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...pace in which the action of addition and the action of multiplication by a scalar are continuous with respect to the given topology in $L$. See [[Topological

...r space with respect to the operations of addition and multiplication by a scalar defined in $E$. A set $L+x$, where $x\in E$, is called a [[linear variety]]

...analysis|vector analysis]] are vectors which are functions of one or more scalar arguments.

...r|Vector]]). These include linear operations, viz. addition of vectors and multiplication of a vector by a number. The operation of multiplication of a vector by a number has the properties:

The application of the Hamilton operator to a scalar function $ f $, which is understood as multiplication of the "vector" $ \nabla $

...ear space, i.e. a vector space with a metric such that addition and scalar multiplication are continuous, then there is an invariant metric $\rho'$ on $X$ that is eq

...the closure of the vector sum. Addition, subtraction, multiplication by a scalar, and a topology are introduced in the space of convex sets, that space beco

...als with the operations of addition of screws, multiplication by a number, scalar and vector products, etc. In this context, the operations of helical calcul For instance, the scalar product of two screws is equal to the product of their complex moduli and t

The convolution has the basic properties of multiplication, namely, ...nd of multiplication by a scalar, with the operation of convolution as the multiplication of elements, and with the norm

while multiplication by a number $ \lambda \in \mathbf R $ and is a Euclidean space with respect to the scalar product

...ion $A$ of a Euclidean space preserving the lengths (or, equivalently, the scalar product) of vectors. Orthogonal transformations and only they can transfer ...orthogonal transformations in a Euclidean space is a group with respect to multiplication of transformations — the [[Orthogonal group|orthogonal group]] of the giv

...more dimensions. However, if one drops the requirement of commutativity of multiplication, then it is possible to construct a number system from the points of $ 4 "basic units" ) and the following multiplication table of the "basic units" :

...contravariant tensors of the same valency and multiplication of them by a scalar. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.

now defines a multiplication operation on the direct sum of linear spaces $ A _ {1} = A \oplus A $, multiplication table

A tensor considered up to multiplication by an arbitrary function (cf. [[Tensor on a vector space|Tensor on a vector is a scalar-valued function. Most frequently (in applications), the function $ \tau $

...h are invariant with respect to a shift, become (under certain conditions) multiplication operators in the image space. In particular, the convolution of two functio and differentiation induces multiplication by the independent variable:

is homogeneous with respect to multiplication by positive numbers and satisfies the condition $ f ( xx ^ {*} ) = f ( x with respect to the scalar product defined by the form $ s $.

of exterior multiplication by $ \Omega $; of interior multiplication by $ \Omega $;

as subspaces, and with a multiplication governed by the rule their (geometric) product splits into a scalar and a so-called bivector part:

...in $\mathbf{E} ^ { n }$. $\mathcal{K} ^ { n }$ with Minkowski addition and multiplication by non-negative scalars is a convex cone. The mapping $K \mapsto h _ { K }$ ($\langle \, .\, ,\, . \, \rangle$ being the scalar product) is the support function, maps this cone isomorphically into the sp

...as a rule, under summation of operators their symbols are added, and under multiplication they are multiplied, either up to terms which are small in some sense or ex of multiplication by one of the coordinates $ x _ {j} $

rings and algebras with an associative multiplication, i.e., sets with two binary operations, addition $+$ and multiplication $\cdot$, that are

and multiplication by a scalar $ \lambda \in \mathbf C $,

among which the most important ones are the order, the addition and the multiplication. In this connection the basic properties of the absolute value must be pres a scalar.

are defined uniquely up to multiplication by a scalar. For all diagrams corresponding to a partition $ \lambda $ denotes the invariant scalar product in $ U _ \lambda $,

which satisfies axioms 1)–5) is called the scalar (or inner) product of $ x $ which preserves the linear operations and the scalar product.

is the operator of scalar multiplication by $ \mu ( h) $.

...tive, then $\phi$ is called a non-degenerate Hermitian form or a Hermitian scalar product on $X$. ...where $\a$ is an invertible element of $R$. The determinant regarded up to multiplication by such elements is called the determinant of the Hermitian form or of the

...Introducing operations of addition, multiplication and multiplication by a scalar for recognition operators allows one to prove that a recognition algorithm

Axioms which describe the operation of multiplication of a vector by a number. Multiplication of a vector $ \mathbf a $

the operations of addition of tensors and of multiplication of a tensor by a scalar from $ k $ ...se operations the corresponding components are added, or multiplied by the scalar. The operation of multiplying tensors of different types is also defined; i

...by introducing on the tensor product $C_1 \tensor_A C_2$ of $A$-modules a multiplication according to the formula [[Matrix multiplication]]), of two matrices $A = [ \alpha_{ij} ]$ and $B$ is the matrix

and multiplication by a scalar are defined [[Pointwise operation|pointwise]] by the formulas

...alf-plane, and that of a unitary operator lies on the unit circle). If the scalar product is not of fixed sign, but its index of indefiniteness $ \kappa $ that is left inverse to the multiplication operator $ X \mapsto A X - X A $(

and if the operations of addition, multiplication by a scalar and the zero section satisfy the natural conditions of analyticity. If $

and the dot denotes multiplication of spinor fields by vector fields which correspond to the above $ C $- has positive scalar curvature, then $ \mathop{\rm ker} D = 0 $(

induced by left-multiplication on the first factor of $ G ( K ) \times A $. with the metric induced by the scalar product on $ V $

is the anti-commutative algebra with multiplication $ [ x , y ] = xy - yx $ as above, the multiplication $ \star $

ring. The operations of addition and multiplication defined on $K$ are $\M_n(K)$. Multiplication of matrices is associative: If $A\in\M_{m,k}(K)$, $B\in\M_{k,n}(K)$ and $C\

and is commutative and associative. The exterior multiplication of differentials on a Riemann surface is distributive with respect to addit In general, exterior multiplication of a differential of order $ k $

such that the scalar product is the sign of exterior multiplication (see [[Exterior product]]). By varying the form of the function $ \chi $

...compilation of parallel programs, is solved differently: expressions with scalar data; expressions over arrays (vectors, matrices, trees, etc.), which can i ...uch as component-wise multiplication and addition of vectors and matrices, scalar and vector products, inversion of matrices, etc. The use of such languages

# $\lVert\lambda x\rVert=\lvert\lambda\rvert\cdot\lVert x\rVert$ for every scalar $\lambda$; ...s complete if $Y$ is. When $X=Y$ is complete, the space $L(X)=L(X,X)$ with multiplication (composition) of operators becomes a [[Banach algebra]], since for the oper

...manifold of smooth functions which satisfy (2) in the norm induced by the scalar product where the round brackets denote the scalar product in $ H ( V) $.

and the semi-group (under [[pointwise multiplication]]) of positive-definite continuous functions on $ \mathbf R ^ {1} $ denotes the scalar product. The facts stated above are also valid for characteristic functions

...of them includes linear substitutions in systems of linear equations, and multiplication of quaternions and elements of a Grassmann algebra; in analytic geometry it ...l059340/l05934028.png" /> with respect to addition and multiplication by a scalar, given by the formulas

An [[Abelian group]] $E$, written additively, in which a multiplication of the elements by scalars is defined, i.e. a [[mapping]] ...closed with respect to the operations of addition and multiplication by a scalar. A subspace, considered apart from its ambient space, is a vector space ove

The addition of Abelian differentials and multiplication by a holomorphic function are defined by natural rules: If After the introduction of the scalar product

...-manifold $M$, for a compact [[Lie group|Lie group]] $G$ with an invariant scalar product on its [[Lie algebra|Lie algebra]]. The connections of interest are ...- } )$ arising from composition of the covariant derivative with Clifford multiplication. The solutions of these equations (these are certain monopoles) are the abs

...[[Locally convex space]]; [[Semi-ordered space]]) in the sense that scalar multiplication is only defined for non-negative real numbers; neither the existence of neg

...d mathematical engineering [[#References|[a11]]]. In such generalizations, scalar-valued functions are often replaced by matrix- or operator-valued functions

multiplication) is called an idempotent semi-ring if $ A $ and a multiplication $ \odot $

The exterior multiplication $ \alpha \wedge \beta $ is the operator of interior multiplication by $ X $:

...[[Lie group]] $G$ acts ''freely and transitively'' (say, by the ''right'' multiplication)<ref>These conditions guarantee that $F$ is diffeomorphic to $G$: choosing Such a bundle (defined by the ''left'' $G$-action of multiplication by $g_{\alpha\beta}$ in the transition maps (T)) is called a [[principal fi

...ntal representations, which is expansive with respect to the corresponding scalar products, this bundle mapping can be realized as the joint characteristic f ...rder differentials on $X$, and the model operators are certain "compressed multiplication operators" by the affine coordinate functions $\lambda _ { 1 }$, $\lambda _

then the associativity axiom reduces to the associativity of multiplication in the semi-group, and the neutral element is the identity in the semi-grou ...case certain generators of lower orders may degenerate, for example, into multiplication by a constant, so that the corresponding equations in (*) contain no useful

by a scalar factor; the automorphism $ \mathop{\rm Ad}\nolimits \ h(i) $ operates by multiplication by $ \overline{z} {} ^{p} z ^{-p} $.

...} $, the action of a morphism $t$ on an element $m \in M$ is given by left multiplication of the corresponding matrix, $T$, with the column vector of right coordinat ...dim } ( \wedge ^ { n } V ) = 1$, the induced action is multiplication by a scalar, $\operatorname { det } ( T )$. When $V$ is $\mathbf{Z}_{2}$-graded and $\o

...$). The shift operator $S$ on $H ^ { 2 }$ is defined to be the operator of multiplication by the coordinate function $z$: $S: f ( z ) \rightarrow z f ( z )$. A close ..._ { 0 }$ operators, for which the operator $T ( \theta )$ (with $\theta$ a scalar inner function) are the building blocks in an analogue of a canonical Jorda

by restricting the operator of multiplication by $ \lambda $ of the operator of multiplication by $ \phi ( \lambda ) $(

A free classical scalar field $ u(x) $, where $ x \stackrel{\text{df}}{=} (x^{0},\mathbf{x}) \in \B ...for real $ f \in {L^{2}}(\Bbb{R}_{x}^{3}) $, and it is called the '''free (scalar) quantum field''' at time $ 0 $. The quantum field at time $ x^{0} $ has th

..._n$, and let $\def\Ph{ {\Phi}}A_\Ph = A_F\otimes \Ph$ be the corresponding scalar extension of the algebra $A$. An element $x=\xi_1 e_1+\cdots+\xi_n e_n\in A ...B$ an element of $B$. Choose a basis $x_1,\dots,x_n$ of $B$ over $k$. Then multiplication with $b$, $b\mapsto ba$, is given by a certain matrix $M_b$. One now define

...akes it possible to obtain results with increased accuracy by accumulating scalar products. Among the direct methods used in practice, the Gauss method requi ...led elimination methods. This name is explained by the fact that for every multiplication by a matrix $ L _ {i} $

...entation $\pi_\L$, the central element $k$ acts as a non-negative integral scalar, also denoted by $k$, which is called the level of $\pi_\L$. The only $\pi_ ...n operator $u(n)$ on $V$ as follows. For $n>0$, $u(-n)$ is the operator of multiplication by $u{(-n)}\in\fh^{(-n)}$; for $n\ge 0$, $u(n)$ is the derivation of $V$ de

...imply connected manifold of dimension $\geq 5$ admits a metric of positive scalar curvature if and only if the index of the spin Dirac operator (in an approp ...{ * } M \cong T M \rightarrow \operatorname { End } ( W )$ is the Clifford multiplication and $e _ { i }$ is a local orthonormal basis (cf. also [[Orthogonal basis|O

such that this product generalizes the usual scalar product in three-dimensional space. A space provided with an inner product ...functions of a certain class by the independent variable. If one considers multiplication by Borel functions, one obtains a representation of a commutative normed al

Here $\rd h$ is the differential of $h$, acting on $v$ as a left multiplication by the Jacobian matrix $\bigl(\frac{\partial h}{\partial x}\bigr)$. Obvious ...explicitly postulated; the classification problem for such forms (modulo a scalar multiple, as before) is equivalent to classification of codimension one fol