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Another machine, propelled by horses with a pillion rider, carries in front of it four scythes mounted on a revolving gear, turned by a shaft driven by the wheels of a cart behind the horses. Leonardo's notebooks also show cannons which he claimed "to hurl small stones like a storm with the smoke of these causing great... | Wikipedia - Science and inventions of Leonardo da Vinci | null | null | null |
Following his detailed drawing, one was constructed by the British Army, but could not be made to fire successfully. In 1481 Leonardo designed a breech-loading, water cooled cannon with three racks of barrels allowed the re-loading of one rack while another was being fired and thus maintaining continuous fire power. Th... | Wikipedia - Science and inventions of Leonardo da Vinci | null | null | null |
Leonardo was the first to sketch the wheel-lock musket c. 1500 AD (the precedent of the flintlock musket which first appeared in Europe by 1547), although as early as the 14th century the Chinese had used a flintlock 'steel wheel' in order to detonate land mines.While Leonardo was working in Venice, he drew a sketch fo... | Wikipedia - Science and inventions of Leonardo da Vinci | null | null | null |
The head was covered by a helmet with two eyeglasses at the front. A breathing tube of bamboo with pigskin joints was attached to the back of the helmet and connected to a float of cork and wood. When the scuba divers tested the suit, they found it to be a workable precursor to a modern diving suit, the cork float acti... | Wikipedia - Science and inventions of Leonardo da Vinci | null | null | null |
Leonardo's study of the motion of water led him to design machinery that utilized its force. Much of his work on hydraulics was for Ludovico il Moro. Leonardo wrote to Ludovico describing his skills and what he could build: ...very light and strong bridges that can easily be carried, with which to pursue, and sometimes... | Wikipedia - Science and inventions of Leonardo da Vinci | null | null | null |
Fortunately, this was too costly to be carried out. He also surveyed Venice and came up with a plan to create a movable dyke for the city's protection against invaders. | Wikipedia - Science and inventions of Leonardo da Vinci | null | null | null |
In 1502, Leonardo produced a drawing of a single span 240 m (720 ft) bridge as part of a civil engineering project for Ottoman Sultan Beyazid II of Istanbul. The bridge was intended to span an inlet at the mouth of the Bosphorus known as the Golden Horn. Beyazid did not pursue the project, because he believed that such... | Wikipedia - Science and inventions of Leonardo da Vinci | null | null | null |
Leonardo's vision was resurrected in 2001 when a smaller bridge based on his design was constructed in Norway. A stone model of the bridge was evaluated in 2019 by MIT researchers. The self-supporting 1:500 scale model was built from 126 3D-printed stone cross-sections, held together without mortar. Researchers conclud... | Wikipedia - Science and inventions of Leonardo da Vinci | null | null | null |
Leonhard Euler published a series of important memoirs on spherical geometry: L. Euler, Principes de la trigonométrie sphérique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9 (1753), 1755, p. 233–257; Opera Omnia, Series 1, vol. XXVII, p. 277–308. | Wikipedia - Spherical geometry | null | null | null |
L. Euler, Eléments de la trigonométrie sphéroïdique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9 (1754), 1755, p. 258–293; Opera Omnia, Series 1, vol. XXVII, p. | Wikipedia - Spherical geometry | null | null | null |
309–339. L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15, 1771, pp. 195–216; Opera Omnia, Series 1, Volume 28, pp. | Wikipedia - Spherical geometry | null | null | null |
142–160. L. Euler, De mensura angulorum solidorum, Acta academiae scientiarum imperialis Petropolitinae 2, 1781, p. 31–54; Opera Omnia, Series 1, vol. | Wikipedia - Spherical geometry | null | null | null |
XXVI, p. 204–223. L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientiarum imperialis Petropolitinae 4, 1783, p. | Wikipedia - Spherical geometry | null | null | null |
91–96; Opera Omnia, Series 1, vol. XXVI, p. 237–242. | Wikipedia - Spherical geometry | null | null | null |
L. Euler, Geometrica et sphaerica quaedam, Mémoires de l'Académie des Sciences de Saint-Pétersbourg 5, 1815, p. 96–114; Opera Omnia, Series 1, vol. XXVI, p. | Wikipedia - Spherical geometry | null | null | null |
344–358. L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientiarum imperialis Petropolitinae 3, 1782, p. 72–86; Opera Omnia, Series 1, vol. | Wikipedia - Spherical geometry | null | null | null |
XXVI, p. 224–236. L. Euler, Variae speculationes super area triangulorum sphaericorum, Nova Acta academiae scientiarum imperialis Petropolitinae 10, 1797, p. | Wikipedia - Spherical geometry | null | null | null |
47–62; Opera Omnia, Series 1, vol. XXIX, p. 253–266. | Wikipedia - Spherical geometry | null | null | null |
Leopards and ladybirds are spotted; angelfish and zebras are striped. These patterns have an evolutionary explanation: they have functions which increase the chances that the offspring of the patterned animal will survive to reproduce. One function of animal patterns is camouflage; for instance, a leopard that is harde... | Wikipedia - Natural patterns | null | null | null |
A young bird may see a warning patterned insect like a ladybird and try to eat it, but it will only do this once; very soon it will spit out the bitter insect; the other ladybirds in the area will remain undisturbed. The young leopards and ladybirds, inheriting genes that somehow create spottedness, survive. But while ... | Wikipedia - Natural patterns | null | null | null |
Leopeard coral groupers are largely piscivores (fish-eating predators). Younger juvenile trout mostly eat crustaceans, especially prawns, which live on or near the reef bottom. However, adults feed upon a variety of reef fish. The most common type of fish eaten is damselfish (family Pomacentridae), particularly the spi... | Wikipedia - Coral trout | null | null | null |
Adult coral trout also eat juveniles of their own species. Individual coral trout usually feed once every 1–3 days, although they may go for many days without feeding. About 90% of a prey item will be digested within 24 hours. | Wikipedia - Coral trout | null | null | null |
This species only feeds during daylight hours, most often at dusk and dawn. Coral trout hunt by ambush and by prowling. They use the ambush method to hunt fish that live among the coral on the reef bottom. | Wikipedia - Coral trout | null | null | null |
The trout hide and remain very still and alert, ready to attack passing prey. The prowling method is used to hunt schooling fish higher up in the water. Here, the trout will move (prowl) slowly towards the prey and attack at great speed. | Wikipedia - Coral trout | null | null | null |
Individual coral trout have different feeding behaviors, possibly explaining the variability in growth and maturity.Coral trout in the southern Great Barrier Reef feed mainly on parrot fish (family Scaridae) and hardyhead bait fish (family Atherinidae). The most common prey items further north are the damselfish (Pomac... | Wikipedia - Coral trout | null | null | null |
This seasonal variation is quite common in the diet of coral trout due to varying abundances of prey at different times of the year. Trout also tend to eat more food in winter, possibly to increase fat stores in preparation for reproduction in spring. They sometimes engage in cooperative hunting with the giant moray (G... | Wikipedia - Coral trout | null | null | null |
Leopold Kronecker (old constructivism, semi-intuitionism) L. E. J. Brouwer (founder of intuitionism) A. A. Markov (forefather of Russian school of constructivism) Arend Heyting (formalized intuitionistic logic and theories) Per Martin-Löf (founder of constructive type theories) Errett Bishop (promoted a version of cons... | Wikipedia - Mathematical constructivism | null | null | null |
Leopold Kronecker remained a strident opponent to Cantor's set theory: Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. God created the integers; all else is the work of man. Reuben Goodstein was another proponent of finitism. Some of his work involved building up to analysis from finitist f... | Wikipedia - Finitism | null | null | null |
Although he denied it, much of Ludwig Wittgenstein's writing on mathematics has a strong affinity with finitism.If finitists are contrasted with transfinitists (proponents of e.g. Georg Cantor's hierarchy of infinities), then also Aristotle may be characterized as a finitist. Aristotle especially promoted the potential... | Wikipedia - Finitism | null | null | null |
Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructi... | Wikipedia - Infinity | null | null | null |
Leopold Löwenheim and Thoralf Skolem obtained the Löwenheim–Skolem theorem, which says that first-order logic cannot control the cardinalities of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable mod... | Wikipedia - Formal logical systems | null | null | null |
Gödel used the completeness theorem to prove the compactness theorem, demonstrating the finitary nature of first-order logical consequence. These results helped establish first-order logic as the dominant logic used by mathematicians. | Wikipedia - Formal logical systems | null | null | null |
In 1931, Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which proved the incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Gödel's incompleteness theorem, establishes severe limitation... | Wikipedia - Formal logical systems | null | null | null |
Hilbert, however, did not acknowledge the importance of the incompleteness theorem for some time.Gödel's theorem shows that a consistency proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibil... | Wikipedia - Formal logical systems | null | null | null |
Lepidosaurs take advantage of their sensory system to assist them in food detection and consumption. Their sensory vision consists of a retina covered with single layer of photoreceptor cells and the nocturnal lepidosaurs also have rod-like cells in their photoreceptor cells. Chemoreception further allows them to detec... | Wikipedia - Lepidosaur herbivory | null | null | null |
Lercanidipine is used in form of the hydrochloride, which is a slightly yellow crystalline powder and melts at 197 to 201 °C (387 to 394 °F) in crystal form I or 207 to 211 °C (405 to 412 °F) in crystal form II. It is readily soluble in chloroform and methanol, but practically insoluble in water. This high lipophilicit... | Wikipedia - Lercanidipine | null | null | null |
Lesinurad is a white to off-white powder and is not hygroscopic. It is a 1:1 racemic mixture of atropisomers. | Wikipedia - Lesinurad | null | null | null |
Lesioning is a test of necessity involves physically damaging a structure so that it loses its function. If lesioning a structure causes a change in the system, then that structure is necessary. Lesioning can look very different across scientific disciplines. | Wikipedia - Biological tests of necessity and sufficiency | null | null | null |
In psychology, information about necessity may be gleaned by observing changes in behavior when a brain region has been destroyed, either by accident or illness in a human or purposefully in a lab animal. Other disciplines may target specific cell types and tag them for degradation. The GAL4/UAS System "reports" a subs... | Wikipedia - Biological tests of necessity and sufficiency | null | null | null |
When combined with an apoptotic gene, such as reaper (rpr), all cells that express the genes of interest along with GAL4/UAS, will initiate cell death, creating a legion specific to a cell type.A classic example of lesioning in chronobiology is the exploration of circadian navigation in monarch butterflies. In a landma... | Wikipedia - Biological tests of necessity and sufficiency | null | null | null |
Lesions – A classic method in which a brain-region of interest is naturally or intentionally destroyed to observe any resulting changes such as degraded or enhanced performance on some behavioral measure. Lesions can be placed with relatively high accuracy "Thanks to a variety of brain 'atlases' which provide a map of ... | Wikipedia - Behavioural neuroscience | null | null | null |
Temporary lesions – Neural tissue is temporarily disabled by cooling or by the use of anesthetics such as tetrodotoxin. Transcranial magnetic stimulation – A new technique usually used with human subjects in which a magnetic coil applied to the scalp causes unsystematic electrical activity in nearby cortical neurons wh... | Wikipedia - Behavioural neuroscience | null | null | null |
These systems utilize G protein-coupled receptors (GPCR) engineered to respond exclusively to synthetic small molecules ligands, like clozapine N-oxide (CNO), and not to their natural ligand(s). RASSL's represent a GPCR-based chemogenetic tool. These synthetic ligands upon activation can decrease neural function by G-p... | Wikipedia - Behavioural neuroscience | null | null | null |
This can with Potassium attenuating neural activity. Psychopharmacological manipulations – A chemical receptor antagonist induces neural activity by interfering with neurotransmission. | Wikipedia - Behavioural neuroscience | null | null | null |
Antagonists can be delivered systemically (such as by intravenous injection) or locally (intracerebrally) during a surgical procedure into the ventricles or into specific brain structures. For example, NMDA antagonist AP5 has been shown to inhibit the initiation of long term potentiation of excitatory synaptic transmis... | Wikipedia - Behavioural neuroscience | null | null | null |
Lesk algorithm: word sense disambiguation Stemming algorithm: a method of reducing words to their stem, base, or root form Sukhotin's algorithm: a statistical classification algorithm for classifying characters in a text as vowels or consonants | Wikipedia - Geometric algorithms | null | null | null |
Leslie Ballentine promoted the ensemble interpretation in his book Quantum Mechanics, A Modern Development. In it, he described what he called the "Watched Pot Experiment". His argument was that, under certain circumstances, a repeatedly measured system, such as an unstable nucleus, would be prevented from decaying by ... | Wikipedia - Ensemble Interpretation | null | null | null |
Less common oxidation states of gold include −1, +2, and +5. The −1 oxidation state occurs in aurides, compounds containing the Au− anion. Caesium auride (CsAu), for example, crystallizes in the caesium chloride motif; rubidium, potassium, and tetramethylammonium aurides are also known. Gold has the highest electron af... | Wikipedia - Gold poisoning | null | null | null |
Gold also has a –1 oxidation state in covalent complexes with the group 4 transition metals, such as in titanium tetraauride and the analogous zirconium and hafnium compounds. These chemicals are expected to form gold-bridged dimers in a manner similar to titanium(IV) hydride.Gold(II) compounds are usually diamagnetic ... | Wikipedia - Gold poisoning | null | null | null |
Originally thought to be a mixed-valence compound, it has been shown to contain Au4+2 cations, analogous to the better-known mercury(I) ion, Hg2+2. A gold(II) complex, the tetraxenonogold(II) cation, which contains xenon as a ligand, occurs in (Sb2F11)2.Gold pentafluoride, along with its derivative anion, AuF−6, and it... | Wikipedia - Gold poisoning | null | null | null |
Well-defined cluster compounds are numerous. In some cases, gold has a fractional oxidation state. A representative example is the octahedral species {Au(P(C6H5)3)}2+6. | Wikipedia - Gold poisoning | null | null | null |
Less commonly, bipolar disorder or a bipolar-like disorder may occur as a result of or in association with a neurological condition or injury including stroke, traumatic brain injury, HIV infection, multiple sclerosis, porphyria, and rarely temporal lobe epilepsy. | Wikipedia - Rapid cycling | null | null | null |
Less studied than above-ground interactions, but proving to be increasingly important, are the below-ground interactions that influence plant defense. There is a complex network of signal transduction pathways involved in plant responses to stimuli, and soil microbes can influence these responses. Certain soil microbes... | Wikipedia - Tritrophic interactions in plant defense | null | null | null |
Many studies have shown both the positive and negative effects that one organism in one environment can have on other organisms in the same or opposite environment, with the plant acting as the intermediary. The colonization of plant roots with mycorhizae typically results in a mutualistic relationship between the plan... | Wikipedia - Tritrophic interactions in plant defense | null | null | null |
The mycorhizal species involved also matters. One common species, Rhizophagus irregularis, has been observed to have a negative effect on the feeding success of chewing herbivores, whereas other species studied have positive effects.The roots of some maize plants produce a defense chemical when roots are damaged by lea... | Wikipedia - Tritrophic interactions in plant defense | null | null | null |
Incorporating these varieties or their genes into commercial maize production could increase the efficacy of nematode treatments.Further studies suggest that the plant-emitted chemicals act as the primary source of attractant to the nematodes. Herbivores are believed to have evolved to evade detection on the part of th... | Wikipedia - Tritrophic interactions in plant defense | null | null | null |
The bacterium Klebsiella aerogenes produces the volatile 2,3-butanediol, which modulates interactions between plants, pathogens, and insects. When maize plants are grown in a soil culture containing the bacterium or the plants are inoculated with the bacterium, the maize is more resistant to the fungus Setosphaeria tur... | Wikipedia - Tritrophic interactions in plant defense | null | null | null |
Lesser grisons are carnivorous, feeding on small to medium rodents, as well as rabbits, birds, frogs, lizards, and snakes. They can also eat fruits, as avocados. They are among the major predators on cavies, including wild guinea pigs, and also of nesting grebes.They are semi-plantigrade, walking partly on the soles of... | Wikipedia - Lesser grison | null | null | null |
They possess anal scent glands that spray a noxious chemical similar to, but probably weaker than, that of skunks. They are monogamous, hunting together when raising their litters of two to five young.Lesser grisons hunt primarily during the day, locating their prey at least partly by scent. They are either solitary, o... | Wikipedia - Lesser grison | null | null | null |
They are said to be particularly fierce, and to play with their food for up to 45 minutes before eating it. During the night, they sleep in hollow trees or natural crevices, or else in excavated burrows. Burrows may be as deep as 4 m (13 ft), and have entrances obscured by leaves. | Wikipedia - Lesser grison | null | null | null |
Let , ∈ π 1 ( S 3 ∖ K ) {\displaystyle ,\in \pi _{1}(\mathbb {S} ^{3}\setminus K)} denote elements corresponding to a preferred longitude and meridian of a tubular neighborhood of K {\displaystyle K} . K {\displaystyle K} has Property P if and only if its Knot group is never trivialised by adjoining a relation of the... | Wikipedia - Property P conjecture | null | null | null |
Let "P" mean "I like chocolate" and Q mean "It's warm outside." AG.P"I will like chocolate from now on, no matter what happens. "EF.P"It's possible I may like chocolate some day, at least for one day. | Wikipedia - Computational tree logic | null | null | null |
"AF.EG.P"It's always possible (AF) that I will suddenly start liking chocolate for the rest of time." (Note: not just the rest of my life, since my life is finite, while G is infinite).EG.AF.P"Depending on what happens in the future (E), it's possible that for the rest of time (G), I'll be guaranteed at least one (AF) ... | Wikipedia - Computational tree logic | null | null | null |
"The two following examples show the difference between CTL and CTL*, as they allow for the until operator to not be qualified with any path operator (A or E): AG(PUQ)"From now until it's warm outside, I will like chocolate every single day. Once it's warm outside, all bets are off as to whether I'll like chocolate any... | Wikipedia - Computational tree logic | null | null | null |
Let ( A , F ) {\displaystyle (A,F)} be an algebraic structure and let ∼ {\displaystyle \sim } be a tolerance relation on A {\displaystyle A} . Suppose that, for each n {\displaystyle n} -ary operation f ∈ F {\displaystyle f\in F} and C 1 , … , C n ∈ A / ∼ {\displaystyle C_{1},\dots ,C_{n}\in A/{\sim }} , there is a uni... | Wikipedia - Tolerance relation | null | null | null |
Therefore, for a variety V {\displaystyle {\mathcal {V}}} of algebraic structures, we may consider the following two conditions. (Tolerance factorability) for any ( A , F ) ∈ V {\displaystyle (A,F)\in {\mathcal {V}}} and any tolerance relation ∼ {\displaystyle \sim } on ( A , F ) {\displaystyle (A,F)} , the uniqueness ... | Wikipedia - Tolerance relation | null | null | null |
Let ( L , ≤ ) {\displaystyle (L,\leq )} be a complete lattice, with infimum and supremum symbolized by ∧ {\displaystyle \wedge } and ∨ {\displaystyle \vee } , respectively. Its universe and least element are symbolized by U and ∅ {\displaystyle \emptyset } , respectively. Moreover, let { X i } {\displaystyle \{X_{i}\}}... | Wikipedia - Mathematical morphology | null | null | null |
I.e.: ⋀ i ε ( X i ) = ε ( ⋀ i X i ) {\displaystyle \bigwedge _{i}\varepsilon (X_{i})=\varepsilon \left(\bigwedge _{i}X_{i}\right)} , ε ( U ) = U {\displaystyle \varepsilon (U)=U} .Dilations and erosions form Galois connections. That is, for every dilation δ {\displaystyle \delta } there is one and only one erosion ε {\... | Wikipedia - Mathematical morphology | null | null | null |
Similarly, for every erosion there is one and only one dilation satisfying the above connection. Furthermore, if two operators satisfy the connection, then δ {\displaystyle \delta } must be a dilation, and ε {\displaystyle \varepsilon } an erosion. Pairs of erosions and dilations satisfying the above connection are cal... | Wikipedia - Mathematical morphology | null | null | null |
Let ( M , a ) {\displaystyle (M,a)} be a Riemannian manifold and b a differential one-form on M with ‖ b ‖ a := a i j b i b j < 1 , {\displaystyle \|b\|_{a}:={\sqrt {a^{ij}b_{i}b_{j}}}<1,} where ( a i j ) {\displaystyle \left(a^{ij}\right)} is the inverse matrix of ( a i j ) {\displaystyle (a_{ij})} and the Einstein no... | Wikipedia - Finsler metric | null | null | null |
Let ( M , g ) {\displaystyle (M,g)} be a Riemannian manifold with Laplace-Beltrami operator Δ M {\displaystyle \Delta _{M}} . An adapted M {\displaystyle M} -valued process X {\displaystyle X} with maximal lifetime ξ {\displaystyle \xi } is called a Brownian motion ( M , g ) {\displaystyle (M,g)} , if for every f ∈ C ∞... | Wikipedia - Stochastic differential geometry | null | null | null |
Let ( M , g ) {\displaystyle (M,g)} be a smooth Riemannian or pseudo-Riemannian manifold of dimension n . {\displaystyle n.} Let h {\displaystyle h} he a smooth symmetric (0,2)-tensor field whose covariant derivative, with respect to the Levi-Civita connection, is completely symmetric. The symmetry condition is an anal... | Wikipedia - Schur's lemma (from Riemannian geometry) | null | null | null |
Let ( R , m ) {\displaystyle (R,{\mathfrak {m}})} be a noetherian local ring and I a m {\displaystyle {\mathfrak {m}}} -primary ideal (i.e., it sits between some power of m {\displaystyle {\mathfrak {m}}} and m {\displaystyle {\mathfrak {m}}} ). Let F ( t ) {\displaystyle F(t)} be the Poincaré series of the associated ... | Wikipedia - Dimension theory (algebra) | null | null | null |
By the Hilbert–Serre theorem, F is a rational function with exactly one pole at t = 1 {\displaystyle t=1} of order d ≤ s {\displaystyle d\leq s} . Since we find that the coefficient of t n {\displaystyle t^{n}} in F ( t ) = ( 1 − t ) d F ( t ) ( 1 − t ) − d {\displaystyle F(t)=(1-t)^{d}F(t)(1-t)^{-d}} is of the form Th... | Wikipedia - Dimension theory (algebra) | null | null | null |
We set d ( R ) = d {\displaystyle d(R)=d} . We also set δ ( R ) {\displaystyle \delta (R)} to be the minimum number of elements of R that can generate an m {\displaystyle {\mathfrak {m}}} -primary ideal of R. Our ambition is to prove the fundamental theorem: Since we can take s to be δ ( R ) {\displaystyle \delta (R)} ... | Wikipedia - Dimension theory (algebra) | null | null | null |
Let p 0 ⊊ ⋯ ⊊ p m {\displaystyle {\mathfrak {p}}_{0}\subsetneq \cdots \subsetneq {\mathfrak {p}}_{m}} be a chain of prime ideals in R. Let D = R / p 0 {\displaystyle D=R/{\mathfrak {p}}_{0}} and x a nonzero nonunit element in D. Since x is not a zero-divisor, we have the exact sequence The degree bound of the Hilbert-S... | Wikipedia - Dimension theory (algebra) | null | null | null |
It now remains to show dim R ≥ δ ( R ) . {\displaystyle \dim R\geq \delta (R).} | Wikipedia - Dimension theory (algebra) | null | null | null |
More precisely, we shall show: (Notice: ( x 1 , … , x d ) {\displaystyle (x_{1},\dots ,x_{d})} is then m {\displaystyle {\mathfrak {m}}} -primary.) The proof is omitted. It appears, for example, in Atiyah–MacDonald. But it can also be supplied privately; the idea is to use prime avoidance. | Wikipedia - Dimension theory (algebra) | null | null | null |
Let ( R , m ) {\displaystyle (R,{\mathfrak {m}})} be a noetherian local ring and put k = R / m {\displaystyle k=R/{\mathfrak {m}}} . Then dim R ≤ dim k m / m 2 {\displaystyle \dim R\leq \dim _{k}{\mathfrak {m}}/{\mathfrak {m}}^{2}} , since a basis of m / m 2 {\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}} lifts... | Wikipedia - Dimension theory (algebra) | null | null | null |
(Krull's principal ideal theorem) The height of the ideal generated by elements x 1 , … , x s {\displaystyle x_{1},\dots ,x_{s}} in a noetherian ring is at most s. Conversely, a prime ideal of height s is minimal over an ideal generated by s elements. (Proof: Let p {\displaystyle {\mathfrak {p}}} be a prime ideal minim... | Wikipedia - Dimension theory (algebra) | null | null | null |
The converse was shown in the course of the proof of the fundamental theorem.) Proof: Let x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} generate a m A {\displaystyle {\mathfrak {m}}_{A}} -primary ideal and y 1 , … , y m {\displaystyle y_{1},\dots ,y_{m}} be such that their images generate a m B / m A B {\displaystyl... | Wikipedia - Dimension theory (algebra) | null | null | null |
The equality is a straightforward application of the going-down property. Q.E.D. | Wikipedia - Dimension theory (algebra) | null | null | null |
Proof: If p 0 ⊊ p 1 ⊊ ⋯ ⊊ p n {\displaystyle {\mathfrak {p}}_{0}\subsetneq {\mathfrak {p}}_{1}\subsetneq \cdots \subsetneq {\mathfrak {p}}_{n}} are a chain of prime ideals in R, then p i R {\displaystyle {\mathfrak {p}}_{i}R} are a chain of prime ideals in R {\displaystyle R} while p n R {\displaystyle {\mathfrak {p... | Wikipedia - Dimension theory (algebra) | null | null | null |
Clearly, R m = R p m {\displaystyle R_{\mathfrak {m}}=R_{\mathfrak {p}}_{\mathfrak {m}}} . Since R m / p R p R m = ( R p / p R p ) m {\displaystyle R_{\mathfrak {m}}/{\mathfrak {p}}R_{\mathfrak {p}}R_{\mathfrak {m}}=(R_{\mathfrak {p}}/{\mathfrak {p}}R_{\mathfrak {p}})_{\mathfrak {m}}} is then a localization of a p... | Wikipedia - Dimension theory (algebra) | null | null | null |
Let ( X , A , μ , T ) {\displaystyle (X,{\mathcal {A}},\mu ,T)} be a measure-preserving dynamical system, with T being the time-evolution or shift operator. The system is said to be strong mixing if, for any A , B ∈ A {\displaystyle A,B\in {\mathcal {A}}} , one has lim n → ∞ μ ( A ∩ T − n B ) = μ ( A ) μ ( B ) . {\disp... | Wikipedia - Topologically mixing | null | null | null |
A dynamical system is said to be weak mixing if one has lim n → ∞ 1 n ∑ k = 0 n − 1 | μ ( A ∩ T − k B ) − μ ( A ) μ ( B ) | = 0. {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=0}^{n-1}\left|\mu (A\cap T^{-k}B)-\mu (A)\mu (B)\right|=0.} In other words, T {\displaystyle T} is strong mixing if μ ( A ∩ T − n B )... | Wikipedia - Topologically mixing | null | null | null |
Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is not true: there exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing. The Chacon system was historically the first example given of a system that is weak... | Wikipedia - Topologically mixing | null | null | null |
Let ( X , T ) {\displaystyle \textstyle (X,T)} be a topological dynamical system consisting of a compact metric space X {\displaystyle \textstyle X} and a homeomorphism T: X → X {\displaystyle \textstyle T:X\rightarrow X} . The topological dynamical system ( X , T ) {\displaystyle \textstyle (X,T)} is called minimal if... | Wikipedia - Rokhlin lemma | null | null | null |
Elon Lindenstrauss proved the following theorem:Theorem: Let ( X , T ) {\displaystyle \textstyle (X,T)} be a topological dynamical system which has an aperiodic minimal factor. Then for integer n ∈ N {\displaystyle \textstyle n\in \mathbb {N} } there is a continuous function f: X → R {\displaystyle \textstyle f\colon X... | Wikipedia - Rokhlin lemma | null | null | null |
Let ( X , c ) {\displaystyle (X,\mathbf {c} )} be a Kuratowski closure space. Then X {\displaystyle X} is a T0-space iff x ≠ y {\displaystyle x\neq y} implies c ( { x } ) ≠ c ( { y } ) {\displaystyle \mathbf {c} (\{x\})\neq \mathbf {c} (\{y\})} ; X {\displaystyle X} is a T1-space iff c ( { x } ) = { x } {\displaystyle ... | Wikipedia - Kuratowski closure axioms | null | null | null |
Let ( X , d ) {\displaystyle (X,d)} be a metric space. For each finite A ⊆ X {\displaystyle A\subseteq X} , let δ ( A ) {\displaystyle \delta (A)} denote the minimum length of a Steiner tree within X connecting elements in A. Then ( X , δ ) {\displaystyle (X,\delta )} is a diversity. | Wikipedia - Diversity (mathematics) | null | null | null |
Let ( X , d ) {\displaystyle (X,d)} be a metric space. Setting δ ( A ) = max a , b ∈ A d ( a , b ) = diam ( A ) {\displaystyle \delta (A)=\max _{a,b\in A}d(a,b)=\operatorname {diam} (A)} for all A ∈ ℘ fin ( X ) {\displaystyle A\in \wp _{\mbox{fin}}(X)} defines a diversity. | Wikipedia - Diversity (mathematics) | null | null | null |
Let ( X , δ ) {\displaystyle (X,\delta )} be a diversity. For all A ∈ ℘ fin ( X ) {\displaystyle A\in \wp _{\mbox{fin}}(X)} define δ ( k ) ( A ) = max { δ ( B ): | B | ≤ k , B ⊆ A } {\displaystyle \delta ^{(k)}(A)=\max \left\{\delta (B)\colon |B|\leq k,B\subseteq A\right\}} . Then if k ≥ 2 {\displaystyle k\geq 2} , ( X... | Wikipedia - Diversity (mathematics) | null | null | null |
Let ( X , τ ) {\displaystyle (X,\tau )} be a topological space with X ≠ ∅ . {\displaystyle X\neq \varnothing .} If F ∈ Filters ( X ) {\displaystyle {\mathcal {F}}\in \operatorname {Filters} (X)} then F {\displaystyle {\mathcal {F}}} is said to converge to a point x ∈ X {\displaystyle x\in X} in ( X , τ ) , {\displays... | Wikipedia - Convergence space | null | null | null |
The set of all x ∈ X {\displaystyle x\in X} such that F → x {\displaystyle {\mathcal {F}}\to x} in ( X , τ ) {\displaystyle (X,\tau )} is denoted by lim ( X , τ ) F , {\displaystyle \lim {}_{(X,\tau )}{\mathcal {F}},} lim X F , {\displaystyle \lim {}_{X}{\mathcal {F}},} or simply lim F , {\displaystyle \lim {\mathcal {... | Wikipedia - Convergence space | null | null | null |
{\displaystyle (X,\tau ).} Equivalently, it is defined by lim ξ τ F := lim ( X , τ ) F {\displaystyle \lim {}_{\xi _{\tau }}{\mathcal {F}}:=\lim {}_{(X,\tau )}{\mathcal {F}}} for all F ∈ Filters ( X ) . {\displaystyle {\mathcal {F}}\in \operatorname {Filters} (X).} A (pre)convergence that is induced by some topology ... | Wikipedia - Convergence space | null | null | null |
Let ( X , ω ) {\displaystyle (X,\omega )} be any symplectic manifold and μ: X → R {\displaystyle \mu :X\to \mathbb {R} } a Hamiltonian on X {\displaystyle X} . Let ϵ {\displaystyle \epsilon } be any regular value of μ {\displaystyle \mu } , so that the level set μ − 1 ( ϵ ) {\displaystyle \mu ^{-1}(\epsilon )} is a smo... | Wikipedia - Symplectic cut | null | null | null |
The symplectic cut is the pair of manifolds X ¯ μ ≤ ϵ {\displaystyle {\overline {X}}_{\mu \leq \epsilon }} and X ¯ μ ≥ ϵ {\displaystyle {\overline {X}}_{\mu \geq \epsilon }} . Sometimes it is useful to view the two halves of the symplectic cut as being joined along their shared submanifold V {\displaystyle V} to produc... | Wikipedia - Symplectic cut | null | null | null |
Let ( X 1 , Σ 1 ) {\displaystyle \left(X_{1},\Sigma _{1}\right)} and ( X 2 , Σ 2 ) {\displaystyle \left(X_{2},\Sigma _{2}\right)} be two measurable spaces. The σ-algebra for the corresponding product space X 1 × X 2 {\displaystyle X_{1}\times X_{2}} is called the product σ-algebra and is defined by Observe that { B 1 ×... | Wikipedia - Sigma algebra | null | null | null |
Let ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} be a given set of observations, where the y i ∈ R {\displaystyle y_{i}\in \mathbb {R} } and the x i {\displaystyle x_{i}} fall in some partially ordered set. For generality, each observation ( x i , y i ) {\displaystyle (x_{i},y_{... | Wikipedia - Isotonic regression | null | null | null |
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