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Like any other bijection, a global isometry has a function inverse. The inverse of a global isometry is also a global isometry. Two metric spaces X and Y are called isometric if there is a bijective isometry from X to Y. The set of bijective isometries from a metric space to itself forms a group with respect to functio... | Wikipedia - Orthonormal transformation | null | null | null |
There is also the weaker notion of path isometry or arcwise isometry: A path isometry or arcwise isometry is a map which preserves the lengths of curves; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. This term is often abridged t... | Wikipedia - Orthonormal transformation | null | null | null |
See also Euclidean group and Euclidean space § Isometries. The map x ↦ | x | {\displaystyle \ x\mapsto |x|\ } in R {\displaystyle \ \mathbb {R} \ } is a path isometry but not a (general) isometry. Note that unlike an isometry, this path isometry does not need to be injective. | Wikipedia - Orthonormal transformation | null | null | null |
Let X → S {\displaystyle X\to S} be an abelian scheme whose geometric fibers have dimension g. Let n be a positive integer that is prime to the residue field of each s in S. For n ≥ 2, a level n-structure is a set of sections σ 1 , … , σ 2 g {\displaystyle \sigma _{1},\dots ,\sigma _{2g}} such that for each geometric p... | Wikipedia - Level structure (algebraic geometry) | null | null | null |
Let X ⊂ P N {\displaystyle X\subset \mathbb {P} ^{N}} be a projective variety. There are at least two equivalent ways to define the degree of X relative to its embedding. The first way is to define it as the cardinality of the finite set # ( X ∩ H 1 ∩ ⋯ ∩ H d ) {\displaystyle \#(X\cap H_{1}\cap \cdots \cap H_{d})} wher... | Wikipedia - Projective embedding | null | null | null |
Indeed, if X is a hypersurface, then the degree of X is the degree of the homogeneous polynomial defining X. The "general positions" can be made precise, for example, by intersection theory; one requires that the intersection is proper and that the multiplicities of irreducible components are all one. The other definit... | Wikipedia - Projective embedding | null | null | null |
If mi denotes the multiplicity of an irreducible component Zi in the intersection (i.e., intersection multiplicity), then the generalization of Bézout's theorem says: ∑ 1 s m i deg Z i = ∏ 1 r deg V i . {\displaystyle \sum _{1}^{s}m_{i}\deg Z_{i}=\prod _{1}^{r}\deg V_{i}.} The intersection multiplicity mi can be de... | Wikipedia - Projective embedding | null | null | null |
In particular, if H ⊂ P N {\displaystyle H\subset \mathbb {P} ^{N}} is a hypersurface not containing X, then ∑ 1 s m i deg Z i = deg ( X ) deg ( H ) {\displaystyle \sum _{1}^{s}m_{i}\deg Z_{i}=\deg(X)\deg(H)} where Zi are the irreducible components of the scheme-theoretic intersection of X and H with multiplicity... | Wikipedia - Projective embedding | null | null | null |
Let Y {\displaystyle Y} be any n × n {\displaystyle n\times n} invertible matrix in G L ( n , R ) {\displaystyle GL(n,\mathbb {R} )} . Typically, one takes Y {\displaystyle Y} to be an approximation to F ′ ( y ) − 1 {\displaystyle F'(y)^{-1}} . Then, define the function G ( x ) = x − Y F ( x ) . | Wikipedia - Numerical certification | null | null | null |
{\displaystyle G(x)=x-YF(x).} We observe that x {\displaystyle x} is a fixed of G {\displaystyle G} if and only if x {\displaystyle x} is a root of F {\displaystyle F} . Therefore the approach above can be used to identify roots of F {\displaystyle F} . | Wikipedia - Numerical certification | null | null | null |
This approach is similar to a multivariate version of Newton's method, replacing the derivative with the fixed matrix Y {\displaystyle Y} . We observe that if J {\displaystyle J} is a compact and convex region and y ∈ J {\displaystyle y\in J} , then, for any x ∈ J {\displaystyle x\in J} , there exist c 1 , … , c n ∈ J ... | Wikipedia - Numerical certification | null | null | null |
Let G ′ ( J ) {\displaystyle G'(J)} be the Jacobian matrix of G {\displaystyle G} evaluated on J {\displaystyle J} . In other words, the entry ( G ′ ( J ) ) i j {\displaystyle (G'(J))_{ij}} consists of the image of ∂ g i ∂ x j {\displaystyle {\frac {\partial g_{i}}{\partial x_{j}}}} over J {\displaystyle J} . It then f... | Wikipedia - Numerical certification | null | null | null |
Then, allowing x {\displaystyle x} to vary in J {\displaystyle J} , it follows that the image of G {\displaystyle G} on J {\displaystyle J} satisfies the following containment: G ( J ) ⊂ G ( y ) + G ′ ( J ) ( J − y ) , {\displaystyle G(J)\subset G(y)+G'(J)(J-y),} where the calculations are, once again, computed using i... | Wikipedia - Numerical certification | null | null | null |
On the other hand, if the maximum matrix norm using the supremum norm for vectors of all matrices in I − F ′ ( J ) {\displaystyle I-F'(J)} is less than 1 {\displaystyle 1} , then G {\displaystyle G} is contractive within J {\displaystyle J} , so G {\displaystyle G} has a unique fixed point. A simpler test, when J {\dis... | Wikipedia - Numerical certification | null | null | null |
Let Z ( 2 ) := { z / n ∣ z , n ∈ Z , n is odd } {\displaystyle \mathbb {Z} _{(2)}:=\{z/n\mid z,n\in \mathbb {Z} ,\,\,n{\text{ is odd}}\}} . Then, the field of fractions of Z ( 2 ) {\displaystyle \mathbb {Z} _{(2)}} is Q {\displaystyle \mathbb {Q} } . For any nonzero element r {\displaystyle r} of Q {\displaystyle \math... | Wikipedia - Discrete valuation ring | null | null | null |
Then Z ( 2 ) {\displaystyle \mathbb {Z} _{(2)}} is the discrete valuation ring corresponding to ν. The maximal ideal of Z ( 2 ) {\displaystyle \mathbb {Z} _{(2)}} is the principal ideal generated by 2, i.e. 2 Z ( 2 ) {\displaystyle 2\mathbb {Z} _{(2)}} , and the "unique" irreducible element (up to units) is 2 (this is ... | Wikipedia - Discrete valuation ring | null | null | null |
Let Z = ( t , B ) {\displaystyle Z=(t,B)} with a d {\displaystyle d} -dimensional Brownian motion B = ( B 1 , … , B d ) {\displaystyle B=(B_{1},\dots ,B_{d})} , then we can show that every maximal solution starting in x 0 {\displaystyle x_{0}} is a flow process to the operator L = A 0 + 1 2 ∑ i = 1 d A i 2 {\displaysty... | Wikipedia - Stochastic analysis on manifolds | null | null | null |
Let Z {\displaystyle Z} be the set of integers and ' − {\displaystyle -} ' be the binary operation 'subtraction'. Then the algebraic structure ( Z , − ) {\displaystyle (Z,-)} obeys the following properties: x − x = 0 {\displaystyle x-x=0} x − 0 = x {\displaystyle x-0=x} 0 − ( x − y ) = y − x {\displaystyle 0-(x-y)=y-x} | Wikipedia - BF-algebra | null | null | null |
Let Z {\displaystyle \mathbb {Z} } be the set of integers. Hereafter, in this section, elements of Z {\displaystyle \mathbb {Z} } will be referred as rational integers to distinguish them from algebraic integers to be defined below. A complex number α {\displaystyle \alpha } is called a quadratic algebraic number if it... | Wikipedia - Gauss composition law | null | null | null |
The norm of the quadratic algebraic number α {\displaystyle \alpha } is defined as N ( α ) = ( b 2 + e 2 d ) / 4 a 2 {\displaystyle N(\alpha )=(b^{2}+e^{2}d)/4a^{2}} .Let Q {\displaystyle \mathbb {Q} } be the field of rational numbers. The smallest field containing Q {\displaystyle \mathbb {Q} } and a quadratic algebra... | Wikipedia - Gauss composition law | null | null | null |
The set of quadratic algebraic integers of radicand d {\displaystyle d} is denoted by O ( d ) {\displaystyle O({\sqrt {d}})} . This set is given by O ( d ) = { { a + b d | a , b ∈ Z } if d ≡ 2 or 3 ( m o d 4 ) { ( a + b d ) / 2 | a , b ∈ Z , a ≡ b m o d 2 ) } if d ≡ 1 ( m o d 4 ) } {\displaystyle O({\sqrt {d}})={\begin... | Wikipedia - Gauss composition law | null | null | null |
Let Z*(X) := Z be the free abelian group on the algebraic cycles of X. Then an adequate equivalence relation is a family of equivalence relations, ∼X on Z*(X), one for each smooth projective variety X, satisfying the following three conditions: (Linearity) The equivalence relation is compatible with addition of cycles.... | Wikipedia - Numerical equivalence | null | null | null |
If α {\displaystyle \alpha } ~X 0, then ( π Y ) ∗ ( β ⋅ ( α × Y ) ) {\displaystyle (\pi _{Y})_{*}(\beta \cdot (\alpha \times Y))} ~Y 0, where π Y: X × Y → Y {\displaystyle \pi _{Y}:X\times Y\to Y} is the projection.The push-forward cycle in the last axiom is often denoted β ( α ) := ( π Y ) ∗ ( β ⋅ ( α × Y ) ) {\displa... | Wikipedia - Numerical equivalence | null | null | null |
Let a ( x ) {\displaystyle a(x)} a polynomial and ω n {\displaystyle \omega _{n}} a principal n {\displaystyle n} th root of unity. We define the DFT of a ( x ) {\displaystyle a(x)} as the n {\displaystyle n} -tuple ( a ^ j ) = ( a ( ω n j ) ) {\displaystyle ({\hat {a}}_{j})=(a(\omega _{n}^{j}))} . In other words, a ^ ... | Wikipedia - Prime-factor FFT algorithm | null | null | null |
Let a composite quantum system have state space H = ⊗ k H k {\displaystyle H=\otimes _{k}H_{k}} and ρ be a density matrix acting on H. The relative entropy of entanglement of ρ is defined by D R E E ( ρ ) = min σ S ( ρ ‖ σ ) {\displaystyle \;D_{\mathrm {REE} }(\rho )=\min _{\sigma }S(\rho \|\sigma )} where the minimum ... | Wikipedia - Quantum relative entropy | null | null | null |
Let a database table exist with the following structure: For this example it is assumed that each book has only one author. A table that conforms to the relational model has a primary key which uniquely identifies a row. Two books could have the same title, but an ISBN uniquely identifies a book, so it can be used as t... | Wikipedia - Normalised data | null | null | null |
Let a demonstration be represented by a sequence, with hypotheses to the left of the turnstile and the conclusion to the right of the turnstile. Then the deduction theorem can be stated as follows: If the sequence ϕ 1 , ϕ 2 , . . . | Wikipedia - Classical propositional logic | null | null | null |
, ϕ n , χ ⊢ ψ {\displaystyle \phi _{1},\ \phi _{2},\ ...,\ \phi _{n},\ \chi \vdash \psi } has been demonstrated, then it is also possible to demonstrate the sequence ϕ 1 , ϕ 2 , . . . | Wikipedia - Classical propositional logic | null | null | null |
, ϕ n ⊢ χ → ψ {\displaystyle \phi _{1},\ \phi _{2},\ ...,\ \phi _{n}\vdash \chi \to \psi } .This deduction theorem (DT) is not itself formulated with propositional calculus: it is not a theorem of propositional calculus, but a theorem about propositional calculus. In this sense, it is a meta-theorem, comparable to theo... | Wikipedia - Classical propositional logic | null | null | null |
In this sense, DT corresponds to the natural conditional proof inference rule which is part of the first version of propositional calculus introduced in this article. The converse of DT is also valid: If the sequence ϕ 1 , ϕ 2 , . . | Wikipedia - Classical propositional logic | null | null | null |
. , ϕ n ⊢ χ → ψ {\displaystyle \phi _{1},\ \phi _{2},\ ...,\ \phi _{n}\vdash \chi \to \psi } has been demonstrated, then it is also possible to demonstrate the sequence ϕ 1 , ϕ 2 , . . | Wikipedia - Classical propositional logic | null | null | null |
. , ϕ n , χ ⊢ ψ {\displaystyle \phi _{1},\ \phi _{2},\ ...,\ \phi _{n},\ \chi \vdash \psi } in fact, the validity of the converse of DT is almost trivial compared to that of DT: If ϕ 1 , . . | Wikipedia - Classical propositional logic | null | null | null |
. , ϕ n ⊢ χ → ψ {\displaystyle \phi _{1},\ ...,\ \phi _{n}\vdash \chi \to \psi } then 1: ϕ 1 , . . | Wikipedia - Classical propositional logic | null | null | null |
. , ϕ n , χ ⊢ χ → ψ {\displaystyle \phi _{1},\ ...,\ \phi _{n},\ \chi \vdash \chi \to \psi } 2: ϕ 1 , . . | Wikipedia - Classical propositional logic | null | null | null |
. , ϕ n , χ ⊢ χ {\displaystyle \phi _{1},\ ...,\ \phi _{n},\ \chi \vdash \chi } and from (1) and (2) can be deduced 3: ϕ 1 , . . | Wikipedia - Classical propositional logic | null | null | null |
. , ϕ n , χ ⊢ ψ {\displaystyle \phi _{1},\ ...,\ \phi _{n},\ \chi \vdash \psi } by means of modus ponens, Q.E.D.The converse of DT has powerful implications: it can be used to convert an axiom into an inference rule. For example, by axiom AND-1 we have, ⊢ ϕ ∧ χ → ϕ , {\displaystyle \vdash \phi \wedge \chi \to \phi ,} w... | Wikipedia - Classical propositional logic | null | null | null |
Let a lattice basis b 1 , b 2 , b 3 ∈ Z 3 {\displaystyle \mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\in \mathbf {Z} ^{3}} , be given by the columns of then the reduced basis is which is size-reduced, satisfies the Lovász condition, and is hence LLL-reduced, as described above. See W. Bosma. for details of the re... | Wikipedia - Lenstra–Lenstra–Lovász lattice basis reduction algorithm | null | null | null |
Let a linear program be given by a canonical tableau. The simplex algorithm proceeds by performing successive pivot operations each of which give an improved basic feasible solution; the choice of pivot element at each step is largely determined by the requirement that this pivot improves the solution. | Wikipedia - Simplex method | null | null | null |
Let a material have a yield strength σ Y {\displaystyle \sigma _{Y}} and a fracture toughness in mode I K I c {\displaystyle K_{Ic}} . Based on fracture mechanics, the material will fail at stress σ fail = K I c / π a {\displaystyle \sigma _{\text{fail}}=K_{Ic}/{\sqrt {\pi a}}} . Based on plasticity, the material will ... | Wikipedia - Crack propagation | null | null | null |
This value of a {\displaystyle a} is called as transition flaw size a t {\displaystyle a_{t}} ., and depends on the material properties of the structure. When the a < a t {\displaystyle a a t {\displaystyle a>a_{t}} the failure is governed by fracture mechanics. The value of a t {\displaystyle a_{t}} for engineering al... | Wikipedia - Crack propagation | null | null | null |
Let a polynomial have degree n {\displaystyle n} . We derive the algorithm's complexity as follows: Due to the binomial theorem ( x − z ) k = ∑ i = 0 k ( k i ) ( − z ) k − i x i {\textstyle (x-z)^{k}=\sum \limits _{i=0}^{k}{\binom {k}{i}}(-z)^{k-i}x^{i}} , we may transition from f ( x ) {\displaystyle f(x)} to f ( x − ... | Wikipedia - Berlekamp–Rabin algorithm | null | null | null |
Taking the gcd {\displaystyle \gcd } of two polynomials via Euclidean algorithm works in O ( n 2 ) {\displaystyle O(n^{2})} .Thus the whole procedure may be done in O ( n 2 log p ) {\displaystyle O(n^{2}\log p)} . Using the fast Fourier transform and Half-GCD algorithm, the algorithm's complexity may be improved to O... | Wikipedia - Berlekamp–Rabin algorithm | null | null | null |
Let a sample of size n of the simultaneously distributed variables A {\displaystyle A} and B {\displaystyle B} for i = 1 , … , r ; j = 1 , … , k {\displaystyle i=1,\ldots ,r;j=1,\ldots ,k} be given by the frequencies n i j = {\displaystyle n_{ij}=} number of times the values ( A i , B j ) {\displaystyle (A_{i},B_{j})} ... | Wikipedia - Cramér's V (statistics) | null | null | null |
j n , {\displaystyle \chi ^{2}=\sum _{i,j}{\frac {(n_{ij}-{\frac {n_{i. }n_{.j}}{n}})^{2}}{\frac {n_{i. }n_{.j}}{n}}}\;,} where n i . | Wikipedia - Cramér's V (statistics) | null | null | null |
= ∑ j n i j {\displaystyle n_{i. }=\sum _{j}n_{ij}} is the number of times the value A i {\displaystyle A_{i}} is observed and n . j = ∑ i n i j {\displaystyle n_{.j}=\sum _{i}n_{ij}} is the number of times the value B j {\displaystyle B_{j}} is observed. | Wikipedia - Cramér's V (statistics) | null | null | null |
Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the minimum dimension minus 1: V = φ 2 min ( k − 1 , r − 1 ) = χ 2 / n min ( k − 1 , r − 1 ) , {\displaystyle V={\sqrt {\frac {\varphi ^{2}}{\min(k-1,r-1)}}}={\sqrt {\frac {\chi ^{2}/n}{\min(k-1,r-1)}}}\;,} wher... | Wikipedia - Cramér's V (statistics) | null | null | null |
Let a screw be an ordered pair S = ( S , V ) , {\displaystyle {\mathsf {S}}=(\mathbf {S} ,\mathbf {V} ),} where S and V are three-dimensional real vectors. The sum and difference of these ordered pairs are computed componentwise. Screws are often called dual vectors. Now, introduce the ordered pair of real numbers â = ... | Wikipedia - Screw theory | null | null | null |
Let the addition and subtraction of these numbers be componentwise, and define multiplication as The multiplication of a screw S = (S, V) by the dual scalar â = (a, b) is computed componentwise to be, Finally, introduce the dot and cross products of screws by the formulas: which is a dual scalar, and which is a screw. ... | Wikipedia - Screw theory | null | null | null |
Let a: I → X {\displaystyle a:I\to X} be a map, also denoted by ( a i ) i ∈ I , {\displaystyle \left(a_{i}\right)_{i\in I},} from some non-empty set I {\displaystyle I} into a Hausdorff abelian topological group X . {\displaystyle X.} Let Finite ( I ) {\displaystyle \operatorname {Finite} (I)} be the collection of al... | Wikipedia - Series (mathematics) | null | null | null |
{\displaystyle V-V\subseteq W.} It follows that the finite partial sums of an unconditionally summable family ( a i ) i ∈ I , {\displaystyle \left(a_{i}\right)_{i\in I},} form a Cauchy net, that is, for every neighborhood W {\displaystyle W} of the origin in X , {\displaystyle X,} there exists a finite subset A 0 {\dis... | Wikipedia - Series (mathematics) | null | null | null |
When X {\displaystyle X} is complete and ( a i ) i ∈ I , {\displaystyle \left(a_{i}\right)_{i\in I},} is unconditionally summable in X , {\displaystyle X,} then for every subset J ⊆ I , {\displaystyle J\subseteq I,} the corresponding subfamily ( a j ) j ∈ J , {\displaystyle \left(a_{j}\right)_{j\in J},} is also uncondi... | Wikipedia - Series (mathematics) | null | null | null |
{\displaystyle X=\mathbb {R} .} If a family ( a i ) i ∈ I {\displaystyle \left(a_{i}\right)_{i\in I}} in X {\displaystyle X} is unconditionally summable then for every neighborhood W {\displaystyle W} of the origin in X , {\displaystyle X,} there is a finite subset A 0 ⊆ I {\displaystyle A_{0}\subseteq I} such that a i... | Wikipedia - Series (mathematics) | null | null | null |
{\displaystyle A_{0}.} If X {\displaystyle X} is a first-countable space then it follows that the set of i ∈ I {\displaystyle i\in I} such that a i ≠ 0 {\displaystyle a_{i}\neq 0} is countable. This need not be true in a general abelian topological group (see examples below). | Wikipedia - Series (mathematics) | null | null | null |
Let again ∑ k = 0 r p k ( n ) y ( n + k ) = f ( n ) {\textstyle \sum _{k=0}^{r}p_{k}(n)\,y(n+k)=f(n)} be a recurrence equation with polynomial coefficients and u ( n ) {\textstyle u(n)} a universal denominator. After substituting y ( n ) = z ( n ) / u ( n ) {\textstyle y(n)=z(n)/u(n)} for an unknown polynomial z ( n ) ... | Wikipedia - Abramov's algorithm | null | null | null |
Let c {\displaystyle c} be the ciphertext to decrypt, where c ∈ Z n 2 ∗ {\displaystyle c\in \mathbb {Z} _{n^{2}}^{*}} Compute the plaintext message as: m = L ( c λ mod n 2 ) ⋅ μ mod n {\displaystyle m=L(c^{\lambda }{\bmod {n}}^{2})\cdot \mu {\bmod {n}}} As the original paper points out, decryption is "essentially one e... | Wikipedia - Paillier cryptosystem | null | null | null |
Let c: → R n {\displaystyle c:\to \mathbb {R} ^{n}} be a differentiable curve. Then the tangent vector to the curve c {\displaystyle c} at t {\displaystyle t} is a vector v {\displaystyle v} at the point c ( t ) {\displaystyle c(t)} whose components are given as: v = ( c 1 ′ ( t ) , … , c n ′ ( t ) ) {\displaystyle v=... | Wikipedia - Calculus on Euclidean space | null | null | null |
Let d1 be the sum of edges in the horizontal direction, and d2 be the sum of edges in the vertical direction. Consider a 7×7 diamond-shaped region centered on the pixel to calculate, using only pixel values from the original, and pixel values added from the diagonal direction. To calculate d1, take the sum of the absol... | Wikipedia - Directional Cubic Convolution Interpolation | null | null | null |
Let d1 be the sum of edges in the up-right direction, and d2 be the sum of edges in the down-right direction. To calculate d1, take the sum of abs(P(X, Y) - P(X - 1, Y + 1)), in the region of X = 1 to 3, and Y = 0 to 2. To calculate d2, take the sum of abs(P(X, Y) - P(X + 1, Y + 1)), in the region of X = 0 to 2, and Y ... | Wikipedia - Directional Cubic Convolution Interpolation | null | null | null |
Let dE(A,B) denote the pseudolength for a given hyperbolic line segment AB. Let a transformation move A to the center of a Poincaré disk with a radius equalling 1. The pseudolength dE(A,B) is the length of this segment in Euclidean geometry. | Wikipedia - Constructions in hyperbolic geometry | null | null | null |
Let demand, D {\displaystyle D} , follow a lognormal distribution with a mean demand of 50, μ {\displaystyle \mu } , and a standard deviation, σ {\displaystyle \sigma } , of 0.2. q opt = F − 1 ( 7 − 5 7 ) = μ e Z − 1 ( 0.285 ) σ = 50 e ( 0.2 ⋅ ( − 0.56595 ) ) = 44.64 ≈ 45. {\displaystyle q_{\text{opt}}=F^{-1}\left({\fr... | Wikipedia - Newsvendor model | null | null | null |
Let demand, D {\displaystyle D} , follow a normal distribution with a mean, μ {\displaystyle \mu } , demand of 50 and a standard deviation, σ {\displaystyle \sigma } , of 20. q opt = F − 1 ( 7 − 5 7 ) = μ + σ Z − 1 ( 0.285 ) = 50 + 20 ( − 0.56595 ) = 38.68 ≈ 39. {\displaystyle q_{\text{opt}}=F^{-1}\left({\frac {7-5}{7}... | Wikipedia - Newsvendor model | null | null | null |
Let demand, D {\displaystyle D} , follow a uniform distribution (continuous) between D min = 50 {\displaystyle D_{\min }=50} and D max = 80 {\displaystyle D_{\max }=80} . q opt = F − 1 ( 7 − 5 7 ) = F − 1 ( 0.285 ) = D min + ( D max − D min ) ⋅ 0.285 = 58.55 ≈ 59. {\displaystyle q_{\text{opt}}=F^{-1}\left({\frac {7-5}{... | Wikipedia - Newsvendor model | null | null | null |
Let e 0 , e 1 , … , e n {\displaystyle e_{0},e_{1},\ldots ,e_{n}} be a basis of V. If a point p in the projective space has homogeneous coordinates ( X 0 , … , X n ) {\displaystyle (X_{0},\ldots ,X_{n})} with respect to this basis, it is on the Hermitian variety if and only if: ∑ i , j = 0 n a i j X i X j θ = 0 {\displ... | Wikipedia - Unital (geometry) | null | null | null |
Since the basis vectors are all constant, vector addition and subtraction will simply be familiar component-wise adding and subtraction. Now, let a = ∑ i a i e i and b = ∑ i b i e i {\displaystyle \mathbf {a} =\sum _{i}a^{i}\mathbf {e} _{i}\quad {\mbox{and}}\quad \mathbf {b} =\sum _{i}b^{i}\mathbf {e} _{i}} where the s... | Wikipedia - Skew coordinates | null | null | null |
{\displaystyle a^{3}={\frac {-\sin(\phi )a_{1}+a_{3}}{\cos ^{2}(\phi )}}.} The dot product in terms of contravariant components is then a ⋅ b = ∑ i a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 + sin ( ϕ ) ( a 1 b 3 + a 3 b 1 ) {\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i}a^{i}b_{i}=a^{1}b^{1}+a^{2}b^{2}+a^{3}b^{3}... | Wikipedia - Skew coordinates | null | null | null |
Let each curve Ct in the family be given as the solution of an equation ft(x, y)=0 (see implicit curve), where t is a parameter. Write F(t, x, y)=ft(x, y) and assume F is differentiable. The envelope of the family Ct is then defined as the set D {\displaystyle {\mathcal {D}}} of points (x,y) for which, simultaneously, ... | Wikipedia - Envelope (mathematics) | null | null | null |
Letting u → t gives the definition above. An important special case is when F(t, x, y) is a polynomial in t. This includes, by clearing denominators, the case where F(t, x, y) is a rational function in t. In this case, the definition amounts to t being a double root of F(t, x, y), so the equation of the envelope can be... | Wikipedia - Envelope (mathematics) | null | null | null |
The equation of Ct is x t + y 11 − t = 1 {\displaystyle {\frac {x}{t}}+{\frac {y}{11-t}}=1} or, clearing fractions, x ( 11 − t ) + y t − t ( 11 − t ) = t 2 + ( − x + y − 11 ) t + 11 x = 0. {\displaystyle x(11-t)+yt-t(11-t)=t^{2}+(-x+y-11)t+11x=0.\,} The equation of the envelope is then ( − x + y − 11 ) 2 − 44 x = ( x −... | Wikipedia - Envelope (mathematics) | null | null | null |
For example, if the family is given by Cθ with an equation of the form u(x, y)cos θ+v(x, y)sin θ=w(x, y), then putting t=eiθ, cos θ=(t+1/t)/2, sin θ=(t-1/t)/2i changes the equation of the curve to u 1 2 ( t + 1 t ) + v 1 2 i ( t − 1 t ) = w {\displaystyle u{1 \over 2}(t+{1 \over t})+v{1 \over 2i}(t-{1 \over t})=w} or (... | Wikipedia - Envelope (mathematics) | null | null | null |
Let expressions may be defined with multiple variables, ( ∃ v ⋯ ∃ w ∃ x E ∧ F ) ⟺ let v , … , w , x: E in F {\displaystyle (\exists v\cdots \exists w\exists xE\land F)\iff \operatorname {let} v,\ldots ,w,x:E\operatorname {in} F} then it can be derived, x ∉ F V ( E ) ⟹ ( ∃ v ⋯ ∃ w ∃ x E ∧ F ) ⟺ ( ∃ v ⋯ ∃ w ( E ∧ ∃ x... | Wikipedia - Let expression | null | null | null |
Let f ( t ) = t d − c d − 1 t d − 1 − . . . − c 0 {\displaystyle f(t)=t^{d}-c_{d-1}t^{d-1}-...-c_{0}} be a monic polynomial of degree d {\displaystyle d} with coefficients in Z {\displaystyle \mathbb {Z} } and suppose that p = f ( 2 m ) {\displaystyle p=f(2^{m})} is a Solinas prime. Given a number n < p 2 {\displaystyl... | Wikipedia - Solinas prime | null | null | null |
Let f ( x ) = ( x − λ 1 ) ( x − λ 2 ) ⋯ ( x − λ n ) {\textstyle f(x)=(x-\lambda _{1})(x-\lambda _{2})\cdots (x-\lambda _{n})} . Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivia... | Wikipedia - Berlekamp–Rabin algorithm | null | null | null |
Let f ( x ) {\displaystyle f(x)} be a global optimization problem, where x {\displaystyle x} is a state in the problem space S {\displaystyle S} . In SCO, each state is called a knowledge point, and the function f {\displaystyle f} is the goodness function. In SCO, there are a population of N c {\displaystyle N_{c}} co... | Wikipedia - Social cognitive optimization | null | null | null |
The algorithm runs in T iterative learning cycles. By running as a Markov chain process, the system behavior in the tth cycle only depends on the system status in the (t − 1)th cycle. The process flow is in follows: ：Initialize the private knowledge point x i {\displaystyle x_{i}} in the memory of each agent i {\displa... | Wikipedia - Social cognitive optimization | null | null | null |
： At each cycle t {\displaystyle t} ( t = 1 , … , T ) {\displaystyle (t=1,\ldots ,T)} ： For each agent i {\displaystyle i} ( i = 1 , … , N c ) {\displaystyle (i=1,\ldots ,N_{c})} ： ：Find a high-quality model point x M {\displaystyle x_{M}} in X ( t ) {\displaystyle X(t)} , normally realized using tournament selection,... | Wikipedia - Social cognitive optimization | null | null | null |
：Share a knowledge point, normally x i ( t + 1 ) {\displaystyle x_{i}(t+1)} , to the social sharing library X {\displaystyle X} . ：Update the private knowledge of agent i {\displaystyle i} , normally replace x i ( t ) {\displaystyle x_{i}(t)} by x i ( t + 1 ) {\displaystyle x_{i}(t+1)} . Some Monte Carlo types might al... | Wikipedia - Social cognitive optimization | null | null | null |
：The social sharing library using all knowledge points submitted by agents to update X ( t ) {\displaystyle X(t)} into X ( t + 1 ) {\displaystyle X(t+1)} . A simple way is one by one tournament selection: for each knowledge point submitted by an agent, replace the worse one among τ W {\displaystyle \tau _{W}} points ra... | Wikipedia - Social cognitive optimization | null | null | null |
With the initialization process, the total number of knowledge points to be generated is N L + N c ∗ ( T + 1 ) {\displaystyle N_{L}+N_{c}*(T+1)} , and is not related too much with N L {\displaystyle N_{L}} if T {\displaystyle T} is large. Compared to traditional swarm algorithms, e.g. particle swarm optimization, SCO c... | Wikipedia - Social cognitive optimization | null | null | null |
Some variants were proposed to guaranteed the global convergence. One can also make a hybrid optimization method using SCO combined with other optimizers. For example, SCO was hybridized with differential evolution to obtain better results than individual algorithms on a common set of benchmark problems. == References ... | Wikipedia - Social cognitive optimization | null | null | null |
Let f ( x ) {\displaystyle f(x)} be a unimodal function on some interval {\displaystyle } . Take any two points m 1 {\displaystyle m_{1}} and m 2 {\displaystyle m_{2}} in this segment: l < m 1 < m 2 < r {\displaystyle l f ( m 2 ) {\displaystyle f(m_{1})>f(m_{2})} , that the situation is similar to the previous, up to ... | Wikipedia - Ternary search | null | null | null |
Let f ( x , t ) {\displaystyle f(\mathbf {x} ,t)} be a physical quantity that is flowing through the body. Let g ( x , t ) {\displaystyle g(\mathbf {x} ,t)} be sources on the surface of the body and let h ( x , t ) {\displaystyle h(\mathbf {x} ,t)} be sources inside the body. Let n ( x , t ) {\displaystyle \mathbf {n} ... | Wikipedia - Deformable body | null | null | null |
Also, let the speed at which the bounding surface ∂ Ω {\displaystyle \partial \Omega } is moving be u n {\displaystyle u_{n}} (in the direction n {\displaystyle \mathbf {n} } ). Then, balance laws can be expressed in the general form d d t = ∫ ∂ Ω f ( x , t ) dA + ∫ ∂ Ω g ( x , t ) dA + ∫ Ω h ( x , t ) dV . {\display... | Wikipedia - Deformable body | null | null | null |
The functions f ( x , t ) {\displaystyle f(\mathbf {x} ,t)} , g ( x , t ) {\displaystyle g(\mathbf {x} ,t)} , and h ( x , t ) {\displaystyle h(\mathbf {x} ,t)} can be scalar valued, vector valued, or tensor valued - depending on the physical quantity that the balance equation deals with. If there are internal boundarie... | Wikipedia - Deformable body | null | null | null |
With respect to the reference configuration (the Lagrangian point of view), the balance laws can be written as ρ det ( F ) − ρ 0 = 0 Balance of Mass ρ 0 x ¨ − ∇ ∘ ⋅ P T − ρ 0 b = 0 Balance of Linear Momentum F ⋅ P T = P ⋅ F T Balance of Angular Momentum ρ 0 e ˙ − P T: F ˙ + ∇ ∘ ⋅ q − ρ 0 s = 0 Balance of Energy. {\disp... | Wikipedia - Deformable body | null | null | null |
Let f 1 , … , f 2 n: { 0 , 1 } k → { 0 , 1 } {\displaystyle f_{1},\dots ,f_{2^{n}}:\{0,1\}^{k}\to \{0,1\}} for k = log n {\displaystyle k=\log n} be the list of all functions from { 0 , 1 } k → { 0 , 1 } {\displaystyle \{0,1\}^{k}\to \{0,1\}} . Then the long code encoding of a message x ∈ { 0 , 1 } k {\displaystyle x... | Wikipedia - Long code (mathematics) | null | null | null |
Since there are only 2 k {\displaystyle 2^{k}} such functions, the block length of the Walsh-Hadamard code is 2 k {\displaystyle 2^{k}} . An equivalent definition of the long code is as follows: The Long code encoding of j ∈ {\displaystyle j\in } is defined to be the truth table of the Boolean dictatorship function on... | Wikipedia - Long code (mathematics) | null | null | null |
Let f 1 , … , f n {\displaystyle f_{1},\dots ,f_{n}} be any system of variables of A {\displaystyle A} ; that is, A = k {\displaystyle A=k} . A derivation of A {\displaystyle A} is called triangular with respect to this system of variables, if ∂ f 1 ∈ k {\displaystyle \partial f_{1}\in k} and ∂ f i ∈ k {\displaystyle... | Wikipedia - Locally nilpotent derivation | null | null | null |
The converse is true for ≤ 2 {\displaystyle \leq 2} by Rentschler's theorem above, but it is not true for n ≥ 3 {\displaystyle n\geq 3} . Bass's exampleThe derivation of k {\displaystyle k} given by x 1 ∂ ∂ x 2 + 2 x 2 x 1 ∂ ∂ x 3 {\displaystyle x_{1}{\tfrac {\partial }{\partial x_{2}}}+2x_{2}x_{1}{\tfrac {\partial }{... | Wikipedia - Locally nilpotent derivation | null | null | null |
Let f = x 11 + 2 x 9 + 2 x 8 + x 6 + x 5 + 2 x 3 + 2 x 2 + 1 ∈ F 3 , {\displaystyle f=x^{11}+2x^{9}+2x^{8}+x^{6}+x^{5}+2x^{3}+2x^{2}+1\in \mathbf {F} _{3},} to be factored over the field with three elements. The algorithm computes first c = gcd ( f , f ′ ) = x 9 + 2 x 6 + x 3 + 2. {\displaystyle c=\gcd(f,f')=x^{9}+2x^... | Wikipedia - Polynomial factorization over finite fields | null | null | null |
After one loop we have y = x + 2, z = x + 1 and R = x + 1 with updates i = 2, w = x + 2 and c = x8 + x7 + x6 + x2+x+1. The second time through the loop gives y = x + 2, z = 1, R = x + 1, with updates i = 3, w = x + 2 and c = x7 + 2x6 + x + 2. The third time through the loop also does not change R. For the fourth time t... | Wikipedia - Polynomial factorization over finite fields | null | null | null |
Since w = 1, we exit the while loop. Since c ≠ 1, it must be a perfect cube. | Wikipedia - Polynomial factorization over finite fields | null | null | null |
The cube root of c, obtained by replacing x3 by x is x2 + 1, and calling the square-free procedure recursively determines that it is square-free. Therefore, cubing it and combining it with the value of R to that point gives the square-free decomposition f = ( x + 1 ) ( x 2 + 1 ) 3 ( x + 2 ) 4 . {\displaystyle f=(x+1)(x... | Wikipedia - Polynomial factorization over finite fields | null | null | null |
Let f and g be continuous, real-valued functions on C4m and C4n, respectively, σ1, σ2, ..., σm be ultraweakly continuous, linear functionals on a von Neumann algebra R acting on the Hilbert space H, and ρ1, ρ2, ..., ρn be bounded linear functionals on R such that, for each a in R, f ( σ 1 ( a ) , σ 1 ( a ∗ ) , σ 1 ( a ... | Wikipedia - Universal representation (C*-algebra) | null | null | null |
Let f be a continuous real-valued function defined on a closed interval . Let F be the function defined, for all x in , by F ( x ) = ∫ a x f ( t ) d t . {\displaystyle F(x)=\int _{a}^{x}f(t)\,dt.} Then, F is continuous on , differentiable on the open interval (a, b), and F ′ ( x ) = f ( x ) {\displaystyle F'(x)=f(x)} f... | Wikipedia - Integrable function | null | null | null |
Let f be a real-valued function defined on a closed interval that admits an antiderivative F on . That is, f and F are functions such that for all x in , f ( x ) = F ′ ( x ) . {\displaystyle f(x)=F'(x).} If f is integrable on then ∫ a b f ( x ) d x = F ( b ) − F ( a ) . {\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).... | Wikipedia - Linearity of integration | null | null | null |
Let f {\displaystyle f} and g {\displaystyle g} be arbitrary functions. The composition of f {\displaystyle f} and g {\displaystyle g} , g ∘ f {\displaystyle g\circ f} , is defined as the relative product f | g {\displaystyle f\,|\,g} , but only if this results in a function such that g ∘ f {\displaystyle g\circ f} is ... | Wikipedia - Mathematical formalization | null | null | null |
Let f {\displaystyle f} be a function satisfying f ( x ) = x r {\displaystyle f(x)=x^{r}} for all x {\displaystyle x} , where r ∈ R {\displaystyle r\in \mathbb {R} } . Then, f ′ ( x ) = r x r − 1 . {\displaystyle f'(x)=rx^{r-1}\,.} | Wikipedia - Power rule | null | null | null |
The power rule for integration states that ∫ x r d x = x r + 1 r + 1 + C {\displaystyle \int \!x^{r}\,dx={\frac {x^{r+1}}{r+1}}+C} for any real number r ≠ − 1 {\displaystyle r\neq -1} . It can be derived by inverting the power rule for differentiation. In this equation C is any constant. | Wikipedia - Power rule | null | null | null |
Let f {\displaystyle f} have a bounded first derivative over , {\displaystyle ,} i.e. f ∈ C 1 ( ) . {\displaystyle f\in C^{1}().} The mean value theorem for f , {\displaystyle f,} where x ∈ {\displaystyle \xi _{x}\in (a,x]} depending on x {\displaystyle x} . If we integrate in x {\displaystyle x} from a {\displaysty... | Wikipedia - Numerical quadrature | null | null | null |
Hence, if we approximate the integral ∫ a b f ( x ) d x {\textstyle \int _{a}^{b}f(x)\,dx} by the quadrature rule ( b − a ) f ( a ) {\displaystyle (b-a)f(a)} our error is no greater than the right hand side of 1. We can convert this into an error analysis for the Riemann sum, giving an upper bound of for the error term... | Wikipedia - Numerical quadrature | null | null | null |
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