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Nagata gave an example of a normal Noetherian local ring that is analytically reducible and therefore not analytically normal. Zariski (1950). Nagata (1958)
Analytically_normal_ring
only one analytic branch at a point. Zariski (1948) proved that if a local ring of an algebraic variety is a normal ring, then it is analytically irreducible
Analytically_irreducible_ring
local ring R is a finite module, then R is analytically unramified. There are two classical theorems of David Rees (1961) that characterize analytically unramified
Analytically_unramified_ring
Concept in commutative algebra
ring is a Nagata ring. Any quasi-excellent reduced local ring is analytically reduced. Any quasi-excellent normal local ring is analytically normal.
Excellent_ring
Theorem of algebraic geometry and commutative algebra
that if x is a normal point of a variety then it is analytically normal; in other words the completion of the local ring at x is a normal integral domain
Zariski's_main_theorem
an ideal appears in a theorem of Rees that characterizes an analytically unramified ring. In C [ x , y ] {\displaystyle \mathbb {C} [x,y]} , x i y d −
Integral_closure_of_an_ideal
from X to a reduced analytic space factors through r. An analytic space is normal if every stalk of the structure sheaf is a normal ring (meaning an integrally
Analytic_space
valuation. analytically This often refers to properties of the completion of a local ring; cf. formally 1. A local ring is called analytically normal if its
Glossary of commutative algebra
Glossary_of_commutative_algebra
Mathematical element
The integral closure of the homogeneous coordinate ring of a normal projective variety X is the ring of sections ⨁ n ≥ 0 H 0 ( X , O X ( n ) ) . {\displaystyle
Integral_element
Construction in commutative algebra
In commutative algebra, the Rees algebra or Rees ring of an ideal I in a commutative ring R is defined to be R [ I t ] = ⨁ n = 0 ∞ I n t n ⊆ R [ t ] .
Rees_algebra
e.g. It may also be computed analytically via a formula, using the normal form method, otherwise. For the storage ring case with distributed sextupoles
Tune_shift_with_amplitude
Topological structure in number theory
p-adic analytic groups were introduced by Lazard (1965). In the special case when the profinite group G is isomorphic to the additive group of the ring of
Iwasawa_algebra
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
_{n}{\color {red}\lrcorner }\\\end{array}}\right]} Example of a matrix in Jordan normal form. All matrix entries not shown are zero. The outlined squares are known
Jordan_normal_form
Torus-shaped vortex in a fluid
A vortex ring, also called a toroidal vortex, is a torus-shaped vortex in a fluid; that is, a region where the fluid mostly spins around an imaginary axis
Vortex_ring
Method of analyzing electrochemical reactions
system. For example, the rotating ring-disk electrode has two distinct and separate working electrodes, a disk, and a ring, which can be used to scan or hold
Voltammetry
disk is a connected component in the Fatou set where the dynamics is analytically conjugate to an irrational rotation. Given a holomorphic endomorphism
Siegel_disc
Set of waveguides including a closed loop
An optical ring resonator, or micro-ring resonator (MRR) is a set of waveguides in which at least one is a closed loop coupled to light input and output
Optical_ring_resonators
Non-perturbative approach to quantum field theory
fields are required to span the ring of functions. In practical calculations, it is usually required that the sums be analytic within some radius of convergence;
Operator_product_expansion
Theorem in algebraic geometry
product of two copies of the multiplicative group C* is isomorphic (analytically but not algebraically) to a non-split extension of any given elliptic
Chevalley's_structure_theorem
Singularities of algebraic varieties
a normal crossing singularity looks locally like a union of coordinate hyperplanes. There are two variants of the concept, a divisor with normal crossings
Normal_crossing_singularity
In mathematics, dimension of a ring
_{k=0}^{\infty }I^{k}/I^{k+1}} be the associated graded ring (geometers call it the ring of the normal cone of I). Then dim gr I ( R ) {\displaystyle \operatorname
Krull_dimension
Generalization of the Riemann zeta function for algebraic number fields
{\displaystyle \zeta _{K}(s)} , is an analytic function that represents information about the ideals in the corresponding number ring, generalizing how the Riemann
Dedekind_zeta_function
Algebraic variety in a projective space
-divisors) not just line bundles; assuming X is normal, the resulting ring is then called a generalized ring of sections. If K X {\displaystyle K_{X}} is
Projective_variety
Concept in algebraic geometry
true for normal complete Gorenstein algebraic spaces). A fundamental theorem of Birkar–Cascini–Hacon–McKernan from 2006 is that the canonical ring of a smooth
Canonical_bundle
Formalism of first-order logic
Together with the normal forms in propositional logic (e.g. disjunctive normal form or conjunctive normal form), it provides a canonical normal form useful
Prenex_normal_form
4th episode of the 1st season of The Lord of the Rings: The Rings of Power
first season of the American fantasy television series The Lord of the Rings: The Rings of Power. The series is based on J. R. R. Tolkien's history of Middle-earth
The Great Wave (The Lord of the Rings: The Rings of Power)
The_Great_Wave_(The_Lord_of_the_Rings:_The_Rings_of_Power)
Doughnut-shaped surface of revolution
coplanar with the circle. The main types of tori include ring tori, horn tori, and spindle tori. A ring torus is sometimes colloquially referred to as a doughnut
Torus
Public university in Tempe, Arizona, US
university in Tempe, Arizona, United States. Founded in 1885 as Territorial Normal School by the 13th Arizona Territorial Legislature, the university is one
Arizona_State_University
Branch of mathematics
maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such
Algebraic_geometry
Traffic intersection
reduced queuing associated with traffic lights. They allow U-turns within the normal flow of traffic, which often are not possible at other forms of junction
Roundabout
2026 Marvel Studios film
maturing the character, after she felt an initial draft had made her too "normal". Sadie Sink as Jean Grey: A young mutant with psionic abilities such as
Spider-Man:_Brand_New_Day
American comedy series
eighth most-streamed series in the U.S. through June 2026. Streaming analytics firm FlixPatrol, which monitors daily updated VOD charts and streaming
Not_Suitable_for_Work
Array of numbers
n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn. If the ring R is commutative, that is
Matrix_(mathematics)
Unsolved problem in geometry
equations. The latter objects can be studied using algebra and the calculus of analytic functions, and this allows one to indirectly understand the broad shape
Hodge_conjecture
Debunked conspiracy theory about alleged child sex ring
that the New York City Police Department (NYPD) had discovered a pedophile ring linked to members of the Democratic Party while searching through Anthony
Pizzagate_conspiracy_theory
1st episode of the 1st season of The Lord of the Rings: The Rings of Power
premiere of the American fantasy television series The Lord of the Rings: The Rings of Power. The series is based on J. R. R. Tolkien's history of Middle-earth
A_Shadow_of_the_Past
Chemical analysis technique
Analytical thermal desorption, known within the analytical chemistry community simply as "thermal desorption" (TD), is a technique that concentrates volatile
Analytical_thermal_desorption
2025 Marvel Studios animated series
Ten Rings alongside him Randall Park as Jimmy Woo: A former FBI agent Feodor Chin as Xu Wenwu: Shang-Chi's father who is the leader of the Ten Rings organization
Marvel_Zombies_(TV_series)
Public university in Carbondale, Illinois, US
from all 50 states and more than 100 countries. Originally founded as a normal college, the university today provides programs in a variety of disciplines
Southern Illinois University Carbondale
Southern_Illinois_University_Carbondale
*-algebra of bounded operators on a Hilbert space
symmetries. Two basic examples of von Neumann algebras are as follows: The ring L ∞ ( R ) {\displaystyle L^{\infty }(\mathbb {R} )} of essentially bounded
Von_Neumann_algebra
American feral child (born 1957)
father used had caused a thick callus and heavy black bruising to form in a ring on her buttocks, which took several weeks to heal. A series of X-rays found
Genie_(feral_child)
off his ring and fight X-23 in his unstable Glartrox form, leaving Anachronism, Nara, and Cammi in the middle. Nara receives Bloodstone's ring and returns
List of Marvel Comics characters: A
List_of_Marvel_Comics_characters:_A
British psychologist and author (born 1958)
Baron-Cohen S, Ring HA, Wheelwright S, Bullmore ET, Brammer MJ, Simmons A, Williams SC (June 1999). "Social intelligence in the normal and autistic brain:
Simon_Baron-Cohen
NFL team in New Orleans, Louisiana
and playoff appearance, going 12–3 in 1987, which had one fewer game than normal due to a players' strike. Another playoff berth would follow during the
New_Orleans_Saints
American online retailer
our border problem. No one goes through the window of a house if they can ring the doorbell and come in the front door." In 2014, Overstock began developing
Bed Bath & Beyond (2023–present)
Bed_Bath_&_Beyond_(2023–present)
Ratio of lengths of fingers
commonly studied digit ratio is that of the 2nd (index finger) and 4th (ring finger), also referred to as the 2D:4D ratio, measured on the palm side.
Digit_ratio
Material with a negative refractive index
determination of the refractive index. For example, in 1967, Victor Veselago analytically determined that light will refract in the reverse direction (negatively)
Negative-index_metamaterial
2026 Marvel Studios television series
terrorist leader the Mandarin before being kidnapped by Xu Wenwu's Ten Rings organization. Executive producer Brad Winderbaum said the series concludes
Wonder_Man_(TV_series)
Semiconductor light source
infrared illumination in night vision uses including security cameras. A ring of LEDs around a video camera, aimed forward into a retroreflective background
Light-emitting_diode
and other Democratic Party officials were involved in a child trafficking ring based out of pizzerias in Washington, D.C. The United States Intelligence
Russian interference in the 2016 United States elections
Russian_interference_in_the_2016_United_States_elections
Fourteenth letter in the Greek alphabet
thermochemistry Unknown stereochemistry or stereocentre configuration in a planar ring system in organic chemistry, as well as uppercase Xi for unknown R/S/E/Z
Xi_(letter)
Aesthetic assessment of physical traits
index and ring fingers. Men have a smaller index-to-ring-finger ratio than women. The gender differences in the ratio between the index and ring fingers
Physical_attractiveness
an incident that Butcher caused when busting Electroshock's prostitution ring. The Seven are the world's premier superhero team, created by Vought-American
List_of_The_Boys_characters
American adult animated series
wanted to write, something that Ellis had found restrictive in working with normal television animation. In adapting the game for the film, Ellis did not want
Castlevania_(TV_series)
Medication for musculoskeletal pain
carbamate bond is not hydrolyzed metabolically; its metabolism is by Phase I ring hydroxylation and O-demethylation, followed by Phase II conjugation. All
Methocarbamol
First-run airings of the ITV medical dramedy
ex-girlfriend Dr. Edith Montgomery. In order to determine if the fetus is normal size, Louisa must provide the last date she had sex with Martin. Louisa
List_of_Doc_Martin_episodes
Study of the deformation of solids that touch each other
bodies' surfaces (known as normal stress) and frictional stresses acting tangentially between the surfaces (shear stress). Normal contact mechanics or frictionless
Contact_mechanics
Normal phase chromatography that uses a supercritical fluid as the mobile phase
Supercritical fluid chromatography (SFC) is a form of normal phase chromatography that uses a supercritical fluid such as carbon dioxide as the mobile
Supercritical fluid chromatography
Supercritical_fluid_chromatography
Theme in Tolkien's fiction
journey to destroy the One Ring in the fires of Mount Doom scarred by multiple weapons, and is unable to settle back into the normal life of his home, the
Psychological journeys of Middle-earth
Psychological_journeys_of_Middle-earth
Proposed orbital infrastructure
Aerospace Google – Project Suncatcher Nvidia OpenAI SpaceX Starcloud Dyson ring Interplanetary Internet Kardashev scale Laser communication in space, microwave
Space-based_data_center
Research and Development Facility in Upton, United States
During normal operations, the electrons in the storage rings lost energy and as such, the rings were re-injected every 12 (X-ray ring) and 4 (VUV ring) hours
National Synchrotron Light Source
National_Synchrotron_Light_Source
Decomposition of periodic functions
convergence to a somewhat arbitrary function. Note the development of the "ringing" (Gibbs phenomenon) at the transitions to/from the vertical sections. The
Fourier_series
Polycyclic organic compound having sterane as a core structure
fused rings: three six-member cyclohexane rings (rings A, B and C in the first illustration) and one five-member cyclopentane ring (the D ring). Steroids
Steroid
Hydrocarbon composed of multiple aromatic rings
class of organic compounds that is composed of multiple fused aromatic rings. Most are produced by the incomplete combustion of organic matter—by engine
Polycyclic aromatic hydrocarbon
Polycyclic_aromatic_hydrocarbon
Phenomenon
its 5-methylcytosine content when compared to its parent normal cell being correct. In normal tissues, the vast majority of CpG islands are completely
CpG_island_hypermethylation
subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ideals in the case of rings, and submodule in the case of modules)
Malcev_algebra
contacted FBI agent Brian O'Conner and became a double agent for Braga's crime ring in order to clear Dominic's charges (presumably hers as well) and allow him
List of Fast & Furious characters
List_of_Fast_&_Furious_characters
Surgery operation in minimal model program
projective over a fixed normal variety Z {\displaystyle Z} terminates after finitely many steps. Atiyah, Michael Francis (1958), "On analytic surfaces with double
Flip_(algebraic_geometry)
Overview of and topical guide to geometry
geometry Information geometry Integral geometry Inversive geometry Inversive ring geometry Klein geometry Lie sphere geometry Non-Euclidean geometry Noncommutative
Outline_of_geometry
American actor (born 1973)
was targeted by a group of fashion-motivated criminals known as "The Bling Ring", who initially targeted them to steal Fox's designer clothes. In June 2010
Brian_Austin_Green
U.S. presidential administration since 2025
Theodore; Kanno-Youngs, Zolan (April 5, 2025). "Trump Family's Cash Registers Ring as Financial Meltdown Plays Out". The New York Times. Retrieved April 19
Second presidency of Donald Trump
Second_presidency_of_Donald_Trump
Great Lakes freighter sunk in Lake Superior
discovered a lone life-saving ring that appeared to have come from Edmund Fitzgerald. It bore markings different from those of rings found at the wreck site
SS_Edmund_Fitzgerald
Public university in San Jose, California, US
Jose, California, United States. Established in 1857 as the state's first normal school, it is the oldest public university in the western United States
San_Jose_State_University
begins to come to terms with the loss of his wife; he takes off his wedding ring, saying "I decided to take Breena's advice. I can hold onto what's important
List_of_NCIS_characters
Diode that emits light from an organic compound
charge-injecting electrodes. The light generated was readily visible in normal lighting conditions though the polymer used had 2 limitations; low conductivity
OLED
of non-aromatic rings. For fluorene, there are 6 apparent double bonds (three pi bonds in each side benzene-like ring); the central ring has one secondary
F_number_(chemistry)
operate. A "busher" refers to someone from the "bush leagues": see subtitle of Ring Lardner's first book, You Know Me Al: A Busher's Letters. A day game on a
Glossary_of_baseball_terms
Interactive street panorama technology
R2: the earliest after Immersive Media. The photos were captured with a ring of eight 11-megapixel CCD sensors with commercial photographic wide-angle
Google_Street_View
All numbers between two given numbers
topology, the algebra of intervals forms a topological ring. The group of units of this ring consists of four quadrants determined by the axes, or ideals
Interval_(mathematics)
2011 film by David Cronenberg
suggests that resistance to transference is symptomatic of the repression of normal, healthy sexual impulses, exhorting Jung to indulge himself with abandon
A_Dangerous_Method
emerald ring that has the power of sucking the coolness out of people. The ring makes him extremely cool, skillful and attractive as the ring causes his
List of Totally Spies! episodes
List_of_Totally_Spies!_episodes
Wakeham, W. A. (October 1986). "The Viscosity and Thermal Conductivity of Normal Hydrogen in the Limit of Zero Density" (PDF). Journal of Physical and Chemical
List of thermal conductivities
List_of_thermal_conductivities
Machine that washes clothes
2020. "Unitary balance ring for a washing machine appliance". "Experimental and Analytical Investigation on a Liquid Balance Ring for Automatic Washing
Washing_machine
the communities are not perfect, they are Lydia's best chance of living a normal life. She is unable to kill her mother, who requested that Lydia kill her
List of The Walking Dead (TV series) characters
List_of_The_Walking_Dead_(TV_series)_characters
group with Byrne Faraday and Calisto Yew in order to take down a smuggling ring whose leader could seemingly not be convicted. Using his privilege as the
List of Ace Attorney characters
List_of_Ace_Attorney_characters
cells tended to exhibit longer relaxation times than their corresponding normal cells and as such stimulated initial interest in the idea of detecting cancer
History of magnetic resonance imaging
History_of_magnetic_resonance_imaging
Biological process of getting older
embryonic status and then used to grow a new tissue or animal without ageing. Normal human cells however die after about 50 cell divisions in laboratory culture
Ageing
Precisely, Matsusaka'a theorem says Let X be a normal irreducible projective scheme over a discrete valuation ring. If the generic fiber is ruled, then each
Degeneration (algebraic geometry)
Degeneration_(algebraic_geometry)
Pipette with its volume, in increments, marked along the tube
"to contain" as per manufacturer and made to be blown out. A set of two rings printed on the upper end of the pipette indicate that it is a "blow out"
Graduated_pipette
Naturally produced monosaccharide
α-D-glucose with para-tolylboronic acid H3C−(C6H4)−B(OH)2 reforms the normal pyranose ring to yield the 4-fold ester α-D-glucofuranose-1,2:3,5-bis(p-tolylboronate)
Glucose
History of research by Augustin-Jean Fresnel
something: when light passes through two similarly oriented calcite crystals at normal incidence, the ordinary ray emerging from the first crystal suffers only
Fresnel's_physical_optics
Concept in geometry
2πt dt, the circumferential length of the ring times its infinitesimal width (one can approximate this ring by a rectangle with width=2πt and height=dt)
Area_of_a_circle
engaging in constitutionally protected public debate. And that alarm should ring even louder when the Government retaliates against those engaged in newsgathering
Twitter_under_Elon_Musk
Extension of the factorial function
(z)>0.} The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which
Gamma_function
Class of chemical compounds
chemical structure formed by attachment of an aromatic benzenoid (phenyl) ring to a hydroxyl (-OH) group (hence the -ol suffix). The term "polyphenol" has
Polyphenol
Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature
matrix whose entries are 2-forms, so it is a matrix over the commutative ring ⋀ even T ∗ M {\textstyle {\bigwedge }^{\text{even}}\,T^{*}M} . Hence the
Chern–Gauss–Bonnet_theorem
generalization) Mean value theorem Multivariate normal distribution (to do) Holomorphic functions are analytic Pythagorean theorem Quadratic equation Quotient
List_of_mathematical_proofs
Set of points equidistant from a center
that contains the normal with the surface will form a curve that is called a normal section, and the curvature of this curve is the normal curvature. For
Sphere
Hallucinogenic class of psychoactive drug
amphetamines, particularly those with methoxy and other substitions on the phenyl ring, are potent serotonin 5-HT2 receptor agonists, including of the serotonin
Psychedelic_drug
Generalization of vector bundles
The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as
Coherent_sheaf
ANALYTICALLY NORMAL-RING
ANALYTICALLY NORMAL-RING
ANALYTICALLY NORMAL-RING
ANALYTICALLY NORMAL-RING
ANALYTICALLY NORMAL-RING
ANALYTICALLY NORMAL-RING
ANALYTICALLY NORMAL-RING
ANALYTICALLY NORMAL-RING
ANALYTICALLY NORMAL-RING