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In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years
Joint_spectral_radius
Largest absolute value of an operator's eigenvalues
mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded
Spectral_radius
Computational problems no algorithm can solve
matrices A 1 , … , A m {\displaystyle A_{1},\dots ,A_{m}} , their joint spectral radius is defined as lim sup N → ∞ max i 1 , … , i N ∈ 1 : m ‖ A i 1 ⋯
List_of_undecidable_problems
American mathematician (born 1934)
Wellesley, MA: Wellesley-Cambridge Press. pp. xii+758. MR 0870634. The Joint spectral radius, introduced by Strang and Rota in the early 60s. Strang splitting
Gilbert_Strang
Topics referred to by the same term
School, Perth, Australia Jai Shri Ram, a Hindu slogan and greeting Joint spectral radius, in mathematics JSR Corporation [ja; es], a Japanese chemical business
JSR
Italian-American mathematician (1932–1999)
inequality Rota's conjecture Rota's basis conjecture Rota–Baxter algebra Joint spectral radius, introduced by Rota in the early 1960s Cyclotomic identity Necklace
Gian-Carlo_Rota
Dynamical system that exhibits continuous and discrete dynamic behavior
mode control Variable structure system Variable structure control Joint spectral radius Cyber-physical system Behavior trees (artificial intelligence, robotics
Hybrid_system
Property of a dynamical system where solutions near an equilibrium point remain so
is asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set { A 1 , … , A m } {\displaystyle \{A_{1},\dots ,A_{m}\}}
Lyapunov_stability
Spherical collection of stars
means of standard radii: the core radius (rc), the half-light radius (rh), and the tidal or Jacobi radius (rt). The radius can be expressed as a physical
Globular_cluster
Subatomic particle with positive charge
energy of the gluon fields that bind the quarks together. The proton charge radius is around 0.841 fm but two different kinds of measurements give slightly
Proton
Aspect of a numerical matrix
are used to study quantum computing. A related concept is the numerical radius, which is the largest absolute value of the numbers in the numerical range
Numerical_range
Sensor for measuring thermal radiation
external radius, and connected to a thin wire at the center, named after its originator Robert Gardon. The foil center and side are the hot- and cold joint of
Gardon_gauge
Random matrix with gaussian entries
GβE(N) ensemble may be defined after the spectral density is defined first, so that any method to motivate the spectral density then motivates the GβE(N) ensemble
Gaussian_ensemble
Star in the constellation Scutum
spectrum and the characteristics of certain spectral features for the latter, such as titanium oxide (TiO) spectral lines. The first calculation of the star's
Stephenson_2_DFK_1
Matrix-valued random variable
rigid, but very close to it. Spectral rigidity has been numerically observed in the zeros of the Riemann zeta function. The joint probability density of the
Random_matrix
Atmospheric modelling gives a temperature of about 1,300 K and a radius of 1.056 RJ. With a spectral type of L8, HIP 99770 b lies at the L/T transition for substellar
HIP_99770_b
lists of smaller objects by observed mean radius. These lists can be sorted according to an object's radius and mass and, for the most massive objects
List of Solar System objects by size
List_of_Solar_System_objects_by_size
American stealth multirole fighter aircraft
Additionally, Lockheed Martin is studying integrating a fiber laser that uses spectral beam combining multiple individual laser modules into a single high-power
Lockheed Martin F-35 Lightning II
Lockheed_Martin_F-35_Lightning_II
Old G-type star with one exoplanet
Earth in the constellation of Sextans. It has a mass of 0.9 solar masses, a radius of 1.3 solar radii and a temperature of 5,842 Kelvin. There is so far one
WASP-127
Triple star system in the constellation Coma Berenices
dwarf companion (HD 114762 B) is classified as an ultra-cool dwarf, with a spectral type around M9. With a visual magnitude of 15 and separated from the primary
HD_114762
Brightest star in the constellation Lyra
"Fundamental stellar photometry for standards of spectral type on the revised system of the Yerkes spectral atlas". Astrophysical Journal. 117: 313–352. Bibcode:1953ApJ
Vega
Mathematical function often applied to matrices
are the (upper) spectral radius and (upper) spectral abscissa, respectively. Both limits exist, although neither the spectral radius nor the determinant
Logarithmic_norm
Hypergiant star in the constellation Aquila
expelled appearing as a fake photosphere, so IRC +10420 appears with a later spectral type as humans see just the expelled dust and gas it has blown out during
IRC_+10420
Generalization of the one-dimensional normal distribution to higher dimensions
multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate)
Multivariate normal distribution
Multivariate_normal_distribution
Star in the constellation of Cetus
exhausted the supply of hydrogen at its core and expanded to 11 times the Sun's radius. It is a red clump giant, which indicates it is on the horizontal branch
81_Ceti
Chemical element with atomic number 17 (Cl)
bromide, but a stronger one than fluoride. It is intermediate in atomic radius between fluorine and bromine, and this leads to many of its atomic properties
Chlorine
Star in the constellation Taurus
Given its large mass, this star, though presently of spectral type K0 III, was formerly of spectral type A that has now evolved off the main sequence into
Epsilon_Tauri
List of largest planets by size
largest exoplanets so far discovered, in terms of physical size, ordered by radius. This list of extrasolar objects may and will change over time due to diverging
List_of_largest_exoplanets
Star in the constellation Pavo
Houk, Nancy; Cowley, A. P. (1979). Michigan catalogue of two-dimensional spectral types for the HD stars. Ann Arbor, Michigan: Dept. of Astronomy, University
HD_175167
Star in the constellation Ursa Minor
fusion of helium. It has 2.7 times the mass of the Sun and a radius about 11 times the radius of the Sun. It radiates 60 times the Sun's luminosity from
HD_120084
Second-largest moon of Jupiter
million km (26.3 times the 71,492 km radius of Jupiter itself). This is significantly larger than the orbital radius—1.07 million km—of the next-closest
Callisto_(moon)
Star in the constellation Ursa Major
which, at its estimated distance, equates to a physical radius of about 14 times the radius of the Sun. It has about three times the mass of the Sun
Omicron_Ursae_Majoris
Star in the constellation of Camelopardalis
1.2 times the mass of the Sun and has expanded to 10.6 times the Sun's radius. It is radiating 51 times the luminosity of the Sun from its enlarged photosphere
HD_104985
American glass and ceramics manufacturer
sunglasses at retail, featuring the exclusive combination of Photochromic and Spectral Control® technologies in the lenses. In July 2008, Corning announced the
Corning_Inc.
Chemical element with atomic number 2 (He)
Janssen. Janssen is often jointly credited with detecting the element, along with Norman Lockyer. Janssen recorded the helium spectral line during the solar
Helium
Binary star in the constellation Triangulum
A is 2.6 times more massive than the Sun, but expanded to 4.4 the Sun's radius and irradiates 60 times more than the Sun. Component B is somewhat smaller
Beta_Trianguli
Comprehensive list of objects beyond Neptune
Hi’iaka and Namaka. EPSC-DPS Joint Meeting 2025. Vol. 18. doi:10.5194/epsc-dps2025-905. Nimmo, Francis; et al. (2017). "Mean radius and shape of Pluto and Charon
List of trans-Neptunian objects
List_of_trans-Neptunian_objects
Okayama Planet Search Program: A new long-period planet, RV astrometry joint analysis, and a multiplicity-metallicity trend around evolved stars". Publications
List of exoplanets and planetary debris around giant stars
List_of_exoplanets_and_planetary_debris_around_giant_stars
radial velocity can be deduced from the displacement in the parent star's spectral lines due to the Doppler effect. The radial-velocity method measures these
Methods of detecting exoplanets
Methods_of_detecting_exoplanets
Star in the constellation Leo
of physical characteristics: it has 87% of the Sun's mass and 91% of the radius of the Sun. It is spinning with a projected rotational velocity of 5.3 km/s
HD_81040
Star in the constellation of Camelopardalis
fusion at the core. It has 1.2 the mass of the Sun but has expanded to a radius of 10.8 R☉. It shines at 46.4 times the luminosity of the Sun from its enlarged
HD_32518
Turkish fighter aircraft
propulsion Supercruise Sufficient and optimized combat radius Advanced and internal multi-spectral sensors (EW and RF/IR) Low observability Sensor fusion
TAI_TF_Kaan
Star in the constellation Ara
HD 152478", Circumstellar Dynamics at High Resolution. Proceedings of a Joint ESP/Brazilian Workshop held at Foz do Iguacu, Brazil, 27 February-2 March
Rho1_Arae
Extrasolar planet orbiting TOI-2431
the host star and be torn apart. TOI-2431 b has around 1.54 times Earth's radius (R🜨) and 6.2 times Earth's mass (M🜨), resulting in a density of 9.2 g/cm3
TOI-2431_b
Elementary particle with negative charge
explained the spectral lines of the hydrogen atom. However, Bohr's model failed to account for the relative intensities of the spectral lines and it was
Electron
Largest dwarf planet in the Solar System
J2000. Surface area derived from the radius r: 4 π r 2 {\displaystyle 4\pi r^{2}} . Volume v derived from the radius r: 4 π r 3 / 3 {\displaystyle 4\pi
Pluto
Star at the centre of the Solar System
central dot: ☉. It is used for such units as M☉ (Solar mass), R☉ (Solar radius) and L☉ (Solar luminosity). The scientific study of the Sun is called heliology
Sun
Region between the orbits of Mars and Jupiter
Palermo, Sicily, found a tiny moving object in an orbit with exactly the radius predicted by this pattern. He dubbed it "Ceres", after the Roman goddess
Asteroid_belt
American unmanned aerial vehicle
meet combat requirements. Its Raytheon AN/AAS-52[citation needed] multi-spectral targeting sensor suite includes a color/monochrome daylight TV, infrared
General_Atomics_MQ-9_Reaper
Star in the constellation Cetus
Two-dimensional Spectral Types for the HD Stars. Volume 4, Declinations -26°.0 to -12°.0". Michigan Catalogue of Two-Dimensional Spectral Types for the
HD_4732
Star in the constellation Dorado
with a stellar classification of G8V. It is a yellow dwarf with a mass and radius similar to the Sun that is fusing hydrogen in its core. The chromosphere
HD_30177
Basic noise model used in information theory
information system. White refers to the idea that it has uniform power spectral density across the frequency band for the information system. It is an
Additive_white_Gaussian_noise
Potentially hazardous near-Earth asteroid
partial resurfacing of the asteroid is possible, which might change its spectral class from a weathered Sq- to an unweathered Q-type. History of close approaches
99942_Apophis
Star system in the constellation of Leo Minor
star of class G0V. The star has a mass 1.11 times that of the Sun and a radius 1.043 times solar. It irradiates 25% more luminosity than the Sun and has
HD_93963
Star in the constellation Delphinus
than double the mass of the Sun, and has expanded to seven times the Sun's radius. It is radiating 34 times the luminosity of the Sun from its photosphere
18_Delphini
Star in the constellation Draco
1.88 times the mass of the Sun and has expanded to 4.49 times the Sun's radius. It is radiating 10.7 times the luminosity of the Sun from its photosphere
HD_167042
Carbonaceous Apollo asteroid
(September 2019). Near-infrared spectral variability on the newly active asteroid (6478) Gault. 2019 EPSC-DPS conference. Epsc-DPS Joint Meeting 2019. Vol. 2019
101955_Bennu
Fifth planet from the Sun
shape of the planet is an oblate spheroid; the radius to the equator is about 7% larger than the radius to its poles. Its internal structure is believed
Jupiter
Irish mathematician (1948–2006)
thick of them. Among the several joint papers between Murphy and West, Finbarr Holland picks out a spectral radius formula for special mention in an
Gerard_Murphy_(mathematician)
Ratio of how much light is reflected back from a body
not only determined by properties of the surface itself, but also by the spectral and angular distribution of solar radiation reaching the Earth's surface
Albedo
Protoplanetary nebula in the constellation Antlia
companion has only been detected from variations in the doppler shift of its spectral lines. It is thought to be a low luminosity main sequence star. Although
HR_4049
Use of multiple antennas in radio
satisfy the radius constraint are found concurrently once the search reaches the bottommost layer. Unlike depth-first search, the sphere radius cannot be
MIMO
Red-clump giant star in the constellation Aquila
interferometry at the Navy Precision Optical Interferometer, which yields a radius 10.17 times that of the Sun. It is radiating 58.5 times the Sun's luminosity
Xi_Aquilae
{\displaystyle \mathbb {R} } and R 2 {\displaystyle \mathbb {R} ^{2}} are spectral if and only if they tile by translation. Goodman's conjecture on the coefficients
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Measure of the "size" of linear operators
conjugate transpose of the matrix A {\displaystyle A} ). In general, the spectral radius of A {\displaystyle A} is bounded above by the operator norm of A {\displaystyle
Operator_norm
Star in the constellation Eridanus
lower level of elements heavier than helium. It is a main-sequence star of spectral class K2, with an effective temperature of about 5,000 K (8,500 °F), giving
Epsilon_Eridani
Star in the constellation Andromeda
helium fusion at its core. The star has expanded to 12.7 times the Sun's radius and is radiating 58 times the Sun's luminosity from its enlarged photosphere
14_Andromedae
Technique to characterize materials using X-ray radiation
other, a wide region of the spectrum can be observed. To observe a large spectral range, three of four different single crystals may be needed. In contrast
X-ray_spectroscopy
Lithography using 13.5 nm UV light
Tin ions in the ionic states from Sn IX to Sn XIV give photon emission spectral peaks around 13.5 nm from 4p64dn – 4p54dn+1 + 4dn−14f ionic state transitions
EUV_lithography
Star in the constellation Corona Borealis
S2CID 118383042. Heard, John Frederick (1956). "The radial velocities, spectral classes and photographic magnitudes of 1041 late-type stars". Publications
HD_145457
Chemical element with atomic number 81 (Tl)
method of flame spectroscopy, in which thallium produces a notable green spectral line. Thallium, from Greek θαλλός, thallós, meaning "green shoot" or "twig"
Thallium
Field of physics that studies the atom
mass v {\displaystyle v} : velocity of the electron r {\displaystyle r} : radius of the orbit ℏ {\displaystyle \hbar } : reduced Planck constant ( ℏ = h
Atomic_physics
Largest type of black hole
volume within its Schwarzschild radius) can be less than the density of water. This is because the Schwarzschild radius ( r s {\displaystyle r_{\text{s}}}
Supermassive_black_hole
Star in the constellation Cetus
the supply of hydrogen at its core and expanded to 10.6 times the Sun's radius, or 0.05 AU. It is a red clump giant, which indicates it is on the horizontal
75_Ceti
Star in the constellation Ursa Major
estimate of 1.1+0.3 −0.2 solar masses. It has expanded to 11 times the Sun's radius and is radiating 55 times as much as the Sun from its outer atmosphere at
41_Lyncis
Surface waves generated by wind on open water
about the sea state can be found from the wave spectra. WHS describes the spectral density of wave height variance ("power") versus wave frequency, with dimension
Wind_wave
F-type dwarf with a planet in Centaurus
core. At present it has 1.59 times the mass of the Sun and 1.94 times the radius of the Sun. It radiates 8.34 times the luminosity of the Sun from its photosphere
CD−34_8618
Minor planet found within the inner Solar System
Originally, spectral designations were based on inferences of an asteroid's composition. However, the correspondence between spectral class and composition
Asteroid
Statistical matching technique
propensity score Radius matching: all matches within a particular radius are used – and reused between treatment units. Kernel matching: same as radius matching
Propensity_score_matching
Constellation near the celestial equator
subgiant star with a spectral type of G9.5 IV and the secondary is a red dwarf. The subgiant primary has three times the radius of the Sun and six times
Aquila_(constellation)
Numerical method for solving physical or engineering problems
element method. Spectral element methods combine the geometric flexibility of finite elements and the acute accuracy of spectral methods. Spectral methods are
Finite_element_method
Chemical element with atomic number 80 (Hg)
lanthanide contraction and relativistic contraction reducing the orbit radius of the outermost electrons, and thus weakening the metallic bonding in mercury
Mercury_(element)
Star in the constellation Andromeda
at the age of three billion years, has expanded to six times the Sun's radius. It is radiating 17–18 times the Sun's luminosity from its enlarged photosphere
HD_5608
Chemical element with atomic number 83 (Bi)
that drive microstructural recrystallisation, reducing a primary cause of joint failure and loss of electrical continuity. Bismuth is used in alloys with
Bismuth
Variable star in the constellation Perseus
Trippe, M. L.; Williams, S. J.; Wingert, D. W.; Zaballa, R. A. (2007). "Joint Hα and X-Ray Observations of Massive X-Ray Binaries. III. The Be X-Ray Binaries
X_Persei
Star in the constellation Aquila
Retrieved 22 November 2024. Buemi, C. S.; et al. (2010). "VISIR/VLT and VLA Joint Imaging Analysis of the Circumstellar Nebula Around IRAS 18576+0341". The
AFGL_2298
Largest moon of Jupiter
(1+e)} . Surface area derived from the radius (r): 4 π r 2 {\displaystyle 4\pi r^{2}} . Volume derived from the radius (r): 4 π r 3 / 3 {\displaystyle 4\pi
Ganymede_(moon)
Star in the constellation Lupus
Sirius A in mass and radius, but slightly cooler, older, and less luminous. It is about twice as massive as the Sun and has a radius of 1.7 R☉, with an
WASP-178
Mathematical model of the Big Bang
spatial metric that is well documented, both as the redshift of prominent spectral absorption or emission lines in the light from distant galaxies, and as
Lambda-CDM_model
Overview of the exploration of the moon Titan
close encounter of Voyager 1 enabled detailed measurements of Titan's mass, radius, atmospheric density, composition, temperature, and surface pressure. These
Exploration_of_Titan
Dwarf planet in the asteroid belt
formula predicted that there ought to be another planet with an orbital radius near 2.8 astronomical units (AU), or 420 million km, from the Sun. The Titius–Bode
Ceres_(dwarf_planet)
Chemical element with atomic number 82 (Pb)
architectural metals in roofing material, cladding, flashing, gutters and gutter joints, roof parapets. Lead is still used in statues and sculptures, including
Lead
Area of the Solar System beyond the planets, comprising small bodies
of a small, sub-kilometre-radius Kuiper belt object in archival Hubble photometry from March 2007. With an estimated radius of 520±60 m or a diameter
Kuiper_belt
Star in the Hydra constellation
Nearby Stars; a later expansion of the original catalogue was compiled jointly by Gliese and Hartmut Jahreiß. The Third Catalogue, the most recent catalogue
GJ_3634
Star in the constellation Leo Minor
an equatorial bulge with the radius at the equator being an estimated 7.4 times the Sun's radius while the polar radius is 6.1 times that of the Sun.
HD_93521
Algorithm for statistical inference on graphical models
convergence condition was formulated by Johnson et al. in 2006, when the spectral radius of the matrix ρ ( I − | D − 1 / 2 A D − 1 / 2 | ) < 1 {\displaystyle
Belief_propagation
System of two stars orbiting each other
observed: visually, by observation; spectroscopically, by periodic changes in spectral lines; photometrically, by changes in brightness caused by an eclipse;
Binary_star
Binary star in the constellation Aquila
it has an oblate shape, with its equatorial radius measuring 2.76 solar radii (R☉) and its polar radius measuring 2.15 R☉. Gravity darkening due to the
Zeta_Aquilae
Largest moon of Saturn
August 27, 2007. Barnes, Jason W.; et al. (2006). "Global-scale surface spectral variations on Titan seen from Cassini/VIMS" (PDF). Icarus. 186 (1): 242–258
Titan_(moon)
Dwarf planet in the Kuiper belt
cover an area of about 350 km2 (140 mi2; equivalent to a ~10 km or 6.2 mi-radius circle). Analysis of JWST spectroscopy in 2025 revealed the presence of
Makemake
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