode:1999SIAMR
Quantum_information_science
Branch of algebraic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry
Arithmetic_geometry
Effect in signal processing
then a discrete Fourier transform (DFT). But the DFT provides only a sparse sampling of the actual discrete-time Fourier transform (DTFT) spectrum. Figure
Spectral_leakage
In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation F ( x , y , y ′ , … ,
Oscillation_theory
Function, homomorphism, or morphism
In mathematics, a map or mapping is a function in its general sense.[vague] These terms may have originated as from the process of making a geographical
Map_(mathematics)
Mathematical space with a notion of closeness
In mathematics, a topological space is, roughly speaking, a space in which closeness is defined but cannot necessarily be measured by a numeric distance
Topological_space
Theorem in mathematical measure theory
In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem provides a way to decompose a measure into two distinct parts based
Lebesgue's decomposition theorem
Lebesgue's_decomposition_theorem
Type of signal in signal processing
\mu {\sqrt {n}}} ; and the power spectrum P {\displaystyle P} will be flat only over the non-zero frequencies. A discrete-time stochastic process W ( n )
White_noise
Branch of applied mathematics
development of mathematical ideas inspired by physics, known as physical mathematics. There are several distinct branches of mathematical physics, and these
Mathematical_physics
Signal with equal energy per octave
various mathematical models to create pink noise. The superposition of exponentially decaying pulses is able to generate a signal with the 1/f-spectrum at
Pink_noise
Zuverlässiges Rechnen) is numerics including mathematically strict error (rounding error, truncation error, discretization error) evaluation, and it is one field
Validated_numerics
Number divisible only by 1 and itself
Textbooks in mathematics. CRC Press. p. 7. ISBN 978-1-4987-0269-0. Bauer, Craig P. (2013). Secret History: The Story of Cryptology. Discrete Mathematics and Its
Prime_number
Basic notion of sameness in mathematics
Grades. IAP. p. 19. ISBN 978-1-64113-847-5. Levin, Oscar (2021). Discrete Mathematics: An Open Introduction (PDF) (3rd ed.). Oscar Levin. p. 5. ISBN 978-1-79290-169-0
Equality_(mathematics)
American mathematician (born 1939)
limits of discrete series. J. Funct. Anal. 80 (1988), no. 2, 451–461. Lectures on the spectrum of L2(Γ\G). Pitman Research Notes in Mathematics Series,
Floyd_Williams
Ukrainian mathematician
"Ukrainian Mathematical Journal", "Algebra and Discrete Mathematics", "Carpathian Mathematical Publications", "Bukovinian Mathematical Journal", and
Rostislav_Grigorchuk
Mathematics prize
Lyttle Satter Prize in Mathematics, also called the Satter Prize, is one of twenty-one prizes given out by the American Mathematical Society (AMS). It is
Ruth Lyttle Satter Prize in Mathematics
Ruth_Lyttle_Satter_Prize_in_Mathematics
American mathematician
Returning to Princeton for doctoral study in mathematics, she completed her Ph.D. in 2009. Her dissertation, Discrete Analogues in Harmonic Analysis, was supervised
Lillian_Pierce
Russian mathematician (1935–2017)
I. Levenshtein, Elements of coding theory, In the book. Discrete mathematics and mathematical questions of cybernetics, Nauka, Moscow, 1974, 207–305.
Vladimir_Levenshtein
Mathematical technique used in data compression and analysis
In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated
Wavelet_transform
mathematics. These include mathematical research, mathematics education, the history and philosophy of mathematics, public outreach, and mathematics contests
List_of_women_in_mathematics
Indian mathematician (1937–2019)
"Asymptotic behaviour of eigen functions on a semisimple Lie group: The discrete spectrum". Acta Mathematica. 129 (1): 237–280. doi:10.1007/bf02392217. Enright
Veeravalli_S._Varadarajan
Topological space in which closed subsets satisfy the descending chain condition
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition
Noetherian_topological_space
Field of mathematics and science based on non-linear systems and initial conditions
Chaos theory is a branch of mathematics and an interdisciplinary area of scientific study. It focuses on underlying patterns and deterministic laws of
Chaos_theory
Method of encoding digital data on multiple carrier frequencies
more robust modulation or error coding to those subcarriers. The term discrete multitone modulation (DMT) denotes OFDM-based communication systems that
Orthogonal frequency-division multiplexing
Orthogonal_frequency-division_multiplexing
American topologist
Weinberger, Shmuel (1996). "Discrete circle actions: a note using non-standard analysis". Israel Journal of Mathematics. 94: 147–155. doi:10.1007/BF02762701
Shmuel_Weinberger
Commutative algebra studies commutative rings, their ideals, and modules over such rings
Regular local ring Localization of a module Valuation (mathematics) Discrete valuation Discrete valuation ring I-adic topology Weierstrass preparation
List of commutative algebra topics
List_of_commutative_algebra_topics
Mathematical set with some added structure
In mathematics, a space is a set (sometimes known as a universe) endowed with a structure defining the relationships among the elements of the set. A
Space_(mathematics)
Proof all ranked voting rules have spoilers
spoilers, creating them even in some situations where they are not mathematically necessary (e.g. in center squeezes). In contrast, majority-rule (Condorcet)
Arrow's_impossibility_theorem
Tool to track locally defined data attached to the open sets of a topological space
Look up sheaf in Wiktionary, the free dictionary. In mathematics, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as sets, abelian
Sheaf_(mathematics)
Mathematical concept
infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. The
Infinity
Sequence of numbers consisting of 1 and -1
In mathematics, a sign sequence, or ±1–sequence or bipolar sequence, is a sequence of numbers, each of which is either 1 or −1. One example is the sequence
Sign_sequence
Mathematical signal manipulation by computers
decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis". Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Digital_signal_processing
Algebraic structure with addition and multiplication
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted
Ring_(mathematics)
American mathematical physicist
Ruskai, Mary Beth (1982). "Absence of discrete spectrum in highly negative ions". Communications in Mathematical Physics. 82 (4): 457–469. Bibcode:1982CMaPh
Mary_Beth_Ruskai
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DISCRETE SPECTRUM-MATHEMATICS
DISCRETE SPECTRUM-MATHEMATICS
DISCRETE SPECTRUM-MATHEMATICS
DISCRETE SPECTRUM-MATHEMATICS
DISCRETE SPECTRUM-MATHEMATICS
DISCRETE SPECTRUM-MATHEMATICS
DISCRETE SPECTRUM-MATHEMATICS