<P> ⟨ ⟩ (\ displaystyle \ langle \ rangle) is the empty tuple (or 0 - tuple). </P> <Tr> <Td> ⟨ ⟩ () </Td> <Td> ⟨ ⟩ (\ displaystyle \ langle \ \ \ rangle \! \,) \ langle \ \ \ rangle \! \, () (\ displaystyle (\ \) \! \,) (\ \) \! \, </Td> <Td> inner product inner product of linear algebra </Td> <Td> ⟨ u v ⟩ means the inner product of u and v, where u and v are members of an inner product space . (u v) means the same . Another variant of the notation is ⟨ u, v ⟩ which is described above . For spatial vectors, the dot product notation, x y is common . For matrices, the colon notation A: B may be used . As ⟨ and ⟩ can be hard to type, the more "keyboard friendly" forms <and> are sometimes seen . These are avoided in mathematical texts . </Td> <Td> </Td> </Tr> <Table> <Tr> <Th> Symbol in HTML </Th> <Th> Symbol in TeX </Th> <Th> Name </Th> <Th> Explanation </Th> <Th> Examples </Th> </Tr> <Tr> <Th> Read as </Th> </Tr> <Tr> <Th> Category </Th> </Tr> <Tr> <Td> * </Td> <Td> ∗ (\ displaystyle * \! \,) </Td> <Td> convolution convolution; convolved with functional analysis </Td> <Td> f ∗ g means the convolution of f and g . </Td> <Td> (f ∗ g) (t) = ∫ 0 t f (τ) g (t − τ) d τ (\ displaystyle (f * g) (t) = \ int _ (0) ^ (t) f (\ tau) g (t - \ tau) \, d \ tau). </Td> </Tr> <Tr> <Td> complex conjugate conjugate complex numbers </Td> <Td> z means the complex conjugate of z . (z _̄ (\ displaystyle (\ bar (z))) can also be used for the conjugate of z, as described below .) </Td> <Td> (3 + 4 i) ∗ = 3 − 4 i (\ displaystyle (3 + 4i) ^ (\ ast) = 3 - 4i). </Td> </Tr> <Tr> <Td> group of units the group of units of ring theory </Td> <Td> R consists of the set of units of the ring R, along with the operation of multiplication . This may also be written R as described above, or U (R). </Td> <Td> (Z / 5 Z) ∗ = ((1), (2), (3), (4)) ≅ C 4 (\ displaystyle (\ begin (aligned) (\ mathbb (Z) / 5 \ mathbb (Z)) ^ (\ ast) & = \ ((1), (2), (3), (4) \) \ \ & \ cong \ mathrm (C) _ (4) \ \ \ end (aligned))) </Td> </Tr> <Tr> <Td> hyperreal numbers the (set of) hyperreals non-standard analysis </Td> <Td> R means the set of hyperreal numbers . Other sets can be used in place of R . </Td> <Td> N is the hypernatural numbers . </Td> </Tr> <Tr> <Td> Hodge dual Hodge dual; Hodge star linear algebra </Td> <Td> ∗ v means the Hodge dual of a vector v. If v is a k - vector within an n - dimensional oriented inner product space, then ∗ v is an (n − k) - vector . </Td> <Td> If (e i) (\ displaystyle \ (e_ (i) \)) are the standard basis vectors of R 5 (\ displaystyle \ mathbb (R) ^ (5)), ∗ (e 1 ∧ e 2 ∧ e 3) = e 4 ∧ e 5 (\ displaystyle * (e_ (1) \ wedge e_ (2) \ wedge e_ (3)) = e_ (4) \ wedge e_ (5)) </Td> </Tr> <Tr> <Td> Kleene star Kleene star computer science, mathematical logic </Td> <Td> Corresponds to the usage of * in regular expressions . If ∑ is a set of strings, then ∑ * is the set of all strings that can be created by concatenating members of ∑ . The same string can be used multiple times, and the empty string is also a member of ∑ * . </Td> <Td> If ∑ = (' a',' b',' c') then ∑ * includes' ',' a',' ab',' aba',' abac', etc . The full set cannot be enumerated here since it is countably infinite, but each individual string must have finite length . </Td> </Tr> <Tr> <Td> ∝ </Td> <Td> ∝ (\ displaystyle \ propto \! \,) \ propto \! \, </Td> <Td> proportionality is proportional to; varies as everywhere </Td> <Td> y ∝ x means that y = kx for some constant k . </Td> <Td> if y = 2x, then y ∝ x . </Td> </Tr> <Tr> <Td> Karp reduction is Karp reducible to; is polynomial - time many - one reducible to computational complexity theory </Td> <Td> A ∝ B means the problem A can be polynomially reduced to the problem B . </Td> <Td> If L ∝ L and L ∈ P, then L ∈ P . </Td> </Tr> <Tr> <Td> ∖ </Td> <Td> ∖ (\ displaystyle \ setminus \! \,) \ setminus </Td> <Td> set - theoretic complement minus; without; throw out; not set theory </Td> <Td> A ∖ B means the set that contains all those elements of A that are not in B . (− can also be used for set - theoretic complement as described above .) </Td> <Td> (1, 2, 3, 4) ∖ (3, 4, 5, 6) = (1, 2) </Td> </Tr> <Tr> <Td> </Td> <Td> (\ displaystyle \! \,) </Td> <Td> conditional event given probability </Td> <Td> P (A B) means the probability of the event A occurring given that B occurs . </Td> <Td> if X is a uniformly random day of the year P (X is May 25 X is in May) = 1 / 31 </Td> </Tr> <Tr> <Td> restriction restriction of...to ...; restricted to set theory </Td> <Td> f means the function f is restricted to the set A, that is, it is the function with domain A ∩ dom (f) that agrees with f . </Td> <Td> The function f: R → R defined by f (x) = x is not injective, but f is injective . </Td> </Tr> <Tr> <Td> such that such that; so that everywhere </Td> <Td> means "such that", see ":" (described below). </Td> <Td> S = ((x, y) 0 <y <f (x)) The set of (x, y) such that y is greater than 0 and less than f (x). </Td> </Tr> <Tr> <Td> ∣ ∤ </Td> <Td> ∣ (\ displaystyle \ mid \! \,) \ mid ∤ (\ displaystyle \ nmid \! \,) \ nmid </Td> <Td> divisor, divides divides number theory </Td> <Td> a ∣ b means a divides b . a ∤ b means a does not divide b . (The symbol ∣ can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar character is often used instead .) </Td> <Td> Since 15 = 3 × 5, it is true that 3 ∣ 15 and 5 ∣ 15 . </Td> </Tr> <Tr> <Td> ∣ ∣ </Td> <Td> ∣ ∣ (\ displaystyle \ mid \ mid \! \,) \ mid \ mid </Td> <Td> exact divisibility exactly divides number theory </Td> <Td> p ∣ ∣ n means p exactly divides n (i.e. p divides n but p does not). </Td> <Td> 2 ∣ ∣ 360 . </Td> </Tr> <Tr> <Td> ∥ ∦ ⋕ </Td> <Td> ‖ (\ displaystyle \ \! \,) \ Requires the viewer to support Unicode: \ unicode (x2225), \ unicode (x2226), and \ unicode (x22D5). \ mathrel (\ rlap (\, \ parallel)) requires \ setmathfont (MathJax). </Td> <Td> parallel is parallel to geometry </Td> <Td> x ∥ y means x is parallel to y . x ∦ y means x is not parallel to y . x ⋕ y means x is equal and parallel to y . (The symbol ∥ can be difficult to type, and its negation is rare, so two regular but slightly longer vertical bar characters are often used instead .) </Td> <Td> If l ∥ m and m ⊥ n then l ⊥ n . </Td> </Tr> <Tr> <Td> incomparability is incomparable to order theory </Td> <Td> x ∥ y means x is incomparable to y . </Td> <Td> (1, 2) ∥ (2, 3) under set containment . </Td> </Tr> <Tr> <Td> #</Td> <Td> #(\ displaystyle \ #\! \,) \ sharp </Td> <Td> cardinality cardinality of; size of; order of set theory </Td> <Td> #X means the cardinality of the set X . (... may be used instead as described above .) </Td> <Td> #(4, 6, 8) = 3 </Td> </Tr> <Tr> <Td> connected sum connected sum of; knot sum of; knot composition of topology, knot theory </Td> <Td> A#B is the connected sum of the manifolds A and B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition . </Td> <Td> A#S is homeomorphic to A, for any manifold A, and the sphere S . </Td> </Tr> <Tr> <Td> primorial primorial number theory </Td> <Td> n #is product of all prime numbers less than or equal to n . </Td> <Td> 12 #= 2 × 3 × 5 × 7 × 11 = 2310 </Td> </Tr> <Tr> <Td>: </Td> <Td>: (\ displaystyle:\! \,) </Td> <Td> such that such that; so that everywhere </Td> <Td>: means "such that", and is used in proofs and the set - builder notation (described below). </Td> <Td> ∃ n ∈ N: n is even . </Td> </Tr> <Tr> <Td> field extension extends; over field theory </Td> <Td> K: F means the field K extends the field F . This may also be written as K ≥ F . </Td> <Td> R: Q </Td> </Tr> <Tr> <Td> inner product of matrices inner product of linear algebra </Td> <Td> A: B means the Frobenius inner product of the matrices A and B . The general inner product is denoted by ⟨ u, v ⟩, ⟨ u v ⟩ or (u v), as described below . For spatial vectors, the dot product notation, x y is common . See also bra--ket notation . </Td> <Td> A: B = ∑ i, j A i j B i j (\ displaystyle A: B = \ sum _ (i, j) A_ (ij) B_ (ij)) </Td> </Tr> <Tr> <Td> index of a subgroup index of subgroup group theory </Td> <Td> The index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G </Td> <Td> G: H = G H (\ displaystyle G: H = (\ frac (G) (H))) </Td> </Tr> <Tr> <Td> division divided by over everywhere </Td> <Td> A: B means the division of A with B (dividing A by B) </Td> <Td> 10: 2 = 5 </Td> </Tr> <Tr> <Td> ⋮ </Td> <Td> ⋮ (\ displaystyle \ vdots \! \,) \ vdots \! \, </Td> <Td> vertical ellipsis vertical ellipsis everywhere </Td> <Td> Denotes that certain constants and terms are missing out (e.g. for clarity) and that only the important terms are being listed . </Td> <Td> P (r, t) = χ ⋮ E (r, t 1) E (r, t 2) E (r, t 3) (\ displaystyle P (r, t) = \ chi \ vdots E (r, t_ (1)) E (r, t_ (2)) E (r, t_ (3))) </Td> </Tr> <Tr> <Td> ≀ </Td> <Td> ≀ (\ displaystyle \ wr \! \,) \ wr \! \, </Td> <Td> wreath product wreath product of...by...group theory </Td> <Td> A ≀ H means the wreath product of the group A by the group H . This may also be written A H . </Td> <Td> S n ≀ Z 2 (\ displaystyle \ mathrm (S) _ (n) \ wr \ mathrm (Z) _ (2)) is isomorphic to the automorphism group of the complete bipartite graph on (n, n) vertices . </Td> </Tr> <Tr> <Td> ↯ ⨳ ⇒ ⇐ </Td> <Td> \ blitza \ lighting: requires \ usepackage (stmaryd). <P> \ smashtimes requires \ usepackage (unicode - math) and \ setmathfont (XITS Math) or another Open Type Math Font . </P> <P> ⇒ ⇐ (\ displaystyle \ Rightarrow \ Leftarrow) \ Rightarrow \ Leftarrow ⊥ (\ displaystyle \ bot) \ bot </P> <P> ↮ (\ displaystyle \ nleftrightarrow) \ nleftrightarrow </P> <P> \ textreferencemark Contradiction! </P> <P> </P> </Td> <Td> downwards zigzag arrow contradiction; this contradicts that everywhere </Td> <Td> Denotes that contradictory statements have been inferred . For clarity, the exact point of contradiction can be appended . </Td> <Td> x + 4 = x − 3 ※ Statement: Every finite, non-empty, ordered set has a largest element . Otherwise, let's assume that X (\ displaystyle X) is a finite, non-empty, ordered set with no largest element . Then, for some x 1 ∈ X (\ displaystyle x_ (1) \ in X), there exists an x 2 ∈ X (\ displaystyle x_ (2) \ in X) with x 1 <x 2 (\ displaystyle x_ (1) <x_ (2)), but then there's also an x 3 ∈ X (\ displaystyle x_ (3) \ in X) with x 2 <x 3 (\ displaystyle x_ (2) <x_ (3)), and so on . Thus, x 1, x 2, x 3,...(\ displaystyle x_ (1), x_ (2), x_ (3), ...) are distinct elements in X (\ displaystyle X). ↯ X (\ displaystyle X) is finite . </Td> </Tr> <Tr> <Td> ⊕ ⊻ </Td> <Td> ⊕ (\ displaystyle \ oplus \! \,) \ oplus \! \, ⊻ (\ displaystyle \ veebar \! \,) \ veebar \! \, </Td> <Td> exclusive or xor propositional logic, Boolean algebra </Td> <Td> The statement A ⊕ B is true when either A or B, but not both, are true . A ⊻ B means the same . </Td> <Td> (¬ A) ⊕ A is always true, A ⊕ A is always false . </Td> </Tr> <Tr> <Td> direct sum direct sum of abstract algebra </Td> <Td> The direct sum is a special way of combining several objects into one general object . (The bun symbol ⊕, or the coproduct symbol ∐, is used; ⊻ is only for logic .) </Td> <Td> Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = (0)) </Td> </Tr> <Tr> <Td> </Td> <Td> ∧ ◯ (\ displaystyle (~ \ wedge \! \! \! \! \! \! \ bigcirc ~)) (~ \ wedge \! \! \! \! \! \! \ bigcirc ~) </Td> <Td> Kulkarni--Nomizu product Kulkarni--Nomizu product tensor algebra </Td> <Td> Derived from the tensor product of two symmetric type (0, 2) tensors; it has the algebraic symmetries of the Riemann tensor . f = g ∧ ◯ h (\ displaystyle f = g (\, \ wedge \! \! \! \! \! \! \ bigcirc \,) h) has components f α β γ δ = g α γ h β δ + g β δ h α γ − g α δ h β γ − g β γ h α δ (\ displaystyle f_ (\ alpha \ beta \ gamma \ delta) = g_ (\ alpha \ gamma) h_ (\ beta \ delta) + g_ (\ beta \ delta) h_ (\ alpha \ gamma) - g_ (\ alpha \ delta) h_ (\ beta \ gamma) - g_ (\ beta \ gamma) h_ (\ alpha \ delta)). </Td> <Td> </Td> </Tr> <Tr> <Td> </Td> <Td> ◻ (\ displaystyle \ Box \! \,) \ Box \! \ </Td> <Td> D'Alembertian; wave operator non-Euclidean Laplacian vector calculus </Td> <Td> It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions . </Td> <Td> ◻ = 1 c 2 ∂ 2 ∂ t 2 − ∂ 2 ∂ x 2 − ∂ 2 ∂ y 2 − ∂ 2 ∂ z 2 (\ displaystyle \ square = (\ frac (1) (c ^ (2))) (\ partial ^ (2) \ over \ partial t ^ (2)) - (\ partial ^ (2) \ over \ partial x ^ (2)) - (\ partial ^ (2) \ over \ partial y ^ (2)) - (\ partial ^ (2) \ over \ partial z ^ (2))) </Td> </Tr> </Table> <Tr> <Th> Symbol in HTML </Th> <Th> Symbol in TeX </Th> <Th> Name </Th> <Th> Explanation </Th> <Th> Examples </Th> </Tr>

What does upside down v mean in math
find me the text answering this question