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In multilinear algebra , applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication .
Let F {\displaystyle F} be a field of characteristic zero, such as R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } .
Let V k {\displaystyle V_{k}} be a finite-dimensional vector space over F {\displaystyle F} , and let A ∈ V 1 ⊗ V 2 ⊗ ⋯ ⊗ V d {\displaystyle {\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}} be an order-d simple tensor , i.e., there exist some vectors v k ∈ V k {\displaystyle \mathbf {v} _{k}\in V_{k}} such that A = v 1 ⊗ v 2 ⊗ ⋯ ⊗ v d {\displaystyle {\mathcal {A}}=\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}} . If we are given a collection of linear maps A k : V k → W k {\displaystyle A_{k}:V_{k}\to W_{k}} , then the multilinear multiplication of A {\displaystyle {\mathcal {A}}} with ( A 1 , A 2 , … , A d ) {\displaystyle (A_{1},A_{2},\ldots ,A_{d})} is defined [ 1 ] as the action on A {\displaystyle {\mathcal {A}}} of the tensor product of these linear maps, [ 2 ] namely
A 1 ⊗ A 2 ⊗ ⋯ ⊗ A d : V 1 ⊗ V 2 ⊗ ⋯ ⊗ V d → W 1 ⊗ W 2 ⊗ ⋯ ⊗ W d , v 1 ⊗ v 2 ⊗ ⋯ ⊗ v d ↦ A 1 ( v 1 ) ⊗ A 2 ( v 2 ) ⊗ ⋯ ⊗ A d ( v d ) {\displaystyle {\begin{aligned}A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d}:V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}&\to W_{1}\otimes W_{2}\otimes \cdots \otimes W_{d},\\\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}&\mapsto A_{1}(\mathbf {v} _{1})\otimes A_{2}(\mathbf {v} _{2})\otimes \cdots \otimes A_{d}(\mathbf {v} _{d})\end{aligned}}}
Since the tensor product of linear maps is itself a linear map, [ 2 ] and because every tensor admits a tensor rank decomposition , [ 1 ] the above expression extends linearly to all tensors. That is, for a general tensor A ∈ V 1 ⊗ V 2 ⊗ ⋯ ⊗ V d {\displaystyle {\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}} , the multilinear multiplication is
B := ( A 1 ⊗ A 2 ⊗ ⋯ ⊗ A d ) ( A ) = ( A 1 ⊗ A 2 ⊗ ⋯ ⊗ A d ) ( ∑ i = 1 r a i 1 ⊗ a i 2 ⊗ ⋯ ⊗ a i d ) = ∑ i = 1 r A 1 ( a i 1 ) ⊗ A 2 ( a i 2 ) ⊗ ⋯ ⊗ A d ( a i d ) {\displaystyle {\begin{aligned}&{\mathcal {B}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})\\[4pt]={}&(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})\left(\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}\right)\\[5pt]={}&\sum _{i=1}^{r}A_{1}(\mathbf {a} _{i}^{1})\otimes A_{2}(\mathbf {a} _{i}^{2})\otimes \cdots \otimes A_{d}(\mathbf {a} _{i}^{d})\end{aligned}}}
where A = ∑ i = 1 r a i 1 ⊗ a i 2 ⊗ ⋯ ⊗ a i d {\textstyle {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}} with a i k ∈ V k {\displaystyle \mathbf {a} _{i}^{k}\in V_{k}} is one of A {\displaystyle {\mathcal {A}}} 's tensor rank decompositions. The validity of the above expression is not limited to a tensor rank decomposition; in fact, it is valid for any expression of A {\displaystyle {\mathcal {A}}} as a linear combination of pure tensors, which follows from the universal property of the tensor product .
It is standard to use the following shorthand notations in the literature for multilinear multiplications: ( A 1 , A 2 , … , A d ) ⋅ A := ( A 1 ⊗ A 2 ⊗ ⋯ ⊗ A d ) ( A ) {\displaystyle (A_{1},A_{2},\ldots ,A_{d})\cdot {\mathcal {A}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})} and A k ⋅ k A := ( Id V 1 , … , Id V k − 1 , A k , Id V k + 1 , … , Id V d ) ⋅ A , {\displaystyle A_{k}\cdot _{k}{\mathcal {A}}:=(\operatorname {Id} _{V_{1}},\ldots ,\operatorname {Id} _{V_{k-1}},A_{k},\operatorname {Id} _{V_{k+1}},\ldots ,\operatorname {Id} _{V_{d}})\cdot {\mathcal {A}},} where Id V k : V k → V k {\displaystyle \operatorname {Id} _{V_{k}}:V_{k}\to V_{k}} is the identity operator .
In computational multilinear algebra it is conventional to work in coordinates. Assume that an inner product is fixed on V k {\displaystyle V_{k}} and let V k ∗ {\displaystyle V_{k}^{*}} denote the dual vector space of V k {\displaystyle V_{k}} . Let { e 1 k , … , e n k k } {\displaystyle \{e_{1}^{k},\ldots ,e_{n_{k}}^{k}\}} be a basis for V k {\displaystyle V_{k}} , let { ( e 1 k ) ∗ , … , ( e n k k ) ∗ } {\displaystyle \{(e_{1}^{k})^{*},\ldots ,(e_{n_{k}}^{k})^{*}\}} be the dual basis, and let { f 1 k , … , f m k k } {\displaystyle \{f_{1}^{k},\ldots ,f_{m_{k}}^{k}\}} be a basis for W k {\displaystyle W_{k}} . The linear map M k = ∑ i = 1 m k ∑ j = 1 n k m i , j ( k ) f i k ⊗ ( e j k ) ∗ {\textstyle M_{k}=\sum _{i=1}^{m_{k}}\sum _{j=1}^{n_{k}}m_{i,j}^{(k)}f_{i}^{k}\otimes (e_{j}^{k})^{*}} is then represented by the matrix M ^ k = [ m i , j ( k ) ] ∈ F m k × n k {\displaystyle {\widehat {M}}_{k}=[m_{i,j}^{(k)}]\in F^{m_{k}\times n_{k}}} . Likewise, with respect to the standard tensor product basis { e j 1 1 ⊗ e j 2 2 ⊗ ⋯ ⊗ e j d d } j 1 , j 2 , … , j d {\displaystyle \{e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}\}_{j_{1},j_{2},\ldots ,j_{d}}} , the abstract tensor A = ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d a j 1 , j 2 , … , j d e j 1 1 ⊗ e j 2 2 ⊗ ⋯ ⊗ e j d d {\displaystyle {\mathcal {A}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}} is represented by the multidimensional array A ^ = [ a j 1 , j 2 , … , j d ] ∈ F n 1 × n 2 × ⋯ × n d {\displaystyle {\widehat {\mathcal {A}}}=[a_{j_{1},j_{2},\ldots ,j_{d}}]\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}} . Observe that A ^ = ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d a j 1 , j 2 , … , j d e j 1 1 ⊗ e j 2 2 ⊗ ⋯ ⊗ e j d d , {\displaystyle {\widehat {\mathcal {A}}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},}
where e j k ∈ F n k {\displaystyle \mathbf {e} _{j}^{k}\in F^{n_{k}}} is the j th standard basis vector of F n k {\displaystyle F^{n_{k}}} and the tensor product of vectors is the affine Segre map ⊗ : ( v ( 1 ) , v ( 2 ) , … , v ( d ) ) ↦ [ v i 1 ( 1 ) v i 2 ( 2 ) ⋯ v i d ( d ) ] i 1 , i 2 , … , i d {\displaystyle \otimes :(\mathbf {v} ^{(1)},\mathbf {v} ^{(2)},\ldots ,\mathbf {v} ^{(d)})\mapsto [v_{i_{1}}^{(1)}v_{i_{2}}^{(2)}\cdots v_{i_{d}}^{(d)}]_{i_{1},i_{2},\ldots ,i_{d}}} . It follows from the above choices of bases that the multilinear multiplication B = ( M 1 , M 2 , … , M d ) ⋅ A {\displaystyle {\mathcal {B}}=(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}} becomes
B ^ = ( M ^ 1 , M ^ 2 , … , M ^ d ) ⋅ ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d a j 1 , j 2 , … , j d e j 1 1 ⊗ e j 2 2 ⊗ ⋯ ⊗ e j d d = ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d a j 1 , j 2 , … , j d ( M ^ 1 , M ^ 2 , … , M ^ d ) ⋅ ( e j 1 1 ⊗ e j 2 2 ⊗ ⋯ ⊗ e j d d ) = ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d a j 1 , j 2 , … , j d ( M ^ 1 e j 1 1 ) ⊗ ( M ^ 2 e j 2 2 ) ⊗ ⋯ ⊗ ( M ^ d e j d d ) . {\displaystyle {\begin{aligned}{\widehat {\mathcal {B}}}&=({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot \sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\&=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot (\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d})\\&=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}({\widehat {M}}_{1}\mathbf {e} _{j_{1}}^{1})\otimes ({\widehat {M}}_{2}\mathbf {e} _{j_{2}}^{2})\otimes \cdots \otimes ({\widehat {M}}_{d}\mathbf {e} _{j_{d}}^{d}).\end{aligned}}}
The resulting tensor B ^ {\displaystyle {\widehat {\mathcal {B}}}} lives in F m 1 × m 2 × ⋯ × m d {\displaystyle F^{m_{1}\times m_{2}\times \cdots \times m_{d}}} .
From the above expression, an element-wise definition of the multilinear multiplication is obtained. Indeed, since B ^ {\displaystyle {\widehat {\mathcal {B}}}} is a multidimensional array, it may be expressed as B ^ = ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d b j 1 , j 2 , … , j d e j 1 1 ⊗ e j 2 2 ⊗ ⋯ ⊗ e j d d , {\displaystyle {\widehat {\mathcal {B}}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},} where b j 1 , j 2 , … , j d ∈ F {\displaystyle b_{j_{1},j_{2},\ldots ,j_{d}}\in F} are the coefficients. Then it follows from the above formulae that
( ( e i 1 1 ) T , ( e i 2 2 ) T , … , ( e i d d ) T ) ⋅ B ^ = ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d b j 1 , j 2 , … , j d ( ( e i 1 1 ) T e j 1 1 ) ⊗ ( ( e i 2 2 ) T e j 2 2 ) ⊗ ⋯ ⊗ ( ( e i d d ) T e j d d ) = ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d b j 1 , j 2 , … , j d δ i 1 , j 1 ⋅ δ i 2 , j 2 ⋯ δ i d , j d = b i 1 , i 2 , … , i d , {\displaystyle {\begin{aligned}&\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot {\widehat {\mathcal {B}}}\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\left((\mathbf {e} _{i_{1}}^{1})^{T}\mathbf {e} _{j_{1}}^{1}\right)\otimes \left((\mathbf {e} _{i_{2}}^{2})^{T}\mathbf {e} _{j_{2}}^{2}\right)\otimes \cdots \otimes \left((\mathbf {e} _{i_{d}}^{d})^{T}\mathbf {e} _{j_{d}}^{d}\right)\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}b_{j_{1},j_{2},\ldots ,j_{d}}\delta _{i_{1},j_{1}}\cdot \delta _{i_{2},j_{2}}\cdots \delta _{i_{d},j_{d}}\\={}&b_{i_{1},i_{2},\ldots ,i_{d}},\end{aligned}}}
where δ i , j {\displaystyle \delta _{i,j}} is the Kronecker delta . Hence, if B = ( M 1 , M 2 , … , M d ) ⋅ A {\displaystyle {\mathcal {B}}=(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}} , then
b i 1 , i 2 , … , i d = ( ( e i 1 1 ) T , ( e i 2 2 ) T , … , ( e i d d ) T ) ⋅ B ^ = ( ( e i 1 1 ) T , ( e i 2 2 ) T , … , ( e i d d ) T ) ⋅ ( M ^ 1 , M ^ 2 , … , M ^ d ) ⋅ ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d a j 1 , j 2 , … , j d e j 1 1 ⊗ e j 2 2 ⊗ ⋯ ⊗ e j d d = ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d a j 1 , j 2 , … , j d ( ( e i 1 1 ) T M ^ 1 e j 1 1 ) ⊗ ( ( e i 2 2 ) T M ^ 2 e j 2 2 ) ⊗ ⋯ ⊗ ( ( e i d d ) T M ^ d e j d d ) = ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d a j 1 , j 2 , … , j d m i 1 , j 1 ( 1 ) ⋅ m i 2 , j 2 ( 2 ) ⋯ m i d , j d ( d ) , {\displaystyle {\begin{aligned}&b_{i_{1},i_{2},\ldots ,i_{d}}=\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot {\widehat {\mathcal {B}}}\\={}&\left((\mathbf {e} _{i_{1}}^{1})^{T},(\mathbf {e} _{i_{2}}^{2})^{T},\ldots ,(\mathbf {e} _{i_{d}}^{d})^{T}\right)\cdot ({\widehat {M}}_{1},{\widehat {M}}_{2},\ldots ,{\widehat {M}}_{d})\cdot \sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}((\mathbf {e} _{i_{1}}^{1})^{T}{\widehat {M}}_{1}\mathbf {e} _{j_{1}}^{1})\otimes ((\mathbf {e} _{i_{2}}^{2})^{T}{\widehat {M}}_{2}\mathbf {e} _{j_{2}}^{2})\otimes \cdots \otimes ((\mathbf {e} _{i_{d}}^{d})^{T}{\widehat {M}}_{d}\mathbf {e} _{j_{d}}^{d})\\={}&\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}m_{i_{1},j_{1}}^{(1)}\cdot m_{i_{2},j_{2}}^{(2)}\cdots m_{i_{d},j_{d}}^{(d)},\end{aligned}}}
where the m i , j ( k ) {\displaystyle m_{i,j}^{(k)}} are the elements of M ^ k {\displaystyle {\widehat {M}}_{k}} as defined above.
Let A ∈ V 1 ⊗ V 2 ⊗ ⋯ ⊗ V d {\displaystyle {\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}} be an order-d tensor over the tensor product of F {\displaystyle F} -vector spaces.
Since a multilinear multiplication is the tensor product of linear maps, we have the following multilinearity property (in the construction of the map): [ 1 ] [ 2 ]
A 1 ⊗ ⋯ ⊗ A k − 1 ⊗ ( α A k + β B ) ⊗ A k + 1 ⊗ ⋯ ⊗ A d = α A 1 ⊗ ⋯ ⊗ A d + β A 1 ⊗ ⋯ ⊗ A k − 1 ⊗ B ⊗ A k + 1 ⊗ ⋯ ⊗ A d {\displaystyle A_{1}\otimes \cdots \otimes A_{k-1}\otimes (\alpha A_{k}+\beta B)\otimes A_{k+1}\otimes \cdots \otimes A_{d}=\alpha A_{1}\otimes \cdots \otimes A_{d}+\beta A_{1}\otimes \cdots \otimes A_{k-1}\otimes B\otimes A_{k+1}\otimes \cdots \otimes A_{d}}
Multilinear multiplication is a linear map : [ 1 ] [ 2 ] ( M 1 , M 2 , … , M d ) ⋅ ( α A + β B ) = α ( M 1 , M 2 , … , M d ) ⋅ A + β ( M 1 , M 2 , … , M d ) ⋅ B {\displaystyle (M_{1},M_{2},\ldots ,M_{d})\cdot (\alpha {\mathcal {A}}+\beta {\mathcal {B}})=\alpha \;(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}+\beta \;(M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {B}}}
It follows from the definition that the composition of two multilinear multiplications is also a multilinear multiplication: [ 1 ] [ 2 ]
( M 1 , M 2 , … , M d ) ⋅ ( ( K 1 , K 2 , … , K d ) ⋅ A ) = ( M 1 ∘ K 1 , M 2 ∘ K 2 , … , M d ∘ K d ) ⋅ A , {\displaystyle (M_{1},M_{2},\ldots ,M_{d})\cdot \left((K_{1},K_{2},\ldots ,K_{d})\cdot {\mathcal {A}}\right)=(M_{1}\circ K_{1},M_{2}\circ K_{2},\ldots ,M_{d}\circ K_{d})\cdot {\mathcal {A}},}
where M k : U k → W k {\displaystyle M_{k}:U_{k}\to W_{k}} and K k : V k → U k {\displaystyle K_{k}:V_{k}\to U_{k}} are linear maps.
Observe specifically that multilinear multiplications in different factors commute,
M k ⋅ k ( M ℓ ⋅ ℓ A ) = M ℓ ⋅ ℓ ( M k ⋅ k A ) = M k ⋅ k M ℓ ⋅ ℓ A , {\displaystyle M_{k}\cdot _{k}\left(M_{\ell }\cdot _{\ell }{\mathcal {A}}\right)=M_{\ell }\cdot _{\ell }\left(M_{k}\cdot _{k}{\mathcal {A}}\right)=M_{k}\cdot _{k}M_{\ell }\cdot _{\ell }{\mathcal {A}},}
if k ≠ ℓ . {\displaystyle k\neq \ell .}
The factor-k multilinear multiplication M k ⋅ k A {\displaystyle M_{k}\cdot _{k}{\mathcal {A}}} can be computed in coordinates as follows. Observe first that
M k ⋅ k A = M k ⋅ k ∑ j 1 = 1 n 1 ∑ j 2 = 1 n 2 ⋯ ∑ j d = 1 n d a j 1 , j 2 , … , j d e j 1 1 ⊗ e j 2 2 ⊗ ⋯ ⊗ e j d d = ∑ j 1 = 1 n 1 ⋯ ∑ j k − 1 = 1 n k − 1 ∑ j k + 1 = 1 n k + 1 ⋯ ∑ j d = 1 n d e j 1 1 ⊗ ⋯ ⊗ e j k − 1 k − 1 ⊗ M k ( ∑ j k = 1 n k a j 1 , j 2 , … , j d e j k k ) ⊗ e j k + 1 k + 1 ⊗ ⋯ ⊗ e j d d . {\displaystyle {\begin{aligned}M_{k}\cdot _{k}{\mathcal {A}}&=M_{k}\cdot _{k}\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}\\&=\sum _{j_{1}=1}^{n_{1}}\cdots \sum _{j_{k-1}=1}^{n_{k-1}}\sum _{j_{k+1}=1}^{n_{k+1}}\cdots \sum _{j_{d}=1}^{n_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \cdots \otimes \mathbf {e} _{j_{k-1}}^{k-1}\otimes M_{k}\left(\sum _{j_{k}=1}^{n_{k}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{k}}^{k}\right)\otimes \mathbf {e} _{j_{k+1}}^{k+1}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d}.\end{aligned}}}
Next, since
F n 1 ⊗ F n 2 ⊗ ⋯ ⊗ F n d ≃ F n k ⊗ ( F n 1 ⊗ ⋯ ⊗ F n k − 1 ⊗ F n k + 1 ⊗ ⋯ ⊗ F n d ) ≃ F n k ⊗ F n 1 ⋯ n k − 1 n k + 1 ⋯ n d , {\displaystyle F^{n_{1}}\otimes F^{n_{2}}\otimes \cdots \otimes F^{n_{d}}\simeq F^{n_{k}}\otimes (F^{n_{1}}\otimes \cdots \otimes F^{n_{k-1}}\otimes F^{n_{k+1}}\otimes \cdots \otimes F^{n_{d}})\simeq F^{n_{k}}\otimes F^{n_{1}\cdots n_{k-1}n_{k+1}\cdots n_{d}},}
there is a bijective map, called the factor- k standard flattening , [ 1 ] denoted by ( ⋅ ) ( k ) {\displaystyle (\cdot )_{(k)}} , that identifies M k ⋅ k A {\displaystyle M_{k}\cdot _{k}{\mathcal {A}}} with an element from the latter space, namely
( M k ⋅ k A ) ( k ) := ∑ j 1 = 1 n 1 ⋯ ∑ j k − 1 = 1 n k − 1 ∑ j k + 1 = 1 n k + 1 ⋯ ∑ j d = 1 n d M k ( ∑ j k = 1 n k a j 1 , j 2 , … , j d e j k k ) ⊗ e μ k ( j 1 , … , j k − 1 , j k + 1 , … , j d ) := M k A ( k ) , {\displaystyle \left(M_{k}\cdot _{k}{\mathcal {A}}\right)_{(k)}:=\sum _{j_{1}=1}^{n_{1}}\cdots \sum _{j_{k-1}=1}^{n_{k-1}}\sum _{j_{k+1}=1}^{n_{k+1}}\cdots \sum _{j_{d}=1}^{n_{d}}M_{k}\left(\sum _{j_{k}=1}^{n_{k}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{k}}^{k}\right)\otimes \mathbf {e} _{\mu _{k}(j_{1},\ldots ,j_{k-1},j_{k+1},\ldots ,j_{d})}:=M_{k}{\mathcal {A}}_{(k)},}
where e j {\displaystyle \mathbf {e} _{j}} is the j th standard basis vector of F N k {\displaystyle F^{N_{k}}} , N k = n 1 ⋯ n k − 1 n k + 1 ⋯ n d {\displaystyle N_{k}=n_{1}\cdots n_{k-1}n_{k+1}\cdots n_{d}} , and A ( k ) ∈ F n k ⊗ F N k ≃ F n k × N k {\displaystyle {\mathcal {A}}_{(k)}\in F^{n_{k}}\otimes F^{N_{k}}\simeq F^{n_{k}\times N_{k}}} is the factor- k flattening matrix of A {\displaystyle {\mathcal {A}}} whose columns are the factor- k vectors [ a j 1 , … , j k − 1 , i , j k + 1 , … , j d ] i = 1 n k {\displaystyle [a_{j_{1},\ldots ,j_{k-1},i,j_{k+1},\ldots ,j_{d}}]_{i=1}^{n_{k}}} in some order, determined by the particular choice of the bijective map
μ k : [ 1 , n 1 ] × ⋯ × [ 1 , n k − 1 ] × [ 1 , n k + 1 ] × ⋯ × [ 1 , n d ] → [ 1 , N k ] . {\displaystyle \mu _{k}:[1,n_{1}]\times \cdots \times [1,n_{k-1}]\times [1,n_{k+1}]\times \cdots \times [1,n_{d}]\to [1,N_{k}].}
In other words, the multilinear multiplication ( M 1 , M 2 , … , M d ) ⋅ A {\displaystyle (M_{1},M_{2},\ldots ,M_{d})\cdot {\mathcal {A}}} can be computed as a sequence of d factor- k multilinear multiplications, which themselves can be implemented efficiently as classic matrix multiplications.
The higher-order singular value decomposition (HOSVD) factorizes a tensor given in coordinates A ∈ F n 1 × n 2 × ⋯ × n d {\displaystyle {\mathcal {A}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}} as the multilinear multiplication A = ( U 1 , U 2 , … , U d ) ⋅ S {\displaystyle {\mathcal {A}}=(U_{1},U_{2},\ldots ,U_{d})\cdot {\mathcal {S}}} , where U k ∈ F n k × n k {\displaystyle U_{k}\in F^{n_{k}\times n_{k}}} are orthogonal matrices and S ∈ F n 1 × n 2 × ⋯ × n d {\displaystyle {\mathcal {S}}\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}} . | https://en.wikipedia.org/wiki/Multilinear_multiplication |
In algebra, a multilinear polynomial [ 1 ] is a multivariate polynomial that is linear (meaning affine ) in each of its variables separately , but not necessarily simultaneously . It is a polynomial in which no variable occurs to a power of 2 {\displaystyle 2} or higher; that is, each monomial is a constant times a product of distinct variables. For example f ( x , y , z ) = 3 x y + 2.5 y − 7 z {\displaystyle f(x,y,z)=3xy+2.5y-7z} is a multilinear polynomial of degree 2 {\displaystyle 2} (because of the monomial 3 x y {\displaystyle 3xy} ) whereas f ( x , y , z ) = x 2 + 4 y {\displaystyle f(x,y,z)=x^{2}+4y} is not. The degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial.
Multilinear polynomials can be understood as a multilinear map (specifically, a multilinear form ) applied to the vectors [1 x], [1 y], etc. The general form can be written as a tensor contraction : f ( x ) = ∑ i 1 = 0 1 ∑ i 2 = 0 1 ⋯ ∑ i n = 0 1 a i 1 i 2 ⋯ i n x 1 i 1 x 2 i 2 ⋯ x n i n {\displaystyle f(x)=\sum _{i_{1}=0}^{1}\sum _{i_{2}=0}^{1}\cdots \sum _{i_{n}=0}^{1}a_{i_{1}i_{2}\cdots i_{n}}x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}}
For example, in two variables: f ( x , y ) = ∑ i = 0 1 ∑ j = 0 1 a i j x i y j = a 00 + a 10 x + a 01 y + a 11 x y = ( 1 x ) ( a 00 a 01 a 10 a 11 ) ( 1 y ) {\displaystyle f(x,y)=\sum _{i=0}^{1}\sum _{j=0}^{1}a_{ij}x^{i}y^{j}=a_{00}+a_{10}x+a_{01}y+a_{11}xy={\begin{pmatrix}1&x\end{pmatrix}}{\begin{pmatrix}a_{00}&a_{01}\\a_{10}&a_{11}\end{pmatrix}}{\begin{pmatrix}1\\y\end{pmatrix}}}
A multilinear polynomial f {\displaystyle f} is linear (affine) when varying only one variable, x k {\displaystyle x_{k}} : f ( x 1 , x 2 , . . . , x k , . . . , x n ) = a x k + b {\displaystyle f(x_{1},x_{2},...,x_{k},...,x_{n})=ax_{k}+b} where a {\displaystyle a} and b {\displaystyle b} do not depend on x k {\displaystyle x_{k}} . Note that b {\displaystyle b} is generally not zero, so f {\displaystyle f} is linear in the "shaped like a line" sense, but not in the "directly proportional" sense of a multilinear map .
All repeated second partial derivatives are zero: ∂ 2 f ∂ x k 2 = 0 {\displaystyle {\frac {\partial ^{2}f}{\partial x_{k}^{2}}}=0} In other words, its Hessian matrix is a symmetric hollow matrix .
In particular, the Laplacian ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} , so f {\displaystyle f} is a harmonic function . This implies f {\displaystyle f} has maxima and minima only on the boundary of the domain .
More generally, every restriction of f {\displaystyle f} to a subset of its coordinates is also multilinear, so ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} still holds when one or more variables are fixed. In other words, f {\displaystyle f} is harmonic on every "slice" of the domain along coordinate axes.
When the domain is rectangular in the coordinate axes (e.g. a hypercube ), f {\displaystyle f} will have maxima and minima only on the vertices of the domain, i.e. the finite set of 2 n {\displaystyle 2^{n}} points with minimal and maximal coordinate values. The value of the function on these points completely determines the function, since the value on the edges of the boundary can be found by linear interpolation , and the value on the rest of the boundary and the interior is fixed by Laplace's equation , ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} . [ 1 ]
The value of the polynomial at an arbitrary point can be found by repeated linear interpolation along each coordinate axis. Equivalently, it is a weighted mean of the vertex values, where the weights are the Lagrange interpolation polynomials . These weights also constitute a set of generalized barycentric coordinates for the hyperrectangle . Geometrically, the point divides the domain into 2 n {\displaystyle 2^{n}} smaller hyperrectangles, and the weight of each vertex is the (fractional) volume of the hyperrectangle opposite it.
Algebraically, the multilinear interpolant on the hyperrectangle [ a i , b i ] i = 1 n {\displaystyle [a_{i},b_{i}]_{i=1}^{n}} is: f ( x ) = ∑ v f ( v ) ∏ i | v i = b i x i − a i b i − a i ∏ i | v i = a i b i − x i b i − a i {\displaystyle f(x)=\sum _{v}f(v)\prod _{i|v_{i}=b_{i}}{\frac {x_{i}-a_{i}}{b_{i}-a_{i}}}\prod _{i|v_{i}=a_{i}}{\frac {b_{i}-x_{i}}{b_{i}-a_{i}}}} where the sum is taken over the vertices v {\displaystyle v} . Equivalently, f ( x ) = 1 V ∑ v f ( v ) ∏ i | v i = b i ( x i − a i ) ∏ i | v i = a i ( b i − x i ) {\displaystyle f(x)={\frac {1}{V}}\sum _{v}f(v)\prod _{i|v_{i}=b_{i}}(x_{i}-a_{i})\prod _{i|v_{i}=a_{i}}(b_{i}-x_{i})} where V is the volume of the hyperrectangle.
The value at the center is the arithmetic mean of the value at the vertices, which is also the mean over the domain boundary, and the mean over the interior. The components of the gradient at the center are proportional to the balance of the vertex values along each coordinate axis.
The vertex values and the coefficients of the polynomial are related by a linear transformation (specifically, a Möbius transform if the domain is the unit hypercube { 0 , 1 } n {\displaystyle \{0,1\}^{n}} , and a Walsh-Hadamard-Fourier transform if the domain is the symmetric hypercube { − 1 , 1 } n {\displaystyle \{-1,1\}^{n}} ).
Multilinear polynomials are the interpolants of multilinear or n-linear interpolation on a rectangular grid, a generalization of linear interpolation , bilinear interpolation and trilinear interpolation to an arbitrary number of variables. This is a specific form of multivariate interpolation , not to be confused with piecewise linear interpolation. The resulting polynomial is not a linear function of the coordinates (its degree can be higher than 1), but it is a linear function of the fitted data values.
The determinant , permanent and other immanants of a matrix are homogeneous multilinear polynomials in the elements of the matrix (and also multilinear forms in the rows or columns).
The multilinear polynomials in n {\displaystyle n} variables form a 2 n {\displaystyle 2^{n}} -dimensional vector space , which is also the basis used in the Fourier analysis of (pseudo-)Boolean functions . Every ( pseudo- ) Boolean function can be uniquely expressed as a multilinear polynomial (up to a choice of domain and codomain).
Multilinear polynomials are important in the study of polynomial identity testing . [ 2 ] | https://en.wikipedia.org/wiki/Multilinear_polynomial |
Multilocus sequence typing ( MLST ) is a technique in molecular biology for the typing of multiple loci , using DNA sequences of internal fragments of multiple housekeeping genes to characterize isolates of microbial species.
The first MLST scheme to be developed was for Neisseria meningitidis , [ 1 ] the causative agent of meningococcal meningitis and septicaemia . Since its introduction for the research of evolutionary history, MLST has been used not only for human pathogens but also for plant pathogens. [ 2 ]
MLST directly measures the DNA sequence variations in a set of housekeeping genes and characterizes strains by their unique allelic profiles. The principle of MLST is simple: the technique involves PCR amplification followed by DNA sequencing . Nucleotide differences between strains can be checked at a variable number of genes depending on the degree of discrimination desired.
The workflow of MLST involves: 1) data collection, 2) data analysis and 3) multilocus sequence analysis. In the data collection step, definitive identification of variation is obtained by nucleotide sequence determination of gene fragments. In the data analysis step, all unique sequences are assigned allele numbers and combined into an allelic profile and assigned a sequence type (ST). If new alleles and STs are found, they are stored in the database after verification. In the final analysis step of MLST, the relatedness of isolates are made by comparing allelic profiles. Researchers do epidemiological and phylogenetical studies by comparing STs of different clonal complexes. A huge set of data is produced during the sequencing and identification process so bioinformatic techniques are used to arrange, manage, analyze and merge all of the biological data.
To strike the balance between the acceptable identification power, time and cost for the strain typing, about seven to eight house-keeping genes are commonly used in the laboratories. Quoting Staphylococcus aureus as an example, seven housekeeping genes are used in MLST typing. These genes include carbamate kinase ( arcC ), shikimate dehydrogenase ( aroE ), glycerol kinase ( glpF ), guanylate kinase ( gmk ), phosphate acetyltransferase ( pta ), triosephosphate isomerase ( tpi ) and acetyl coenzyme A acetyltransferase ( yqiL ) as specified by the MLST website. However, it is not uncommon for up to ten housekeeping genes to be used. For Vibrio vulnificus , the housekeeping genes used are glucose-6-phosphate isomerase ( glp ), DNA gyrase, subunit B ( gyrB ), malate-lactate dehydrogenase ( mdh ), methionyl-tRNA synthetase ( metG ), phosphoribosylaminoimidazole synthetase ( purM ), threonine dehydrogenase ( dtdS ), diaminopimelate decarboxylase ( lysA ), transhydrogenase alpha subunit ( pntA ), dihydroorotase ( pyrC ) and tryptophanase ( tnaA ). Thus both the number and type of housekeeping genes interrogated by MLST may differ from species to species.
For each of these housekeeping genes, the different sequences are assigned as alleles and the alleles at the loci provide an allelic profile. A series of profiles can then be the identification marker for strain typing. Sequences that differ at even a single nucleotide are assigned as different alleles and no weighting is given to take into account the number of nucleotide differences between alleles, as we cannot distinguish whether differences at multiple nucleotide sites are a result of multiple point mutations or a single recombinational exchange. The large number of potential alleles at each of the loci provides the ability to distinguish billions of different allelic profiles, and a strain with the most common allele at each locus would only be expected to occur by chance approximately once in 10,000 isolates. [ citation needed ] Despite MLST providing high discriminatory power, the accumulation of nucleotide changes in housekeeping genes is a relatively slow process and the allelic profile of a bacterial isolate is sufficiently stable over time for the method to be ideal for global epidemiology.
The relatedness of isolates is displayed as a dendrogram constructed using the matrix of pairwise differences between their allelic profiles, eBURST or a minimum spanning tree (MST). The dendrogram is only a convenient way of displaying those isolates that have identical or very similar allelic profiles that can be assumed to be derived from a common ancestor; the relationships between isolates that differ at more than three out of seven loci are likely to be unreliable and should not be taken to infer their phylogeny. [ 3 ] [ 4 ] The MST connects all samples in such a way that the summed distance of all branches of the tree is minimal. [ 5 ]
Alternatively, the relatedness of isolates can also be analysed with MultiLocus Sequence Analysis ( MLSA ). This does not use the assigned alleles, but instead concatenates the sequences of the gene fragments of the housekeeping genes and uses this concatenated sequence to determine phylogenetic relationships. In contrast to MLST, this analysis does assign a higher similarity between sequences differing only a single nucleotide and a lower similarity between sequences with multiple nucleotide differences. As a result, this analysis is more suitable for organisms with a clonal evolution and less suitable for organisms in which recombinational events occur very often. It can also be used to determine phylogenetic relationships between closely related species. [ 6 ] The terms MLST and MLSA are very often considered interchangeable. This is however not correct as each analysis method has its distinctive features and uses. Care should be taken to use the correct term.
Earlier serological typing approaches had been established for differentiating bacterial isolates, but immunological typing has drawbacks such as reliance on few antigenic loci and unpredictable reactivities of antibodies with different antigenic variants. Several molecular typing schemes have been proposed to determine the relatedness of pathogens such as pulsed-field gel electrophoresis ( PFGE ), ribotyping, and PCR-based fingerprinting . But these DNA banding-based subtyping methods do not provide meaningful evolutionary analyses. Despite PFGE being considered by many researchers as the “gold standard”, many strains are not typable by this technique due to the degradation of the DNA during the process (gel smears).
The approach of MLST is distinct from Multi locus enzyme electrophoresis (MLEE), which is based on different electrophoretic mobilities (EM) of multiple core metabolic enzymes. The alleles at each locus define the EM of their products, as different amino acid sequences between enzymes result in different mobilities and distinct bands when run on a gel. The relatedness of isolates can then be visualized with a dendrogram generated from the matrix of pairwise differences between the electrophoretic types. This method has a lower resolution than MLST for several reasons, all arising from the fact that enzymatic phenotype diversity is merely a proxy for DNA sequence diversity. First, enzymes may have different amino acid sequences without having sufficiently different EM to give distinct bands. Second, "silent mutations" may alter the DNA sequence of a gene without altering the encoded amino acids. Thirdly, the phenotype of the enzyme can easily be altered in response to environmental conditions and badly affect the reproducibility of MLEE results - common modifications of enzymes are phosphorylation, cofactor binding and cleavage of transport sequences. This also limits comparability of MLEE data obtained by different laboratories, whereas MLST provides portable and comparable DNA sequence data and has great potential for automation and standardization.
MLST should not be confused with DNA barcoding . The latter is a taxonomic method that uses short genetic markers to recognize particular species of eukaryotes. It is based on the fact that mitochondrial DNA (mtDNA) or some parts of the ribosomal DNA cistron have relatively fast mutation rates, which give significant variation in sequences between species. mtDNA methods are only possible in eukaryotes (as prokaryotes lack mitochondria), whereas MLST, although initially developed for prokaryotes, is now finding application in eukaryotes and in principle could be applied to any kingdom.
MLST is highly unambiguous and portable. Materials required for ST determination can be exchanged between laboratories. Primer sequences and protocols can be accessed electronically. It is reproducible and scalable. MLST is automated, combines advances in high throughput sequencing and bioinformatics with established population genetics techniques. MLST data can be used to investigate evolutionary relationships among bacteria. MLST provides good discriminatory power to differentiate isolates.
The application of MLST is huge, and provides a resource for the scientific, public health, and veterinary communities as well as the food industry. The following are examples of MLST applications.
Campylobacter is the common causative agent for bacterial infectious intestinal diseases, usually arising from undercooked poultry or unpasteurised milk. However, its epidemiology is poorly understood since outbreaks are rarely detected, so that the sources and transmission routes of outbreak are not easily traced. In addition, Campylobacter genomes are genetically diverse and unstable with frequent inter- and intragenomic recombination, together with phase variation, which complicates the interpretation of data from many typing methods. Until recently, with the application of MLST technique, Campylobacter typing has achieved a great success and added onto the MLST database. As at 1 May 2008, the Campylobacter MLST database contains 3516 isolates and about 30 publications that use or mention MLST in research on Campylobacter ( http://pubmlst.org/campylobacter/ ).
MLST has provided a more richly textured picture of bacteria within human populations and on strain variants that may be pathogenic to human, plants and animals. MLST technique was first used by Maiden et al. (1) to characterize Neisseria meningitidis using six loci. The application of MLST has clearly resolved the major meningococcal lineages known to be responsible for invasive disease around the world. To improve the level of discriminatory power between the major invasive lineages, seven loci are now being used and have been accepted by many laboratories as the method of choice for characterizing meningococcal isolates. It is a well known fact [ 7 ] that recombinational exchanges commonly occur in N. meningitidis , leading to rapid diversification of meningococcal clones. MLST has successfully provided a reliable method for characterization of clones within other bacterial species in which the rates of clonal diversification are generally lower.
S. aureus causes a number of diseases. Methicillin-resistant S. aureus ( MRSA ) has generated growing concerns over its resistance to almost all antibiotics except vancomycin. However, most serious S. aureus infections in the community, and many in hospitals, are caused by methicillin-susceptible isolates (MSSA) and there have been few attempts to identify the hypervirulent MSSA clones associated with serious disease. MLST was therefore developed to provide an unambiguous method of characterizing MRSA clones and for the identification of the MSSA clones associated with serious disease.
S. pyogenes causes diseases ranging from pharyngitis to life-threatening impetigo including necrotizing fasciitis. An MLST scheme for S. pyogenes has been developed. At present, the database ( mlst.net ) [ 8 ] contains the allelic profiles of isolates that represent the worldwide diversity of the organism and isolates from serious invasive disease. [ 9 ]
C. albicans is a fungal pathogen of humans and is responsible for hospital-acquired bloodstream infections. MLST technique has used to characterize C. albicans isolates. Combination of the alleles at the different loci results in unique diploid sequence types that can be used to discriminate strains. MLST has been shown successfully applied to study the epidemiology of C. albicans in the hospital as well as the diversity of C. albicans isolates obtained from diverse ecological niches including human and animal hosts.
The genus Cronobacter is composed of 7 species. Before 2007, the single species name Enterobacter sakazakii was applied to these organisms. The Cronobacter MLST was initially applied to distinguish between C. sakazakii and C. malonaticus because 16S rDNA sequencing is not always accurate enough, and biotyping is too subjective. [ 10 ] The Cronobacter MLST scheme uses 7 alleles; atpD , fusA , glnS , gltB , gyrB , infB and ppsA giving a concatenated sequence of 3036 bp for phylogenetic analysis (MLSA) and comparative genomics . [ 11 ] MLST has also been used in the formal recognition of new Cronobacter species. [ 12 ] The method has revealed a strong association between one genetic lineage, sequence type 4 (ST4), and cases of neonatal meningitis., [ 13 ] The Cronobacter MLST site is at http://www.pubMLST.org/cronobacter .
MLST appears best in population genetic study but it is expensive. Due to the sequence conservation in housekeeping genes, MLST sometimes lacks the discriminatory power to differentiate bacterial strains, which limits its use in epidemiological investigations. To improve the discriminatory power of MLST, a multi-virulence-locus sequence typing (MVLST) approach has been developed using Listeria monocytogenes . [ 14 ] MVLST broadens the benefits of MLST but targets virulence genes, which may be more polymorphic than housekeeping genes. Population genetics is not the only relevant factor in an epidemic. Virulence factors are also important in causing disease, and population genetic studies struggle to monitor these. This is because the genes involved are often highly recombining and mobile between strains in comparison with the population genetic framework. Thus, for example in Escherichia coli , identifying strains carrying toxin genes is more important than having a population genetics-based evaluation of prevalent strains.
The advent of second-generation sequencing technologies has made it possible to obtain sequence information across the entire bacterial genome at relatively modest cost and effort, and MLST can now be assigned from whole-genome sequence information, rather than sequencing each locus separately as was the practice when MLST was first developed. [ 15 ] Whole-genome sequencing provides richer information for differentiating bacterial strains (MLST uses approximately 0.1% of the genomic sequence to assign type while disregarding the rest of the bacterial genome). For example, whole-genome sequencing of numerous isolates has revealed the single MLST lineage ST258 of Klebsiella pneumoniae comprises two distinct genetic clades, [ 16 ] providing additional information about the evolution and spread of these multi-drug resistant organisms, and disproving the previous hypothesis of a single clonal origin for ST258. [ 17 ]
MLST databases contain the reference allele sequences and sequence types for each organism, and also isolate epidemiological data. The websites contain interrogation and analysis software which allow users to query their allele sequences and sequence types. MLST is widely used as a tool for researchers and public healthcare workers.
The majority of MLST databases are hosted at web server currently located in Oxford University ( pubmlst.org ).
The database hosted at the site hold the organism specific reference allele sequences and lists of STs for individual organisms.
To assist the gathering and formatting of the utilized sequences a simple and free plug-in for Firefox has been developed ( link Archived 2014-02-22 at the Wayback Machine ). | https://en.wikipedia.org/wiki/Multilocus_sequence_typing |
Multimedia refers to the integration of multiple forms of content, such as text, audio, images, video, and interactive elements into a single digital platform or application. This integration allows for a more immersive and engaging experience compared to traditional single-medium content. Multimedia is utilized in various fields, including education, entertainment, communication, game design, and digital art, reflecting its broad impact on modern technology and media.
Multimedia encompasses various types of content, each serving different purposes:
Multimedia can be recorded for playback on computers, laptops , smartphones , and other electronic devices. In the early years of multimedia, the term "rich media" was synonymous with interactive multimedia . Over time, hypermedia extensions brought multimedia to the World Wide Web, and streaming services became more common.
The term multimedia was coined by singer and artist Bob Goldstein (later ' Bobb Goldsteinn ') to promote the July 1966 opening of his "Lightworks at L'Oursin" show in Southampton, New York , Long Island. [ 1 ] Goldstein was perhaps aware of an American artist named Dick Higgins , who had two years previously discussed a new approach to art-making he called " intermedia ". [ 2 ]
On August 10, 1966, Richard Albarino of Variety borrowed the terminology, reporting: "Brainchild of song scribe-comic Bob (' Washington Square ') Goldstein, the 'Lightworks' is the latest multi-media music-cum-visuals to debut as discothèque fare." [ 3 ] Two years later, in 1968, the term "multimedia" was re-appropriated to describe the work of a political consultant, David Sawyer, the husband of Iris Sawyer—one of Goldstein's producers at L'Oursin.
In the intervening forty years, the word has taken on different meanings. In the late 1970s, the term referred to presentations consisting of multi-projector slide shows timed to an audio track. However, by the 1990s, 'multimedia' had taken on its current meaning.
In the 1993 first edition of Multimedia: Making It Work , Tay Vaughan declared, "Multimedia is any combination of text, graphic art, sound, animation, and video that is delivered by computer. When you allow the user – the viewer of the project – to control what and when these elements are delivered, it is interactive multimedia . When you provide a structure of linked elements through which the user can navigate, interactive multimedia becomes hypermedia ." [ 4 ] This book contained the Tempra Show software. [ 5 ] This was a later, rebranded version of the 1985 DOS multimedia software VirtulVideo Producer, about which the Smithsonian declared, "It is one of the first, if not the first, multi-media authoring systems on the market." [ 6 ]
The German language society Gesellschaft für deutsche Sprache recognized the word's significance and ubiquitousness in the 1990s by awarding it the title of German 'Word of the Year' in 1995. [ 7 ] The institute summed up its rationale by stating, "[Multimedia] has become a central word in the wonderful new media world". [ 8 ]
In common usage, multimedia refers to the usage of multiple media of communication, including video, still images, animation, audio, and text, in such a way that they can be accessed interactively. Video, still images, animation, audio, and written text are the building blocks on which multimedia takes shape. In the 1990s, some computers were called "multimedia computers" because they represented advances in graphical and audio quality, such as the Amiga 1000, which could produce 4096 colors (12-bit color), outputs for TVs and VCRs, and four-voice stereo audio. [ 9 ] Multimedia used to be saved on CD-ROMs, which could store roughly 700 MB of data, and floppy disks, which could only store 1.44 MB. Multimedia files are now easier to keep and retrieve because of the widespread usage of USB devices, cloud storage, and solid-state drives (SSDs), which provide significantly greater speed and space. [ 10 ] Greater storage allowed for larger digital media files and therefore more complex multimedia.
The term "video," if not used exclusively to describe motion photography, is ambiguous in multimedia terminology. Video is often used to describe the file format, delivery format, or presentation format instead of " footage ," which is used to distinguish motion photography from " animation " of rendered motion imagery. Multiple forms of information content are often not considered modern forms of presentation, such as audio or video. Likewise, single forms of information content with single methods of information processing (e.g., non-interactive audio) are often called multimedia, perhaps to distinguish static media from active media. In the fine arts , for example, Leda Luss Luyken 's ModulArt brings two key elements of musical composition and film into the world of painting: variation of a theme and movement of and within a picture, making ModulArt an interactive multimedia form of art. Performing arts may also be considered multimedia, considering that performers and props are multiple forms of both content and media.
In modern times, a multimedia device can be referred to as an electronic device, such as a smartphone, a video game system, or a computer. Each and every one of these devices has a main function but also has other uses beyond their intended purpose, such as reading, writing, recording video and audio, listening to music, and playing video games. This has led them to be called "multimedia devices." While previous media was always local, many are now handled through web-based solutions, particularly streaming.
Multimedia presentations are presentations featuring multiple types of media. The different types of media can include text , graphics , audio , video , and animations . These different types of media convey information to their target audience and effectively communicate with them. Videos are a great visual example to use in multimedia presentations because they can create visual aids to the presenter's ideas. They are commonly used among education and many other industries to benefit students and workers, as they effectively retain chunks of information in a limited amount of time and can be stored easily. Another example is charts and graphs, as the presenters can show their audience the trends using data associated with their researches. This provides the audience a visual idea of a company's capabilities and performances. [ 11 ] Audio also helps people understand the message being presented, as most modern videos are combined with audio to increase their efficiency, while animations are made to simplify things from the presenter's perspective. These technological methods allow efficient communication and understanding across a wide range of audiences (with an even wider range of abilities) throughout different fields, which create a more engaging experience for the audience. With the creation of Zoom and other such platforms, the immersive interaction is even more widely available to more people. Now people can be the audience for a presentation without having to travel to see it. This makes conferences more accessible to more people. "With the increasing remote working trends in today’s digital age, virtual meetings have become integral to business and social communication. For people with disabilities, these virtual meetings on digital platforms can sometimes pose accessibility challenges." [ 12 ]
Multimedia games and simulations may be used in a physical environment with special effects, with multiple users in an online network , or locally with an offline computer, game system , simulator , virtual reality , or augmented reality .
The various formats of technological or digital multimedia may be intended to enhance the users' experience, for example, to make it easier and faster to convey information. Or in entertainment or art, combine an array of artistic insights that include elements from different art forms to engage, inspire, or captivate an audience.
Enhanced levels of interactivity are made possible by combining multiple forms of media content. Online multimedia is increasingly becoming object-oriented and data-driven, enabling applications with collaborative end-user innovation and personalization on multiple forms of content over time. Examples of these range from multiple forms of content on websites, like photo galleries with both images (pictures) and titles (text) user-updated, to simulations whose coefficients, events, illustrations, animations, or videos are modifiable, allowing the multimedia "experience" to be altered without reprogramming. In addition to seeing and hearing, haptic technology enables virtual objects to be felt. Emerging technology involving illusions of taste and smell may also enhance the multimedia experience.
Multimedia may be broadly divided into linear and non-linear categories:
Multimedia presentations can be live or recorded:
Multimedia finds its application in various areas, including, but not limited to, advertisements , art , education , entertainment , engineering , medicine , mathematics , business , scientific research , and spatial temporal applications . Several examples are as follows:
Creative industries use multimedia for various purposes, ranging from fine arts, entertainment, commercial art, and journalism , to media and software services provided for any of the industries listed below. An individual multimedia designer may cover the spectrum throughout their career. Requests for their skills range from technical to analytical to creative. Multimedia, but more impressively in the modern day, the interactivity of the multimedia created forms the foundation for which most creative endeavors that take place online. Microsoft is one of the biggest computer industries in the world, and the core foundation of its success relies on the ability of multimedia designers to optimize user experience through interacting with their products.
Marketing and commercial practices increasingly rely on interactive multimedia, allowing for more sophisticated tactics and increased customer retention. Advertising companies heavily utilize social media, online interfaces, and television to promote products, while ads and websites that utilize pop-ups need shorter, more concise methods to be as efficient and pleasing to potential customers as possible. These platforms can be used by commercial businesses to specifically target their desired audience with a message, advertisement, or promotion. External and internal office communications are often developed by hired creative service firms to display information in a variety of situations. This can range from providing more engaging presentations to educating trainees or new workers on a company's policies or process. Commercial multimedia developers may also be hired to design for governmental services or nonprofit service applications, usually in the form of campaign websites and commercials aimed at the general public. Data mining within multimedia platforms can also allow advertisers to adjust their marketing techniques to quickly and efficiently understand the demographic of their target audience. [ 13 ] : Recently developed techniques include digital billboards, often placed on the side of buildings and wrapped around the edge or corner. Clips can then be added at differing angles to create a three-dimensional optical illusion , which is more likely to draw the attention of an observer.
Multimedia is heavily used in the entertainment industry, especially to develop special effects in movies and animations (VFX, 3D animation, etc.). Multimedia games are a popular pastime and are software programs available either as CD-ROMs or online. Video games are considered multimedia, as they meld animation, audio, and interactivity to give the player an immersive experience. While video games can vary in terms of animation style or audio type, the element of interactivity makes them a striking example of interactive multimedia . Interactive multimedia refers to multimedia applications that allow users to actively participate instead of just sitting by as passive recipients of information.
In the arts , there are multimedia artists who blend techniques using different media that in some way incorporate interaction with the viewer. Another approach entails the creation of multimedia that can be displayed in a traditional fine arts arena, such as an art gallery . Video has become an intrinsic part of many concerts and theatrical productions in the modern era and has spawned content creation opportunities for many media professionals. The first interactive music-only CD, No World Order by Todd Rundgren , was released in 1993. The CD acted as a customizable album. Although multimedia display material may be volatile, the survivability of the content is as strong as any traditional medium.
In education , multimedia is used to produce computer-based training courses (popularly called CBTs) and reference books like encyclopedias and almanacs. A CBT lets the user go through a series of presentations, text about a particular topic, and associated illustrations in various information formats.
Learning theory in the past decade has expanded dramatically because of the introduction of multimedia. Several lines of research have evolved, e.g., cognitive load and multimedia learning .
From multimedia learning (MML) theory, David Roberts has developed a large group lecture practice using PowerPoint based on the use of full-slide images in conjunction with a reduction of visible text (all text can be placed in the notes view' section of PowerPoint). [ 14 ] The method has been applied and evaluated in 9 disciplines. In each experiment, students' engagement and active learning have been approximately 66% greater than with the same material being delivered using bullet points, text, and speech, corroborating a range of theories presented by multimedia learning scholars like Sweller and Mayer . [ 15 ] The idea of media convergence is also becoming a major factor in education, particularly higher education. Defined as separate technologies such as voice (and telephony features), data (and productivity applications), and video that now share resources and interact with each other, media convergence is rapidly changing the curriculum in universities all over the world. Higher education has been implementing the use of social media applications such as Twitter, YouTube, Facebook, etc. to increase student collaboration and develop new processes in how information can be conveyed to students. [ 16 ]
Multimedia provides students with an alternate means of acquiring knowledge designed to enhance teaching and learning through various media and platforms. [ citation needed ] In the 1960s, technology began to expand into classrooms through devices such as screens and telewriters. [ 17 ] This technology allows students to learn at their own pace and gives teachers the ability to observe the individual needs of each student. The capacity for multimedia to be used in multi-disciplinary settings is structured around the idea of creating a hands-on learning environment through the use of technology. [ citation needed ] Lessons can be tailored to the subject matter as well as personalized to the students' varying levels of knowledge on the topic. Learning content can be managed through activities that utilize and take advantage of multimedia platforms. [ citation needed ] This kind of usage of modern multimedia encourages interactive communication between students and teachers and opens feedback channels, introducing an active learning process, especially with the prevalence of new media and social media . [ 18 ] Technology has impacted multimedia as it is largely associated with the use of computers or other electronic devices and digital media due to its capabilities concerning research, communication, problem-solving through simulations, and feedback opportunities. [ 19 ] The innovation of technology in education through the use of multimedia allows for diversification among classrooms to enhance the overall learning experience for students. [ 20 ]
Within education, video games, specifically fast-paced action games, are able to play a big role in improving cognitive abilities involving attention, task switching, and resistance to distractors. Research also shows that, though video games may take time away from schoolwork, implementing games into the school curriculum has an increased probability of moving attention from games to curricular goals. [ 21 ]
Multimedia is a robust education methodology within the social work context. The five different types of multimedia that support the education process are narrative media , interactive media , communicative media, adaptive media, and productive media. Contrary to long-standing belief, multimedia technology in social work education existed before the prevalence of the internet. It takes the form of images, audio, and video into the curriculum.
First introduced to social work education by Seabury & Maple in 1993, multimedia technology is utilized to teach social work practice skills, including interviewing, crisis intervention, and group work. In comparison with conventional teaching methods, including face-to-face courses, multimedia education shortens transportation time, increases knowledge and confidence in a richer and more authentic context for learning, generates interaction between online users, and enhances understanding of conceptual materials for novice students.
In an attempt to examine the impact of multimedia technology on students' studies, A. Elizabeth Cauble & Linda P. Thurston conducted research in which Building Family Foundations (BFF), an interactive multimedia training platform, was utilized to assess social work students' reactions to multimedia technology on variables of knowledge, attitudes, and self-efficacy . The results state that respondents show a substantial increase in academic knowledge, confidence, and attitude. Multimedia also benefits students because it brings experts online, fits students' schedule, and allows students to choose courses that suit them.
Mayer's Cognitive Theory of Multimedia Learning suggests that "people learn more from words and pictures than from words alone." According to Mayer and other scholars, multimedia technology stimulates people's brains by implementing visual and auditory effects and thereby assists online users to learn efficiently. Researchers suggest that when users establish dual channels while learning, they tend to understand and memorize better. The mixed literature of this theory is still present in the fields of multimedia and social work. [ 22 ] [ 23 ] [ 24 ]
With the spread and development of the English language around the world, multimedia has become an important way of communicating between different people and cultures. Multimedia technology creates a platform where language can be taught. The traditional form of teaching English as a Second Language in classrooms has drastically changed with the prevalence of technology, making it easier for students to obtain language learning skills. Multimedia motivates students to learn more languages through audio, visual, and animation support. It also helps create English contexts since an important aspect of learning a language is developing their grammar, vocabulary, and knowledge of pragmatics and genres. In addition, cultural connections in terms of forms, contexts, meanings, and ideologies have to be constructed. [ citation needed ] By improving thought patterns, multimedia develops students' communicative competence by improving their capacity to understand the language. [ 25 ] One of the studies, carried out by Izquierdo, Simard and Pulido, presented the correlation between "Multimedia Instruction (MI) and learners' second language (L2)" [ 26 ] and its effects on learning behavior. Their findings, based on Gardner 's theory of the " socio-educational model of learner motivation and attitudes," show that there is easier access to language learning materials as well as increased motivation with MI along with the use of computer-assisted language learning .
Newspaper companies all over the world are trying to embrace the new phenomenon by implementing its practices in their work. While some have been slow to come around, other major newspapers like The New York Times , USA Today , and The Washington Post are setting a precedent for the positioning of the newspaper industry in a globalized world. To keep up with the changing world of multimedia, journalistic practices are adopting and utilizing different multimedia functions through the inclusion of visuals such as varying audio, video, text, etc. in their writings. [ 27 ]
News reporting is not limited to traditional media outlets. Freelance journalists can use different new media to produce multimedia pieces for their news stories. It engages global audiences and tells stories with technology, which develops new communication techniques for both media producers and consumers. The Common Language Project, later renamed The Seattle Globalist , is an example of this type of multimedia journalism production.
Multimedia reporters who are mobile (usually driving around a community with cameras, audio and video recorders, and laptop computers) are often referred to as mojos , or mobile journalists.
Software engineers may use multimedia in computer simulations for anything from entertainment to training , such as military or industrial training. Multimedia for software interfaces is often done as a collaboration between creative professionals and software engineers. Multimedia helps expand the teaching practices that can be found in engineering to allow for more innovative methods to not only educate future engineers but to help evolve the scope of understanding of where multimedia can be used in specialized engineer careers like software engineers. [ 28 ]
Multimedia is also allowing major car manufacturers, such as Ford and General Motors , to expand the design and safety standards of their cars. By using a game engine and virtual reality glasses, these companies are able to test the safety features and the design of the car before a prototype is even made. Building a car virtually reduces the time it takes to produce new vehicles, cutting down on the time needed to test designs and allowing the designers to make changes in real time. It also reduces expenses since, with a virtual car, making real-world prototypes is no longer needed. [ 29 ]
In mathematical and scientific research , multimedia is mainly used for modeling and simulation with binary code. For example, a scientist can look at a molecular model of a particular substance and manipulate it to arrive at a new substance. Representative research can be found in journals such as the Journal of Multimedia . One well-known example of this being applied would be in the movie Interstellar , where Executive Director Kip Thorne helped create one of the most realistic depictions of a black hole in film. The visual effects team under Paul Franklin took Kip Thorne's mathematical data and applied it into their own visual effects engine called "Double Negative Gravitational Renderer," a.k.a. "Gargantua," to create a "real" black hole used in the final cut. Later on, the visual effects team went on to publish a black hole study. [ citation needed ]
Medical professionals and students have a wide variety of ways to learn new techniques and procedures through interactive media, online courses, and lectures. The methods of conveying information to students have drastically evolved with the help of multimedia. From the 1800s to today, lessons are commonly taught using chalkboards. Projected aids, such as the epidiascope and slide projectors, were introduced into classrooms around the 1960s. [ 30 ] With the growing use of computers, the medical field has begun to incorporate new devices and procedures to assist in teaching students, performing procedures, and analyzing patient data. As well as providing that data in a meaningful way to the patients. [ 31 ]
Virtual reality is a technology that creates a simulated environment, often using computer-generated imagery or a combination of real and virtual content, to immerse users in an interactive and lifelike experience. The aim of virtual reality is to make users feel as if they are physically present in a different environment, even though they are typically still physically located in the real world. Virtual reality finds applications across various fields, including gaming, education, healthcare, training, and entertainment. In gaming, users can be transported to fantastical worlds, experiencing games in a more immersive way. In education, VR can provide realistic simulations for training purposes, allowing users to practice skills in a risk-free environment. Healthcare professionals use VR for therapeutic purposes and medical training. The U.S. Air Force has shown using VR for training programs for their new pilots to simulate piloting an aircraft. People have also used VR in a court room to show the judge how an interaction took place to try and get a clearer understanding of why the client might have done what they did. [ 32 ] This allows new pilots to learn in a safe environment and get comfortable before getting in a real aircraft.
Head-mounted display (HMD): Users wear a headset that covers their eyes and ears, providing visual and auditory stimuli. These headsets are equipped with screens that display the virtual environment, and some may also have built-in speakers or headphones for audio.
Motion tracking: Sensors track the user's movements, allowing them to interact with the virtual world. This can include head movements, hand gestures, and sometimes even full-body movements, enhancing the sense of immersion.
Input devices: Controllers or other input devices are used to interact with the virtual environment. These devices can simulate hands or tools, enabling users to manipulate objects or navigate within the virtual space.
Computer processing: Powerful computers or gaming consoles are often required to generate and render the complex graphics and simulations needed for a convincing virtual experience.
Augmented reality overlays digital content or output onto the real world using media such as audio, animation, and text. Augmented reality became widely popular only in the 21st century; however, some of the earlier versions of such were things like the Sega Genesis Activator Controller back in 1992, which allowed users to literally stand in an octagon and control in-game movement with physical movement, or to stretch back even further, the R.O.B. NES Robot back in 1984, which, with its array of accessories, was able to also provide users with the sensation of holding a firearm. These multimedia input devices are among the earliest of the augmented reality devices, allowing users to input commands to facilitate a different user experience. A more modern example of augmented reality is Pokémon GO , a mobile game released on July 6, 2016, which allows users to see a Pokémon in a real-world environment. | https://en.wikipedia.org/wiki/Multimedia |
Multimedia Broadcast Multicast Services ( MBMS ) is a point-to-multipoint interface specification for existing 3GPP cellular networks , which is designed to provide efficient delivery of broadcast and multicast services , both within a cell as well as within the core network . For broadcast transmission across multiple cells, it defines transmission via single-frequency network configurations. The specification is referred to as Evolved Multimedia Broadcast Multicast Services ( eMBMS ) when transmissions are delivered through an LTE (Long Term Evolution) network. eMBMS is also known as LTE Broadcast . [ 1 ]
Target applications include mobile TV and radio broadcasting, live streaming video services, as well as file delivery and emergency alerts.
Questions remain whether the technology is an optimization tool for the operator or if an operator can generate new revenues with it. Several studies have been published on the domain identifying both cost savings and new revenues. [ 2 ]
In 2013, [ 3 ] Verizon announced that it would launch eMBMS services in 2014, over its nationwide (United States) LTE networks. AT&T subsequently announced plans to use the 700 MHz Lower D and E Block licenses it acquired in 2011 from Qualcomm for an LTE Broadcast service. [ 4 ]
Several major operators worldwide have been lining-up to deploy and test the technology. The frontrunners being Verizon in the United States, [ 5 ] Kt and Reliance [ 6 ] in Asia , and recently EE [ 7 ] and Vodafone in Europe . [ 8 ]
In January 2014, Korea’s Kt launched the first commercial LTE Broadcast service. [ 9 ] The solution includes Kt’s internally developed eMBMS Bearer Service, and Samsung mobile devices fitted with the ENENSYS Expway Middleware as the eMBMS User Service.
In February 2014, Verizon demonstrated the potential of LTE Broadcast during Super Bowl XLVIII , using Samsung Galaxy Note 3s , fitted with ENENSYS Expway's eMBMS User Service. [ 10 ]
In July 2014, Nokia demonstrated the use of LTE Broadcast to replace Traditional Digital TV. [ 11 ] This use case remains controversial as some study are doubting about the capability of LTE Broadcast to address this use case efficiently in its current version. [ 12 ]
Also in July 2014, BBC Research & Development and EE demonstrated LTE Broadcast during the XX Commonwealth Games in Glasgow , Scotland using equipment from Huawei and Qualcomm. [ 13 ] [ 14 ]
In August 2014, Ericsson and Polkomtel successfully tested LTE Broadcast technology by streaming the opening game of the 2014 World Volleyball Championship to hundreds of guests at Warsaw’s National Stadium in Poland on August 30. [ 15 ]
In June 2015, BBC Research & Development and EE demonstrated LTE Broadcast during the FA Cup final in the U.K. [ 16 ] [ 17 ]
In September 2015, Verizon demonstrated eMBMS by broadcasting INDYCAR races. [ 18 ]
In October 2015, Verizon commercially launched their Go90 eMBMS service. Go90 offers both On-Demand and LiveTV, in both Unicast and Broadcast, and supports more than 10 different LTE Broadcast mobile devices. [ 19 ] [ 20 ] [ 21 ] Verizon ceased operating the go90 service on July 31, 2018. [ 22 ]
In February 2016, Akamai demonstrated with ENENSYS Expway, delivery of video streams across LTE networks with live on the fly switching from unicast to broadcast, at Mobile World Congress 2016. [ 23 ]
In April 2016, Verizon, Telstra , KT and EE launched the LTE Broadcast Alliance. [ 24 ]
As of January 2019, the Global Mobile Suppliers Association had identified 41 operators that have invested in eMBMS (including those considering/testing/trialling, deploying or piloting and those that have deployed or launched eMBMS). Five operators state they have now deployed eMBMS or launched some sort of commercial service using eMBMS. [ 1 ]
The range of chipsets available that can support eMBMS has been steadily growing, with three mobile processors/platforms released since March 2018. GSA has identified 69 chipsets supporting eMBMS, and there are at least 59 devices that support eMBMS (in some instances after operator-specific upgrades). [ 25 ]
Main competing technologies of MBMS include DVB-H / DVB-T , DVB-SH , DMB , ESM- DAB , and MediaFLO . However, due to spectrum scarcity and the cost of building new broadcast infrastructure some of these technologies may not be viable. MediaFLO has been deployed commercially in the US by Verizon Wireless through their relationship with MediaFLO USA, Inc. (a subsidiary of Qualcomm) however the service was shut down in early 2011. [ 26 ] DMB and DVB-H trials have been ongoing for more than a year now, like those during the football 2006 championships in Germany.
Huawei's proprietary CMB is a precursor to the Multimedia Broadcast Multicast Service. It was specified in 3GPP R6 and is using existing UMTS infrastructure. Huawei says that CMB is based on existing UMTS infrastructure and real time streaming application protocol.
The most significant competition is from services that stream individual video feeds to users over unicast data connections. While less efficient in certain situations, particularly the traditional case where everyone watches the same stream simultaneously, the user convenience of individual streaming has taken over the vast majority of the mobile media streaming market.
The MBMS feature is split into the MBMS Bearer Service and the MBMS User Service and has been defined to be offered over both UTRAN (i.e. WCDMA , TD-CDMA and TD-SCDMA) and LTE (where it is often referred to as eMBMS). The MBMS Bearer Service includes a Unicast and a Broadcast Mode. MBMS Operation On-Demand (MOOD) allows dynamic switching between Unicast and Broadcast over LTE, based on configured triggers. The MBMS Bearer Service uses IP multicast addresses for the IP flows. The advantage of the MBMS Bearer Service compared to unicast bearer services (interactive, streaming, etc.) is that the transmission resources in the core and radio networks are shared. [ 27 ] One MBMS packet flow is replicated by GGSN , SGSN and RNCs. MBMS may use an advanced counting scheme to decide, whether or not zero, one or more dedicated (i.e. unicast) radio channels lead to a more efficient system usage than one common (i.e. broadcast) radio channel.
The MBMS User Service is basically the MBMS Service Layer and offers two different data Delivery Methods:
MBMS has been standardized in various groups of 3GPP (Third Generation Partnership Project), and the first phase standards are found in UMTS release 6. As Release 6 was functionally frozen by the 3rd quarter of 2004, practical network implementations may be expected by the end of 2007, and the first functional mobile terminals supporting MBMS are estimated to be available by also end of 2007.
eMBMS has been standardized in various groups of 3GPP as part of LTE release 9. The LTE version of MBMS, referred to as Multicast-broadcast single-frequency network (MBSFN), supports broadcast only services and is based on a Single Frequency Network (SFN) based OFDM waveform and so is functional similar to other broadcast solutions such as DVB-H, -SH and -NGH. In Release 14, the 3GPP enhanced the specifications for eMBMS with a view to making the technology more attractive for deployment by operators and broadcasters. The 3GPP’s work on the next generation of technology in Release 16 includes a study on LTE-based broadcast on 5G networks, [ 28 ] MBMS APIs for mission-critical services and MBMS user services for IoT. [ 1 ]
MBMS Bearer Service (Distribution Layer):
MBMS User Service (Service Layer): | https://en.wikipedia.org/wiki/Multimedia_Broadcast_Multicast_Service |
Multimedia City ( Polish : Miasteczko Multimedialne ) an innovative project, that has been realized in Nowy Sącz , in southern Poland . It has started in 2006, on the initiative of leaders and alumnus from WSB-NLU (Wyższa Szkoła Biznesu — National Luis University), with Krzysztof Pawlowski and Krzysztof Wnek at the head of. The Multimedia City's main goal is to initiate the cooperation between science and business in testing, incubating and commercialization of innovative projects of new technologies idea to economy.
The aim of the Multimedia City Project is to establish in Nowy Sącz Center of Innovation working in the field of multimedia and informative system . The strategic goal of Multimedia City is to become one of the most innovative centers in the world, which are working on application of multimedia in education, business and entertainment. The individual elements of Multimedia City are enabling to implement innovation to economy in accordance with the following stages of innovation chain: fresh ideas and innovative know-how , testing ideas through research and development phase, and implementation and adaptation of the innovative solutions in enterprises. [ 1 ]
The following elements of Multimedia City currently operate: [ when? ]
Numerous companies like Microsoft , CISCO or Lewiatan Business Angels already cooperate with the innovative project from Nowy Sacz. [ 4 ]
The following are elements of Multimedia City that will operate with the realization of the project: [ 1 ]
The following are areas of multimedia technologies that the project strongly supports: [ 1 ]
On 24 August 2010, Multimedia City signed an agreement with the Polish Agency for Enterprise Development for a bailout to build the Technology Park. Multimedia City will be given 25 mln € for its infrastructure. Thanks to the European funds from the Innovative Economy Operation Program (indicative list) the modern research and development infrastructure of the Technology park will be built till 2012. That investment will provide a chance for Nowy Sącz to become the Polish Innovation Valley.
In September 2010, building of a complex infrastructure for the Technology Park (16 thousand square meter) has started. In Multimedia City's main building will be found specialist technology laboratories of amongst others: 3D and special effects , virtual reality , post-production , sound, motion capture as well as modern offices surface. [ 2 ] | https://en.wikipedia.org/wiki/Multimedia_City |
The Multimedia PC ( MPC ) is a recommended configuration for a personal computer (PC) with a CD-ROM drive. The standard was set and named by the Multimedia PC Marketing Council (MPMC), which was a working group of the Software Publishers Association (SPA, now the Software and Information Industry Association ). The MPMC comprised companies including Microsoft , Creative Labs , Dell , Gateway , and Fujitsu . Any PC with the required standards could be called an "MPC" by licensing the use of the logo from the SPA.
CD-ROM drives were just coming to market in 1990, and it was difficult to concisely communicate to a consumer all the hardware requirements for using "multimedia software", which mostly meant "displaying video synced with audio on a PC via a CD-ROM drive". The MPC standard was supposed to communicate this concisely, so a consumer buying hardware or software could simply look for the MPC logo and be assured of compatibility.
The MPC program had mixed results primarily because of the vast number of PCs sold under different brands, and once Windows became ubiquitous on PCs, specifying minimum or recommended Windows versions and features was often clearer to consumers than the MPC nomenclature. As the standardized term failed to catch on, and as the Software Publishers Association turned away from consumer software in the late 1990s, interest in the MPC standard vanished. The problem of software labeling continues, especially in the field of computer games , where a multitude of 3D video cards has been manufactured with an extremely wide range of capabilities, and no common industry labeling standard to let consumers know whether their card is powerful enough to run a particular game.
No further specification beyond MPC Level 3 in 1996 was published; the Multimedia PC Marketing Council dissolved not too long afterward. [ 1 ]
The first MPC minimum standard, set in 1991, was:
In 1993, an MPC Level 2 minimum standard was announced:
In 1996, MPC Level 3 was announced: | https://en.wikipedia.org/wiki/Multimedia_PC |
A multimedia computer is a computer that is optimized for multimedia performance.
Early home computers lacked the power and storage necessary for true multimedia. The games for these systems, along with the demo scene , were able to achieve high sophistication and technical polish using only simple, blocky graphics and digitally generated sound.
The Amiga 1000 from Commodore International has been called the first multimedia computer. [ 1 ] Its groundbreaking animation, graphics and sound technologies enabled multimedia content to flourish. Famous demos such as the Boing Ball [ 2 ] and Juggler [ 3 ] showed off the Amiga's abilities. Later the Atari ST series and Apple Macintosh II extended the concept; the Atari integrated a MIDI port and was the first computer under US$1000 to have 1 megabyte of RAM , which is a realistic minimum for multimedia content, and the Macintosh was the first computer able to display true photorealistic graphics as well as integrating a CD-ROM drive, whose high capacity was essential for delivering multimedia content in the pre- Internet era.
While the Commodore machines had the hardware to present multimedia of the kinds listed above, they lacked a way to create it easily. One of the earliest authoring systems on the market for creating and deploying multimedia content can be found in the archives of the Smithsonian Institution and is called VirtualVideo . [ 4 ] It consisted of a standard PC with an added digital imaging board, an added digital audio capture board (that was sold as a phone answering device), and the DOS authoring software, VirtualVideo Producer . The system stored content on a local hard drive but could use networked computer storage as well. The name for the software was used because at the time, the mid-1980s, the term multimedia was used to describe slide shows with sound. This software was later sold as Tempra, [ 5 ] and in 1993 was included with Tay Vaugh's first edition of Multimedia: Making It Work .
Multimedia capabilities were not common on IBM PC compatibles until the advent of Windows 3.0 and the MPC standards in the early 1990s. The original PCs were devised as "serious" business machines and colorful graphics and powerful sound abilities weren't a priority. The few games available suffered from slow video hardware, PC speaker sound and limited color palette when compared to its contemporaries. But as PCs penetrated the home market in the late 1980s, a thriving industry arose to equip PCs to take advantage of the latest sound, graphics and animation technologies. Creative 's SoundBlaster series of sound cards, as well as video cards from ATI , Nvidia and Matrox , soon became standard equipment for most PCs sold.
As of 2021 [update] , most PCs have good multimedia features. They have dual or more core CPUs clocked at 2.0 GHz or faster, at least 4 GB of RAM and an integrated graphics processing unit . Popular graphics cards include Nvidia GeForce or AMD Radeon . The Intel Core and AMD Ryzen platform, and Microsoft Windows 10 and Windows 11 are some of today's products that excel at multimedia computing.
More recently, high-performance devices have become more compact, and multimedia computer capabilities are found in mobile devices such as the Apple iPhone and many Android phones , featuring DVD-like video quality, multi- megapixel cameras, music and video players, and internet call (VoIP) functionality. | https://en.wikipedia.org/wiki/Multimedia_computer |
A Multimedia database ( MMDB ) is a collection of related for multimedia data . [ 1 ] The multimedia data include one or more primary media data types such as text , images , graphic objects (including drawings , sketches and illustrations ) animation sequences, audio and video .
A Multimedia Database Management System ( MMDBMS ) is a framework that manages different types of data potentially represented in a wide diversity of formats on a wide array of media sources. It provides support for multimedia data types , and facilitate for creation, storage, access, query and control of a multimedia database. [ 2 ]
A Multimedia Database (MMDB) hosts one or more multimedia data types [ 3 ] (i.e. text, images, graphic objects, audio, video, animation sequences).
These data types are broadly categorized into three classes :
Additionally, a Multimedia Database (MMDB) needs to manage additional information pertaining to the actual multimedia data.
The information is about the following:
The last three types are called metadata as they describe several different aspects of the media data. The media keyword data and media feature data are used as indices for searching purpose. The media format data is used to present the retrieved information.
Like the traditional databases , Multimedia databases should address the following requirements:
Multimedia databases should have the ability to uniformly query data (media data, textual data ) represented in different formats and have the ability to simultaneously query different media sources and conduct classical database operations across them. ( Query support)
They should have the ability to retrieve media objects from a local storage device in a good manner. (Storage support)
They should have the ability to take the response generated by a query and develop a presentation of that response in terms of audio-visual media and have the ability to deliver this presentation. (Presentation and delivery support)
Examples of multimedia database application areas: | https://en.wikipedia.org/wiki/Multimedia_database |
Multimedia fugacity model is a model in environmental chemistry that summarizes the processes controlling chemical behavior in environmental media by developing and applying of mathematical statements or "models" of chemical fate. [ 1 ]
Most chemicals have the potential to migrate from the medium to medium. Multimedia fugacity models are utilized to study and predict the behavior of chemicals in different environmental compartments. [ 1 ] [ 2 ]
The models are formulated using the concept of fugacity , which was introduced by Gilbert N. Lewis in 1901 as a criterion of equilibrium and convenient method of calculating multimedia equilibrium partitioning.
The fugacity of chemicals is a mathematical expression that describes the rates at which chemicals diffuse , or are transported between phases. The transfer rate is proportional to the fugacity difference that exists between the source and destination phases.
For building the model, the initial step is to set up a mass balance equation for each phase in question that includes fugacities, concentrations, fluxes and amounts. The important values are the proportionality constant, called fugacity capacity expressed as Z-values (SI unit: mol/m 3 Pa) for a variety of media, and transport parameters expressed as D-values (SI unit: mol/Pa h) for processes such as advection , reaction and intermedia transport. The Z-values are calculated using the equilibrium partitioning coefficients of the chemicals, Henry's law constant and other related physical-chemical properties. [ 1 ] [ 3 ]
There are four levels of multimedia fugacity Models applied for prediction of fate and transport of organic chemicals in the multicompartmental environment: [ 1 ] [ 4 ] [ 5 ] [ 6 ]
Depending on the number of phases and complexity of processes different level models are applied. Many of the models apply to steady-state conditions and can be reformulated to describe time-varying conditions by using differential equations. The concept has been used to assess the relative propensity for chemicals to transform from temperate zones and “condense out” at the polar regions.
The multicompartmental approach has been applied to the “quantitative water air sediment interaction" or "QWASI" model designed to assist in understanding chemical fate in lakes. [ 7 ] Another application found in POPCYCLING-BALTIC model, which is describing fate of persistent organic pollutants in Baltic region. [ 8 ] | https://en.wikipedia.org/wiki/Multimedia_fugacity_model |
Multimedia search enables information search using queries in multiple data types including text and other multimedia formats.
Multimedia search can be implemented through multimodal search interfaces, i.e., interfaces that allow to submit search queries not only as textual requests, but also through other media.
We can distinguish two methodologies in multimedia search:
Search is made using the layers in metadata which contain information of the content of a multimedia file. Metadata search is easier, faster and effective because instead of working with complex material, such as an audio, a video or an image, it searches using text.
There are three processes which should be done in this method:
In query by example , the element used to search is a multimedia content (image, audio, video). In other words, the query is a media. Often, it's used audiovisual indexing . It will be necessary to choose the criteria we are going to use for creating metadata. The process of search can be divided in three parts:
There are two big search families, in function of the content:
Inside this family we can distinguish two topics: image search and video search
There are different methods of audio searching : | https://en.wikipedia.org/wiki/Multimedia_search |
The 3GPP / NGN IP Multimedia Subsystem (IMS) multimedia telephony service ( MMTel ) is a global standard based on the IMS, offering converged, fixed and mobile real-time multimedia communication using the media capabilities such as voice, real-time video, text, file transfer and sharing of pictures, audio and video clips. With MMTel, users have the capability to add and drop media during a session. You can start with chat, add voice (for instance Mobile VoIP ), add another caller, add video, share media and transfer files, and drop any of these without losing or having to end the session. MMTel is one of the registered ICSI (IMS Communication Service Identifier) feature tags. [ 1 ]
The MMTel standard is a joint project between the 3GPP and ETSI / TISPAN standardization bodies. The MMTel standard is today the only global standard that defines an evolved telephony service that enables real-time multimedia communication with the characteristics of a telephony service over both fixed broadband, fixed narrowband and mobile access types. MMTel also provides a standardized network-to-network interface (NNI). This allow operators to interconnect their networks which in turn enables users belonging to different operators to communicate with each other, using the full set of media capabilities and supplementary services defined within the MMTel service definition.
One of the main differences with the MMTel standard is that, in contrast to legacy circuit switched telephony services, IP transport is used over the mobile access. This means that the mobile access technologies that are in main focus for MMTel are access types such as high-speed packet access (HSPA), 3GPP long-term evolution (LTE) and EDGE Evolution that all are developed with efficient IP transport in mind.
MMTel allows a single SIP session to control virtually all MMTel supplementary services and MMTel media. All available media components can easily be accessed or activated within the session. Employing a single session for all media parts means that no additional sessions need to be set up to activate video, to add new users, or to start transferring a file. Even though it is possible to manage single-session user scenarios with several sessions – for instance, using a circuit-switched voice service that is complemented with a packet-switched video session, a messaging service or both – there are some concrete benefits to MMTel’s single-session approach. A single SIP session in an all-IP environment benefits conferencing; in particular, lip synchronization, which is quite complex when the voice part is carried over a circuit-switched service and the video part is carried over a packet-switched service. In fixed-mobile convergence scenarios, the single-session approach enables all media parts of the multimedia communication solution to interoperate. | https://en.wikipedia.org/wiki/Multimedia_telephony |
A multimeter (also known as a multi-tester, volt-ohm-milliammeter , volt-ohmmeter or VOM, avometer or ampere-volt-ohmmeter ) [ 1 ] is a measuring instrument that can measure multiple electrical properties. [ 2 ] [ 3 ] A typical multimeter can measure voltage , resistance , and current , [ 4 ] in which case can be used as a voltmeter , ohmmeter , and ammeter . Some feature the measurement of additional properties such as temperature and capacitance .
Analog multimeters use a microammeter with a moving pointer to display readings. [ 5 ] Digital multimeters (DMMs) have numeric displays and are more precise than analog multimeters as a result. Meters will typically include probes that temporarily connect the instrument to the device or circuit under test, and offer some intrinsic safety features to protect the operator if the instrument is connected to high voltages that exceed its measurement capabilities.
Multimeters vary in size, features, and price. [ 6 ] They can be portable handheld devices or highly-precise bench instruments. [ 7 ]
Multimeters are used in diagnostic operations to verify the correct operation of a circuit or to test passive components for values in tolerance with their specifications.
The first attested usage of the word "multimeter" listed by the Oxford English Dictionary is from 1907. [ 8 ]
The first moving-pointer current-detecting device was the galvanometer in 1820. These were used to measure resistance and voltage by using a Wheatstone bridge , and comparing the unknown quantity to a reference voltage or resistance. While useful in the lab, the devices were very slow and impractical in the field. These galvanometers were bulky and delicate.
The D'Arsonval–Weston meter movement uses a moving coil which carries a pointer and rotates on pivots or a taut band ligament. The coil rotates in a permanent magnetic field and is restrained by fine spiral springs which also serve to carry current into the moving coil. It gives proportional measurement rather than just detection, and deflection is independent of the orientation of the meter. Instead of balancing a bridge, values could be directly read off the instrument's scale, which made measurement quick and easy.
The basic moving coil meter is suitable only for direct current measurements, usually in the range of 10 μA to 100 mA. It is easily adapted to read heavier currents by using shunts (resistances in parallel with the basic movement) or to read voltage using series resistances known as multipliers. To read alternating currents or voltages, a rectifier is needed. One of the earliest suitable rectifiers was the copper oxide rectifier developed and manufactured by Union Switch & Signal Company, Swissvale, Pennsylvania, later part of Westinghouse Brake and Signal Company, from 1927. [ 9 ]
The invention of the first multimeter is attributed to British Post Office engineer, Donald Macadie, who became dissatisfied with the need to carry many separate instruments required for maintenance of telecommunication circuits . [ 10 ] Macadie invented an instrument which could measure amperes (amps), volts and ohms , so the multifunctional meter was then named Avometer . [ 11 ] The meter comprised a moving coil meter, voltage and precision resistors, and switches and sockets to select the range.
The first Avometer had a sensitivity of 60 Ω/V, three direct current ranges (12 mA, 1.2 A, and 12 A), three direct voltage ranges (12, 120, and 600 V or optionally 1,200 V), and a 10,000 Ω resistance range. An improved version of 1927 increased this to 13 ranges and 166.6 Ω/V (6 mA) movement. A "Universal" version having additional alternating current and alternating voltage ranges was offered from 1933 and in 1936 the dual-sensitivity Avometer Model 7 offered 500 and 100 Ω/V. [ 12 ] Between the mid-1930s until the 1950s, 1,000 Ω/V became a de facto standard of sensitivity for radio work and this figure was often quoted on service sheets. However, some manufacturers such as Simpson, Triplett and Weston, all in the US, produced 20,000 Ω/V VOMs before the Second World War and some of these were exported. After 1945–46, 20,000 Ω/V became the expected standard for electronics, but some makers offered even more sensitive instruments. For industrial and other "heavy-current" use low sensitivity multimeters continued to be produced and these were considered more robust than the more sensitive types.
The Automatic Coil Winder and Electrical Equipment Company (ACWEECO), founded in 1923, was set up to manufacture the Avometer and a coil winding machine also designed and patented by MacAdie. Although a shareholder of ACWEECO, Mr MacAdie continued to work for the Post Office until his retirement in 1933. His son, Hugh S. MacAdie, joined ACWEECO in 1927 and became Technical Director. [ 13 ] [ 14 ] [ 11 ] The first AVO was put on sale in 1923, and many of its features remained almost unaltered through to the last Model 8.
Pocket-watch-style meters were in widespread use in the 1920s. The metal case was typically connected to the negative connection, an arrangement that caused numerous electric shocks. The technical specifications of these devices were often crude, for example the one illustrated has a resistance of just 25 Ω/V, a non-linear scale and no zero adjustment on both ranges.
Vacuum tube voltmeters or valve voltmeters (VTVM, VVM) were used for voltage measurements in electronic circuits where high input impedance was necessary. The VTVM had a fixed input impedance of typically 1 MΩ or more, usually through use of a cathode follower input circuit, and thus did not significantly load the circuit being tested. VTVMs were used before the introduction of electronic high-impedance analog transistor and field effect transistor voltmeters (FETVOMs). Modern digital meters (DVMs) and some modern analog meters also use electronic input circuitry to achieve high input impedance—their voltage ranges are functionally equivalent to VTVMs. The input impedance of some poorly designed DVMs (especially some early designs) would vary over the course of a sample-and-hold internal measurement cycle, causing disturbances to some sensitive circuits under test.
The first digital multimeter was manufactured in 1955 by Non Linear Systems. [ 15 ] [ 16 ]
It is claimed that the first handheld digital multimeter was developed by Frank Bishop of Intron Electronics in 1977, [ 17 ] which at the time presented a major breakthrough for servicing and fault finding in the field.
Any meter will load the circuit under test to some extent. For example, a multimeter using a moving coil movement with full-scale deflection current of 50 microamps (μA), the highest sensitivity commonly available, must draw at least 50 μA from the circuit under test for the meter to reach the top end of its scale. This may load a high-impedance circuit so much as to affect the circuit, thereby giving a low reading. The full-scale deflection current may also be expressed in terms of "ohms per volt" (Ω/V). The ohms per volt figure is often called the "sensitivity" of the instrument. Thus a meter with a 50 μA movement will have a "sensitivity" of 20,000 Ω/V. "Per volt" refers to the fact that the impedance the meter presents to the circuit under test will be 20,000 Ω multiplied by the full-scale voltage to which the meter is set. For example, if the meter is set to a range of 300 V full scale, the meter's impedance will be 6 MΩ. 20,000 Ω/V is the best (highest) sensitivity available for typical analog multimeters that lack internal amplifiers. For meters that do have internal amplifiers (VTVMs, FETVMs, etc.), the input impedance is fixed by the amplifier circuit.
Additional scales such as decibels , and measurement functions such as capacitance , transistor gain , frequency , duty cycle , display hold, and continuity which sounds a buzzer when the measured resistance is small have been included on many multimeters. While multimeters may be supplemented by more specialized equipment in a technician's toolkit, some multimeters include additional functions for specialized applications (temperature with a thermocouple probe, inductance , connectivity to a computer , speaking measured value, etc.).
Contemporary multimeters can measure many values. [ 18 ] [ 19 ] The most common are:
Additionally, some multimeters also measure:
Digital multimeters may also include circuits for:
Various sensors can be attached to (or included in) multimeters to take measurements such as:
An un-amplified analog multimeter combines a meter movement, range resistors and switches; VTVMs are amplified analog meters and contain active circuitry. For an analog meter movement, DC voltage is measured with a series resistor connected between the meter movement and the circuit under test. A switch (usually rotary) allows greater resistance to be inserted in series with the meter movement to read higher voltages. The product of the basic full-scale deflection current of the movement, and the sum of the series resistance and the movement's own resistance, gives the full-scale voltage of the range. As an example, a meter movement that required 1 mA for full-scale deflection, with an internal resistance of 500 Ω, would, on a 10 V range of the multimeter, have 9,500 Ω of series resistance. [ 20 ] For analog current ranges, matched low-resistance shunts are connected in parallel with the meter movement to divert most of the current around the coil. Again for the case of a hypothetical 1 mA, 500 Ω movement on a 1 A range, the shunt resistance would be just over 0.5 Ω.
Moving coil instruments can respond only to the average value of the current through them. To measure alternating current, which changes up and down repeatedly, a rectifier is inserted in the circuit so that each negative half cycle is inverted; the result is a varying and nonzero DC voltage whose maximum value will be half the AC peak to peak voltage, assuming a symmetrical waveform. Since the rectified average value and the root mean square (RMS) value of a waveform are only the same for a square wave, simple rectifier-type circuits can only be calibrated for sinusoidal waveforms. Other wave shapes require a different calibration factor to relate RMS and average value. This type of circuit usually has fairly limited frequency range. Since practical rectifiers have non-zero voltage drop, accuracy and sensitivity is poor at low AC voltage values. [ 21 ]
To measure resistance, switches arrange for a small battery within the instrument to pass a current through the device under test and the meter coil. Since the current available depends on the state of charge of the battery which changes over time, a multimeter usually has an adjustment for the ohm scale to zero it. In the usual circuits found in analog multimeters, the meter deflection is inversely proportional to the resistance, so full-scale will be 0 Ω, and higher resistance will correspond to smaller deflections. The ohms scale is compressed, so resolution is better at lower resistance values.
Amplified instruments simplify the design of the series and shunt resistor networks. The internal resistance of the coil is decoupled from the selection of the series and shunt range resistors; the series network thus becomes a voltage divider . Where AC measurements are required, the rectifier can be placed after the amplifier stage, improving precision at low range.
The meter movement in a moving pointer analog multimeter is practically always a moving-coil galvanometer of the d'Arsonval type, using either jeweled pivots or taut bands to support the moving coil. In a basic analog multimeter the current to deflect the coil and pointer is drawn from the circuit being measured; it is usually an advantage to minimize the current drawn from the circuit, which implies delicate mechanisms. The sensitivity of an analog multimeter is given in units of ohms per volt. For example, a very low-cost multimeter with a sensitivity of 1,000 Ω/V would draw 1 mA from a circuit at full-scale deflection. [ 22 ] More expensive, (and mechanically more delicate) multimeters typically have sensitivities of 20,000 ohms per volt and sometimes higher, with 50,000 ohms per volt (drawing 20 microamperes at full scale) being about the upper limit for a portable, general purpose, non-amplified analog multimeter.
To avoid the loading of the measured circuit by the current drawn by the meter movement, some analog multimeters use an amplifier inserted between the measured circuit and the meter movement. While this increases the expense and complexity of the meter, by use of vacuum tubes or field effect transistors the input resistance can be made very high and independent of the current required to operate the meter movement coil. Such amplified multimeters are called VTVMs (vacuum tube voltmeters), [ 23 ] TVMs (transistor volt meters), FET-VOMs, and similar names.
Analog meters are intuitive where the trend of a measurement was more important than an exact value obtained at a particular moment. A change in angle or in a proportion is easier to interpret than a change in the value of a digital readout. For this reason, some digital multimeters additionally have a bar graph as a second display, typically with a more rapid sampling rate than used for the primary readout. These fast sampling rate bar graphs have a superior response than the physical pointer of analog meters, obsoleting the older technology. With rapidly fluctuating DC, AC or a combination of both, advanced digital meters are able to track and display fluctuations better than analog meters whilst also having the ability to separate and simultaneously display DC and AC components. [ 24 ]
Because of the absence of amplification, ordinary analog multimeter are typically less susceptible to radio frequency interference , and so continue to have a prominent place in some fields even in a world of more accurate and flexible electronic multimeters. [ 25 ]
Analog meter movements are inherently more fragile physically and electrically than digital meters. Many analog multimeters feature a range switch position marked "off" to protect the meter movement during transportation which places a low resistance across the meter movement, resulting in dynamic braking . Meter movements as separate components may be protected in the same manner by connecting a shorting or jumper wire between the terminals when not in use. Meters which feature a shunt across the winding such as an ammeter may not require further resistance to arrest uncontrolled movements of the meter needle because of the low resistance of the shunt.
High-quality analog multimeters continue to be made by several manufacturers, including Chauvin Arnoux (France), Gossen Metrawatt (Germany), and Simpson and Triplett (USA). [ citation needed ]
Digital instruments, which necessarily incorporate amplifiers, use the same principles as analog instruments for resistance readings. For resistance measurements, usually a small constant current is passed through the device under test and the digital multimeter reads the resultant voltage drop; this eliminates the scale compression found in analog meters, but requires a source of precise current. An autoranging digital multimeter can automatically adjust the scaling network so the measurement circuits use the full precision of the A/D converter.
In a digital multimeter the signal under test is converted to a voltage and an amplifier with electronically controlled gain preconditions the signal. A digital multimeter displays the quantity measured as a number, which eliminates parallax errors.
Modern digital multimeters may have an embedded computer , which provides a wealth of convenience features. Measurement enhancements available include:
Modern meters may be interfaced with a personal computer by IrDA links, RS-232 connections, USB , Bluetooth , or an instrument bus such as IEEE-488 . The interface allows the computer to record measurements as they are made. Some DMMs can store measurements and upload them to a computer. [ 31 ]
A multimeter can use many different test probes to connect to the circuit or device under test. Crocodile clips , retractable hook clips, and pointed probes are the three most common types. Tweezer probes are used for closely spaced test points, as for instance surface-mount devices . The connectors are attached to flexible, well insulated leads terminated with connectors appropriate for the meter. Probes are connected to portable meters typically by shrouded or recessed banana jacks , while benchtop meters may use banana jacks or BNC connectors . 2 mm plugs and binding posts have also been used at times, but are less commonly used today. Indeed, safety ratings now require shrouded banana jacks.
The banana jacks are typically placed with a standardized center-to-center distance of 3 ⁄ 4 in (19 mm), to allow standard adapters or devices such as voltage multiplier or thermocouple probes to be plugged in.
Clamp meters clamp around a conductor carrying a current to measure without the need to connect the meter in series with the circuit, or make metallic contact at all. Those for AC measurement use the transformer principle; clamp-on meters to measure small current or direct current require more exotic sensors, such as; hall effect based systems that measure the nonchanging magnetic field to determine the current.
Analog meters can measure voltage and current by using power from the test circuit, but require a supplementary internal voltage source for resistance testing, while electronic meters always require an internal power supply to run their internal circuitry. Hand-held meters use batteries, while bench meters usually use mains power; either arrangement allows the meter to test devices. Testing often requires that the component under test be isolated from the circuit in which they are mounted, as otherwise stray or leakage current paths may distort measurements. In some cases, the voltage from the multimeter may turn active devices on, distorting a measurement, or in extreme cases even damage an element in the circuit being investigated.
Most multimeters include a fuse , or two fuses, which will sometimes prevent damage to the multimeter from a current overload on the highest current range. (For added safety, test leads with fuses built in are available.) A common error when operating a multimeter is to set the meter to measure resistance or current, and then connect it directly to a low-impedance voltage source. Unfused meters are often quickly destroyed by such errors; fused meters often survive. Fuses used in meters must carry the maximum measuring current of the instrument, but are intended to disconnect if operator error exposes the meter to a low-impedance fault. Meters with inadequate or unsafe fusing were not uncommon; this situation has led to the creation of the IEC61010 categories to rate the safety and robustness of meters.
Digital meters are rated into four categories based on their intended application, as set forth by IEC 61010-1 [ 32 ] and echoed by country and regional standards groups such as the CEN EN61010 standard. [ 33 ]
Each Category rating also specifies maximum safe transient voltages for selected measuring ranges in the meter. [ 34 ] [ 35 ] Category-rated meters also feature protections from over-current faults. [ 36 ] On meters that allow interfacing with computers, optical isolation may be used to protect attached equipment against high voltage in the measured circuit.
Good quality multimeters designed to meet Category II and above standards include high rupture capacity (HRC) ceramic fuses typically rated at more than 20 A capacity; these are much less likely to fail explosively than more common glass fuses. They will also include high energy overvoltage MOV (Metal Oxide Varistor ) protection, and circuit over-current protection in the form of a Polyswitch . [ citation needed ]
Meters intended for testing in hazardous locations or for use on blasting circuits may require use of a manufacturer-specified battery to maintain their safety rating. [ citation needed ]
The resolution of a multimeter is the smallest part of the scale which can be shown, which is scale dependent. On some digital multimeters it can be configured, with higher resolution measurements taking longer to complete. For example, a multimeter that has a 1 mV resolution on a 10 V scale can show changes in measurements in 1 mV increments. Absolute accuracy is the error of the measurement compared to a perfect measurement. Relative accuracy is the error of the measurement compared to the device used to calibrate the multimeter. Most multimeter datasheets provide relative accuracy. To compute the absolute accuracy from the relative accuracy of a multimeter add the absolute accuracy of the device used to calibrate the multimeter to the relative accuracy of the multimeter. [ 37 ]
The resolution of a multimeter is often specified in the number of decimal digits resolved and displayed. If the most significant digit cannot take all values from 0 to 9 it is generally, and confusingly, termed a fractional digit. For example, a multimeter which can read up to 19999 (plus an embedded decimal point) is said to read 4 + 1 ⁄ 2 digits. By convention, if the most significant digit can be either 0 or 1, it is termed a half-digit; if it can take higher values without reaching 9 (often 3 or 5), it may be called three-quarters of a digit. A 5 + 1 ⁄ 2 -digit multimeter would display one "half digit" that could only display 0 or 1, followed by five digits taking all values from 0 to 9. [ 38 ] Such a meter could show positive or negative values from 0 to 199999. A 3 + 3 ⁄ 4 -digit meter can display a quantity from 0 to 3999 or 5999, depending on the manufacturer. While a digital display can easily be extended in resolution , the extra digits are of no value if not accompanied by care in the design and calibration of the analog portions of the multimeter. Meaningful (i.e., high-accuracy) measurements require a good understanding of the instrument specifications, good control of the measurement conditions, and traceability of the calibration of the instrument. However, even if its resolution exceeds the accuracy , a meter can be useful for comparing measurements. For example, a meter reading 5 + 1 ⁄ 2 stable digits may indicate that one nominally 100 kΩ resistor is about 7 Ω greater than another, although the error of each measurement is 0.2% of reading plus 0.05% of full-scale value. Specifying "display counts" is another way to specify the resolution. Display counts give the largest number, or the largest number plus one (to include the display of all zeros) the multimeter's display can show, ignoring the decimal separator . For example, a 5 + 1 ⁄ 2 -digit multimeter can also be specified as a 199999 display count or 200000 display count multimeter. Often the display count is just called the 'count' in multimeter specifications. The accuracy of a digital multimeter may be stated in a two-term form, such as "±1% of reading +2 counts", reflecting the different sources of error in the instrument. [ 39 ]
Analog meters are older designs, but despite being technically surpassed by digital meters with bar graphs, may still be preferred [ according to whom? ] by engineers [ which? ] and troubleshooters. [ original research? ] One reason given is that analog meters are more sensitive (or responsive) to changes in the circuit that is being measured. [ citation needed ] A digital multimeter samples the quantity being measured over time, and then displays it. Analog multimeters continuously read the test value. If there are slight changes in readings, the needle of an analog multimeter will attempt to track it, as opposed to the digital meter having to wait until the next sample, giving delays between each discontinuous reading (plus the digital meter may additionally require settling time to converge on the value). The digital display value as opposed to an analog display is subjectively more difficult to read. This continuous tracking feature becomes important when testing capacitors or coils, for example. A properly functioning capacitor should allow current to flow when voltage is applied, then the current slowly decreases to zero and this "signature" is easy to see on an analog multimeter but not on a digital multimeter. This is similar when testing a coil, except the current starts low and increases. Resistance measurements on an analog meter, in particular, can be of low precision due to the typical resistance measurement circuit which compresses the scale heavily at the higher resistance values. Inexpensive analog meters may have only a single resistance scale, seriously restricting the range of precise measurements. Typically, an analog meter will have a panel adjustment to set the zero-ohms calibration of the meter, to compensate for the varying voltage of the meter battery, and the resistance of the meter's test leads.
Digital multimeters generally take measurements with accuracy superior to their analog counterparts. Standard analog multimeters measure with typically ±3% accuracy, [ 40 ] though instruments of higher accuracy are made. Standard portable digital multimeters are specified to have an accuracy of typically ±0.5% on the DC voltage ranges. Mainstream bench-top multimeters are available with specified accuracy of better than ±0.01%. Laboratory grade instruments can have accuracies of a few parts per million . [ 41 ]
Accuracy figures need to be interpreted with care. The accuracy of an analog instrument usually refers to full-scale deflection; a measurement of 30 V on the 100 V scale of a 3% meter is subject to an error of 3 V, 10% of the reading. Digital meters usually specify accuracy as a percentage of reading plus a percentage of full-scale value, sometimes expressed in counts rather than percentage terms.
Quoted accuracy is specified as being that of the lower millivolt (mV) DC range, and is known as the "basic DC volts accuracy" figure. Higher DC voltage ranges, current, resistance, AC and other ranges will usually have a lower accuracy than the basic DC volts figure. AC measurements only meet specified accuracy within a specified range of frequencies .
Manufacturers can provide calibration services so that new meters may be purchased with a certificate of calibration indicating the meter has been adjusted to standards traceable to, for example, the US National Institute of Standards and Technology (NIST), or other national standards organization .
Test equipment tends to drift out of calibration over time, and the specified accuracy cannot be relied upon indefinitely. For more expensive equipment, manufacturers and third parties provide calibration services so that older equipment may be recalibrated and recertified. The cost of such services is disproportionate for inexpensive equipment; however extreme accuracy is not required for most routine testing. Multimeters used for critical measurements may be part of a metrology program to assure calibration.
A multimeter can be assumed to be "average responding" to AC waveforms unless stated as being a "true RMS" type. An average responding multimeter will only meet its specified accuracy on AC volts and amps for purely sinusoidal waveforms. A True RMS responding multimeter on the other hand will meet its specified accuracy on AC volts and current with any waveform type up to a specified crest factor ; RMS performance is sometimes claimed for meters which report accurate RMS readings only at certain frequencies (usually low) and with certain waveforms (essentially always sine waves).
A meter's AC voltage and current accuracy may have different specifications at different frequencies.
When used for measuring voltage, the input impedance of the multimeter must be very high compared to the impedance of the circuit being measured; otherwise circuit operation may be affected and the reading will be inaccurate. Meters with electronic amplifiers (all digital multimeters and some analog meters) have a fixed input impedance that is high enough not to disturb most circuits. This is often either one or ten megohms ; the standardization of the input resistance allows the use of external high-resistance probes which form a voltage divider with the input resistance to extend voltage range up to tens of thousands of volts. High-end multimeters generally provide an input impedance greater than 10 GΩ for ranges less than or equal to 10 V. Some high-end multimeters provide >10 Gigaohms of impedance to ranges greater than 10 V. [ 37 ] Most analog multimeters of the moving-pointer type are unbuffered , and draw current from the circuit under test to deflect the meter pointer. The impedance of the meter varies depending on the basic sensitivity of the meter movement and the range which is selected. For example, a meter with a typical 20,000 Ω/V sensitivity will have an input resistance of 2 MΩ on the 100 V range (100 V × 20,000 Ω/V = 2,000,000 Ω). On every range, at full-scale voltage of the range, the full current required to deflect the meter movement is taken from the circuit under test. Lower sensitivity meter movements are acceptable for testing in circuits where source impedances are low compared to the meter impedance, for example, power circuits; these meters are more rugged mechanically. Some measurements in signal circuits require higher sensitivity movements so as not to load the circuit under test with the meter impedance. [ 42 ] [ 43 ]
Sensitivity should not be confused with resolution of a meter, which is defined as the lowest signal change (voltage, current, resistance and so on) that can change the observed reading. [ 43 ]
For general-purpose digital multimeters, the lowest voltage range is typically several hundred millivolts AC or DC, but the lowest current range may be several hundred microamperes, although instruments with greater current sensitivity are available. Multimeters designed for (mains) "electrical" use instead of general electronics engineering use will typically forego the microamps current ranges. Measurement of low resistance requires lead resistance (measured by touching the test probes together) to be subtracted for best accuracy. This can be done with the "delta", "zero", or "null" feature of many digital multimeters. Contact pressure to the device under test and cleanliness of the surfaces can affect measurements of very low resistances. Some meters offer a four wire test where two probes supply the source voltage and the others take measurement. Using a very high impedance allows for very low voltage drop in the probes and resistance of the source probes is ignored resulting in very accurate results. The upper end of multimeter measurement ranges varies considerably; measurements over perhaps 600 volts, 10 amperes, or 100 megohms may require a specialized test instrument.
Every inline series-connected ammeter, including a multimeter in a current range, has a certain resistance. Most multimeters inherently measure voltage, and pass a current to be measured through a shunt resistance , measuring the voltage developed across it. The voltage drop is known as the burden voltage, specified in volts per ampere. The value can change depending on the range the meter sets, since different ranges usually use different shunt resistors. [ 44 ] The burden voltage can be significant in very low-voltage circuit areas. To check for its effect on accuracy and on external circuit operation the meter can be switched to different ranges; the current reading should be the same and circuit operation should not be affected if burden voltage is not a problem. If this voltage is significant it can be reduced (also reducing the inherent accuracy and precision of the measurement) by using a higher current range.
Since the basic indicator system in either an analog or digital meter responds to DC only, a multimeter includes an AC to DC conversion circuit for making alternating current measurements. Basic meters utilize a rectifier circuit to measure the average or peak absolute value of the voltage, but are calibrated to show the calculated root mean square (RMS) value for a sinusoidal waveform ; this will give correct readings for alternating current as used in power distribution. User guides for some such meters give correction factors for some simple non- sinusoidal waveforms , to allow the correct root mean square (RMS) equivalent value to be calculated. More expensive multimeters include an AC to DC converter that measures the true RMS value of the waveform within certain limits; the user manual for the meter may indicate the limits of the crest factor and frequency for which the meter calibration is valid. RMS sensing is necessary for measurements on non-sinusoidal periodic waveforms, such as found in audio signals and variable-frequency drives .
A quality general-purpose electronics digital multimeter is generally considered adequate for measurements at signal levels greater than 1 mV or 1 μA, or below about 100 MΩ; these values are far from the theoretical limits of sensitivity, and are of considerable interest in some circuit design situations. Other instruments—essentially similar, but with higher sensitivity—are used for accurate measurements of very small or very large quantities. These include nanovoltmeters, electrometers (for very low currents, and voltages with very high source resistance, such as 1 TΩ) and picoammeters . Accessories for more typical multimeters permit some of these measurements, as well. Such measurements are limited by available technology, and ultimately by inherent thermal noise . | https://en.wikipedia.org/wiki/Multimeter |
Multimodal search is a type of search that uses different methods to get relevant results. They can use any kind of search, search by keyword , search by concept , search by example , etc.
A multimodal search engine is designed to imitate the flexibility and agility of how the human mind works to create, process and refuse irrelevant ideas. So, the more elements you have in the input of the search engine to compare, the more accurate the results can be.
Multimodal search engines use different inputs of different nature and methods of search at the same time with the possibility of combining the results by merging all of the input elements of the search. There are also engines that can use a feedback of the results with the evaluation of the user to perform a more appropriate and relevant search.
Nowadays, mobile devices have been developed to a point that they can perform infinite functions from any place at any time, thanks to the internet and GPS connections. Touch screens, motion sensors and voice recognition are now featured on mobile devices called smartphones . All the features and functions make it possible to execute multimodal searches from any place in the world at any time.
The use of text is an option, as well as multimedia searching , image , video , audio , and voice search . Even the location of the user can help the search engine to perform a more effective search, adaptable to every situation.
Nowadays, different ways to interact with a search engine are being discovered, in terms of input elements of the search and in the variety of results obtained.
Many queries from mobiles are location-based (LBS), that use the location of the user to interact with the applications. If available, the browser uses the device GPS, or computes an approximate location based on cell tower triangulation, with the permission of the user, who must be agree to share his/her location with the application in the download.
Therefore, multimodal searches use not only audiovisual content that the user provides directly, but also the context where the user is, like his/her location, language, time at the moment, web site or document where the user is surfing, or other elements that can help to improve of a search in every situation.
The multimodal search engine works in parallel, whilst at the same time, performs a search of more to less relevance of every element introduced directly or indirectly (personal context). Afterwards, it provides a combination of all the results, merging every element with its associated weight for every descriptor.
The engine analyzes every element and tags them, so a comparison of the tags can be made with existent indexed information in databases. A classification of the results proceeds, to show them from more to less relevance.
It’s necessary to define the importance of every input element. There are search engines that do this automatically, however there are also engines where the user can do it manually, giving more or less weight to every element of the search.
It’s also important that the user provides the appropriate and essential information for the search; too much information can confuse the system and provide unsatisfactory results.
With multimodal searches users can get better results than with a simple search, but multimodal searches must process more input information. It can also spend more time to process it and require more memory space.
An efficient search engine interprets the query of the users, realizes his/her intention and applies a strategy to use an appropriate search, i.e. the engine adapts to every input query and also to the combination of the elements and methods.
Nowadays, existing multimodal search engines are not very complex, and some of them are in an experimental phase. Some of the more simple engines are Google Images [1] or Bing [2] , web interfaces that use text and images as inputs to find images in the output.
MMRetrieval [3] is a multimodal experimental search engine that uses multilingual and multimedia information through a web interface. The engine searches the different inputs in parallel and merges all the results by different chosen methods. The engine also provides different multistage retrieval, as well as a single text index baseline to be able to compare all the different phases of search.
There are a lot of applications for mobile devices, using the context of the user, like based-location services, and using also text, images, audios or videos that the user provides at the moment or with saved files, or even interacting with the voice. | https://en.wikipedia.org/wiki/Multimodal_search |
In mathematics , the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials .
For any positive integer m and any non-negative integer n , the multinomial theorem describes how a sum with m terms expands when raised to the n th power: ( x 1 + x 2 + ⋯ + x m ) n = ∑ k 1 + k 2 + ⋯ + k m = n k 1 , k 2 , ⋯ , k m ≥ 0 ( n k 1 , k 2 , … , k m ) x 1 k 1 ⋅ x 2 k 2 ⋯ x m k m {\displaystyle (x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{\begin{array}{c}k_{1}+k_{2}+\cdots +k_{m}=n\\k_{1},k_{2},\cdots ,k_{m}\geq 0\end{array}}{n \choose k_{1},k_{2},\ldots ,k_{m}}x_{1}^{k_{1}}\cdot x_{2}^{k_{2}}\cdots x_{m}^{k_{m}}} where ( n k 1 , k 2 , … , k m ) = n ! k 1 ! k 2 ! ⋯ k m ! {\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}} is a multinomial coefficient . [ 1 ] The sum is taken over all combinations of nonnegative integer indices k 1 through k m such that the sum of all k i is n . That is, for each term in the expansion, the exponents of the x i must add up to n . [ 2 ] [ a ]
In the case m = 2 , this statement reduces to that of the binomial theorem . [ 2 ]
The third power of the trinomial a + b + c is given by ( a + b + c ) 3 = a 3 + b 3 + c 3 + 3 a 2 b + 3 a 2 c + 3 b 2 a + 3 b 2 c + 3 c 2 a + 3 c 2 b + 6 a b c . {\displaystyle (a+b+c)^{3}=a^{3}+b^{3}+c^{3}+3a^{2}b+3a^{2}c+3b^{2}a+3b^{2}c+3c^{2}a+3c^{2}b+6abc.} This can be computed by hand using the distributive property of multiplication over addition and combining like terms, but it can also be done (perhaps more easily) with the multinomial theorem. It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. For example, the term a 2 b 0 c 1 {\displaystyle a^{2}b^{0}c^{1}} has coefficient ( 3 2 , 0 , 1 ) = 3 ! 2 ! ⋅ 0 ! ⋅ 1 ! = 6 2 ⋅ 1 ⋅ 1 = 3 {\displaystyle {3 \choose 2,0,1}={\frac {3!}{2!\cdot 0!\cdot 1!}}={\frac {6}{2\cdot 1\cdot 1}}=3} , the term a 1 b 1 c 1 {\displaystyle a^{1}b^{1}c^{1}} has coefficient ( 3 1 , 1 , 1 ) = 3 ! 1 ! ⋅ 1 ! ⋅ 1 ! = 6 1 ⋅ 1 ⋅ 1 = 6 {\displaystyle {3 \choose 1,1,1}={\frac {3!}{1!\cdot 1!\cdot 1!}}={\frac {6}{1\cdot 1\cdot 1}}=6} , and so on.
The statement of the theorem can be written concisely using multiindices :
where
and
This proof of the multinomial theorem uses the binomial theorem and induction on m .
First, for m = 1 , both sides equal x 1 n since there is only one term k 1 = n in the sum. For the induction step, suppose the multinomial theorem holds for m . Then
by the induction hypothesis. Applying the binomial theorem to the last factor,
which completes the induction. The last step follows because
as can easily be seen by writing the three coefficients using factorials as follows:
The numbers
appearing in the theorem are the multinomial coefficients . They can be expressed in numerous ways, including as a product of binomial coefficients or of factorials :
The substitution of x i = 1 for all i into the multinomial theorem
gives immediately that
The number of terms in a multinomial sum, # n , m , is equal to the number of monomials of degree n on the variables x 1 , …, x m :
The count can be performed easily using the method of stars and bars .
The largest power of a prime p that divides a multinomial coefficient may be computed using a generalization of Kummer's theorem .
By Stirling's approximation , or equivalently the log-gamma function 's asymptotic expansion, log ( k n n , n , ⋯ , n ) = k n log ( k ) + 1 2 ( log ( k ) − ( k − 1 ) log ( 2 π n ) ) − k 2 − 1 12 k n + k 4 − 1 360 k 3 n 3 − k 6 − 1 1260 k 5 n 5 + O ( 1 n 6 ) {\displaystyle \log {\binom {kn}{n,n,\cdots ,n}}=kn\log(k)+{\frac {1}{2}}\left(\log(k)-(k-1)\log(2\pi n)\right)-{\frac {k^{2}-1}{12kn}}+{\frac {k^{4}-1}{360k^{3}n^{3}}}-{\frac {k^{6}-1}{1260k^{5}n^{5}}}+O\left({\frac {1}{n^{6}}}\right)} so for example, ( 2 n n ) ∼ 2 2 n n π {\displaystyle {\binom {2n}{n}}\sim {\frac {2^{2n}}{\sqrt {n\pi }}}}
The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing n distinct objects into m distinct bins, with k 1 objects in the first bin, k 2 objects in the second bin, and so on. [ 3 ]
In statistical mechanics and combinatorics , if one has a number distribution of labels, then the multinomial coefficients naturally arise from the binomial coefficients. Given a number distribution { n i } on a set of N total items, n i represents the number of items to be given the label i . (In statistical mechanics i is the label of the energy state.)
The number of arrangements is found by
Multiplying the number of choices at each step results in:
Cancellation results in the formula given above.
The multinomial coefficient
is also the number of distinct ways to permute a multiset of n elements, where k i is the multiplicity of each of the i th element. For example, the number of distinct permutations of the letters of the word MISSISSIPPI, which has 1 M, 4 Is, 4 Ss, and 2 Ps, is
One can use the multinomial theorem to generalize Pascal's triangle or Pascal's pyramid to Pascal's simplex . This provides a quick way to generate a lookup table for multinomial coefficients. | https://en.wikipedia.org/wiki/Multinomial_theorem |
Multinucleate cells (also known as multinucleated cells or polynuclear cells ) are eukaryotic cells that have more than one nucleus , i.e., multiple nuclei share one common cytoplasm . Mitosis in multinucleate cells can occur either in a coordinated, synchronous manner where all nuclei divide simultaneously or asynchronously where individual nuclei divide independently in time and space. Certain organisms may have a multinuclear stage of their life cycle. For example, slime molds have a vegetative, multinucleate life stage called a plasmodium . [ 1 ]
Multinucleate cells, depending on the mechanism by which they are formed, can be divided into [ 2 ] [ 3 ] " syncytia " (formed by cell fusion ) or " coenocytes " (formed by nuclear division not being followed by cytokinesis ). [ 4 ]
Some bacteria , such as Mycoplasma pneumoniae , a pathogen of the respiratory tract , may display multinuclear filaments as a result of a delay between genome replication and cellular division . [ 5 ]
Some biologists use the term "acellular" to refer to multinucleate cell forms ( syncitia and plasmodia ), such as to differentiate "acellular" slime molds from the purely "cellular" ones (which do not form such structures). [ 6 ] [ 7 ] [ 8 ] This usage is incorrect and highly misleading to laymen , and as such it is discouraged.
Some use the term "syncytium" in a wide sense, to mean any type of multinucleate cell, [ 9 ] while others differentiate the terms for each type. [ 10 ]
Syncytia are multinuclear cells that can form either through normal biological processes, such as the mammalian placenta , or under the influence of certain pathogens, such as HIV , via fusion of the plasma membrane. [ 11 ] [ 12 ] Other examples include the skeletal muscle cells of mammals , the tapetal cells of plants , and the storage cells of Douglas-fir seeds. [ 13 ] The polymorphonuclear leukocytes of mammals are not polynuclear cells, although the lobes of their nuclei are so deeply bifurcated that they can appear so under non-optimal microscopy.
Osteoclasts are multinuclear cells that are found commonly in the human body that aid in the maintenance and repair of the bones by secreting acid that dissolves bone matter. They are typically found to have 5 nuclei per cell, due to the fusion of preosteoclasts.
The chlorarachniophytes form multinucleate cells by fusion, being syncytia and not coenocytes. This syncytia is called plasmodium , in the sense of a multinucleate protoplast without a cell wall which exhibits amoeboid movement . [ 14 ] Other examples include some plasmodiophorids , some haplosporidians , [ 15 ] and the grex of cellular slime moulds ( dictyostelids and acrasids ).
The placenta , a temporary organ that transports nutrients, oxygen, waste, and other materials between a mother and a developing fetus, is partially composed of a syncytial layer that forms the interface between the foetus and the mother. [ 16 ] In addition to performing simple interface duties, the placental syncytia also acts as a barrier to infection from viruses , bacteria , and protozoa , which is likely due to unique cytoskeletal properties of these cells. [ 16 ]
Furthermore, multinucleate cells are produced from specialized cell cycles in which nuclear division occurs without cytokinesis, thus leading to large coenocytes or plasmodia. In filamentous fungi , multinucleate cells may extend over hundreds of meters so that different regions of a single cell experience dramatically different microenvironments. Other examples include, the plasmodia of plasmodial slime molds ( myxogastrids ) and the schizont of the Plasmodium parasite which causes malaria .
Multinucleated cells can also occur under pathological conditions as the consequence of a disturbed cell cycle control (e.g., some binucleated cells and metastasizing tumor cells).
As previously mentioned, syncytia may be induced through the actions of HIV, where T-cells are fused by the action of virus-derived proteins on the cell membrane . [ 12 ] During viral replication in T lymphoid cells , large amounts of viral envelope Glycoprotein ( Env ) are synthesized and trafficked to the cell membrane where they can be incorporated into new virus particles. However, some of the Env molecules interact with neighboring T-cell receptors , which brings the cells into close enough proximity to enable trigger events culminating in the fusion of two host cells, likely due to the close contact of the two plasma membranes. [ 17 ] This interaction is likely specific to CD4+ T-cells , as cells lacking this receptor were unable to form syncytia in laboratory conditions. [ 18 ]
Although not normally viewed as a case of multinucleation, plant cells share a common cytoplasm by plasmodesmata , and most cells in animal tissues are in communication with their neighbors via gap junctions . [ 19 ]
Although endosymbiosis among eukaryotes can result in multiple nuclei in a cell, these are not considered "multinucleate" because the nuclei are not surrounded by the same cytoplasm. These extra nuclei are called nucleomorphs . | https://en.wikipedia.org/wiki/Multinucleate |
Multiomics , multi-omics , integrative omics , "panomics" or "pan-omics" is a biological analysis approach in which the data consists of multiple " omes ", such as the genome , epigenome , transcriptome , proteome , metabolome , exposome , and microbiome (i.e., a meta-genome and/or meta-transcriptome , depending upon how it is sequenced); [ 1 ] [ 2 ] [ 3 ] in other words, the use of multiple omics technologies to study life in a concerted way . By combining these "omes", scientists can analyze complex biological big data to find novel associations between biological entities, pinpoint relevant biomarkers and build elaborate markers of disease and physiology. In doing so, multiomics integrates diverse omics data to find a coherently matching geno-pheno-envirotype relationship or association. [ 4 ] The OmicTools service lists more than 99 pieces of software related to multiomic data analysis, as well as more than 99 databases on the topic.
Systems biology approaches are often based upon the use of multiomic analysis data. [ 5 ] [ 6 ] The American Society of Clinical Oncology (ASCO) defines panomics as referring to "the interaction of all biological
functions within a cell and with other body functions, combining data collected by targeted tests ... and global assays (such as genome sequencing) with other patient-specific information." [ 7 ]
Combined multiomic data collection approaches have evolved to address the limitations of traditional multiomics research, which typically requires separate sample processing for different molecular classes then subsequent computational integration, introducing variability and increasing costs. Early advances in this field include sequential extraction, [ 8 ] TRIzol-based sequential isolation methods, which demonstrated that a reagent traditionally used for RNA isolation could simultaneously extract DNA, RNA, proteins, metabolites, and lipids from a single sample. Similar approaches like the Metabolite, Protein, and Lipid extraction (MPLEx) [ 9 ] and the "Three-in-One" [ 10 ] method adapted biphasic fractionation to extract proteins, metabolites, and lipids for LC-MS/MS analysis. More recent technological developments include the Multi-Omic Single-Shot Technology (MOST), [ 11 ] which integrates proteome and lipidome analysis in a single LC-MS run using one reverse-phase column and a binary mobile phase system, and the Bead-enabled Accelerated Monophasic Multi-omics (BAMM) [ 12 ] method that combines n-butanol-based monophasic extraction with magnetic beads and accelerated protein digestion for the separate analysis of metabolites, lipids, and proteins. One of the most comprehensive technologies in this space is Dalton Bioanalytics Inc.'s Omni-MS®, a multiomic assay that uses its proprietary method to simultaneously profile proteins, lipids, electrolytes, metabolites, and other small molecules in a single preparation and single LC-MS analysis. This platform has been applied to biomarker discovery, identifying potential biomarkers across multiple molecular classes and across various conditions and diseases [ 13 ] [ 14 ] including COVID severity during pregnancy, [ 15 ] 22q11.2 deletion syndrome, [ 16 ] and hereditary angioedema. [ 17 ] These integrated approaches significantly reduce sample requirements, processing time, and technical variation while improving correlation analysis across different molecular classes, making them increasingly valuable for precision medicine and systems biology research.
A branch of the field of multiomics is the analysis of multilevel single-cell data , called single-cell multiomics. [ 18 ] [ 19 ] This approach gives us an unprecedent resolution to look at multilevel transitions in health and disease at the single cell level. An advantage in relation to bulk analysis is to mitigate confounding factors derived from cell to cell variation, allowing the uncovering of heterogeneous tissue architectures. [ 18 ]
Methods for parallel single-cell genomic and transcriptomic analysis can be based on simultaneous amplification [ 20 ] or physical separation of RNA and genomic DNA. [ 21 ] They allow insights that cannot be gathered solely from transcriptomic analysis, as RNA data do not contain non-coding genomic regions and information regarding copy-number variation , for example. An extension of this methodology is the integration of single-cell transcriptomes to single-cell methylomes, combining single-cell bisulfite sequencing [ 22 ] [ 23 ] to single cell RNA-Seq. [ 24 ] Other techniques to query the epigenome, as single-cell ATAC-Seq [ 25 ] and single-cell Hi-C [ 26 ] also exist.
A different, but related, challenge is the integration of proteomic and transcriptomic data. [ 27 ] [ 28 ] One approach to perform such measurement is to physically separate single-cell lysates in two, processing half for RNA, and half for proteins. [ 27 ] The protein content of lysates can be measured by proximity extension assays (PEA), for example, which use DNA-barcoded antibodies. [ 29 ] A different approach uses a combination of heavy-metal RNA probes and protein antibodies to adapt mass cytometry for multiomic analysis. [ 28 ]
Related to Single-cell multiomics is the field of Spatial Omics which assays tissues through omics readouts that preserve the relative spatial orientation of the cells in the tissue. The number of Spatial Omics methods published still lags behind the number of methods published for Single-Cell multiomics, but the numbers are catching up ( Single-cell and Spatial methods ).
In parallel to the advances in high-throughput biology, machine learning applications to biomedical data analysis are flourishing. The integration of multi-omics data analysis and machine learning has led to the discovery of new biomarkers . [ 30 ] [ 31 ] [ 32 ] For example, one of the methods of the mixOmics project implements a method based on sparse Partial Least Squares regression for selection of features (putative biomarkers). [ 33 ] A unified and flexible statistical framewok for heterogeneous data integration called "Regularized Generalized Canonical Correlation Analysis" (RGCCA [ 34 ] [ 35 ] [ 36 ] [ 37 ] ) enables identifying such putative biomarkers. This framework is implemented and made freely available within the RGCCA R package .
Multiomics currently holds a promise to fill gaps in the understanding of human health and disease, and many researchers are working on ways to generate and analyze disease-related data. [ 38 ] The applications range from understanding host-pathogen interactions and infectious diseases, [ 39 ] [ 40 ] cancer, [ 41 ] to understanding better chronic and complex non-communicable diseases [ 42 ] and improving personalized medicine. [ 43 ]
The second phase of the $170 million Human Microbiome Project was focused on integrating patient data to different omic datasets, considering host genetics, clinical information and microbiome composition. [ 44 ] [ 45 ] The phase one focused on characterization of communities in different body sites. Phase 2 focused in the integration of multiomic data from host & microbiome to human diseases. Specifically, the project used multiomics to improve the understanding of the interplay of gut and nasal microbiomes with type 2 diabetes , [ 46 ] gut microbiomes and inflammatory bowel disease [ 47 ] and vaginal microbiomes and pre-term birth. [ 48 ]
The complexity of interactions in the human immune system has prompted the generation of a wealth of immunology-related multi-scale omic data. [ 49 ] Multi-omic data analysis has been employed to gather novel insights about the immune response to infectious diseases, such as pediatric chikungunya , [ 50 ] as well as noncommunicable autoimmune diseases . [ 51 ] Integrative omics has also been employed strongly to understand effectiveness and side effects of vaccines , a field called systems vaccinology. [ 52 ] For example, multiomics was essential to uncover the association of changes in plasma metabolites and immune system transcriptome on response to vaccination against herpes zoster . [ 53 ]
The Bioconductor project curates a variety of R packages aimed at integrating omic data:
The RGCCA package implements a versatile framework for data integration. This package is freely available on the Comprehensive R Archive Network (CRAN) .
The OmicTools [ 59 ] database further highlights R packages and othertools for multi omic data analysis:
A major limitation of classical omic studies is the isolation of only one level of biological complexity. For example, transcriptomic studies may provide information at the transcript level, but many different entities contribute to the biological state of the sample ( genomic variants , post-translational modifications , metabolic products, interacting organisms, among others). With the advent of high-throughput biology , it is becoming increasingly affordable to make multiple measurements, allowing transdomain (e.g. RNA and protein levels) correlations and inferences. These correlations aid the construction or more complete biological networks , filling gaps in our knowledge.
Integration of data, however, is not an easy task. To facilitate the process, groups have curated database and pipelines to systematically explore multiomic data: | https://en.wikipedia.org/wiki/Multiomics |
In theoretical computer science , multiparty communication complexity is the study of communication complexity in the setting where there are more than 2 players.
In the traditional two–party communication game , introduced by Yao (1979) , [ 1 ] two players, P 1 and P 2 attempt to compute a Boolean function
Player P 1 knows the value of x 2 , P 2 knows the value of x 1 , but P i does not know the value of x i , for i = 1, 2.
In other words, the players know the other's variables, but not their own. The minimum number of bits that must be communicated by the players to compute f is the communication complexity of f , denoted by κ ( f ).
The multiparty communication game, defined in 1983, [ 2 ] is a powerful generalization of the 2–party case: Here the players know all the others' input, except their own. Because of this property, sometimes this model is called "numbers on the forehead" model, since if the players were seated around a round table, each wearing their own input on the forehead, then every player would see all the others' input, except their own.
The formal definition is as follows: k {\displaystyle k} players: P 1 , P 2 , . . . , P k {\displaystyle P_{1},P_{2},...,P_{k}} intend to compute a Boolean function
On set S = { x 1 , x 2 , . . . , x n } {\displaystyle S=\{x_{1},x_{2},...,x_{n}\}} of variables there is a fixed partition A {\displaystyle A} of k {\displaystyle k} classes A 1 , A 2 , . . . , A k {\displaystyle A_{1},A_{2},...,A_{k}} , and player P 1 {\displaystyle P_{1}} knows every variable, except those in A i {\displaystyle A_{i}} , for i = 1 , 2 , . . . , k {\displaystyle i=1,2,...,k} . The players have unlimited computational power, and they communicate with the help of a blackboard, viewed by all players.
The aim is to compute f ( x 1 , x 2 , . . . , x n {\displaystyle f(x_{1},x_{2},...,x_{n}} ), such that at the end of the computation, every player knows this value. The cost of the computation is the number of bits written onto the blackboard for the given input x = ( x 1 , x 2 , . . . , x n ) {\displaystyle x=(x_{1},x_{2},...,x_{n})} and partition A = ( A 1 , A 2 , . . . , A x n ) {\displaystyle A=(A_{1},A_{2},...,Ax_{n})} . The cost of a multiparty protocol is the maximum number of bits communicated for any x {\displaystyle x} from the set {0,1} n and the given partition A {\displaystyle A} . The k {\displaystyle k} -party communication complexity, C A ( k ) ( f ) {\displaystyle C_{A}^{(k)}(f)} of a function f {\displaystyle f} , with respect to partition A {\displaystyle A} , is the minimum of costs of those k {\displaystyle k} -party protocols which compute f {\displaystyle f} . The k {\displaystyle k} -party symmetric communication complexity of f {\displaystyle f} is defined as
where the maximum is taken over all k -partitions of set x = ( x 1 , x 2 , . . . , x n ) {\displaystyle x=(x_{1},x_{2},...,x_{n})} .
For a general upper bound both for two and more players, let us suppose that A 1 is one of the smallest classes of the partition A 1 , A 2 ,..., A k . Then P 1 can compute any Boolean function of S with | A 1 | + 1 bits of communication: P 2 writes down the | A 1 | bits of A 1 on the blackboard, P 1 reads it, and computes and announces the value f ( x ) {\displaystyle f(x)} . So, the following can be written:
The Generalized Inner Product function (GIP) [ 3 ] is defined as follows:
Let y 1 , y 2 , . . . , y k {\displaystyle y_{1},y_{2},...,y_{k}} be n {\displaystyle n} -bit vectors, and let Y {\displaystyle Y} be the n {\displaystyle n} times k {\displaystyle k} matrix, with k {\displaystyle k} columns as the y 1 , y 2 , . . . , y k {\displaystyle y_{1},y_{2},...,y_{k}} vectors. Then G I P ( y 1 , y 2 , . . . , y k ) {\displaystyle GIP(y_{1},y_{2},...,y_{k})} is the number of the all-1 rows of matrix Y {\displaystyle Y} , taken modulo 2. In other words, if the vectors y 1 , y 2 , . . . , y k {\displaystyle y_{1},y_{2},...,y_{k}} correspond to the characteristic vectors of k {\displaystyle k} subsets of an n {\displaystyle n} element base-set, then GIP corresponds to the parity of the intersection of these k {\displaystyle k} subsets.
It was shown [ 3 ] that
with a constant c > 0.
An upper bound on the multiparty communication complexity of GIP shows [ 4 ] that
with a constant c > 0.
For a general Boolean function f , one can bound the multiparty communication complexity of f by using its L 1 norm [ 5 ] as follows: [ 6 ]
A construction of a pseudorandom number generator was based on the BNS lower bound for the GIP function. [ 3 ] | https://en.wikipedia.org/wiki/Multiparty_communication_complexity |
Multiple-pass or long path absorption cells are commonly used in spectroscopy to measure low-concentration components or to observe weak spectra in gases or liquids. Several important advances were made in this area beginning in the 1930s, and research into a wide range of applications continues to the present day.
Generally the goal of this type of sample cell is to improve detection sensitivity by increasing the total optical path length that travels through a small, constant sample volume. In principle, a longer path length results in greater detection sensitivity. Focusing mirrors must be used to redirect the beam at each reflection point, resulting in the beam being restricted to a predefined space along a controlled path until it exits the optical cavity . The output of the cell is the input of an optical detector (a specialized type of transducer ), which senses specific changes in the properties of the beam that occur during interaction with the test sample . For instance, the sample may absorb energy from the beam, resulting in an attenuation of the output that is detectable by the transducer. Two conventional multipass cells are called the White cell and Herriott cell. [ 1 ]
In the late 1930s August Pfund used a triple-pass cell like the one shown above for atmospheric study. The cell, which became known as the Pfund cell, is constructed using two identical spherical mirrors, each having a hole carefully machined into its center. The separation distance between the mirrors is equal to the mirror focal length. A source enters from a hole in either mirror, is redirected twice at two reflection points, and then exits the cell through the other mirror on the third pass. The Pfund cell was one of the earliest examples of this type of spectroscopic technique and is noted for having used multiple passes. [ 2 ]
The White cell was first described in 1942 by John U. White in his paper Long Optical Paths of Large Aperture , [ 3 ] and was a significant improvement over previous long path spectroscopic measurement techniques. A White cell is constructed using three spherical, concave mirrors having the same radius of curvature. The mirrors are separated by a distance equal to their radii of curvature. The animation on the right shows a White Cell in which a beam makes eight reflective passes or traversals. The number of traversals can be changed quite easily by making slight rotational adjustments to either M2 or M3; however, the total number of traversals must always occur in multiples of four. The entering and exiting beams do not change position as traversals are added or removed, while the total number of traversals can be increased many times without changing the volume of the cell, and therefore the total optical path length can be made large compared to the volume of the sample under test. The spots from various passes can overlap on mirrors M2 and M3 but must be distinct on mirror M1. If the input beam is focused in the plane of M1, then each round trip will also be focused in this plane. The tighter the focus, the more nonoverlapping spots there can be on M1 and thus the higher the maximum pathlength.
At present the White cell is still the most commonly used multipass cell and provides many advantages. [ 4 ] For example,
White cells are available with path lengths ranging from less than a meter to many hundreds of meters. [ 5 ]
The Herriott cell first appeared in 1965 when Donald R. Herriott and Harry J. Schulte published Folded Optical Delay Lines while at Bell Laboratories . [ 6 ] The Herriott cell is made up of two opposing spherical mirrors. A hole is machined into one of the mirrors to allow the input and output beams to enter and exit the cavity. Alternatively, the beam may exit through a hole in the opposite mirror. In this fashion the Herriott cell can support multiple light sources by providing multiple entrance and exit holes in either of the mirrors. Unlike the White cell, the number of traversals is controlled by adjusting the separation distance D between the two mirrors. This cell is also commonly used and has some advantages [ 4 ] over the White cell:
However, the Herriot cell does not accept high numerical aperture beams. In addition, larger sized mirrors must be used when longer path lengths are needed.
Another category of multipass cells is generally referred to as circular multipass reflection cells. They were first introduced by Thoma and co-workers in 1994. [ 7 ] Such cells rely on a circular arrangement of mirrors. The beam enters the cell under an angle and propagates on a star-shaped pattern (see picture on the right). The path length in circular multipass cells can be varied by adjusting the incidence angle of the beam. An advantage lies in their robustness towards mechanical stress such as vibrations or temperature changes. Furthermore, circular multipass cells stand out because of the small detection volumes they provide. [ 8 ] A stable beam propagation is achieved by shaping individual reflection points to form a non-concentric mirror-arrangement. [ 9 ] [ 10 ]
In a special case, a circular mirror is used, allowing continuous adjustment of the angle of incidence. A drawback of this circular cell configuration is the inherent concentric mirror arrangement which leads to imperfect imaging after a large number of reflections. | https://en.wikipedia.org/wiki/Multipass_spectroscopic_absorption_cells |
A multiphase flow meter is a device used to measure the individual phase flow rates of constituent phases in a given flow (for example in oil and gas industry ) where oil, water and gas mixtures are initially co-mingled together during the oil production processes.
Knowledge of the individual fluid flow rates of a producing oil well is required to facilitate reservoir management, field development, operational control, flow assurance and production allocation . [ 1 ]
Conventional solutions [ buzzword ] concerning two- and three-phase metering systems require expensive and cumbersome test separators , with associated high maintenance, and field personnel intervention. These conventional solutions [ buzzword ] do not lend themselves to continuous automated monitoring or metering. Moreover, with diminishing oil resources, oil companies are now frequently confronted with the need to recover hydrocarbons from marginally economical reservoirs. [ 2 ] In order to ensure economic viability of these accumulations, the wells may have to be completed subsea, or crude oil from several wells sent to a common production facility with excess processing capacity. The economic constraints on such developments do not lend themselves to the continued deployment of three-phase separators as the primary measurement devices. Consequently, viable alternatives to three-phase separators are essential. Industry’s response is the multiphase flow meter (MPFM).
The oil and gas industry began to be interested in developing MPFMs in the early 1980s, as measurement technology improved, and wellhead separators were costly. Depleting oil reserves, (More water and gas in the produced oil) along with smaller, deeper wells with higher water contents, saw the advent of increasingly frequent occurrences of multiphase flow where the single-phase meters were unable to provide accurate answers. After a lengthy gestation period, MPFMs capable of performing the required measurements became commercially available. Much of the early research was done at the Christian Michelsen research center in Bergen, Norway, [ 3 ] and this work spawned a number of spin off companies in Norway leading to the Roxar / Emerson, Schlumberger, Framo, and MPM meters. ENI and Shell supported the development in Italy of the Pietro Fiorentini meter. Haimo introduced a meter with partial separation, making accurate measurement simpler, but at the expense of a physically larger device. Norway has remained a technology center for MPFM with the Norwegian Society for Oil and Gas Measurement (NFOGM) providing an academic and educational role. [ 4 ] Since 1994, MPFM installation numbers have steadily increased as technology in the field has advanced, with substantial growth witnessed from 1999 onwards. [ 5 ] A recent study estimated that there were approximately 2,700 MPFM applications including field allocation, production optimisation and mobile well testing in 2006. [ 6 ]
A number of factors have instigated the recent rapid uptake of multiphase measurement technology: improved meter performances, decreases in meter costs, more compact meters enabling deployment of mobile systems, the need for sub sea metering, increases in oil prices and a wider assortment of operators. As the initial interest in multiphase flow metering came from the offshore industry, most of the multiphase metering activity was concentrated in the North Sea . However, the present distribution of multiphase flow meters is much more diverse.
Most modern meters combine a venturi flow rate meter, with a gamma densitometer, and some meters have additional measurements for water salinity. The meter measures the flow rates at line pressures, which are typically orders of magnitude greater than atmospheric pressure, but the meter must report the oil and gas volumes at standard (atmospheric) pressure and temperature. The meter must thus know the Pressure / Volume / Temperature properties of the oil, to add to the measured gas rate at line pressure the additional gas that would be liberated from the oil at atmospheric pressure, and also know the loss in oil volume from the release of that gas in conversion to standard conditions. With co-mingled flow from oil zones with differing PVT response, and different water salinities and hence densities, this PVT uncertainty may be the largest source of error in the measurement.
The introduction of the multiport selector valve (MSV) also facilitated the automation of the use of MPFMs, but this can also be achieved with conventional valving designs for well tests. MSVs are particularly suitable for onshore pad drilling, and where many nearby wells have similar pressures, and allow MPFMs to be shared between groups of wells. Subsea meters typically use conventional subsea valve designs, to ensure maintainability.
Measurement and interpretation of 2 and 3 phase multiphase flow can also be achieved by using alternative flow measurement technologies such as SONAR . SONAR meters apply the principles of underwater acoustics to measure flow regimes and; can be clamped on to wellheads and flow lines to measure the bulk (mean) fluid velocity of the total mixture which is then post-processed and analyzed along with wellbore compositional information and process conditions to infer the flow rates of each individual phase. This approached can be used in various applications such as black oil , gas condensate and wet gas.
Industry experts have forecast that MPFMs will become feasible on an installation per well basis when their capital cost falls to around US$40,000 – US$60,000. The cost of MPFMs today remains in the range of US$100,000 – US$500,000 (varying with onshore/offshore, topside/subsea, the physical dimensions of the meter and the number of units ordered). Installation of these MPFMs can cost up to 25% of the hardware cost and associated operating costs are estimated at between US$20,000 and $40,000 per year. [ 7 ]
A number of novel multiphase metering techniques, employing a variety of technologies, have been developed which eliminate the need for three-phase separator deployment. These MPFMs offer substantial economic and operating advantages over their phase separating predecessor. Nevertheless, it is still widely recognised that no single MPFM on the market can meet all multiphase metering requirements. [ 8 ] | https://en.wikipedia.org/wiki/Multiphase_flow_meter |
A multiphase flow system is one characterized by the simultaneous presence of several phases , the two-phase system being the simplest case. The term ‘two-component’ is sometimes used to describe flows in which the phases consist of different chemical substances. However, since the same mathematics describes two-phase and two-component flows, the two expressions can be treated as synonymous.
Analysis of multiphase systems can include consideration of multiphase flow and multiphase heat transfer. The former occurs only if all parts are at the same temperature, but interphase heat transfer also occurs when the temperatures of the individual phases are different.
If different phases of the same pure substance are present in a multiphase system, interphase heat transfer will result in a change of phase, which is always accompanied by interphase mass transfer.
A multiphase flow system is one characterized by the simultaneous presence of several phases , the two-phase system being the simplest case. The term ‘two-component’ is sometimes used to describe flows in which the phases consist of different chemical substances. For example, steam-water flows are two-phase, while air-water flows are two-component. Some two-component flows (mostly liquid-liquid) technically consist of a single phase but are identified as two-phase flows in which the term “phase” is applied to each of the components. Since the same mathematics describes two-phase and two-component flows, the two expressions can be treated as synonymous.
The analysis of multiphase systems can include consideration of multiphase flow and multiphase heat transfer. When all of the phases in a multiphase system exist at the same temperature, multiphase flow is the only concern. However, when the temperatures of the individual phases are different, interphase heat transfer also occurs.
If different phases of the same pure substance are present in a multiphase system, interphase heat transfer will result in a change of phase, which is always accompanied by interphase mass transfer. The combination of heat transfer with mass transfer during phase change makes multiphase systems distinctly more challenging than simpler systems. Based on the phases that are involved in the system, phase change problems can be classified as: (1) solid–liquid phase change ( melting and solidification ), (2) solid–vapor phase change ( sublimation and deposition ), and (3) liquid–vapor phase change ( boiling / evaporation and condensation ). Melting and sublimation are also referred to as fluidification because both liquid and vapor are regarded as fluids. | https://en.wikipedia.org/wiki/Multiphase_heat_transfer |
A multiphasic liquid is a mixture consisting of more than two immiscible liquid phases . Biphasic mixtures consisting of two immiscible phases are very common and usually consist of an organic solvent and an aqueous phase ("oil and water").
Multiphasic liquids can be used for selective liquid–liquid extractions or for decorative purposes, e.g. in cosmetics .
While it is possible to get multilayered phases by layering nonpolar and aqueous phases of decreasing densities on top of each other, these phases will not separate after mixing like true multiphasic liquids.
The following types of multiphasic liquids exist:
A system with eight phases is known. In addition to a hydrocarbon and an aqueous phase, it includes a silicone oil , an aniline and a fluorous phase, and molten phosphorus , gallium and mercury . [ 2 ] | https://en.wikipedia.org/wiki/Multiphasic_liquid |
In computational modelling , multiphysics simulation (often shortened to simply "multiphysics") is defined as the simultaneous simulation of different aspects of a physical system or systems and the interactions among them. [ 1 ] For example, simultaneous simulation of the physical stress on an object, the temperature distribution of the object and the thermal expansion which leads to the variation of the stress and temperature distributions would be considered a multiphysics simulation. [ 2 ] Multiphysics simulation is related to multiscale simulation, which is the simultaneous simulation of a single process on either multiple time or distance scales. [ 3 ]
As an interdisciplinary field, multiphysics simulation can span many science and engineering disciplines. Simulation methods frequently include numerical analysis , partial differential equations and tensor analysis . [ 4 ]
The implementation of a multiphysics simulation follows a typical series of steps: [ 1 ]
Mathematical models used in multiphysics simulations are generally a set of coupled equations. The equations can be divided into three categories according to the nature and intended role: governing equation , auxiliary equations and boundary/initial conditions . A governing equation describes a major physical mechanism or process. Multiphysics simulations are numerically implemented with discretization methods such as the finite element method , finite difference method , or finite volume method . [ 5 ]
Generally speaking, multiphysics simulation is much harder than that for individual aspects of the physical processes.
The main extra issue is how to integrate the multiple aspects of the processes with proper handling of the interactions among them.
Such issues become quite difficult when different types of numerical methods are used for the simulations of individual physical aspects.
For example, when simulating a fluid-structure interaction problem with typical Eulerian finite volume method for flow
and Lagrangian finite element method for structure dynamics. | https://en.wikipedia.org/wiki/Multiphysics_simulation |
In chemical engineering , a multiple-effect evaporator is an apparatus for efficiently using the heat from steam to evaporate water. [ 1 ] Water is boiled in a sequence of vessels , each held at a lower pressure than the last. Because the boiling temperature of water decreases as pressure decreases, the vapor boiled off in one vessel can be used to heat the next, and only the first vessel (at the highest pressure) requires an external source of heat. The multiple-effect evaporator was invented by the American ( Louisiana Creole ) engineer Norbert Rillieux . Although he may have designed the apparatus during the 1820s and constructed a prototype in 1834, he did not build the first industrially practical evaporator until 1845. Originally designed for concentrating sugar in sugar cane juice, it has since become widely used in all industrial applications where large volumes of water must be evaporated, such as salt production and water desalination .
Multiple-effect evaporation commonly uses sensible heat in the condensate to preheat liquor to be flashed. In practice the design liquid flow paths can be somewhat complicated in order to extract the most recoverable heat and to obtain the highest evaporation rates from the equipment. While in theory, evaporators may be built with an arbitrarily large number of stages, evaporators with more than four stages are rarely practical except in certain applications. Multiple-effect evaporation plants in sugar beet factories have up to eight effects; sextuple-effect evaporators are common in the recovery of black liquor in the kraft process for making wood pulp.
Entrainment of the product in the solvent causes a number of issues including firstly decrease in the amount of product recovered, secondly potential damage or fouling of the evaporation lines and steam chest of the next stage, and thirdly problems dealing or disposing of the solvent.
In forward feed the condensate and the feed-concentrate move in the same direction. In this case the vacuum pressure can be sufficient to pull the feed through the system. [ 2 ]
Here the concentrate flows in the opposite direction to the condensate. Pumps are required between stages, however the design has less heating costs.
In the simplest version of this design the feed goes to each stage.
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multiple-effect_evaporator |
The first description of multiple-prism arrays, and multiple-prism dispersion, was given by Isaac Newton in his book Opticks , also introducing prisms as beam expanders. [ 1 ] [ 2 ] Prism pair expanders were introduced by David Brewster in 1813. [ 3 ] A modern mathematical description of the single-prism dispersion was given by Max Born and Emil Wolf in 1959. [ 4 ] The generalized multiple-prism dispersion theory was introduced by F. J. Duarte and Piper [ 5 ] [ 6 ] in 1982.
The generalized mathematical description of multiple-prism dispersion, as a function of the angle of incidence, prism geometry, prism refractive index, and number of prisms, was introduced as a design tool for multiple-prism grating laser oscillators by Duarte and Piper, [ 5 ] [ 6 ] and is given by
∂ ϕ 2 , m ∂ λ = H 2 , m ∂ n m ∂ λ + ( k 1 , m k 2 , m ) − 1 ( H 1 , m ∂ n m ∂ λ ± ∂ ϕ 2 , ( m − 1 ) ∂ λ ) {\displaystyle {\frac {\partial \phi _{2,m}}{\partial \lambda }}=H_{2,m}{\frac {\partial n_{m}}{\partial \lambda }}+(k_{1,m}k_{2,m})^{-1}{\bigg (}H_{1,m}{\frac {\partial n_{m}}{\partial \lambda }}\pm \ {\frac {\partial \phi _{2,(m-1)}}{\partial \lambda }}{\bigg )}}
which can also be written as
∇ λ ϕ 2 , m = H 2 , m ∇ λ n m + ( k 1 , m k 2 , m ) − 1 ( H 1 , m ∇ λ n m ± ∇ λ ϕ 2 , ( m − 1 ) ) {\displaystyle \nabla _{\lambda }\phi _{2,m}=H_{2,m}\nabla _{\lambda }n_{m}+(k_{1,m}k_{2,m})^{-1}{\bigg (}H_{1,m}\nabla _{\lambda }n_{m}\pm \nabla _{\lambda }\phi _{2,(m-1)}{\bigg )}}
using
∇ λ = ∂ ∂ λ {\displaystyle \nabla _{\lambda }={\frac {\partial }{\partial \lambda }}}
Also,
k 1 , m = cos ψ 1 , m cos ϕ 1 , m k 2 , m = cos ϕ 2 , m cos ψ 2 , m H 1 , m = tan ϕ 1 , m n m H 2 , m = tan ϕ 2 , m n m {\displaystyle {\begin{aligned}k_{1,m}={\frac {\cos \psi _{1,m}}{\cos \phi _{1,m}}}&\qquad k_{2,m}={\frac {\cos \phi _{2,m}}{\cos \psi _{2,m}}}\\[8pt]H_{1,m}={\frac {\tan \phi _{1,m}}{n_{m}}}&\qquad H_{2,m}={\frac {\tan \phi _{2,m}}{n_{m}}}\end{aligned}}}
Here, ϕ 1 , m {\displaystyle \phi _{1,m}} is the angle of incidence, at the m th prism, and ψ 1 , m {\displaystyle \psi _{1,m}} its corresponding angle of refraction. Similarly, ϕ 2 , m {\displaystyle \phi _{2,m}} is the exit angle and ψ 2 , m {\displaystyle \psi _{2,m}} its corresponding angle of refraction. The two main equations give the first order dispersion for an array of m prisms at the exit surface of the m th prism. The plus sign in the second term in parentheses refers to a positive dispersive configuration while the minus sign refers to a compensating configuration. [ 5 ] [ 6 ] The k factors are the corresponding beam expansions, and the H factors are additional geometrical quantities. It can also be seen that the dispersion of the m th prism depends on the dispersion of the previous prism ( m – 1) .
These equations can also be used to quantify the angular dispersion in prism arrays, as described in Isaac Newton 's book Opticks , and as deployed in dispersive instrumentation such as multiple-prism spectrometers. A comprehensive review on practical multiple-prism beam expanders and multiple-prism angular dispersion theory, including explicit and ready to apply equations (engineering style), is given by Duarte. [ 8 ]
More recently, the generalized multiple-prism dispersion theory has been extended to include positive and negative refraction . [ 9 ] Also, higher order phase derivatives have been derived using a Newtonian iterative approach. [ 10 ] This extension of the theory enables the evaluation of the Nth higher derivative via an elegant mathematical framework. Applications include further refinements in the design of prism pulse compressors and nonlinear optics .
For a single generalized prism ( m = 1 ), the generalized multiple-prism dispersion equation simplifies to [ 4 ] [ 11 ]
∂ ϕ 2 , 1 ∂ λ = sin ψ 2 , 1 cos ϕ 2 , 1 ∂ n 1 ∂ λ + cos ψ 2 , 1 cos ϕ 2 , 1 tan ψ 1 , 1 ∂ n 1 ∂ λ {\displaystyle {\frac {\partial \phi _{2,1}}{\partial \lambda }}={\frac {\sin \psi _{2,1}}{\cos \phi _{2,1}}}{\frac {\partial n_{1}}{\partial \lambda }}+{\frac {\cos \psi _{2,1}}{\cos \phi _{2,1}}}\tan \psi _{1,1}{\frac {\partial n_{1}}{\partial \lambda }}}
If the single prism is a right-angled prism with the beam exiting normal to the output face, that is ϕ 2 , m {\displaystyle \phi _{2,m}} equal to zero, this equation reduces to [ 8 ]
∂ ϕ 2 , 1 ∂ λ = tan ψ 1 , 1 ∂ n 1 ∂ λ {\displaystyle {\frac {\partial \phi _{2,1}}{\partial \lambda }}=\tan \psi _{1,1}{\frac {\partial n_{1}}{\partial \lambda }}}
The first application of this theory was to evaluate the laser linewidth in multiple-prism grating laser oscillators. [ 5 ] The total intracavity angular dispersion plays an important role in the linewidth narrowing of pulsed tunable lasers through the equation [ 5 ] [ 8 ]
where Δ θ {\displaystyle \Delta \theta } is the beam divergence and the overall intracavity angular dispersion is the quantity in parentheses (elevated to –1). Although originally classical in origin, in 1992 it was shown that this laser cavity linewidth equation can also be derived from interferometric quantum principles . [ 12 ]
For the special case of zero dispersion from the multiple-prism beam expander, the single-pass laser linewidth is given by [ 8 ] [ 11 ]
where M is the beam magnification provided by the beam expander that multiplies the angular dispersion provided by the diffraction grating. In practice, M can be as high as 100-200. [ 8 ] [ 11 ]
When the dispersion of the multiple-prism expander is not equal to zero, then the single-pass linewidth is given by [ 5 ] [ 8 ]
where the first differential refers to the angular dispersion from the grating and the second differential refers to the overall dispersion from the multiple-prism beam expander (given in the section above). [ 8 ] [ 11 ]
In 1987 the multiple-prism angular dispersion theory was extended to provide explicit second order equations directly applicable to the design of prismatic pulse compressors . [ 13 ] The generalized multiple-prism dispersion theory is applicable to: | https://en.wikipedia.org/wiki/Multiple-prism_dispersion_theory |
A multiple-vehicle collision (colloquially known as a pileup or multi-car collision ), [ 1 ] is a road traffic collision involving many vehicles . Generally occurring on high-capacity and high-speed routes such as freeways , they are one of the deadliest forms of traffic collisions. The most disastrous pileups have involved more than a hundred vehicles.
A chain collision can be defined as "an accident involving three or more vehicles in which one vehicle has only rear impact damage (i.e., the "lead" vehicle); one vehicle has only frontal damage; and all other vehicles have frontal and rear impact damage (these are the "middle" vehicles)". [ 2 ]
In Great Britain, statistics are available on the number of vehicles involved in crashes.
In America, statistics are available from the Fatality Analysis Reporting System (FARS). [ 4 ]
Pile-ups generally occur in low-visibility conditions as drivers on freeways are following too closely and unable to adjust to road conditions.
Chain-reaction crashes can also occur in conditions of good visibility, when black ice or other road hazards are encountered unexpectedly as drivers round a curve or crest a hill. [ 1 ]
Multiple vehicle collisions can also occur when a third vehicle is too close to an initial collision to avoid hitting one or both of the vehicles. Due to the high traffic speeds on the road, if one car develops a problem and suddenly halts, ones behind it cannot stop in time and may hit it. Considering that these roads often have high traffic volumes, more cars are forced into braking and skidding, darting into other lanes and in front of other traffic; more vehicles become involved, creating a chain reaction effect.
Determining the cause of such collisions is difficult for investigators and it is often impossible to tell if negligence caused the crash. In spite of their frequency, little formal research has been done in the United States regarding their causes. [ 6 ]
Multiple-vehicle collisions are particularly deadly as the mass of crumpled vehicles makes escape for survivors difficult. Even if survivors are able to exit their vehicles, other cars may strike them. Individual vehicles in a multiple-vehicle collision are often hit multiple times at high speed, increasing the risk of injury to passengers who may have survived the first impact with the benefit of now-discharged protective airbags . Collisions after the initial collision may occur from the side of the vehicle, where the passenger compartment is more vulnerable.
A fire in one part of the collision can quickly spread via spilled gasoline and cover the entire crash area. Multiple-vehicle collisions can also overwhelm local emergency services making speedy rescues more difficult. If the collision takes place in a remote area, getting medical help to the scene can be difficult.
The destruction and intense heat of fires can also damage roadways, particularly by melting and burning the asphalt or spalling concrete surfaces. The structural steel of bridges and overpasses can also be weakened by the heat. A fiery pileup inside a tunnel is the most serious, as there is little means to escape the poisonous fumes and the confined heat may damage structural supports.
The large scale of these collisions can close highway routes for several days, or even longer if highway support structures are damaged.
A NHTSA report suggests that a vehicle fit with a center high-mounted stop lamps has 23.7% less risk to be involved as a lead vehicle in a chain collision while it has 16.0% less risk to be involved as a middle vehicle in such a chain collision. [ 2 ]
In the French ASFA motorway network, out of all vehicle crashes with injuries, single vehicle crashes make up 42% of the total, dual vehicle crashes make up 41%, and crashes involving three or more vehicles make up 17%. [ 7 ]
On Korean expressways, chain crashes represent 10.7% of crashes with injuries involving truck drivers. On Korean rural roads, chain crashes represent 0.6% of crashes with injuries involving truck drivers. [ 8 ]
Multiple vehicle collisions can occur in the restricted courses used in motorsports as well, most commonly after a green flag (on road courses) being waved following a warm-up lap during the start of the race. Reporters and fans apply subjective guidelines as to what threshold needs to be crossed before a simple on-track incident can be described as such. NASCAR fans, for example, talk about the " Big One ", where many cars can be or are involved in a wreck while running close together. | https://en.wikipedia.org/wiki/Multiple-vehicle_collision |
Multiple Expectation maximizations for Motif Elicitation (MEME) is a tool for discovering motifs in a group of related DNA or protein sequences. [ 1 ]
A motif is a sequence pattern that occurs repeatedly in a group of related protein or DNA sequences and is often associated with some biological function. MEME represents motifs as position-dependent letter-probability matrices which describe the probability of each possible letter at each position in the pattern. Individual MEME motifs do not contain gaps. Patterns with variable-length gaps are split by MEME into two or more separate motifs.
MEME takes as input a group of DNA or protein sequences (the training set) and outputs as many motifs as requested. It uses statistical modeling techniques to automatically choose the best width, number of occurrences, and description for each motif.
MEME is the first of a collection of tools for analyzing motifs called the MEME suite .
The MEME algorithm could be understood from two different perspectives. From a biological point of view, MEME identifies and characterizes shared motifs in a set of unaligned sequences. From the computer science aspect, MEME finds a set of non-overlapping, approximately matching substrings given a starting set of strings. [ citation needed ]
MEME can be used to find similar biological functions and structures in different sequences. It is necessary to take into account that the sequences variation can be significant and that the motifs are sometimes very small. It is also useful to take into account that the binding sites for proteins are very specific. This makes it easier to reduce wet-lab experiments (saving cost and time). Indeed, to better discover the motifs relevant from a biological point it is necessary to carefully choose: the best width of motifs, the number of occurrences in each sequence, and the composition of each motif.
The algorithm uses several types of well known functions:
However, one often doesn't know where the starting position is. Several possibilities exist: exactly one motif per sequence, or one or zero motif per sequence, or any number of motifs per sequence. | https://en.wikipedia.org/wiki/Multiple_EM_for_Motif_Elicitation |
Multiple Michael/aldol reaction (or domino Michael/aldol reaction ) is a consecutive series of reactions composed of either Michael addition reactions or aldol reactions . More than two steps of reaction are usually involved. This reaction has been used for synthesis of large macrocyclic or polycyclic ring structures.
Gary Posner and co-workers were the first to report using multiple Michael/aldol reactions to construct macrolide structures. [ 1 ] [ 2 ] Their method utilized a Michael-Michael-Michael-ring closure (MIMI-MIRC) or a Michael-Michael-aldol-ring closure annulation sequences to assemble acrylates and/or aldehydes together to form substituted 9-, 10-, and 11-membered macrolide structures. Besides synthesis of complex ring structures, multiple Michael/aldol reaction can also be used for rapid production of complex compound libraries.
Aldolases have been used to mediate multiple aldol reactions. Chi-Huey Wong and co-workers had shown that 2-deoxyribose-5-phosphate aldolase and fructose-1, 6-diphosphate aldolase could be used together in a one-pot reaction to connect two aldehydes and one ketone together through sequential aldol reactions. [ 3 ] This reaction could be used to generate a variety of carbohydrate derivatives. | https://en.wikipedia.org/wiki/Multiple_Michael/aldol_reaction |
A multiple birth is the culmination of a multiple pregnancy , wherein the mother gives birth to two or more babies. A term most applicable to vertebrate species, multiple births occur in most kinds of mammals , with varying frequencies. Such births are often named according to the number of offspring, as in twins and triplets . In non-humans, the whole group may also be referred to as a litter , and multiple births may be more common than single births. Multiple births in humans are the exception and can be exceptionally rare in the largest mammals.
A multiple pregnancy may be the result of the fertilization of a single egg that then splits to create identical fetuses , or it may be the result of the fertilization of multiple eggs that create fraternal ("non-identical") fetuses, or it may be a combination of these factors. A multiple pregnancy from a single zygote is called monozygotic , from two zygotes is called dizygotic , or from three or more zygotes is called polyzygotic .
Similarly, the siblings themselves from a multiple birth may be referred to as monozygotic if they are identical or as dizygotic (in cases of twins) or polyzygotic (for three or more siblings) if they are fraternal, i.e., non-identical.
Each fertilized ovum ( zygote ) may produce a single embryo , or it may split into two or more embryos, each carrying the same genetic material. Fetuses resulting from different zygotes are called fraternal and share only 50% of their genetic material, as ordinary full siblings from separate births do. Fetuses resulting from the same zygote share 100% of their genetic material and hence are called identical . [ 1 ] Identical twins are always the same sex.
Terms used for the number of offspring in a multiple birth, where a number higher than three ends with the suffix -uplet :
Terms used for multiple births or the genetic relationships of their offspring are based on the zygosity of the pregnancy:
Multiple pregnancies are also classified by how the fetuses are surrounded by one or more placentas ( chorionicity ) and amniotic sacs ( amnionicity ). [ 2 ]
In humans, the average length of pregnancy (2 weeks fewer than gestation ) is 38 weeks with a single fetus. This average decreases for each additional fetus: to 36 weeks for twin births, 32 weeks for triplets, and 30 weeks for quadruplets. [ 3 ] With the decreasing gestation time, the risks from immaturity at birth and subsequent viability increase with the size of the sibling group. Only as of the twentieth century have more than four all survived infancy.
Recent history has also seen increasing numbers of multiple births. In the United States, it has been estimated that by 2011, 36% of twin births, and 78% of triplet and higher-order births resulted from conception by assisted reproductive technology . [ 4 ]
Twins are by far the most common form of multiple births in humans. The U.S. Centers for Disease Control and Prevention report more than 132,000 sets of twins out of 3.9 million births of all kinds each year, about 3.4%, or 1 in 30. [ 5 ] Compared to other multiple births, twin births account for 97% of them in the US. [ 6 ] Without fertility treatments, the probability is about 1 in 60; with fertility treatments, it can be as high as 20-25%. [ 7 ]
Dizygotic (fraternal) twins can be caused by a hyperovulation gene in the mother. [ 8 ] Although the father's genes do not influence the woman's chances of having twins, he could influence his children's chances of having twins by passing on a copy of the hyperovulation gene to them.
Monozygotic (identical) twins do not run in families. The twinning is random, due to the egg splitting, so all parents have an equal chance of conceiving identical twins. [ 9 ]
Triplets can be either fraternal, identical, or a combination of both. The most common are strictly fraternal triplets, which come from a polyzygotic pregnancy of three eggs. Less common are triplets from a dizygotic pregnancy, where one zygote divides into two identical fetuses, and the other does not. Least common are identical triplets, three fetuses from one egg. In this case, sometimes the original zygote divides into two and then one of those two zygotes divides again but the other does not, or the original zygote divides into three. [ 11 ]
Triplets are far less common than twins, according to the U.S. Centers for Disease Control and Prevention, accounting for only about 4,300 sets in 3.9 million births, just a little more than .1%, or 1 in 1,000. [ 5 ] According to the American Society of Reproductive Medicine , only about 10% of these are identical triplets: about 1 in ten thousand. [ 5 ] Nevertheless, only 4 sets of identical triplets were reported in the U.S. during 2015, about one in a million. [ 5 ] According to Victor Khouzami, Chairman of Obstetrics at Greater Baltimore Medical Center , "No one really knows the incidence". [ 5 ]
Identical triplets or quadruplets are very rare and result when the original fertilized egg splits and then one of the resultant cells splits again (for triplets) or, even more rarely, a further split occurs (for quadruplets). The odds of having identical triplets is unclear. News articles and other non-scientific organizations give odds from one in 60,000 to one in 200 million pregnancies. [ 5 ] [ 12 ] [ 13 ] [ 14 ] [ 15 ] [ 16 ]
Quadruplets are much rarer than twins or triplets. As of 2007, there were approximately 3,556 sets recorded worldwide. Quadruplet births are becoming increasingly common due to fertility treatments . There are around 70 sets of all-identical quadruplets worldwide. Many sets of quadruplets contain a mixture of identical and fraternal siblings, such as three identical and one fraternal, two identical and two fraternal, or two pairs of identicals. One famous set of identical quadruplets was the Genain quadruplets , all of whom developed schizophrenia . [ 17 ] Quadruplets are sometimes referred to as "quads" in Britain. [ 18 ]
Quintuplets occur naturally in 1 in 55,000,000 births. [ 19 ] The first quintuplets known to survive infancy were the identical female Canadian Dionne quintuplets , born in 1934. Quintuplets are sometimes referred to as "quins" in the UK [ 20 ] and "quints" in North America. [ 21 ] A famous set of all-girl quintuplets are the Busby quints from the TV series OutDaughtered .
Born in Liverpool , England , on November 18, 1983, the Walton sextuplets were the world's first all-female surviving sextuplets, and the world's fourth known set of surviving sextuplets. Another well-known set of sextuplets is the Gosselin sextuplets , born on May 10, 2004, in Hershey, Pennsylvania . [ 22 ] Reality television shows Jon & Kate Plus 8 and later Kate Plus 8 have chronicled the lives of these sextuplets. Other shows of this nature include Table for 12 and Sweet Home Sextuplets .
In 1997, the McCaughey septuplets , born in Des Moines, Iowa , became the first septuplets known to survive infancy. The first surviving set of octuplets on record are the Suleman octuplets , born in 2009 in Bellflower, California . [ 23 ] [ 24 ] In 2019, all 8 children celebrated their 10th birthday. [ 25 ] Multiple births of as many as 9 babies have been born alive; In May 2021, the Cissé nonuplets were born in Morocco to Halima Cissé , a 25-year-old woman from Mali . [ 26 ] As of May 2023 [update] , [ 27 ] two years since their births, all 9 are still living and reportedly in good health.
The list of multiple births covers notable examples.
The frequency of N multiple births from natural pregnancies has been given as approximately 1:89 N −1 ( Hellin's law ) and as about 1:80 N −1 . [ 28 ] This gives:
North American dizygotic twinning occurs about once in 83 conceptions, and triplets about once in 8000 conceptions. US figures for 2010 were: [ 29 ] [ 30 ]
Human multiple births can occur either naturally (the woman ovulates multiple eggs or the fertilized egg splits into two) or as the result of infertility treatments such as in vitro fertilization (several embryos are often transferred to compensate for lower quality) or fertility drugs (which can cause multiple eggs to mature in one ovulatory cycle).
For reasons that are not yet known, the older a woman is, the more likely she is to have a multiple birth naturally. It is theorized that this is due to the higher level of follicle-stimulating hormone that older women sometimes have as their ovaries respond more slowly to FSH stimulation. [ 31 ]
The number of multiple births has increased since the late 1970s. For example, in Canada between 1979 and 1999, the number of multiple birth babies increased 35%. Before the advent of ovulation-stimulating drugs, triplets were quite rare (approximately 1 in 8000 births) and higher-order births much rarer still. [ citation needed ] Much of the increase can probably be attributed to the impact of fertility treatments, such as in-vitro fertilization . Younger patients who undergo treatment with fertility medication containing artificial FSH , followed by intrauterine insemination , are particularly at risk for multiple births of higher order.
Certain factors appear to increase the likelihood that a woman will naturally conceive multiples. These include:
Women conceiving multiples over the age of 35 increase the risk of having fetuses with certain conditions and complications that are not as common in women who are pregnant.
The increasing use of fertility drugs and consequent increased rate of multiple births has made the phenomenon of multiples more frequent and hence more visible. In 2004 the birth of sextuplets, six children, to Pennsylvania couple Kate and Jon Gosselin helped them to launch their television series, originally Jon & Kate Plus 8 and (following their divorce) Kate Plus 8 , which became the highest-rated show on the TLC network.
Babies born from multiple-birth pregnancies are much more likely to result in premature birth than those from single pregnancies. 51% of twins and 91% of triplets are born preterm, compared to 9.4% in singletons. [ 33 ] 14% of twins and 41% of triplets are even born very preterm , compared to 1.7% in singletons. [ 33 ]
Drugs known as betamimetics can be used to relax the muscles of the uterus and delay birth in singleton pregnancies. [ 34 ] There is some evidence that these drugs can also reduce the risk of preterm birth for twin pregnancies, but existing studies are small. More data is required before solid conclusions can be drawn. [ 35 ] Likewise, existing studies are too small to determine if a cervical suture is effective for reducing prematurity in cases of multiple birth. [ 36 ]
As a result of preterm birth , multiples tend to have lower birth weight than singletons. Exceptions are possible, however, as with the Kupresak triplets, born in 2008 in Mississauga , Ontario , Canada. Their combined weight was 17 lbs, 2.7 oz, which set a world record. Two of the triplets were similar in size and, as expected, moderately low birth weight. The two combined weighed 9 lbs, 2.7 oz. The third triplet, however, was much larger and weighed 8 lbs. individually. [ 37 ]
Cerebral palsy is more common among multiple births than single births, being 2.3 per 1,000 survivors in singletons, 13 in twins, and 45 in triplets in North West England . [ 38 ] This is likely a side effect of premature birth and low birth weight.
Premature birth is associated with a higher risk for a breadth of behavioral and socioemotional difficulties that begin in childhood and continue through teenagehood and often into adulthood. Conditions where the risk is greatest include attention deficit hyperactivity disorder , autism spectrum disorder , and anxiety disorders. [ 39 ]
Multiples may be monochorionic , sharing the same chorion , with resultant risk of twin-to-twin transfusion syndrome . Monochorionic multiples may even be monoamniotic , sharing the same amniotic sac , resulting in risk of umbilical cord compression and nuchal cord . In very rare cases, there may be conjoined twins , possibly impairing function of internal organs.
Multiples are also known to have a higher mortality rate. It is more common for multiple births to be stillborn, while for singletons the risk is not as high. A literary review on multiple pregnancies shows a study done on one set each of septuplets and octuplets, two sets of sextuplets, 8 sets of quintuplets, 17 sets of quadruplets, and 228 sets of triplets. By doing this study, Hammond found that the mean gestational age (how many weeks when birthed) at birth was 33.4 weeks for triplets and 31 weeks for quadruplets. This shows that stillbirth happens usually 3–5 weeks before the woman reaches full term and also that for sextuplets or higher it almost always ends in death of the fetuses. [ 40 ] Though multiples are at a greater risk of being stillborn, there is inconclusive evidence whether the actual mortality rate is higher in multiples than in singletons.
Today many multiple pregnancies are the result of in vitro fertilisation (IVF). In a 1997 study of 2,173 embryo transfers performed as part of in vitro fertilisation (IVF), 34% were successfully delivered pregnancies. [ 41 ] The overall multiple pregnancy rate was 31.3% (24.7% twins, 5.8% triplets, and .08% quadruplets). [ 41 ] Because IVFs are producing more multiples, a number of efforts are being made to reduce the risk of multiple births- specifically triplets or more. Medical practitioners are doing this by limiting the number of embryos per embryo transfer to one or two. That way, the risks for the mother and fetuses are decreased.
The appropriate number of embryos to be transferred depends on the age of the woman, whether it is the first, second or third full IVF cycle attempt and whether there are top-quality embryos available. According to a guideline from The National Institute for Health and Care Excellence (NICE) in 2013, the number of embryos transferred in a cycle should be chosen as in following table: [ 42 ]
Also, it is recommended to use single embryo transfer in all situations if a top-quality blastocyst is available. [ 42 ]
Bed rest has not been found to change outcomes and therefore is not generally recommended outside of a research study. [ 43 ]
Selective reduction is the practice of reducing the number of fetuses in a multiple pregnancy; it is also called "multifetal reduction". [ 44 ]
The procedure generally takes two days; the first day for testing in order to select which fetuses to remove, and the second day for the procedure itself, in which potassium chloride is injected into the heart of each selected fetus under the guidance of ultrasound imaging. [ 45 ] Risks of the procedure include bleeding requiring transfusion, rupture of the uterus, retained placenta , infection, a miscarriage, and prelabor rupture of membranes . Each of these appears to be rare. [ 46 ] There are also ethical concerns about this procedure, since it is a form of abortion , and also because of concerns over which fetuses are terminated and why. [ 47 ] [ 48 ]
Selective reduction was developed in the mid-1980s, as people in the field of assisted reproductive technology became aware of the risks that multiple pregnancies carried for the mother and for the fetuses. [ 49 ] [ 50 ]
Women with a multiple pregnancy are usually seen more regularly by midwives or doctors than those with singleton pregnancies because of the higher risks of complications. [ 51 ] However, there is currently no evidence to suggest that specialised antenatal services produce better outcomes for mother or babies than 'normal' antenatal care. [ 51 ] Women with a multiple pregnancy are also encouraged after 24 weeks to be on bed rest. This recommendation is not a requirement for women with a multiple pregnancy, but it has been used as a method to prevent complications. Some doctors may prescribe this method to be on the safe side and if they believe it is necessary.
As preterm birth is such a risk for women with multiple pregnancies, it has been suggested that these women should be encouraged to follow a high-calorie diet to increase the birth weights of the babies. [ 52 ] Evidence around this subject is not yet good enough to advise women to do this because the long term effects of the high-calorie diets on the mother are not known. [ 53 ]
A study in 2013 involving 106 participating centers in 25 countries came to the conclusion that, in a twin pregnancy of a gestational age between 32 weeks 0 days and 38 weeks 6 days, and the first twin is in cephalic presentation , planned Cesarean section does not significantly decrease or increase the risk of fetal or neonatal death or serious neonatal disability , as compared with planned vaginal delivery. [ 54 ] In this study, 44% of the women planned for vaginal delivery still ended up having Cesarean section for unplanned reasons such as pregnancy complications . In comparison, it has been estimated that 75% of twin pregnancies in the United States were delivered by Cesarean section in 2008. [ 55 ] Also in comparison, the rate of Cesarean section for all pregnancies in the general population varies between 40% and 14%. [ 56 ]
Fetal position (the way the babies are lying in the womb) usually determines if they are delivered by caesarean section or vaginally. A review of good quality research on this subject found that if the twin that will be born first (i.e. is lowest in the womb) is head down there is no good evidence that caesarean section will be safer than a vaginal birth for the mother or babies. [ 57 ]
Monoamniotic twins (twins that form after the splitting of a fertilised egg and share the same amniotic fluid sac) are at more risk of complications than twins that have their own sacs. There is also insufficient evidence around whether to deliver the babies early by caesarean section or to wait for labour to start naturally while running checks on the babies' wellbeing. [ 58 ] The birth of this type of twins should therefore be decided with the mother and her family and should take into account the need for good neonatal care services. [ 58 ]
Cesarean delivery is needed when first twin is in non cephalic presentation or when it is a monoamniotic twin pregnancy.
Multiple-birth infants are usually admitted to neonatal intensive care or a special care nursery in the hospital immediately after being born. The records for all the triplet pregnancies managed and delivered from 1992 to 1996 were looked over to see what the neonatal statistics were. Kaufman found from reviewing these files that during a five-year period, 55 triplet pregnancies (i.e. 165 babies) were delivered. Of the 165 babies 149 were admitted to neonatal intensive care after the delivery. [ 59 ]
In Iran, the Iranian government provides free housing to families that have birthed quintuplets. [ 60 ]
A study by the U.S. Agency for Healthcare Research and Quality found that, in 2011, pregnant women covered by private insurance in the United States were older and more likely to have multiple gestation than women covered by Medicaid . [ 61 ]
Certain cultures consider multiple births a portent of either good or evil. [ 62 ]
Mayan culture saw twins as a blessing, and was fascinated by the idea of two bodies looking alike. The Mayans used to believe that twins were one soul that had fragmented. [ citation needed ]
In Ancient Rome, the legend of the twin brothers who founded the city ( Romulus and Remus ) made the birth of twin boys a blessing, while twin girls were seen as an unlucky burden, since both would have to be provided with an expensive dowry at about the same time.
In Greek mythology, fraternal twins Castor and Polydeuces , and Heracles and Iphicles , are sons of two different fathers. One of the twins ( Polydeuces , Heracles ) is the illegitimate son of the god Zeus ; his brother is the son of their mother's mortal husband. A similar pair of twin sisters are Helen (of Troy ) and Clytemnestra (who are also sisters of Castor and Polydeuces ). The theme occurs in other mythologies as well, and is called superfecundation .
In certain medieval European chivalric romances , such as Marie de France 's Le Fresne , a woman cites a multiple birth (often to a lower-class woman) as proof of adultery on her part; while this may reflect a widespread belief, it is invariably treated as malicious slander, to be justly punished by the accuser having a multiple birth of her own, and the events of the romance are triggered by her attempt to hide one or more of the children. [ 63 ] : 244, 295 A similar effect occurs in the Knight of the Swan romance, in the Beatrix variants of the Swan-Children; her taunt is punished by giving birth to seven children at once, and her wicked mother-in-law returns her taunt before exposing the children. [ 63 ] : 239, 243
In vitro fertilization (IVF) was first successfully completed in the 1970s as a form of assisted reproductive technology. Out of all the assisted reproductive technology available that is currently in practice, in vitro fertilization has the highest chance of producing multiple offspring. Per each female egg, IVF currently has a 60–70% chance of conceiving. Fertilization is made possible by administering a fertility drug to the eggs or by directly injecting semen into the eggs. [ 32 ] There is an increased chance for women over the age of 35 to have multiple births. [ 31 ] IVF is a common genetic and ethical topic. Through IVF individuals can produce offspring successfully when natural procreation is not viable. However, in vitro can become genetically specific and allow for the selection of particular genes or expressible traits to be dominantly present in the formed embryo. [ 64 ] Ethical dilemmas arise when determining health care coverage and the deviation from natural selection and gene variations. In regards to multiple births different ethical concerns arise from the use of in vitro fertilization. Overall, multiple pregnancies can cause potential harm to the mother and children due to potential complications. Such complications can include uterine bleeding and children not receiving equal nutrients. [ 65 ] IVF has also revealed some pre term deliveries and lower birth weights in babies. [ 66 ] While some view medically assisted procreation as a saving grace to have children, others consider these procedure to be unnatural and costly to the community.
Multifetal pregnancy reduction is the reduction of one or more embryos from the bearing woman. Selective reduction usually occurs for pregnancies assisted by assisted reproductive technology (ART). The first multifetal pregnancy reductions to occur in a clinical setting took place in the 1980s. The procedure aims to reduce pregnancies down to approximately one to two fetuses. The overarching purpose of the procedure is not primarily to simply terminate life, but to increase the survival and success of the mother and babies. However, multifetal pregnancy reduction raises some ethical questions. The main argument is similar to abortion ethics in reduction of fetus versus fetus life. The protection of maternal well being versus harm of newly formed fetal life is an extension of the aforementioned ethical question. [ 67 ] It can be viewed that all life is important and that no life should be terminated without consent from the life that is being terminated. A polar opposite viewpoint advocates for the right of choice, that being the choice to terminate a pregnancy due to desire or pregnancy risks. [ 68 ] Overall, most multifetal pregnancy reductions that occur as a result of ART are being done for the protection of the child-bearer's health and to maximize the health of the remaining fetuses. [ 61 ] | https://en.wikipedia.org/wiki/Multiple_birth |
A multiple cloning site ( MCS ), also called a polylinker , is a short segment of DNA which contains many (up to ~20) restriction sites —a standard feature of engineered plasmids . [ 1 ] Restriction sites within an MCS are typically unique, occurring only once within a given plasmid. The purpose of an MCS in a plasmid is to allow a piece of DNA to be inserted into that region. [ 2 ]
MCSs are found in a variety of vectors, including cloning vectors to increase the number of copies of target DNA, and in expression vectors to create a protein product. [ 3 ] In expression vectors, the MCS is located downstream of a promoter to enable gene transcription. [ 2 ] The MCS is often inserted within a non-essential gene, such as lacZα, facilitating blue-white screening for recombinant selection. [ 4 ] By including recognition sequences for a variety of restriction enzymes, the MCS greatly enhances flexibility and efficiency in molecular cloning workflows, allowing for precise DNA insertion in synthetic biology, genetic engineering, and transgenic organism development. [ 1 ] [ 4 ]
In some instances, a vector may not contain an MCS. Rather, an MCS can be added to a vector. [ 5 ] The first step is designing complementary oligonucleotide sequences that contain restriction enzyme sites along with additional bases on the end that are complementary to the vector after digesting. Then the oligonucleotide sequences can be annealed and ligated into the digested and purified vector. The digested vector is cut with a restriction enzyme that complements the oligonucleotide insert overhangs. After ligation, transform the vector into bacteria and verify the insert by sequencing. This method can also be used to add new restriction sites to a multiple cloning site.
Multiple cloning sites are strategically designed to maximize flexibility while maintaining vector integrity. The primary consideration is the inclusion of unique restriction enzyme recognition sequences, meaning each restriction site appears only once in the plasmid backbone to prevent off-target cleavage during digestion. These enzymes are chosen to offer a range of cutting options (e.g., blunt vs. sticky ends), and their recognition sites are confirmed not to appear in essential vector elements or the insert sequence itself to avoid undesired fragmentation. [ 6 ]
Directional cloning is another key design strategy, using two different restriction enzymes flanking the insert region to ensure the gene is integrated in a single orientation. This method minimizes the need to screen for orientation, which would otherwise be necessary in non-directional cloning using a single restriction site. [ 7 ]
For expression vectors, MCSs are often placed in frame with N- or C-terminal tags, ensuring that inserted coding sequences remain in the correct open reading frame. Extra nucleotides may be added to maintain reading frame continuity, especially in fusion protein constructs. Additionally, short buffer sequences flanking restriction sites help improve enzyme activity by giving enough DNA for proper binding and cleavage. [ 8 ]
Multiple cloning sites are a feature that allows for the insertion of foreign DNA without disrupting the rest of the plasmid which makes it extremely useful in biotechnology , bioengineering , and molecular genetics . [ 1 ] MCS can aid in making transgenic organisms, more commonly known as a genetically modified organism (GMO) using genetic engineering. To take advantage of the MCS in genetic engineering, a gene of interest has to be added to the vector during production when the MCS is cut open. [ 9 ] After the MCS is made and ligated it will include the gene of interest and can be amplified to increase gene copy number in a bacterium-host. After the bacterium replicates, the gene of interest can be extracted out of the bacterium. In some instances, an expression vector can be used to create a protein product. After the products are isolated, they have a wide variety of uses such as the production of insulin , the creation of vaccines , production of antibiotics , and creation of gene therapies.
Despite their utility, multiple cloning sites present several limitations. A common issue is the limited availability of unique restriction sites compatible with both the vector and the insert. If an insert gene contains internal recognition sequences for enzymes in the MCS, it may be cut during digestion, necessitating site-directed mutagenesis or alternative cloning strategies. [ 10 ]
Additionally, the presence of multiple adjacent restriction sites can sometimes form secondary structures such as hairpins or stem-loops in the mRNA transcript. These structures, especially in the 5’ untranslated region (UTR), may interfere with ribosome binding and reduce translation efficiency of the inserted gene. [ 11 ]
Another limitation lies in sequence context variability. Inserts cloned at different restriction sites within the same MCS may end up with different upstream or downstream untranslated regions, affecting mRNA stability, transcription termination, or protein expression. [ 12 ] Some MCS designs have also been found to unintentionally introduce cryptic regulatory sequences or frameshifts, especially when modifying or stacking inserts, which can hinder reproducibility across experiments.
MCSs have distinct structural features depending on the type of vector in which they are used.
In cloning vectors , MCSs are typically placed within a selection marker , such as the lacZα gene in pUC vectors. This configuration allows for efficient screening for recombinant plasmids because the insertion of foreign DNA into the MCS inactivates the marker gene, allowing for blue-white screening or other selection methods. [ 4 ] The MCS in this region provides many restriction sites that can be utilized to enable the insertion of foreign DNA into the vector, upon this insertion, the continuity of selection marker is disrupted, making it non-functional, and allowing selection for insertion. [ 4 ] A gene of interest is inserted into the MCS in this type of vector, and is then subsequently transferred into bacteria to be reproduced and propagated. This vector allows for rapid accumulation of specific DNA sequences for experimental use. [ 13 ]
In expression vectors , MCSs are placed between a promoter and a terminator in order to regulate gene expression . The upstream promoter can be either constitutive or inducible and can respond to specific chemical inducers, while the downstream terminator facilitates proper termination of transcription and enhances plasmid stability. [ 4 ] The MCS in this type of vector allows the insertion of a gene of interest for subsequent expression ( transcription and translation ). Being between a promoter and terminator sequence, the placement of the MCS between facilitates the creation of a functionally expressed gene, with initiation and termination sequences. [ 4 ] These vectors are prepared and inserted into prokaryotic or eukaryotic cells to induce expression of particular genes within that cell. [ 14 ]
In reporter vectors , an MCS is typically placed near a reporter gene , for example, a fluorescent protein ( GFP ), luciferase , or lacZ . This allows promoter sequences to be added to the MCS to facilitate the study of promoter activity and gene regulation by monitoring reporter gene expression. [ 4 ] The MCS allows for easy and flexible insertions of different promoter types, which can then be used to study levels of expression of the reporter gene to understand dynamics of specific promoters. [ 4 ]
Multiple cloning sites are widely used in viral vectors to facilitate the insertion of therapeutic genes. In adeno-associated virus (AAV) vectors, an MCS is typically inserted between the promoter and polyadenylation signal within the inverted terminal repeats (ITRs), allowing seamless cloning of genes of interest for tissue-specific expression. [ 16 ]
Lentiviral vectors, commonly used for stable gene delivery in dividing and non-dividing cells, often feature a central MCS to accommodate diverse inserts, including fluorescent reporters, transcription factors , or shRNA expression cassettes . [ 17 ] The presence of an MCS allows rapid customization of lentiviral backbones without requiring vector redesign, enhancing the versatility of gene therapy approaches.
Additionally, adenoviral shuttle plasmids contain MCS regions that replace deleted E1 or E3 regions of the viral genome, facilitating insertion of transgenes. These modular plasmids are recombined with adenoviral backbones to generate replication-deficient vectors. [ 18 ] The MCS thereby functions as a universal insertion site across diverse viral delivery systems.
Stanley N. Cohen and Herbert W. Boyer conducted experiments in 1973 demonstrating that genes from a different species could be inserted into bacterial cells and expressed. [ 19 ] They employed the plasmid pSC101 , a naturally occurring plasmid from the bacterial species Salmonella panama , which they altered to have a single EcoRI restriction site and a tetracycline resistance gene. [ 19 ] The pSC101 plasmid did not have a formal multiple cloning site, but its design highlighted the utility of restriction sites for gene insertion, being the foundation for expanding restriction enzyme constructs in vectors to increase flexibility of cloning sites. [ 20 ]
From previous research, in 1977, scientists Francisco Bolivar and Raymond L. Rodriguez , built the pBR322 plasmid. [ 21 ] While not being officially a multiple cloning site, this plasmid was one of the first vectors to have more than one unique restriction site. [ 21 ] The sites were inserted into a strategic location in its sequence, such as in antibiotic resistance gene or lacZ , where they could be used for insertional inactivation to identify recombinant clones . [ 21 ] [ 22 ] This introduced greater flexibility in the insertion of foreign DNA fragments and marked a considerable progression toward having designated MCS regions in future vectors, such as the pUC sites.
The pUC vector series was built in 1982 by Jeffrey Vieira and Joachim Messing, starting from M13mp7. [ 23 ] The vectors had a particular region defined as having numerous independent restriction sites and formally defining the concept of the MCS. [ 23 ] The pUC vectors facilitated sequencing and cloning, significantly enhancing the molecular cloning procedure. [ 23 ]
Present advancements in MCS design have made cloning more efficient, flexible, and easy to perform, which makes them frequently used in molecular biology.
Modern synthetic biology has driven innovations in MCS architecture to enable modular and scalable genetic assembly. The BioBrick standard pioneered the use of flanking restriction sites (e.g., EcoRI, XbaI, SpeI, PstI) to standardize part junctions, allowing genetic elements like promoters, genes, and terminators to be interchangeably cloned into a common framework. Building on this, newer systems such as Modular Cloning (MoClo) and Golden Gate Assembly use Type IIS restriction enzymes that cut outside their recognition sites, enabling scarless and directional insertion of multiple parts in a single reaction. [ 24 ] These systems typically design MCS regions with flanking BsaI or BsmBI sites to enable seamless multi-part assembly.
Some platforms have introduced universal MCSs, optimized short sequences flanked by homology arms or unique sites that enable compatibility across vector backbones, expression hosts, or assembly methods. [ 25 ] Such modularity streamlines cloning workflows, reduces errors, and improves standardization across synthetic biology labs.
Restriction site placement and strategic selection in MCSs optimize flexibility and compatibility and reduce potential cloning issues. Also facilitating greater workability and versatility for accommodating a vast range of experiments and applications. In an MCS, the occurrence of multiple unique restriction sites in proximity allow for less constraints on enzyme selection. [ 4 ] Such a design enables enzymatic cleavage at specific positions, enabling specific MCS insertions and changes for various applications. [ 4 ] Widely used plasmids such as pUC19 and other pUC plasmids have purposefully placed restriction sites in the MCS to facilitate effective cloning, contributing to the flexibility of MCS in molecular use. [ 26 ]
Improvements in bioinformatics and molecular techniques enable identification and elimination of undesirable restriction sites, thus streamlining the process of cloning. Bioinformatic tools assist in screening and identification of unwanted restriction sites in an MCS or vector backbone. [ 4 ] Unwanted restriction sites have the potential to cause significant variation in protein expression depending on their position, and some impose very sharp restriction of expression of a gene of interest if located in the MCS. [ 4 ] By hitting and removing these problematic sites, researchers are able to create MCSs to obtain uniform and effective protein expression. [ 4 ]
Modern MCSs are designed with modularity in mind, making it easier to integrate and swap out genetic elements, which is particularly beneficial in the field of synthetic biology. Standardized flanking genetic sequences in MCS construction allow easy replacement of the genetic elements. [ 4 ] Modular design enables rapid assembly and customization of vectors for specific research needs. [ 4 ] The MoClo system allows for efficient assembly of DNA fragments into multigene constructs with a modular design. [ 27 ] [ 28 ] The system allows ease of coupling DNA fragments (unwanted sequences-free) to make the process size-effective and efficient. [ 27 ] For example, the MoClo system allows for efficient coupling of two genes, e.g., a promoter sequence and a coding sequence, each from different MCSs, into a single multigene construct without incorporating unwanted sequences. [ 27 ]
Contemporary MCSs are designed to be compatible with advanced cloning technologies, enhancing the accuracy and efficiency of the genetic manipulations. Contemporary MCS designs facilitate such technologies as Gibson Assembly , Golden Gate cloning and Universal MCS that have advantages over the traditional restriction enzyme-based protocols, such as parallel assembly of multiple DNA pieces. [ 4 ] [ 29 ] Such compatibility enhances the efficiency and usefulness of the MCS region. This compatibility is essential in genetic engineering and synthetic biology, in which the effective and precise assembly of various genetic components is essential to construct intricate genetic circuits or metabolic pathways. [ 4 ]
Strict attention to sequence context near MCSs ensures effective cloning and proper gene expression. Elimination of secondary structures that are possible between DNA components in the MCS (such as promoters and open reading frames ) prevents interference with restriction enzyme activity and optimizes MCS functionality. [ 26 ] Secondary structures within the 5' untranslated region , for instance, can prevent ribosome binding, reducing translation efficiency of an inserted gene. [ 26 ] In addition, preventing the additional placement of early stop codons and preventing the interruption of reading frames eliminates undesirable transcription or translational errors, such as premature stop codons that may truncate the protein product, further optimizing MCS reliability in vectors. [ 26 ]
Despite optimizations and advancements, certain challenges persist in MCS design, necessitating ongoing research and innovation. One of these issues is the occurrence of internal restriction sites in genes or vectors, which can interfere with restriction enzyme activity and cloning efficiency. [ 26 ] Furthermore, the structural environment of MCSs can confer unintended regulatory properties. [ 26 ] For instance, MCSs inserted too far away from promoter regions can result in secondary structure formation, interfering with expression of a gene in an MCS. [ 26 ] In addition, sequence context variability (e.g., differences in proximity to promoters or proximal regulatory elements) can lead to uneven gene expression. [ 4 ] This variability arises as a result of variations in the surrounding sequences, which can interfere with transcription efficiency, mRNA stability, or initiation of translation. [ 4 ] In order to minimize such variability, innovations beyond are needed to engineer standardized MCS designs that yield uniform performance in diverse genetic constructs. [ 4 ]
One bacterial plasmid used in genetic engineering as a plasmid cloning vector is pUC18. Its polylinker region is composed of several restriction enzyme recognition sites, that have been engineered into a single cluster (the polylinker). It has restriction sites for various restriction enzymes, including EcoRI , BamHI , and PstI . Another vector used in genetic engineering is pUC19 , which is similar to pUC18, but its polylinker region is reversed. E.coli is also commonly used as the bacterial host because of the availability, quick growth rate, and versatility. [ 30 ] An example of a plasmid cloning vector which modifies the inserted protein is pFUSE-Fc plasmid .
In order to genetically engineer insulin, the first step is to cut the MCS in the plasmid being used. [ 31 ] Once the MCS is cut, the gene for human insulin can be added making the plasmid genetically modified. After that, the genetically modified plasmid is put into the bacterial host and allowed to divide. To make the large supply that is demanded, the host cells are put into a large fermentation tank that is an optimal environment for the host. The process is finished by filtering out the insulin from the host. Purification can then take place so the insulin can be packaged and distributed to individuals with diabetes.
Multiple cloning sites play a central role in CRISPR gene editing systems. Many CRISPR- Cas9 plasmids feature MCSs immediately downstream of the U6 promoter, flanked by Type IIS restriction sites like BbsI or BsmBI. This design allows efficient insertion of synthetic oligonucleotides encoding 20-nucleotide single-guide RNA for targeted editing. [ 32 ]
CRISPR multiplexing often relies on tandem MCSs, where multiple single-guide RNA expression cassettes are cloned into a single vector. Using Golden Gate or Gibson Assembly methods, multiple guides can be inserted via standardized MCSs, enabling simultaneous gene knockout or regulation of several loci . [ 33 ]
Additionally, CRISPR plasmids used for homology-directed repair (HDR) frequently contain MCSs for inserting donor templates, such as fluorescent reporters or epitope tags. The presence of an MCS enables precise control over insert composition and position, making it easier to customize genome editing vectors for complex experiments. | https://en.wikipedia.org/wiki/Multiple_cloning_site |
Multiple description coding (MDC) in computing is a coding technique that fragments a single media stream into n substreams ( n ≥ 2) referred to as descriptions. The packets of each description are routed over multiple, (partially) disjoint paths. In order to decode the media stream, any description can be used, however, the quality improves with the number of descriptions received in parallel. The idea of MDC is to provide error resilience to media streams. Since an arbitrary subset of descriptions can be used to decode the original stream, network congestion or packet loss — which are common in best-effort networks such as the Internet — will not interrupt the stream but only cause a (temporary) loss of quality. The quality of a stream can be expected to be roughly proportional to data rate sustained by the receiver.
MDC is a form of data partitioning, thus comparable to layered coding as it is used in MPEG-2 and MPEG-4 . Yet, in contrast to MDC, layered coding mechanisms generate a base layer and n enhancement layers. The base layer is necessary for the media stream to be decoded, enhancement layers are applied to improve stream quality. However, the first enhancement layer depends on the base layer and each enhancement layer n + 1 depends on its subordinate layer n , thus can only be applied if n was already applied. Hence, media streams using the layered approach are interrupted whenever the base layer is missing and, as a consequence, the data of the respective enhancement layers is rendered useless. The same applies for missing enhancement layers. In general, this implies that in lossy networks the quality of a media stream is not proportional to the amount of correctly received data.
Besides increased fault tolerance, MDC allows for rate-adaptive streaming: Content providers send all descriptions of a stream without paying attention to the download limitations of clients. Receivers that cannot sustain the data rate only subscribe to a subset of these streams, thus freeing the content provider from sending additional streams at lower data rates.
The vast majority of state-of-the art codecs uses single description (SD) video coding. This approach does not partition any data at all. Despite the aforementioned advantages of MDC, SD codecs are still predominant. The reasons are probably the comparingly high complexity of codec development, the loss of some compression efficiency as well as the caused transmission overhead.
Though MDC has its practical roots in media communication, it is widely researched in the area of information theory .
A related technology is layered coding , which also produces multiple compressed streams, but with a hierarchy between these streams. | https://en.wikipedia.org/wiki/Multiple_description_coding |
Multiple displacement amplification (MDA) is a DNA amplification technique. This method can rapidly amplify minute amounts of DNA samples to a reasonable quantity for genomic analysis. The reaction starts by annealing random hexamer primers to the template: DNA synthesis is carried out by a high fidelity enzyme , preferentially Φ29 DNA polymerase . Compared with conventional PCR amplification techniques, MDA does not employ sequence-specific primers but amplifies all DNA, generates larger-sized products with a lower error frequency, and works at a constant temperature. MDA has been actively used in whole genome amplification (WGA) and is a promising method for application to single cell genome sequencing and sequencing-based genetic studies.
Many biological and forensic cases involving genetic analysis require sequencing of DNA from minute amounts of sample, such as DNA from uncultured single cells or trace amounts of tissue collected from crime scenes. Conventional Polymerase Chain Reaction ( PCR )-based DNA amplification methods require sequence-specific oligonucleotide primers and heat-stable (usually Taq ) polymerase , and can be used to generate significant amounts of DNA from minute amounts of DNA. However, this is not sufficient for modern techniques which use sequencing-based DNA analysis. Therefore, a more efficient non-sequence-specific method to amplify minute amounts of DNA is necessary, especially in single-cell genomic studies.
Bacteriophage Φ29 DNA polymerase is a high-processivity enzyme that can produce DNA amplicons greater than 70 kilobase pairs. [ 1 ] Its high fidelity and 3’–5' proofreading activity reduces the amplification error rate to 1 in 10 6 −10 7 bases compared to conventional Taq polymerase with a reported error rate of 1 in 9,000. [ 2 ] The reaction can be carried out at a moderate isothermal condition of 30 °C and therefore does not require a thermocycler . It has been actively used in cell-free cloning, which is the enzymatic method of amplifying DNA in vitro without cell culturing and DNA extraction . The large fragment of Bst DNA polymerase is also used in MDA, but Ф29 is generally preferred due to its sufficient product yield and proofreading activity. [ 3 ]
Hexamer primers are sequences composed of six random nucleotides . For MDA applications, these primers are usually thiophosphate-modified at their 3’ end to convey resistance to the 3’–5’ exonuclease activity of Ф29 DNA polymerase . MDA reactions start with the annealing of such primers to the DNA template followed by polymerase-mediated chain elongation. Increasing numbers of primer annealing events happen along the amplification reaction.
The amplification reaction initiates when multiple primer hexamers anneal to the template. When DNA synthesis proceeds to the next starting site, the polymerase displaces the newly produced DNA strand and continues its strand elongation. The strand displacement generates a newly synthesized single-stranded DNA template for more primers to anneal. Further primer annealing and strand displacement on the newly synthesized template results in a hyper-branched DNA network. The sequence debranching during amplification results in a high yield of the products. To separate the DNA branching network, S1 nucleases are used to cleave the fragments at displacement sites. The nicks on the resulting DNA fragments are repaired by DNA polymerase I .
MDA can generate 1–2 μg of DNA from single cell with genome coverage of up to 99%. [ 4 ] Products also have lower error rate and larger sizes compared to PCR based Taq amplification. [ 4 ] [ 5 ]
General work flow of MDA: [ 6 ]
MDA generates sufficient yield of DNA products. It is a powerful tool of amplifying DNA molecules from samples, such as uncultured microorganism or single cells to the amount that would be sufficient for sequencing studies. The large size of MDA-amplified DNA products also provides desirable sample quality for identifying the size of polymorphic repeat alleles. Its high fidelity also makes it reliable to be used in the single-nucleotide polymorphism (SNP) allele detection. Due to its strand displacement during amplification, the amplified DNA has sufficient coverage of the source DNA molecules, which provides a high-quality product for genomic analysis. The products of displaced strands can be subsequently cloned into vectors to construct library for subsequent sequencing reactions.
ADO is defined as the random non-amplification of one of the alleles present in a heterozygous sample. Some studies have reported the ADO rate of the MDA products to be 0–60%. [ 7 ] This drawback decreases the accuracy of genotyping of single sample and misdiagnosis in other MDA involved applications. ADO appears to be independent of the fragment sizes and has been reported to have a similar rate in other single-cell techniques. Possible solutions are the use of different lysis conditions or to carry out multiple rounds of amplifications from the diluted MDA products since PCR mediated amplification from cultured cells has been reported to give lower ADO rates.
'Preferential amplification' is over-amplification of one of the alleles in comparison to the other. Most studies on MDA have reported this issue. The amplification bias is currently observed to be random. It might affect the analysis of small stretches of genomic DNA in identifying Short Tandem Repeats (STR) alleles.
Endogenous template-independent primer-primer interaction is due to the random design of hexamer primers. One possible solution is to design constrained-randomized hexanucleotide primers that do not cross-hybridize.
Single cells of uncultured bacteria, archaea and protists, as well as individual viral particles and single fungal spores have been sequenced with the help of MDA. [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] [ 13 ] [ 14 ] [ 15 ] [ 16 ] [ 17 ] [ 18 ]
The ability to sequence individual cells is also useful in combating human disease. Genomes from single human embryonic cells have been successfully amplified for sequencing using MDA, allowing preimplantation genetic diagnosis (PGD): screening for genetic health issues in an early-stage embryo before implantation . [ 19 ] Diseases with heterogeneous properties, such as cancer , also benefit from MDA-based genome sequencing's ability to study mutations in individual cells.
The MDA products from a single cell have also been successfully used in array-comparative genomic hybridization experiments, which usually require a relatively large amount of amplified DNA.
Chromatin Immunoprecipitation results in production of complex mixtures of relatively short DNA fragments, which is challenging to amplify with MDA without causing a bias in the fragment representation. A method to circumvent this problem was proposed, which is based on conversion of these mixtures to circular concatemers using ligation, followed by Φ29 DNA polymerase-mediated MDA. [ 20 ]
The trace amount of samples collected from crime scenes can be amplified by MDA to the quantity that is enough for forensic DNA analysis, which is commonly used in identifying victims and suspects. | https://en.wikipedia.org/wiki/Multiple_displacement_amplification |
Multiple drug resistance ( MDR ), multidrug resistance or multiresistance is antimicrobial resistance shown by a species of microorganism to at least one antimicrobial drug in three or more antimicrobial categories. [ 1 ] Antimicrobial categories are classifications of antimicrobial agents based on their mode of action and specific to target organisms. [ 1 ] The MDR types most threatening to public health are MDR bacteria that resist multiple antibiotics ; other types include MDR viruses , parasites (resistant to multiple antifungal , antiviral , and antiparasitic drugs of a wide chemical variety). [ 2 ]
Recognizing different degrees of MDR in bacteria, the terms extensively drug-resistant ( XDR ) and pandrug-resistant ( PDR ) have been introduced. Extensively drug-resistant (XDR) is the non-susceptibility of one bacteria species to all antimicrobial agents except in two or less antimicrobial categories. Within XDR, pandrug-resistant (PDR) is the non-susceptibility of bacteria to all antimicrobial agents in all antimicrobial categories. [ 1 ] The definitions were published in 2011 in the journal Clinical Microbiology and Infection and are openly accessible. [ 1 ]
Common multidrug-resistant organisms, typically bacteria, include: [ 3 ]
Overlapping with MDRGN, a group of Gram-positive and Gram-negative bacteria of particular recent importance have been dubbed as the ESKAPE group ( Enterococcus faecium , Staphylococcus aureus , Klebsiella pneumoniae , Acinetobacter baumannii , Pseudomonas aeruginosa and Enterobacter species). [ 4 ]
Various microorganisms have survived for thousands of years by their ability to adapt to antimicrobial agents. They do so via spontaneous mutation or by DNA transfer . This process enables some bacteria to oppose the action of certain antibiotics, rendering the antibiotics ineffective. [ 5 ] These microorganisms employ several mechanisms in attaining multi-drug resistance:
Many different bacteria now exhibit multi-drug resistance, including staphylococci , enterococci , gonococci , streptococci , salmonella , as well as numerous other Gram-negative bacteria and Mycobacterium tuberculosis . Antibiotic resistant bacteria are able to transfer copies of DNA that code for a mechanism of resistance to other bacteria even distantly related to them, which then are also able to pass on the resistance genes, resulting in generations of antibiotics resistant bacteria. [ 11 ] This initial transfer of DNA is called horizontal gene transfer . [ 12 ]
Phage-resistant bacteria variants have been observed in human studies. As for antibiotics, horizontal transfer of phage resistance can be acquired by plasmid acquisition. [ 13 ]
Yeasts such as Candida species can become resistant under long-term treatment with azole preparations, requiring treatment with a different drug class. Lomentospora prolificans infections are often fatal because of their resistance to multiple antifungal agents. [ 14 ]
HIV is the prime example of MDR against antivirals, as it mutates rapidly under monotherapy. Influenza virus has become increasingly MDR; first to amantadines, then to neuraminidase inhibitors such as oseltamivir , (2008-2009: 98.5% of Influenza A tested resistant), also more commonly in people with weak immune systems. Cytomegalovirus can become resistant to ganciclovir and foscarnet under treatment, especially in immunosuppressed patients. Herpes simplex virus rarely becomes resistant to acyclovir preparations, mostly in the form of cross-resistance to famciclovir and valacyclovir , usually in immunosuppressed patients. [ 15 ]
The prime example for MDR against antiparasitic drugs is malaria . Plasmodium vivax has become chloroquine and sulfadoxine-pyrimethamine resistant a few decades ago, and as of 2012 artemisinin -resistant Plasmodium falciparum has emerged in western Cambodia and western Thailand. [ 16 ] Toxoplasma gondii can also become resistant to artemisinin , as well as atovaquone and sulfadiazine , but is not usually MDR [ 17 ] Antihelminthic resistance is mainly reported in the veterinary literature, for example in connection with the practice of livestock drenching [ 18 ] and has been recent focus of FDA regulation.
To limit the development of antimicrobial resistance, it has been suggested to: [ citation needed ]
The medical community relies on education of its prescribers, and self-regulation in the form of appeals to voluntary antimicrobial stewardship , which at hospitals may take the form of an antimicrobial stewardship program. It has been argued that depending on the cultural context government can aid in educating the public on the importance of restrictive use of antibiotics for human clinical use, but unlike narcotics , there is no regulation of its use anywhere in the world at this time. Antibiotic use has been restricted or regulated for treating animals raised for human consumption with success, in Denmark for example. [ 19 ]
Infection prevention is the most efficient strategy of prevention of an infection with a MDR organism within a hospital, because there are few alternatives to antibiotics in the case of an extensively resistant or panresistant infection; if an infection is localized, removal or excision can be attempted (with MDR-TB the lung for example), but in the case of a systemic infection only generic measures like boosting the immune system with immunoglobulins may be possible. The use of bacteriophages (viruses which kill bacteria) is a developing area of possible therapeutic treatments. [ 20 ]
It is necessary to develop new antibiotics over time since the selection of resistant bacteria cannot be prevented completely. This means with every application of a specific antibiotic, the survival of a few bacteria which already have a resistance gene against the substance is promoted, and the concerning bacterial population amplifies. Therefore, the resistance gene is farther distributed in the organism and the environment, and a higher percentage of bacteria means they no longer respond to a therapy with this specific antibiotic. In addition to developing new antibiotics, new strategies entirely must be implemented in order to keep the public safe from the event of total resistance. New strategies are being tested such as UV light treatments and bacteriophage utilization, however more resources must be dedicated to this cause. | https://en.wikipedia.org/wiki/Multiple_drug_resistance |
A multiple hearth furnace also known as a vertical calciner, is used for continuous preparation and calcining of materials.
The multiple hearth furnaces consist of several circular hearths or kilns superimposed on each other. Material is fed from the top and is moved by the action of rotating "rabble arms", and the revolving mechanical rabbles attached to the arms move over the surface of each hearth to continuously shift the ore. The arms are attached to a rotating central shaft that passes through the center of the roaster. As the material is moved, the ore that is charged at the top hearth gradually moves downward as it passes through windows in the floor of each hearth or through alternate passages around the shaft and the periphery until it finally emerges at the bottom.
The oxidizing gases flow upward, i.e., counter-current to the descending charge. In a well-insulated roaster, external heating is unnecessary except when the charge is highly moist. The hearth at the top of the roaster dries and heats the charge. Ignition and oxidation of the charge occur lower down. [ 1 ]
The hearths may be individually heated and the number, temperature, rotation rate, and size of each hearth determine the residence time and conditions for the calcining powder in order to achieve the desired final properties.
The individual hearths are lined with refractory brick, and the rabble arms are typically a force-cooled metal alloy. The entire structure is enclosed in a cylindrical brick-lined steel shell. | https://en.wikipedia.org/wiki/Multiple_hearth_furnace |
In mathematics (specifically multivariable calculus ), a multiple integral is a definite integral of a function of several real variables , for instance, f ( x , y ) or f ( x , y , z ) .
Integrals of a function of two variables over a region in R 2 {\displaystyle \mathbb {R} ^{2}} (the real-number plane) are called double integrals , and integrals of a function of three variables over a region in R 3 {\displaystyle \mathbb {R} ^{3}} (real-number 3D space) are called triple integrals . [ 1 ] For repeated antidifferentiation of a single-variable function, see the Cauchy formula for repeated integration .
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x -axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where z = f ( x , y ) ) and the plane which contains its domain . [ 1 ] If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
Multiple integration of a function in n variables: f ( x 1 , x 2 , ..., x n ) over a domain D is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign: [ 2 ]
Since the concept of an antiderivative is only defined for functions of a single real variable, the usual definition of the indefinite integral does not immediately extend to the multiple integral.
For n > 1 , consider a so-called "half-open" n -dimensional hyperrectangular domain T , defined as
Partition each interval [ a j , b j ) into a finite family I j of non-overlapping subintervals i j α , with each subinterval closed at the left end, and open at the right end.
Then the finite family of subrectangles C given by
is a partition of T ; that is, the subrectangles C k are non-overlapping and their union is T .
Let f : T → R be a function defined on T . Consider a partition C of T as defined above, such that C is a family of m subrectangles C m and
We can approximate the total ( n + 1) -dimensional volume bounded below by the n -dimensional hyperrectangle T and above by the n -dimensional graph of f with the following Riemann sum :
where P k is a point in C k and m( C k ) is the product of the lengths of the intervals whose Cartesian product is C k , also known as the measure of C k .
The diameter of a subrectangle C k is the largest of the lengths of the intervals whose Cartesian product is C k . The diameter of a given partition of T is defined as the largest of the diameters of the subrectangles in the partition. Intuitively, as the diameter of the partition C is restricted smaller and smaller, the number of subrectangles m gets larger, and the measure m( C k ) of each subrectangle grows smaller. The function f is said to be Riemann integrable if the limit
exists, where the limit is taken over all possible partitions of T of diameter at most δ . [ 3 ]
If f is Riemann integrable, S is called the Riemann integral of f over T and is denoted
Frequently this notation is abbreviated as
where x represents the n -tuple ( x 1 , ..., x n ) and d n x is the n -dimensional volume differential .
The Riemann integral of a function defined over an arbitrary bounded n -dimensional set can be defined by extending that function to a function defined over a half-open rectangle whose values are zero outside the domain of the original function. Then the integral of the original function over the original domain is defined to be the integral of the extended function over its rectangular domain, if it exists.
In what follows the Riemann integral in n dimensions will be called the multiple integral .
Multiple integrals have many properties common to those of integrals of functions of one variable (linearity, commutativity, monotonicity, and so on). One important property of multiple integrals is that the value of an integral is independent of the order of integrands under certain conditions. This property is popularly known as Fubini's theorem . [ 4 ]
In the case of T ⊆ R 2 {\displaystyle T\subseteq \mathbb {R} ^{2}} , the integral
is the double integral of f on T , and if T ⊆ R 3 {\displaystyle T\subseteq \mathbb {R} ^{3}} the integral
is the triple integral of f on T .
Notice that, by convention, the double integral has two integral signs, and the triple integral has three; this is a notational convention which is convenient when computing a multiple integral as an iterated integral, as shown later in this article.
The resolution of problems with multiple integrals consists, in most cases, of finding a way to reduce the multiple integral to an iterated integral , a series of integrals of one variable, each being directly solvable. For continuous functions, this is justified by Fubini's theorem . Sometimes, it is possible to obtain the result of the integration by direct examination without any calculations.
The following are some simple methods of integration: [ 1 ]
When the integrand is a constant function c , the integral is equal to the product of c and the measure of the domain of integration. If c = 1 and the domain is a subregion of R 2 , the integral gives the area of the region, while if the domain is a subregion of R 3 , the integral gives the volume of the region.
Example. Let f ( x , y ) = 2 and
in which case
since by definition we have:
When the domain of integration is symmetric about the origin with respect to at least one of the variables of integration and the integrand is odd with respect to this variable, the integral is equal to zero, as the integrals over the two halves of the domain have the same absolute value but opposite signs. When the integrand is even with respect to this variable, the integral is equal to twice the integral over one half of the domain, as the integrals over the two halves of the domain are equal.
Example 1. Consider the function f ( x , y ) = 2 sin( x ) − 3 y 3 + 5 integrated over the domain
a disc with radius 1 centered at the origin with the boundary included.
Using the linearity property, the integral can be decomposed into three pieces:
The function 2 sin( x ) is an odd function in the variable x and the disc T is symmetric with respect to the y -axis, so the value of the first integral is 0. Similarly, the function 3 y 3 is an odd function of y , and T is symmetric with respect to the x -axis, and so the only contribution to the final result is that of the third integral. Therefore the original integral is equal to the area of the disk times 5, or 5 π .
Example 2. Consider the function f ( x , y , z ) = x exp( y 2 + z 2 ) and as integration region the ball with radius 2 centered at the origin,
The "ball" is symmetric about all three axes, but it is sufficient to integrate with respect to x -axis to show that the integral is 0, because the function is an odd function of that variable.
This method is applicable to any domain D for which:
Such a domain will be here called a normal domain . Elsewhere in the literature, normal domains are sometimes called type I or type II domains, depending on which axis the domain is fibred over. In all cases, the function to be integrated must be Riemann integrable on the domain, which is true (for instance) if the function is continuous.
If the domain D is normal with respect to the x -axis, and f : D → R is a continuous function ; then α ( x ) and β ( x ) (both of which are defined on the interval [ a , b ] ) are the two functions that determine D . Then, by Fubini's theorem: [ 5 ]
If D is normal with respect to the y -axis and f : D → R is a continuous function; then α ( y ) and β ( y ) (both of which are defined on the interval [ a , b ] ) are the two functions that determine D . Again, by Fubini's theorem:
If T is a domain that is normal with respect to the xy -plane and determined by the functions α ( x , y ) and β ( x , y ) , then
This definition is the same for the other five normality cases on R 3 . It can be generalized in a straightforward way to domains in R n .
The limits of integration are often not easily interchangeable (without normality or with complex formulae to integrate). One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae. To do so, the function must be adapted to the new coordinates.
Example 1a. The function is f ( x , y ) = ( x − 1) 2 + √ y ; if one adopts the substitution u = x − 1 , v = y therefore x = u + 1 , y = v one obtains the new function f 2 ( u , v ) = ( u ) 2 + √ v .
There exist three main "kinds" of changes of variable (one in R 2 , two in R 3 ); however, more general substitutions can be made using the same principle.
In R 2 if the domain has a circular symmetry and the function has some particular characteristics one can apply the transformation to polar coordinates (see the example in the picture) which means that the generic points P ( x , y ) in Cartesian coordinates switch to their respective points in polar coordinates. That allows one to change the shape of the domain and simplify the operations.
The fundamental relation to make the transformation is the following:
Example 2a. The function is f ( x , y ) = x + y and applying the transformation one obtains
Example 2b. The function is f ( x , y ) = x 2 + y 2 , in this case one has:
using the Pythagorean trigonometric identity (can be useful to simplify this operation).
The transformation of the domain is made by defining the radius' crown length and the amplitude of the described angle to define the ρ , φ intervals starting from x , y .
Example 2c. The domain is D = { x 2 + y 2 ≤ 4} , that is a circumference of radius 2; it's evident that the covered angle is the circle angle, so φ varies from 0 to 2 π , while the crown radius varies from 0 to 2 (the crown with the inside radius null is just a circle).
Example 2d. The domain is D = { x 2 + y 2 ≤ 9, x 2 + y 2 ≥ 4, y ≥ 0} , that is the circular crown in the positive y half-plane (please see the picture in the example); φ describes a plane angle while ρ varies from 2 to 3. Therefore the transformed domain will be the following rectangle :
The Jacobian determinant of that transformation is the following:
which has been obtained by inserting the partial derivatives of x = ρ cos( φ ) , y = ρ sin( φ ) in the first column respect to ρ and in the second respect to φ , so the dx dy differentials in this transformation become ρ dρ dφ .
Once the function is transformed and the domain evaluated, it is possible to define the formula for the change of variables in polar coordinates:
φ is valid in the [0, 2π] interval while ρ , which is a measure of a length, can only have positive values.
Example 2e. The function is f ( x , y ) = x and the domain is the same as in Example 2d. From the previous analysis of D we know the intervals of ρ (from 2 to 3) and of φ (from 0 to π ). Now we change the function:
Finally let's apply the integration formula:
Once the intervals are known, you have
In R 3 the integration on domains with a circular base can be made by the passage to cylindrical coordinates ; the transformation of the function is made by the following relation:
The domain transformation can be graphically attained, because only the shape of the base varies, while the height follows the shape of the starting region.
Example 3a. The region is D = { x 2 + y 2 ≤ 9, x 2 + y 2 ≥ 4, 0 ≤ z ≤ 5} (that is the "tube" whose base is the circular crown of Example 2d and whose height is 5); if the transformation is applied, this region is obtained:
(that is, the parallelepiped whose base is similar to the rectangle in Example 2d and whose height is 5).
Because the z component is unvaried during the transformation, the dx dy dz differentials vary as in the passage to polar coordinates: therefore, they become ρ dρ dφ dz .
Finally, it is possible to apply the final formula to cylindrical coordinates:
This method is convenient in case of cylindrical or conical domains or in regions where it is easy to individuate the z interval and even transform the circular base and the function.
Example 3b. The function is f ( x , y , z ) = x 2 + y 2 + z and as integration domain this cylinder : D = { x 2 + y 2 ≤ 9, −5 ≤ z ≤ 5} . The transformation of D in cylindrical coordinates is the following:
while the function becomes
Finally one can apply the integration formula:
developing the formula you have
In R 3 some domains have a spherical symmetry, so it's possible to specify the coordinates of every point of the integration region by two angles and one distance. It's possible to use therefore the passage to spherical coordinates ; the function is transformed by this relation:
Points on the z -axis do not have a precise characterization in spherical coordinates, so θ can vary between 0 and 2 π .
The better integration domain for this passage is the sphere.
Example 4a. The domain is D = x 2 + y 2 + z 2 ≤ 16 (sphere with radius 4 and center at the origin); applying the transformation you get the region
The Jacobian determinant of this transformation is the following:
The dx dy dz differentials therefore are transformed to ρ 2 sin( φ ) dρ dθ dφ .
This yields the final integration formula:
It is better to use this method in case of spherical domains and in case of functions that can be easily simplified by the first fundamental relation of trigonometry extended to R 3 (see Example 4b); in other cases it can be better to use cylindrical coordinates (see Example 4c).
The extra ρ 2 and sin φ come from the Jacobian.
In the following examples the roles of φ and θ have been reversed.
Example 4b. D is the same region as in Example 4a and f ( x , y , z ) = x 2 + y 2 + z 2 is the function to integrate. Its transformation is very easy:
while we know the intervals of the transformed region T from D :
We therefore apply the integration formula:
and, developing, we get
Example 4c. The domain D is the ball with center at the origin and radius 3 a ,
and f ( x , y , z ) = x 2 + y 2 is the function to integrate.
Looking at the domain, it seems convenient to adopt the passage to spherical coordinates, in fact, the intervals of the variables that delimit the new T region are:
However, applying the transformation, we get
Applying the formula for integration we obtain:
which can be solved by turning it into an iterated integral. ∭ T ρ 4 sin 3 θ d ρ d θ d φ = ∫ 0 3 a ρ 4 d ρ ⏟ I ∫ 0 π sin 3 θ d θ ⏟ I I ∫ 0 2 π d φ ⏟ I I I {\displaystyle \iiint _{T}\rho ^{4}\sin ^{3}\theta \,d\rho \,d\theta \,d\varphi =\underbrace {\int _{0}^{3a}\rho ^{4}d\rho } _{I}\,\underbrace {\int _{0}^{\pi }\sin ^{3}\theta \,d\theta } _{II}\,\underbrace {\int _{0}^{2\pi }d\varphi } _{III}} .
I = ∫ 0 3 a ρ 4 d ρ = ρ 5 5 | 0 3 a = 243 5 a 5 {\displaystyle I=\left.\int _{0}^{3a}\rho ^{4}d\rho ={\frac {\rho ^{5}}{5}}\right\vert _{0}^{3a}={\frac {243}{5}}a^{5}} ,
I I = ∫ 0 π sin 3 θ d θ = − ∫ 0 π sin 2 θ d ( cos θ ) = ∫ 0 π ( cos 2 θ − 1 ) d ( cos θ ) = cos 3 θ 3 | 0 π − cos θ | 0 π = 4 3 {\displaystyle II=\int _{0}^{\pi }\sin ^{3}\theta \,d\theta =-\int _{0}^{\pi }\sin ^{2}\theta \,d(\cos \theta )=\int _{0}^{\pi }(\cos ^{2}\theta -1)\,d(\cos \theta )=\left.{\frac {\cos ^{3}\theta }{3}}\right|_{0}^{\pi }-\left.\cos \theta \right|_{0}^{\pi }={\frac {4}{3}}} ,
I I I = ∫ 0 2 π d φ = 2 π {\displaystyle III=\int _{0}^{2\pi }d\varphi =2\pi } .
Collecting all parts,
∭ T ρ 4 sin 3 θ d ρ d θ d φ = I ⋅ I I ⋅ I I I = 243 5 a 5 ⋅ 4 3 ⋅ 2 π = 648 5 π a 5 {\displaystyle \iiint _{T}\rho ^{4}\sin ^{3}\theta \,d\rho \,d\theta \,d\varphi =I\cdot II\cdot III={\frac {243}{5}}a^{5}\cdot {\frac {4}{3}}\cdot 2\pi ={\frac {648}{5}}\pi a^{5}} .
Alternatively, this problem can be solved by using the passage to cylindrical coordinates. The new T intervals are
the z interval has been obtained by dividing the ball into two hemispheres simply by solving the inequality from the formula of D (and then directly transforming x 2 + y 2 into ρ 2 ). The new function is simply ρ 2 . Applying the integration formula
Then we get:
Thanks to the passage to cylindrical coordinates it was possible to reduce the triple integral to an easier one-variable integral.
See also the differential volume entry in nabla in cylindrical and spherical coordinates .
Let us assume that we wish to integrate a multivariable function f over a region A :
From this we formulate the iterated integral
The inner integral is performed first, integrating with respect to x and taking y as a constant, as it is not the variable of integration . The result of this integral, which is a function depending only on y , is then integrated with respect to y .
We then integrate the result with respect to y .
In cases where the double integral of the absolute value of the function is finite, the order of integration is interchangeable, that is, integrating with respect to x first and integrating with respect to y first produce the same result. That is Fubini's theorem . For example, doing the previous calculation with order reversed gives the same result:
Consider the region (please see the graphic in the example):
Calculate
This domain is normal with respect to both the x - and y -axes. To apply the formulae it is required to find the functions that determine D and the intervals over which these functions are defined. In this case the two functions are:
while the interval is given by the intersections of the functions with x = 0, so the interval is [ a , b ] = [0, 1] (normality has been chosen with respect to the x -axis for a better visual understanding).
It is now possible to apply the formula:
(at first the second integral is calculated considering x as a constant). The remaining operations consist of applying the basic techniques of integration:
If we choose normality with respect to the y -axis we could calculate
and obtain the same value.
Using the methods previously described, it is possible to calculate the volumes of some common solids.
This is in agreement with the formula for the volume of a prism
In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the double improper integral or the triple improper integral .
Fubini's theorem states that if [ 4 ]
that is, if the integral is absolutely convergent, then the multiple integral will give the same result as either of the two iterated integrals:
In particular this will occur if | f ( x , y ) | is a bounded function and A and B are bounded sets .
If the integral is not absolutely convergent, care is needed not to confuse the concepts of multiple integral and iterated integral , especially since the same notation is often used for either concept. The notation
means, in some cases, an iterated integral rather than a true double integral. In an iterated integral, the outer integral
is the integral with respect to x of the following function of x :
A double integral, on the other hand, is defined with respect to area in the xy -plane. If the double integral exists, then it is equal to each of the two iterated integrals (either " dy dx " or " dx dy ") and one often computes it by computing either of the iterated integrals. But sometimes the two iterated integrals exist when the double integral does not, and in some such cases the two iterated integrals are different numbers, i.e., one has
This is an instance of rearrangement of a conditionally convergent integral.
On the other hand, some conditions ensure that the two iterated integrals are equal even though the double integral need not exist. By the Fichtenholz – Lichtenstein theorem, if f is bounded on [0, 1] × [0, 1] and both iterated integrals exist, then they are equal. Moreover, existence of the inner integrals ensures existence of the outer integrals. [ 6 ] [ 7 ] [ 8 ] The double integral need not exist in this case even as Lebesgue integral , according to Sierpiński . [ 9 ]
The notation
may be used if one wishes to be emphatic about intending a double integral rather than an iterated integral.
Triple integral was demonstrated by Fubini's theorem. [ 10 ] Drichlet theorem and Liouville 's extension theorem on Triple integral.
Quite generally, just as in one variable, one can use the multiple integral to find the average of a function over a given set. Given a set D ⊆ R n and an integrable function f over D , the average value of f over its domain is given by
where m ( D ) is the measure of D .
Additionally, multiple integrals are used in many applications in physics . The examples below also show some variations in the notation.
In mechanics , the moment of inertia is calculated as the volume integral (triple integral) of the density weighed with the square of the distance from the axis:
The gravitational potential associated with a mass distribution given by a mass measure dm on three-dimensional Euclidean space R 3 is [ 11 ]
If there is a continuous function ρ ( x ) representing the density of the distribution at x , so that dm ( x ) = ρ ( x ) d 3 x , where d 3 x is the Euclidean volume element , then the gravitational potential is
In electromagnetism , Maxwell's equations can be written using multiple integrals to calculate the total magnetic and electric fields. [ 12 ] In the following example, the electric field produced by a distribution of charges given by the volume charge density ρ ( r → ) is obtained by a triple integral of a vector function:
This can also be written as an integral with respect to a signed measure representing the charge distribution. | https://en.wikipedia.org/wiki/Multiple_integral |
Multiple layered plasmonics use electronically responsive media to change and manipulate the plasmonic properties of plasmons . The properties typically being manipulated can include the directed scattering of light and light absorption. [ 1 ] [ 2 ] The use of these to use “changeable” [ 3 ] plasmonics is currently undergoing development in the academic community by allowing them to have multiple sets of functions that are dependent on how they are being manipulated or excited. Under these new manipulations, such as multiple layers that respond to different resonant frequencies, their new functions were designed to accomplish multiple objectives in a single application.
This article provides an overview of current developing medical usage of multiple layered plasmonics, more specifically those developed by the Halas Group at Rice University [ 4 ]
In addition to the bio-medical applications purposed, several other uses will be briefly described below.
Gold shelled nanoparticles, which are spherical nanoparticles with silica cores and gold shells, are used in cancer therapy and bio imaging enhancement. Theranostic probes – capable of detection and treatment of cancer in a single treatment - are nanoparticles that have binding sites on their shell that allow them to attach to a desired location (typically cancerous cells) then can be imaged through dual modality imagery (an imaging strategy that uses x-rays and radionuclide imaging ) and through near-infrared fluorescence. [ 5 ] The reason gold nanoparticles are used is due to their vivid optical properties which are controlled by their size, geometry, and their surface plasmons. Gold nanoparticles (such as AuNPs) have the benefit of being biocompatible and the flexibility to have multiple different molecules and fundamental materials, attached to their shell (almost anything that can normally be attached to gold can be attached to the gold nano-shell, helping in identifying and treating cancer). The treatment of cancer is possible only because of the scattering and absorption that occurs for plasmonics . Under scattering, the gold plated nanoparticles become visible to imaging processes that are tuned to the correct wavelength which is dependent upon the size and geometry of the particles. Under absorption, photothermal ablation occurs, which heats the nanoparticles and their immediate surroundings to temperatures capable of killing the surrounding cells. Additionally, these nanoparticles can be made to release antisense DNA oligonucleotides when under photo-activation. These oligonucleotides are used in conjunction with the photo-thermal ablation treatments to perform gene-therapy. This is accomplished because nanoparticle complexes are delivered inside of cells then undergo light induced release of DNA from their surface. This will allow for the internal manipulation of a cell and provide a means for monitoring a group cells return to equilibrium. [ 6 ]
Another example of multiple layered plasmonics involves placing drugs inside of the nanoparticle and using it as a vehicle to deliver toxic drugs to cancerous sites only. [ 7 ] This is accomplished by coating the outside of a nanoparticle with iron oxide (allowing for easy tracking with an MRI machine ) then once the area of the tumor is coated with the drug filled nanoparticles, the nanoparticles can be activated using resonant light waves to release the drug.
Multiple layered plasmonics can be coated in nanoparticles to modify or drive a reaction near a metallic surface when properly excited. [ 8 ] Additionally, the scattering of light from these plasmonics can be controlled and even directed based on the surface particles, geometry, and size.
Multiple layered plasmonics can be used in harvesting solar radiation for energy applications. This is accomplished by redirecting incident light into the waveguide and evanescent surface modes of thin film photovoltaic devices.
Using multiple layered plasmons to purify water is also being investigated.
For more information on the research behind energy applications, and the collaborations behind this research, please visit the Halas group website listed below in the external links. | https://en.wikipedia.org/wiki/Multiple_layered_plasmonics |
Multiple loci VNTR analysis ( MLVA ) is a method employed for the genetic analysis of particular microorganisms , such as pathogenic bacteria , that takes advantage of the polymorphism of tandemly repeated DNA sequences . A " VNTR " is a "variable-number tandem repeat". This method is well known in forensic science since it is the basis of DNA fingerprinting in humans. When applied to bacteria, it contributes to forensic microbiology through which the source of a particular strain might eventually be traced back, making it a useful technique for outbreak surveillance .
In a typical MLVA, a number of well-selected and characterised (in terms of mutation rate and diversity) loci are amplified by polymerase chain reaction ( PCR ), so that the size of each locus can be measured, usually by electrophoresis of the amplification products together with reference DNA fragments (a so-called DNA size marker). Different electrophoresis equipment can be used depending on the required size estimate accuracy, and the local laboratory set-up, from basic agarose gel electrophoresis up to the more sophisticated and high-throughput capillary electrophoresis devices. [ 1 ] From this size estimate, the number of repeat units at each locus can be deduced. The resulting information is a code which can be easily compared to reference databases once the assay has been harmonised and standardised. [ 2 ] [ 3 ]
MLVA has become a major first line typing tool in a number of pathogens where such an harmonisation could be achieved, including Mycobacterium tuberculosis , [ 4 ] Bacillus anthracis , [ 5 ] Brucella . [ 6 ] [ 7 ] | https://en.wikipedia.org/wiki/Multiple_loci_VNTR_analysis |
Multiple rule-based problems are problems containing various conflicting rules and restrictions. [ 1 ] Such problems typically have an " optimal " solution, found by striking a balance between the various restrictions, without directly defying any of the aforementioned restrictions.
Solutions to such problems can either require complex, non-linear thinking processes, or can instead require mathematics-based solutions in which an optimal solution is found by setting the various restrictions as equations, and finding an appropriate maximum value when all equations are added. These problems may thus require more working information as compared to causal relationship problem solving or single rule-based problem solving. The multiple rule-based problem solving is more likely to increase cognitive load than are the other two types of problem solving.
This mathematical analysis –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multiple_rule-based_problems |
Multiple satellite imaging is the process of using multiple satellites to gather more information than a single satellite so that a better estimate of the desired source is possible. Something that cannot be resolved with one telescope might be visible with two or more telescopes.
Interferometry is the process of combining waves in such a way that they constructively interfere. When two or more independent sources detect a signal at the same given frequency those signals can be combined and the result is better than each one individually. An overview of Astronomical interferometers and a History of astronomical interferometry can be referenced from their respective pages.
The NASA Origins Program was created in the 1990s to ultimately search for the origin of the universe . The theory that the Origins Program is based on is: since light travels at a constant speed until it is absorbed by something; there is still light that was part of the first light ever created traveling about the universe and ultimately some of that light is coming in the general direction of Earth. So a satellite system capable of collecting light from the beginning of the universe would be able to tell us more about where we came from.
There is also the constant search for life in other worlds . A satellite system using the interferometric technologies mentioned above would be able to have a much higher resolution than any of the current deep space imaging systems.
NASA is currently focused on the Vision for Space Exploration and has reduced current funding for scientific unmanned space exploration in favor of human exploration. These budget cuts have slowed the multiple satellite imaging development and relevant scientific missions as Project Prometheus and Terrestrial Planet Finder have ended as well but research continues. | https://en.wikipedia.org/wiki/Multiple_satellite_imaging |
Multiple scattering theory (MST) is the mathematical formalism that is used to describe the propagation of a wave through a collection of scatterers. Examples are acoustical waves traveling through porous media, light scattering from water droplets in a cloud, or x-rays scattering from a crystal. A more recent application is to the propagation of quantum matter waves like electrons or neutrons through a solid.
As pointed out by Jan Korringa , [ 1 ] the origin of this theory can be traced back to an 1892 paper by Lord Rayleigh . An important mathematical formulation of the theory was made by Paul Peter Ewald . [ 2 ] Korringa and Ewald acknowledged the influence on their work of the 1903 doctoral dissertation of Nikolai Kasterin , portions of which were published in German in the Proceedings of the Royal Academy of Sciences in Amsterdam under the sponsorship of Heike Kamerlingh Onnes . [ 3 ] The MST formalism is widely used for electronic structure calculations as well as diffraction theory , and is the subject of many books. [ 4 ] [ 5 ]
The multiple-scattering approach is the best way to derive one-electron Green's functions . These functions differ from the Green's functions used to treat the many-body problem , but they are the best starting point for calculations of the electronic structure of condensed matter systems that cannot be treated with band theory.
The terms "multiple scattering" and "multiple scattering theory" are often used in other contexts. For examples, Molière's theory of the scattering of fast charged particles in matter, [ 6 ] or Glauber multiple scattering theory [ 7 ] for high-energy particle multiple-scattering off nucleons in a nucleus are also denominated that way.
The MST equations can be derived with different wave equations, but one of the simplest and most useful ones is the Schrödinger equation for an electron moving in a solid. With the help of density functional theory , this problem can be reduced to the solution of a one-electron equation
where the effective one-electron potential, V ( r ) {\displaystyle V({\bf {r}})} , is a functional of the density of the electrons in the system.
In the Dirac notation, the wave equation can be written as an inhomogeneous equation, ( E − H 0 ) | ψ ⟩ = V | ψ ⟩ {\displaystyle \left({E-{H_{0}}}\right)\left|\psi \right\rangle =V\left|\psi \right\rangle } , where H 0 {\displaystyle {H_{0}}} is the kinetic energy operator. The solution of the homogeneous equation is | ϕ ⟩ {\displaystyle \left|\phi \right\rangle } , where ( E − H 0 ) | ϕ ⟩ = 0 {\displaystyle \left({E-{H_{0}}}\right)\left|\phi \right\rangle =0} . A formal solution of the inhomogeneous equation is the sum of the solution of the homogeneous equation with a particular solution of the inhomogeneous equation | ψ ⟩ = | ϕ ⟩ + G 0 + V | ψ ⟩ {\displaystyle \left|\psi \right\rangle =\left|\phi \right\rangle +{G_{0+}}V\left|\psi \right\rangle } , where G 0 + = lim ε → 0 ( E − H 0 + i ε ) − 1 {\displaystyle {G_{0+}}={\lim _{\varepsilon \to 0}}{\left({E-{H_{0}}+i\varepsilon }\right)^{-1}}} .
This is the Lippmann–Schwinger equation , which can also be written | ψ ⟩ = ( 1 + G 0 + T ) | ϕ ⟩ {\displaystyle \left|\psi \right\rangle =\left({1+{G_{0+}}T}\right)\left|\phi \right\rangle } . The t-matrix is defined by T = V + V G 0 + V + V G 0 + V G 0 + V + . . . {\displaystyle T=V+V{G_{0+}}V+V{G_{0+}}V{G_{0+}}V+...} .
Suppose that the potential V {\displaystyle V} is the sum of N {\displaystyle N} non-overlapping potentials, V = ∑ i = 1 N v i {\displaystyle V=\sum \limits _{i=1}^{N}{v_{i}}} . The physical meaning of this is that it describes the interaction of the electron with a cluster of N {\displaystyle N} atoms having nuclei located at positions R i {\displaystyle {{\bf {R}}_{i}}} . Define an operator Q i {\displaystyle {Q_{i}}} so that T {\displaystyle T} can be written as a sum T = ∑ i = 1 N Q i {\displaystyle T=\sum \limits _{i=1}^{N}{Q_{i}}} . Inserting the expressions for V {\displaystyle V} and T {\displaystyle T} into the definition of T {\displaystyle T} leads to
so Q i = t i ( 1 + G 0 + ∑ j ≠ i Q j ) {\displaystyle {Q_{i}}={t_{i}}(1+{G_{0+}}\sum \limits _{j\neq i}{Q_{j}})} , where t i = ( 1 − v i G 0 + ) − 1 v i {\displaystyle {t_{i}}={\left({1-{v_{i}}{G_{0+}}}\right)^{-1}}{v_{i}}} is the scattering matrix for one atom. Iterating this equation leads to
The solution of the Lippmann-Schwinger equation can thus be written as the sum of an incoming wave on any site i {\displaystyle i} and the outgoing wave from that site
The site i {\displaystyle i} that we have chosen to focus on can be any of the sites in the cluster. The incoming wave on this site is the incoming wave on the cluster and the outgoing waves from all the other sites
The outgoing wave from the site i {\displaystyle i} is defined as
These last two equations are the fundamental equations of multiple scattering.
To apply this theory to x-ray or neutron diffraction we go back to the Lippmann–Schwinger equation , | ψ ⟩ = | ϕ ⟩ + G 0 + T | ϕ ⟩ = | ϕ ⟩ + ∑ j = 1 N | ϕ j o u t ⟩ {\displaystyle \left|\psi \right\rangle =\left|\phi \right\rangle +{G_{0+}}T\left|\phi \right\rangle =\left|\phi \right\rangle +\sum \limits _{j=1}^{N}{\left|{\phi _{j}^{out}}\right\rangle }} . The scattering from a site is assumed to be very small, so | ϕ i o u t ⟩ = G 0 + t i | ϕ ⟩ {\displaystyle \left|{\phi _{i}^{out}}\right\rangle ={G_{0+}}{t_{i}}\left|\phi \right\rangle } or T = ∑ i = 1 N t i {\displaystyle T=\sum \limits _{i=1}^{N}{t_{i}}} . The Born approximation is used to calculate the t-matrix, which simply means that t i {\displaystyle {t_{i}}} is replaced with v i {\displaystyle {v_{i}}} . A plane wave impinges on a site, and a spherical wave exits it. The outgoing wave from the crystal is determined by the constructive interference of the waves from the sites. Advances to this theory involve the inclusion of higher-order terms in the total scattering matrix T {\displaystyle T} , such as ∑ i t i G 0 + ∑ j ≠ i t j {\displaystyle \sum \limits _{i}{t_{i}}{G_{0+}}\sum \limits _{j\neq i}{t_{j}}} . These terms are particularly important in the scattering of charged particles treated by Molière.
In 1947, Korringa pointed out that the multiple scattering equations can be used to calculate stationary states in a crystal for which the number of scatterers N {\displaystyle N} goes to infinity. [ 8 ] Setting the incoming wave on the cluster and the outgoing wave from the cluster to zero, he wrote the first multiple scattering as
A simple description of this process is that the electrons scatter from one atom to the other ad infinitum.
Since the v i ( r ) {\displaystyle {v_{i}}({\bf {r}})} are bounded in space and do not overlap, there is an interstitial region between them within which the potential is a constant, usually taken to be zero. In this region, the Schrödinger equation becomes [ ∇ 2 + α 2 ] ψ i ( r ) = 0 {\displaystyle \left[{{\nabla ^{2}}+{\alpha ^{2}}}\right]{\psi _{i}}\left({\bf {r}}\right)=0} , where α = 2 m E / ℏ {\displaystyle \alpha ={\sqrt {2mE}}/\hbar } . The incoming wave on site i {\displaystyle i} can thus be written in the position representation
where the d l m i {\displaystyle d_{lm}^{i}} are undetermined coefficients and r i = r − R i {\displaystyle {{\bf {r}}_{i}}={\bf {r}}-{{\bf {R}}_{i}}} . The Green's function may be expanded in the interstitial region
and the outgoing Hankel function can be written
This leads to a set of homogeneous simultaneous equations that determines the unknown coefficients d l m i {\displaystyle d_{lm}^{i}}
which is a solution in principle of the multiple scattering equations for stationary states. This theory is very important for studies in condensed matter physics. [ 4 ] [ 5 ]
The calculation of stationary states is simplified considerably for periodic solids in which all of the potentials v i ( r i ) {\displaystyle {v_{i}}({{\bf {r}}_{i}})} are the same, and the nuclear positions R i {\displaystyle {{\bf {R}}_{i}}} form a periodic array. [ 8 ] Bloch's theorem holds for such a system, which means that the solutions of the Schrödinger equation may be written as a Bloch wave ψ k ( r + R i ) = e i k ⋅ R i ψ k ( r ) {\displaystyle {\psi _{\bf {k}}}\left({{\bf {r}}+{{\bf {R}}_{i}}}\right)={e^{i{\bf {k}}\cdot {{\bf {R}}_{i}}}}{\psi _{\bf {k}}}\left({\bf {r}}\right)} .
It is more convenient to deal with a symmetric matrix for the coefficients, and this can be done by defining
These coefficients satisfy the set of linear equations ∑ j , l ′ m ′ M l m , l ′ m ′ i j c l ′ m ′ j = 0 {\displaystyle \sum \limits _{j,l'm'}{M_{lm,l'm'}^{ij}c_{l'm'}^{j}=0}} , with the elements of the matrix M {\displaystyle {\bf {M}}} being
and the m l m , l ′ m ′ i {\displaystyle m_{lm,l'm'}^{i}} are the elements of the inverse of the t-matrix.
For a Bloch wave the coefficients depend on the site only through a phase factor, c l ′ m ′ j = e − i k ⋅ R j c l ′ m ′ ( E , k ) {\displaystyle c_{l'm'}^{j}={e^{-i{\bf {k}}\cdot {{\bf {R}}_{j}}}}{c_{l'm'}}\left(E,{\bf {k}}\right)} , and the c l ′ m ′ ( E , k ) {\displaystyle {c_{l'm'}}\left(E,{\bf {k}}\right)} satisfy the homogeneous equations
where M l m , l ′ m ′ ( E , k ) = m l m , l ′ m ′ ( E ) − A l m , l ′ m ′ ( E , k ) {\displaystyle {M_{lm,l'm'}}\left({E,{\bf {k}}}\right)={m_{lm,l'm'}}\left(E\right)-{A_{lm,l'm'}}\left({E,{\bf {k}}}\right)} and A l m , l ′ m ′ ( E , k ) = ∑ j e i k ⋅ R i j g l m , l ′ m ′ ( E , R i j ) {\displaystyle {A_{lm,l'm'}}\left({E,{\bf {k}}}\right)=\sum \limits _{j}{e^{i{\bf {k}}\cdot {{\bf {R}}_{ij}}}}{g_{lm,l'm'}}\left({E,{{\bf {R}}_{ij}}}\right)} .
Walter Kohn and Norman Rostoker derived this same theory using the Kohn variational method. It is called the Korringa–Kohn–Rostoker method (KKR method) for band theory calculations. Ewald derived a mathematically sophisticated summation process that makes it possible to calculate the structure constants, A l m , l ′ m ′ ( E , k ) {\displaystyle {A_{lm,l'm'}}\left({E,{\bf {k}}}\right)} . The energy eigenvalues of the periodic solid for a particular k {\displaystyle {\bf {k}}} , E b ( k ) {\displaystyle {E_{b}}\left({\bf {k}}\right)} , are the roots of the equation det M ( E , k ) = 0 {\displaystyle \det {\bf {M}}\left({E,{\bf {k}}}\right)=0} . The eigenfunctions are found by solving for the c l , m ( E , k ) {\displaystyle {c_{l,m}}\left({E,k}\right)} with E = E b ( k ) {\displaystyle E={E_{b}}\left({\bf {k}}\right)} . The dimension of these matrix equations is technically infinite, but by ignoring all contributions that correspond to an angular momentum quantum number l {\displaystyle l} greater than l max {\displaystyle {l_{\max }}} , they have dimension ( l max + 1 ) 2 {\displaystyle {\left({{l_{\max }}+1}\right)^{2}}} . The justification for this approximation is that the matrix elements of the t-matrix t l m , l ′ m ′ {\displaystyle {t_{lm,l'm'}}} are very small when l {\displaystyle l} and l ′ {\displaystyle l'} are greater than l max {\displaystyle {l_{\max }}} , and the elements of the inverse matrix m l m , l ′ m ′ {\displaystyle {m_{lm,l'm'}}} are very large.
In the original derivations of the KKR method, spherically symmetric muffin-tin potentials were used. Such potentials have the advantage that the inverse of the scattering matrix is diagonal in l {\displaystyle l}
where δ l ( E ) {\displaystyle {\delta _{l}}\left(E\right)} is the scattering phase shift that appears in the partial wave analysis in scattering theory. It is also easier to visualize the waves scattering from one atom to another, and l max = 3 {\displaystyle {l_{\max }}=3} can be used in many applications. The muffin-tin approximation is adequate for most metals in a close-packed arrangement. It cannot be used for calculating forces between atoms, or for important systems like semiconductors.
It is now known that the KKR method can be used with space-filling non-spherical potentials. [ 4 ] [ 9 ] It can be extended to treat crystals with any number of atoms in a unit cell. There are versions of the theory that can be used to calculate surface states . [ 10 ]
The arguments that lead to a multiple scattering solution for the single-particle orbital ψ ( r ) {\displaystyle \psi ({\bf {r}})} can also be used to formulate a multiple scattering version of the single-particle Green's function G ( E , r , r ′ ) {\displaystyle G(E,{\bf {r}},{\bf {r'}})} which is a solution of the equation
The potential V ( r ) {\displaystyle V({\bf {r}})} is the same one from density functional theory that was used in the preceding discussion. With this Green's function and the Korringa–Kohn–Rostoker method , the Korringa–Kohn–Rostoker coherent potential approximation (KKR-CPA) is obtained. [ 11 ] The KKR-CPA is used to calculate the electronic states for substitutional solid-solution alloys, for which Bloch's theorem does not hold. The electronic states for an even wider range of condensed matter structures can be found using the locally self-consistent multiple scattering (LSMS) method, which is also based on the single-particle Green's function. [ 12 ] | https://en.wikipedia.org/wiki/Multiple_scattering_theory |
The Multiple Streams Framework (MSF) is a prominent approach for analyzing public policymaking processes . It emphasizes the unpredictable and complex nature of policy development, proposing that three distinct, yet interconnected streams influence the process:
The MSF highlights that policy change occurs when elements from each stream converge in a policy window.
The MSF was first proposed by John W. Kingdon to describe the agenda setting stage of the policy making process. [ 1 ] In developing his framework Kingdon took inspiration from the garbage can model of organizational choice, [ 2 ] which views organizations as anarchical processes resulting from the interaction of four streams: 1) choices, 2) problems, 3) solutions, and 4) energy from participants. Kingdon adapted this general idea to understand agenda setting in the federal government. While his MSF only includes three streams (problems, policies, politics), the general logic is similar: Separate streams run through the organization (of government), each with a life of its own. Solutions are developed whether or not they respond to a problem. Likewise sudden changes can occur in the political landscape, potentially leaving the policy community and existing problems unaddressed. According to Kingdon, the key to understanding agenda and policy change is the coupling of streams, again an idea taken from the garbage can model. At critical times, a problem is recognized, a solution is available, and the political climate allows for action. Kingdon called this situation a "policy window", which is open only for a short time. Only then, the three stream can be coupled in just the right way to yield a significant agenda change. The agents that achieve such coupling have been called policy entrepreneurs by Kingdon.
Whereas Kingdon himself did little to encourage others to apply the MSF in different settings, [ 3 ] the framework has since been widely adopted and refined by other policy scholars. It was taken up in the field of comparative policy analysis, generalized to policy making systems different from the United States, and combined with the concept of the policy cycle , allowing it to be used to study not only agenda setting but all stages of the policy process, including policy formulation, decision-making, implementation, and evaluation. [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] In terms of actors and agency, Kingdon mainly focused on the role of the policy entrepreneur , raising questions about the role of other actors, especially collective policy actors. [ 9 ] As a consequence, leading scholars now describe the different streams in the multiple-streams framework as being populated by distinct collective actors, most notably epistemic communities in the problem stream, policy instrument constituencies in the policy stream, and advocacy coalitions in the politics stream. [ 9 ] [ 10 ] [ 11 ] | https://en.wikipedia.org/wiki/Multiple_streams_framework |
The possibility that there might be more than one dimension of time has occasionally been discussed in physics and philosophy . Similar ideas appear in folklore and fantasy literature.
Speculative theories with more than one time dimension have been explored in physics. The additional dimensions may be similar to conventional time, [ 1 ] compactified like the additional spatial dimensions in string theory , [ 2 ] or components of a complex time (sometimes referred to as kime). [ 3 ]
Itzhak Bars has proposed models of a two-time physics, noting in 2001 that "The 2T-physics approach
in d + 2 dimensions offers a highly symmetric and unified version of the phenomena described by 1T-physics in d dimensions." [ 4 ] [ 5 ]
F-theory , a branch of modern string theory , describes a 12-dimensional spacetime having two dimensions of time, giving it the metric signature (10,2). [ 6 ]
The existence of a well-posed initial value problem for the ultrahyperbolic equation (a wave equation in more than one time dimension) demonstrates that initial data on a mixed (spacelike and timelike) hypersurface, obeying a particular nonlocal constraint, evolves deterministically in the remaining time dimension. [ 1 ]
Like other complex number variables, complex time is two-dimensional, comprising one real time dimension and one imaginary time dimension, changing time from a real number line into a complex plane. [ 3 ] Introducing it into Minkowski spacetime allows a generalization of Kaluza–Klein theory . [ 7 ]
Max Tegmark has argued that, if there is more than one time dimension, then the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations . In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover protons and electrons would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.) [ 8 ]
Multiple time dimensions appear to allow the breaking or re-ordering of cause-and-effect in the flow of any one dimension of time. This and conceptual difficulties with multiple physical time dimensions have been raised in modern analytic philosophy . [ 9 ]
As a solution to the problem of the subjective passage of time, J. W. Dunne proposed an infinite hierarchy of time dimensions, inhabited by a similar hierarchy of levels of consciousness. Dunne suggested that, in the context of a "block" spacetime as modelled by General Relativity , a second dimension of time was needed in order to measure the speed of one's progress along one's own timeline. This in turn required a level of the conscious self existing at the second level of time. But the same arguments then applied to this new level, requiring a third level, and so on in an infinite regress . At the end of the regress was a "superlative general observer" who existed in eternity . [ 10 ] He published his theory in relation to precognitive dreams in his 1927 book An Experiment with Time and went on to explore its relevance to contemporary physics in The Serial Universe (1934). His infinite regress was criticised as logically flawed and unnecessary, although writers such as J. B. Priestley acknowledged the possibility of his second time dimension. [ 11 ] [ 12 ]
The Esoteric J. G. Bennett described three dimensions or aspects of time: a) Time – Causal or determinate influences on the present moment, b) Eternity – The influences of forms and values, c) Hyparxis – The influences of the Will (freedom) to choose within the present Moment. The physical world, life and consciousness lie in intermediate zones between these dimensions. [ 13 ] Physicist David Bohm corresponded with Bennett and they influenced each other's ideas. [ 14 ]
Multiple independent timeframes, in which time passes at different rates, have long been a feature of stories. [ 15 ] Fantasy writers such as J. R. R. Tolkien and C. S. Lewis have made use of these and other multiple time dimensions, such as those proposed by Dunne, in some of their most well-known stories. [ 15 ] It has been argued that Tolkien borrowed his ideas for Lórien time in The Lord of the Rings , [ 15 ] and that Lewis adopted them for his Chronicles of Narnia . [ 16 ]
Science fiction author H. Beam Piper , in his Paratime series of short stories and novel that multiple timelines exist as "worlds of alternate
probability on the lateral dimension of time." [ 17 ] | https://en.wikipedia.org/wiki/Multiple_time_dimensions |
In physics and particularly in particle physics , a multiplet is the state space for 'internal' degrees of freedom of a particle ; that is, degrees of freedom associated to a particle itself, as opposed to 'external' degrees of freedom such as the particle's position in space. Examples of such degrees of freedom are the spin state of a particle in quantum mechanics , or the color , isospin and hypercharge state of particles in the Standard Model of particle physics. Formally, we describe this state space by a vector space which carries the action of a group of continuous symmetries.
Mathematically, multiplets are described via representations of a Lie group or its corresponding Lie algebra , and is usually used to refer to irreducible representations (irreps, for short).
At the group level, this is a triplet ( V , G , ρ ) {\displaystyle (V,G,\rho )} where
At the algebra level, this is a triplet ( V , g , ρ ) {\displaystyle (V,{\mathfrak {g}},\rho )} , where
The symbol ρ {\displaystyle \rho } is used for both Lie algebras and Lie groups as, at least in finite dimension, there is a well understood correspondence between Lie groups and Lie algebras.
In mathematics, it is common to refer to the homomorphism ρ {\displaystyle \rho } as the representation, for example in the sentence 'consider a representation ρ {\displaystyle \rho } ', and the vector space V {\displaystyle V} is referred to as the 'representation space'. In physics sometimes the vector space is referred to as the representation, for example in the sentence 'we model the particle as transforming in the singlet representation', or even to refer to a quantum field which takes values in such a representation, and the physical particles which are modelled by such a quantum field.
For an irreducible representation, an n {\displaystyle n} - plet refers to an n {\displaystyle n} dimensional irreducible representation. Generally, a group may have multiple non-isomorphic representations of the same dimension, so this does not fully characterize the representation. An exception is SU ( 2 ) {\displaystyle {\text{SU}}(2)} which has exactly one irreducible representation of dimension n {\displaystyle n} for each non-negative integer n {\displaystyle n} .
For example, consider real three-dimensional space, R 3 {\displaystyle \mathbb {R} ^{3}} . The group of 3D rotations SO(3) acts naturally on this space as a group of 3 × 3 {\displaystyle 3\times 3} matrices. This explicit realisation of the rotation group is known as the fundamental representation ρ fund {\displaystyle \rho _{\text{fund}}} , so R 3 {\displaystyle \mathbb {R} ^{3}} is a representation space. The full data of the representation is ( R 3 , SO(3) , ρ fund ) {\displaystyle (\mathbb {R} ^{3},{\text{SO(3)}},\rho _{\text{fund}})} . Since the dimension of this representation space is 3, this is known as the triplet representation for SO ( 3 ) {\displaystyle {\text{SO}}(3)} , and it is common to denote this as 3 {\displaystyle \mathbf {3} } .
For applications to theoretical physics, we can restrict our attention to the representation theory of a handful of physically important groups. Many of these have well understood representation theory:
These groups all appear in the theory of the Standard model. For theories which extend these symmetries, the representation theory of some other groups might be considered:
In quantum physics, the mathematical notion is usually applied to representations of the gauge group . For example, an SU ( 2 ) {\displaystyle {\text{SU}}(2)} gauge theory will have multiplets which are fields whose representation of SU ( 2 ) {\displaystyle {\text{SU}}(2)} is determined by the single half-integer number s =: n / 2 {\displaystyle s=:n/2} , the isospin. Since irreducible SU ( 2 ) {\displaystyle {\text{SU}}(2)} representations are isomorphic to the n {\displaystyle n} th symmetric power of the fundamental representation, every field has n {\displaystyle n} symmetrized internal indices.
Fields also transform under representations of the Lorentz group SO ( 1 , 3 ) {\displaystyle {\text{SO}}(1,3)} , or more generally its spin group Spin ( 1 , 3 ) {\displaystyle {\text{Spin}}(1,3)} which can be identified with SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} due to an exceptional isomorphism . Examples include scalar fields , commonly denoted ϕ {\displaystyle \phi } , which transform in the trivial representation, vector fields A μ {\displaystyle A_{\mu }} (strictly, this might be more accurately labelled a covector field), which transforms as a 4-vector, and spinor fields ψ α {\displaystyle \psi _{\alpha }} such as Dirac or Weyl spinors which transform in representations of SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} . A right-handed Weyl spinor transforms in the fundamental representation, C 2 {\displaystyle \mathbb {C} ^{2}} , of SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} .
Beware that besides the Lorentz group, a field can transform under the action of a gauge group. For example, a scalar field ϕ ( x ) {\displaystyle \phi (x)} , where x {\displaystyle x} is a spacetime point, might have an isospin state taking values in the fundamental representation C 2 {\displaystyle \mathbb {C} ^{2}} of SU ( 2 ) {\displaystyle {\text{SU}}(2)} . Then ϕ ( x ) {\displaystyle \phi (x)} is a vector valued function of spacetime, but is still referred to as a scalar field, as it transforms trivially under Lorentz transformations.
In quantum field theory different particles correspond one to one with gauged fields transforming in irreducible representations of the internal and Lorentz group. Thus, a multiplet has also come to describe a collection of subatomic particles described by these representations.
The best known example is a spin multiplet , which describes symmetries of a group representation of an SU(2) subgroup of the Lorentz algebra , which is used to define spin quantization. A spin singlet is a trivial representation, a spin doublet is a fundamental representation and a spin triplet is in the vector representation or adjoint representation .
In QCD , quarks are in a multiplet of SU(3) , specifically the three-dimensional fundamental representation.
In spectroscopy, particularly Gamma spectroscopy and X-ray spectroscopy , a multiplet is a group of related or unresolvable spectral lines . Where the number of unresolved lines is small, these are often referred to specifically as doublet or triplet peaks, while multiplet is used to describe groups of peaks in any number. | https://en.wikipedia.org/wiki/Multiplet |
In the biological sciences, a multiplex assay is a type of immunoassay that uses magnetic beads to simultaneously measure multiple analytes in a single experiment. [ 1 ] A multiplex assay is a derivative of an ELISA using beads for binding the capture antibody. Multiplex assays are still more common in research than in clinical settings. [ 2 ]
In a multiplex assay, microspheres of designated colors are coated with antibodies of defined binding specificities. The results can be read by flow cytometry because the beads are distinguishable by fluorescent signature. The number of analytes measured is determined by the number of different bead colors. [ 3 ]
Multiplex assays within a given application area or class of technology can be further stratified based on how many analytes can be measured per assay, where "multiplex" refers to those with the highest number of analyte measurements per assay (up to millions) and "low-plex" or "mid-plex" refers to procedures that process fewer (10s to 1000s), though there are no formal guidelines for calling a procedure multi-, mid-, or low-plex based on number of analytes measured. [ citation needed ] Single-analyte assays or low-to-mid-plex procedures typically predate the rise of their multiplex versions, which often require specialized technologies or miniaturization to achieve a higher degree of parallelization. | https://en.wikipedia.org/wiki/Multiplex_(assay) |
Multiplex ligation-dependent probe amplification ( MLPA ) is a variation of the multiplex polymerase chain reaction that permits amplification of multiple targets with only a single primer pair. [ 1 ] It detects copy number changes at the molecular level, and software programs are used for analysis. Identification of deletions or duplications can indicate pathogenic mutations, thus MLPA is an important diagnostic tool used in clinical pathology laboratories worldwide.
Multiplex ligation-dependent probe amplification was invented by Jan Schouten , a Dutch scientist. [ 1 ] The method was first described in 2002 in the scientific journal Nucleic Acid Research . [ 2 ] The first applications included the detection of exon deletions in the human genes BRCA1 , MSH2 and MLH1 , which are linked to hereditary breast and colon cancer. Now MLPA is used to detect hundreds of hereditary disorders, as well as for tumour profiling.
MLPA quantifies the presence of particular sequences in a sample of DNA, using a specially designed probe pair for each target sequence of interest. The process consists of multiple steps: [ 3 ]
Each probe pair consists of two oligonucleotides, with sequence that recognizes adjacent sites of the target DNA , a PCR priming site, and optionally a "stuffer" to give the PCR product a unique length when compared to other probe pairs in the MLPA assay. Each complete probe pair must have a unique length, so that its resulting amplicons can be uniquely identified during quantification, avoiding the resolution limitations of multiplex PCR . Because the forward primer used for probe amplification is fluorescently labeled, each amplicon generates a fluorescent peak which can be detected by a capillary sequencer. Comparing the peak pattern obtained on a given sample with that obtained on various reference samples, the relative quantity of each amplicon can be determined. This ratio is a measure for the ratio in which the target sequence is present in the sample DNA.
Various techniques including DGGE ( Denaturing Gradient Gel Electrophoresis ), DHPLC ( Denaturing High Performance Liquid Chromatography ), and SSCA (Single Strand Conformation Analysis) effectively identify SNPs and small insertions and deletions. MLPA, however, is one of the only accurate, time-efficient techniques to detect genomic deletions and insertions (one or more entire exons), which are frequent causes of cancers such as hereditary non-polyposis colorectal cancer ( HNPCC ), breast, and ovarian cancer. MLPA can successfully and easily determine the relative copy number of all exons within a gene simultaneously with high sensitivity.
An important use of MLPA is to determine relative ploidy . For example, probes may be designed to target various regions of chromosome 21 of a human cell. The signal strengths of the probes are compared with those obtained from a reference DNA sample known to have two copies of the chromosome. If an extra copy is present in the test sample, the signals are expected to be 1.5 times the intensities of the respective probes from the reference. If only one copy is present the proportion is expected to be 0.5. If the sample has two copies, the relative probe strengths are expected to be equal.
Dosage quotient analysis is the usual method of interpreting MLPA data. [ 4 ] If a and b are the signals from two amplicons in the patient sample, and A and B are the corresponding amplicons in the experimental control, then the dosage quotient DQ = (a/b) / (A/B). Although dosage quotients may be calculated for any pair of amplicons, it is usually the case that one of the pair is an internal reference probe.
MLPA facilitates the amplification and detection of multiple targets with a single primer pair. In a standard multiplex PCR reaction, each fragment needs a unique amplifying primer pair. These primers being present in a large quantity result in various problems such as dimerization and false priming. With MLPA, amplification of probes can be achieved. Thus, many sequences (up to 40) can be amplified and quantified using just a single primer pair. MLPA reaction is fast, inexpensive and very simple to perform.
MLPA has a variety of applications [ 5 ] including detection of mutations and single nucleotide polymorphisms , [ 6 ] analysis of DNA methylation , [ 7 ] relative mRNA quantification, [ 8 ] chromosomal characterisation of cell lines and tissue samples, [ 9 ] detection of gene copy number, [ 10 ] detection of duplications and deletions in human cancer predisposition genes such as BRCA1 , BRCA2 , hMLH1 and hMSH2 [ 11 ] and aneuploidy determination. [ 12 ] MLPA has potential application in prenatal diagnosis both invasive [ 13 ] and noninvasive . [ 14 ]
Recent studies have shown that MLPA (as well as another variants such as iMLPA) is a robust technique for inversion characterisation. [ 15 ]
Giner-Delgado, Carla, et al. described a variant of MLPA combining it with iPCR. They call these new method iMLPA [ 15 ] and its procedure is the same as MLPA but there are necessary two additional steps at the beginning:
The probe design is quite similar. Each probe will be formed by two parts that have at least: a target sequence , which is a region that contains the sequence complementary to the region of interest, so that the correct hybridization can occur. And a primer sequence at the end, it is a sequence whose design varies and is what will allow the design of primers and subsequent fragment amplification. In addition, one of the parts of the probe usually contains a stuffer between the target sequence and the primer sequence. The use of different stuffers allows the identification of probes with the same primer sequences but different target sequences, that is key for multiple amplification of several different fragments in a single reaction.
The next step continues with the typical MLPA protocol. [ 1 ] | https://en.wikipedia.org/wiki/Multiplex_ligation-dependent_probe_amplification |
Multiplex polymerase chain reaction ( Multiplex PCR ) refers to the use of polymerase chain reaction to amplify several different DNA sequences simultaneously (as if performing many separate PCR reactions all together in one reaction). This process amplifies DNA in samples using multiple primers and a temperature-mediated DNA polymerase in a thermal cycler . The primer design for all primers pairs has to be optimized so that all primer pairs can work at the same annealing temperature during PCR.
Multiplex-PCR was first described in 1988 as a method to detect deletions in the dystrophin gene. [ 1 ] It has also been used with the steroid sulfatase gene. [ 2 ] In 2008, multiplex-PCR was used for analysis of microsatellites and SNPs . [ 3 ] In 2020, RT-PCR multiplex assays were designed that combined multiple gene targets from the Center for Diseases and Control in a single reaction to increase molecular testing accessibility and throughput for SARS-CoV-2 diagnostics. [ 4 ]
Multiplex-PCR consists of multiple primer sets within a single PCR mixture to produce amplicons of varying sizes that are specific to different DNA sequences. By targeting multiple sequences at once, additional information may be gained from a single test run that otherwise would require several times the reagents and more time to perform. Annealing temperatures for each of the primer sets must be optimized to work correctly within a single reaction, and amplicon sizes, i.e., their base pair length, should be different enough to form distinct bands when visualized by gel electrophoresis . Alternatively, if amplicon sizes overlap, the different amplicons may be differentiated and visualised using primers that have been dyed with different colour fluorescent dyes. Commercial multiplexing kits for PCR are available and used by many forensic laboratories to amplify degraded DNA samples.
Some of the applications of multiplex PCR include: | https://en.wikipedia.org/wiki/Multiplex_polymerase_chain_reaction |
Multiplexin is a family of collagens . [ 1 ]
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multiplexin |
Multiplication is the process in Western alchemy used to increase the potency of the philosopher's stone , elixir or projection powder. It occurs near the end of the magnum opus in order to increase the gains in the subsequent projection . George Ripley gives the following definition of multiplication: [ 1 ]
Multiplication was also used to describe the facet of alchemy chiefly concerned with the reproduction of physical gold and silver. Such is the case in Henry IV's 1404 statute against the craft of multiplication ( 5 Hen. 4 . c. 4). [ 2 ] Henry VI began to issue patents for the practice of alchemy, but the act of Parliament against multipliers was not repealed until 1689.
In his discussion of multiplication, Ripley goes on to compare the medicine or elixir to fire. He notes that the methods of multiplication are related to the processes of congelation , cibation, and fermentation. In doing so he also hints at multiplication's internal significance.
And why may you multiply this medicine infinitely, Forsooth the cause is this, For it is fire, which kindled will never die, Dwelling with you, as fire does in houses, Of which one spark may make more fire this way, As musk in pigments and other spices more, In virtue multiplied, and our medicine right so.[...]
Like Ripley, other sources describe multiplication as subjecting the philosopher's stone to further maturation by reiterating the same processes as was used to originally make it. [ 4 ] Sendivogius writes, "Having made an End with the Composition of the Lapis, there remains its Multiplication in infinitum which is effected by the same way and with the same operations the Lapis was made; only that instead of dissolved Gold or Silver, you lay in only so much of the Lapis as you laid in before of the said Gold or Silver for the first Confection of the Lapis." [ 5 ] Comparisons to plant and animal reproduction were also made. | https://en.wikipedia.org/wiki/Multiplication_(alchemy) |
The multiplication sign ( × ), also known as the times sign or the dimension sign , is a mathematical symbol used to denote the operation of multiplication , which results in a product . [ 1 ]
The symbol is also used in botany , in botanical hybrid names .
The form is properly a four-fold rotationally symmetric saltire . [ 2 ] The multiplication sign × is similar to a lowercase X ( x ).
The earliest known use of the × symbol to indicate multiplication appears in an anonymous appendix to the 1618 edition of John Napier 's Mirifici Logarithmorum Canonis Descriptio . [ 3 ] This appendix has been attributed to William Oughtred , [ 3 ] who used the same symbol in his 1631 algebra text, Clavis Mathematicae , stating:
Multiplication of species [i.e. unknowns] connects both proposed magnitudes with the symbol 'in' or × : or ordinarily without the symbol if the magnitudes be denoted with one letter. [ 4 ]
Other works have been identified in which crossed diagonals appear in diagrams involving multiplied numbers, such as Robert Recorde 's The Ground of Arts [ 5 ] [ 6 ] and Oswald Schreckenfuchs 's 1551 edition of Almagest , but these are not symbolizations. [ 3 ]
In mathematics , the symbol × has a number of uses, including
In biology , the multiplication sign is used in a botanical hybrid name , for instance Ceanothus papillosus × impressus (a hybrid between C. papillosus and C. impressus ) or Crocosmia × crocosmiiflora (a hybrid between two other species of Crocosmia ). However, the communication of these hybrid names with a Latin letter "x" is common, especially when the actual "×" symbol is not readily available.
The multiplication sign is also used by historians for an event between two dates . When employed between two dates – for example 1225 and 1232 – the expression "1225×1232" means "no earlier than 1225 and no later than 1232". [ 8 ]
A monadic × symbol is used by the APL programming language to denote the sign function .
The lower-case Latin letter x is sometimes used in place of the multiplication sign. This is considered incorrect in mathematical writing.
In algebraic notation, widely used in mathematics, a multiplication symbol is usually omitted wherever it would not cause confusion: " a multiplied by b " can be written as ab or a b . [ 1 ]
Other symbols can also be used to denote multiplication, often to reduce confusion between the multiplication sign × and the common variable x . In some countries, such as Germany , the primary symbol for multiplication is the " dot operator " ⋅ (as in a ⋅ b ). This symbol is also used in compound units of measurement , e.g., N⋅m (see International System of Units § Lexicographic conventions ). In algebra, it is a notation to resolve ambiguity (for instance, " b times 2 " may be written as b ⋅2 , to avoid being confused with a value called b 2 ). This notation is used wherever multiplication should be written explicitly, such as in " ab = a ⋅2 for b = 2 "; this usage is also seen in English-language texts. In some languages, the use of full stop as a multiplication symbol, such as a . b , is common when the symbol for decimal point is comma .
Historically, computer language syntax was restricted to the ASCII character set, and the asterisk * became the de facto symbol for the multiplication operator. This selection is reflected in the numeric keypad on English-language keyboards, where the arithmetic operations of addition, subtraction, multiplication and division are represented by the keys + , - , * and / , respectively.
Other variants and related characters: | https://en.wikipedia.org/wiki/Multiplication_sign |
In mathematics , the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function . For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises.
The multiplication theorem takes two common forms. In the first case, a finite number of terms are added or multiplied to give the relation. In the second case, an infinite number of terms are added or multiplied. The finite form typically occurs only for the gamma and related functions, for which the identity follows from a p-adic relation over a finite field . For example, the multiplication theorem for the gamma function follows from the Chowla–Selberg formula , which follows from the theory of complex multiplication . The infinite sums are much more common, and follow from characteristic zero relations on the hypergeometric series.
The following tabulates the various appearances of the multiplication theorem for finite characteristic; the characteristic zero relations are given further down. In all cases, n and k are non-negative integers. For the special case of n = 2, the theorem is commonly referred to as the duplication formula .
The duplication formula and the multiplication theorem for the gamma function are the prototypical examples. The duplication formula for the gamma function is
It is also called the Legendre duplication formula [ 1 ] or Legendre relation , in honor of Adrien-Marie Legendre . The multiplication theorem is
for integer k ≥ 1, and is sometimes called Gauss's multiplication formula , in honour of Carl Friedrich Gauss . The multiplication theorem for the gamma functions can be understood to be a special case, for the trivial Dirichlet character , of the Chowla–Selberg formula .
Formally similar duplication formulas hold for the sine function, which are rather simple consequences of the trigonometric identities . Here one has the duplication formula
and, more generally, for any integer k , one has
The polygamma function is the logarithmic derivative of the gamma function, and thus, the multiplication theorem becomes additive, instead of multiplicative:
for m > 1 {\displaystyle m>1} , and, for m = 1 {\displaystyle m=1} , one has the digamma function :
The polygamma identities can be used to obtain a multiplication theorem for harmonic numbers .
The Hurwitz zeta function generalizes the polygamma function to non-integer orders, and thus obeys a very similar multiplication theorem:
where ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function . This is a special case of
and
Multiplication formulas for the non-principal characters may be given in the form of Dirichlet L-functions .
The periodic zeta function [ 2 ] is sometimes defined as
where Li s ( z ) is the polylogarithm . It obeys the duplication formula
As such, it is an eigenvector of the Bernoulli operator with eigenvalue 2 1− s . The multiplication theorem is
The periodic zeta function occurs in the reflection formula for the Hurwitz zeta function, which is why the relation that it obeys, and the Hurwitz zeta relation, differ by the interchange of s → 1− s .
The Bernoulli polynomials may be obtained as a limiting case of the periodic zeta function, taking s to be an integer, and thus the multiplication theorem there can be derived from the above. Similarly, substituting q = log z leads to the multiplication theorem for the polylogarithm.
The duplication formula takes the form
The general multiplication formula is in the form of a Gauss sum or discrete Fourier transform :
These identities follow from that on the periodic zeta function, taking z = log q .
The duplication formula for Kummer's function is
and thus resembles that for the polylogarithm, but twisted by i .
For the Bernoulli polynomials , the multiplication theorems were given by Joseph Ludwig Raabe in 1851:
and for the Euler polynomials ,
and
The Bernoulli polynomials may be obtained as a special case of the Hurwitz zeta function, and thus the identities follow from there.
The Bernoulli map is a certain simple model of a dissipative dynamical system , describing the effect of a shift operator on an infinite string of coin-flips (the Cantor set ). The Bernoulli map is a one-sided version of the closely related Baker's map . The Bernoulli map generalizes to a k-adic version, which acts on infinite strings of k symbols: this is the Bernoulli scheme . The transfer operator L k {\displaystyle {\mathcal {L}}_{k}} corresponding to the shift operator on the Bernoulli scheme is given by
Perhaps not surprisingly, the eigenvectors of this operator are given by the Bernoulli polynomials. That is, one has that
It is the fact that the eigenvalues k − m < 1 {\displaystyle k^{-m}<1} that marks this as a dissipative system: for a non-dissipative measure-preserving dynamical system , the eigenvalues of the transfer operator lie on the unit circle.
One may construct a function obeying the multiplication theorem from any totally multiplicative function . Let f ( n ) {\displaystyle f(n)} be totally multiplicative; that is, f ( m n ) = f ( m ) f ( n ) {\displaystyle f(mn)=f(m)f(n)} for any integers m , n . Define its Fourier series as
Assuming that the sum converges, so that g ( x ) exists, one then has that it obeys the multiplication theorem; that is, that
That is, g ( x ) is an eigenfunction of Bernoulli transfer operator, with eigenvalue f ( k ). The multiplication theorem for the Bernoulli polynomials then follows as a special case of the multiplicative function f ( n ) = n − s {\displaystyle f(n)=n^{-s}} . The Dirichlet characters are fully multiplicative, and thus can be readily used to obtain additional identities of this form.
The multiplication theorem over a field of characteristic zero does not close after a finite number of terms, but requires an infinite series to be expressed. Examples include that for the Bessel function J ν ( z ) {\displaystyle J_{\nu }(z)} :
where λ {\displaystyle \lambda } and ν {\displaystyle \nu } may be taken as arbitrary complex numbers. Such characteristic-zero identities follow generally from one of many possible identities on the hypergeometric series. | https://en.wikipedia.org/wiki/Multiplication_theorem |
In number theory, the multiplicative digital root of a natural number n {\displaystyle n} in a given number base b {\displaystyle b} is found by multiplying the digits of n {\displaystyle n} together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of n {\displaystyle n} . [ 1 ] [ 2 ] The multiplicative digital root for the first few positive integers are:
Multiplicative digital roots are the multiplicative equivalent of digital roots .
Let n {\displaystyle n} be a natural number. We define the digit product for base b > 1 {\displaystyle b>1} F b : N → N {\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} } to be the following:
where k = ⌊ log b n ⌋ + 1 {\displaystyle k=\lfloor \log _{b}{n}\rfloor +1} is the number of digits in the number in base b {\displaystyle b} , and
is the value of each digit of the number. A natural number n {\displaystyle n} is a multiplicative digital root if it is a fixed point for F b {\displaystyle F_{b}} , which occurs if F b ( n ) = n {\displaystyle F_{b}(n)=n} .
For example, in base b = 10 {\displaystyle b=10} , 0 is the multiplicative digital root of 9876, as
All natural numbers n {\displaystyle n} are preperiodic points for F b {\displaystyle F_{b}} , regardless of the base. This is because if n ≥ b {\displaystyle n\geq b} , then
and therefore
If n < b {\displaystyle n<b} , then trivially
Therefore, the only possible multiplicative digital roots are the natural numbers 0 ≤ n < b {\displaystyle 0\leq n<b} , and there are no cycles other than the fixed points of 0 ≤ n < b {\displaystyle 0\leq n<b} .
The number of iterations i {\displaystyle i} needed for F b i ( n ) {\displaystyle F_{b}^{i}(n)} to reach a fixed point is the multiplicative persistence of n {\displaystyle n} . The multiplicative persistence is undefined if it never reaches a fixed point.
In base 10 , it is conjectured that there is no number with a multiplicative persistence i > 11 {\displaystyle i>11} : this is known to be true for numbers n ≤ 10 20585 {\displaystyle n\leq 10^{20585}} . [ 3 ] [ 4 ] The smallest numbers with persistence 0, 1, ... are:
The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be sorted, and, except for the first two digits, all digits must be 7, 8, or 9. There are also additional restrictions on the first two digits.
Based on these restrictions, the number of candidates for k {\displaystyle k} -digit numbers with record-breaking persistence is only proportional to the square of k {\displaystyle k} , a tiny fraction of all possible k {\displaystyle k} -digit numbers. However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist. [ 3 ]
The multiplicative digital root can be extended to the negative integers by use of a signed-digit representation to represent each integer.
The example below implements the digit product described in the definition above to search for multiplicative digital roots and multiplicative persistences in Python . | https://en.wikipedia.org/wiki/Multiplicative_digital_root |
In number theory , a multiplicative function is an arithmetic function f {\displaystyle f} of a positive integer n {\displaystyle n} with the property that f ( 1 ) = 1 {\displaystyle f(1)=1} and f ( a b ) = f ( a ) f ( b ) {\displaystyle f(ab)=f(a)f(b)} whenever a {\displaystyle a} and b {\displaystyle b} are coprime .
An arithmetic function is said to be completely multiplicative (or totally multiplicative ) if f ( 1 ) = 1 {\displaystyle f(1)=1} and f ( a b ) = f ( a ) f ( b ) {\displaystyle f(ab)=f(a)f(b)} holds for all positive integers a {\displaystyle a} and b {\displaystyle b} , even when they are not coprime.
Some multiplicative functions are defined to make formulas easier to write:
The above functions are all completely multiplicative.
Other examples of multiplicative functions include many functions of importance in number theory, such as:
An example of a non-multiplicative function is the arithmetic function r 2 ( n ) {\displaystyle r_{2}(n)} , the number of representations of n {\displaystyle n} as a sum of squares of two integers, positive , negative , or zero , where in counting the number of ways, reversal of order is allowed. For example:
and therefore r 2 ( 1 ) = 4 ≠ 1 {\displaystyle r_{2}(1)=4\neq 1} . This shows that the function is not multiplicative. However, r 2 ( n ) / 4 {\displaystyle r_{2}(n)/4} is multiplicative.
In the On-Line Encyclopedia of Integer Sequences , sequences of values of a multiplicative function have the keyword "mult". [ 1 ]
See arithmetic function for some other examples of non-multiplicative functions.
A multiplicative function is completely determined by its values at the powers of prime numbers , a consequence of the fundamental theorem of arithmetic . Thus, if n is a product of powers of distinct primes, say n = p a q b ..., then f ( n ) = f ( p a ) f ( q b ) ...
This property of multiplicative functions significantly reduces the need for computation, as in the following examples for n = 144 = 2 4 · 3 2 : d ( 144 ) = σ 0 ( 144 ) = σ 0 ( 2 4 ) σ 0 ( 3 2 ) = ( 1 0 + 2 0 + 4 0 + 8 0 + 16 0 ) ( 1 0 + 3 0 + 9 0 ) = 5 ⋅ 3 = 15 {\displaystyle d(144)=\sigma _{0}(144)=\sigma _{0}(2^{4})\,\sigma _{0}(3^{2})=(1^{0}+2^{0}+4^{0}+8^{0}+16^{0})(1^{0}+3^{0}+9^{0})=5\cdot 3=15} σ ( 144 ) = σ 1 ( 144 ) = σ 1 ( 2 4 ) σ 1 ( 3 2 ) = ( 1 1 + 2 1 + 4 1 + 8 1 + 16 1 ) ( 1 1 + 3 1 + 9 1 ) = 31 ⋅ 13 = 403 {\displaystyle \sigma (144)=\sigma _{1}(144)=\sigma _{1}(2^{4})\,\sigma _{1}(3^{2})=(1^{1}+2^{1}+4^{1}+8^{1}+16^{1})(1^{1}+3^{1}+9^{1})=31\cdot 13=403} σ ∗ ( 144 ) = σ ∗ ( 2 4 ) σ ∗ ( 3 2 ) = ( 1 1 + 16 1 ) ( 1 1 + 9 1 ) = 17 ⋅ 10 = 170 {\displaystyle \sigma ^{*}(144)=\sigma ^{*}(2^{4})\,\sigma ^{*}(3^{2})=(1^{1}+16^{1})(1^{1}+9^{1})=17\cdot 10=170}
Similarly, we have: φ ( 144 ) = φ ( 2 4 ) φ ( 3 2 ) = 8 ⋅ 6 = 48 {\displaystyle \varphi (144)=\varphi (2^{4})\,\varphi (3^{2})=8\cdot 6=48}
In general, if f ( n ) is a multiplicative function and a , b are any two positive integers, then
Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.
If f and g are two multiplicative functions, one defines a new multiplicative function f ∗ g {\displaystyle f*g} , the Dirichlet convolution of f and g , by ( f ∗ g ) ( n ) = ∑ d | n f ( d ) g ( n d ) {\displaystyle (f\,*\,g)(n)=\sum _{d|n}f(d)\,g\left({\frac {n}{d}}\right)} where the sum extends over all positive divisors d of n .
With this operation, the set of all multiplicative functions turns into an abelian group ; the identity element is ε . Convolution is commutative, associative, and distributive over addition.
Relations among the multiplicative functions discussed above include:
The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring .
The Dirichlet convolution of two multiplicative functions is again multiplicative. A proof of this fact is given by the following expansion for relatively prime a , b ∈ Z + {\displaystyle a,b\in \mathbb {Z} ^{+}} : ( f ∗ g ) ( a b ) = ∑ d | a b f ( d ) g ( a b d ) = ∑ d 1 | a ∑ d 2 | b f ( d 1 d 2 ) g ( a b d 1 d 2 ) = ∑ d 1 | a f ( d 1 ) g ( a d 1 ) × ∑ d 2 | b f ( d 2 ) g ( b d 2 ) = ( f ∗ g ) ( a ) ⋅ ( f ∗ g ) ( b ) . {\displaystyle {\begin{aligned}(f\ast g)(ab)&=\sum _{d|ab}f(d)g\left({\frac {ab}{d}}\right)\\&=\sum _{d_{1}|a}\sum _{d_{2}|b}f(d_{1}d_{2})g\left({\frac {ab}{d_{1}d_{2}}}\right)\\&=\sum _{d_{1}|a}f(d_{1})g\left({\frac {a}{d_{1}}}\right)\times \sum _{d_{2}|b}f(d_{2})g\left({\frac {b}{d_{2}}}\right)\\&=(f\ast g)(a)\cdot (f\ast g)(b).\end{aligned}}}
More examples are shown in the article on Dirichlet series .
An arithmetical function f is said to be a rational arithmetical function of order ( r , s ) {\displaystyle (r,s)} if there exists completely multiplicative functions g 1 ,..., g r , h 1 ,..., h s such that f = g 1 ∗ ⋯ ∗ g r ∗ h 1 − 1 ∗ ⋯ ∗ h s − 1 , {\displaystyle f=g_{1}\ast \cdots \ast g_{r}\ast h_{1}^{-1}\ast \cdots \ast h_{s}^{-1},} where the inverses are with respect to the Dirichlet convolution. Rational arithmetical functions of order ( 1 , 1 ) {\displaystyle (1,1)} are known as totient functions, and rational arithmetical functions of order ( 2 , 0 ) {\displaystyle (2,0)} are known as quadratic functions or specially multiplicative functions. Euler's function φ ( n ) {\displaystyle \varphi (n)} is a totient function, and the divisor function σ k ( n ) {\displaystyle \sigma _{k}(n)} is a quadratic function.
Completely multiplicative functions are rational arithmetical functions of order ( 1 , 0 ) {\displaystyle (1,0)} . Liouville's function λ ( n ) {\displaystyle \lambda (n)} is completely multiplicative. The Möbius function μ ( n ) {\displaystyle \mu (n)} is a rational arithmetical function of order ( 0 , 1 ) {\displaystyle (0,1)} .
By convention, the identity element ε {\displaystyle \varepsilon } under the Dirichlet convolution is a rational arithmetical function of order ( 0 , 0 ) {\displaystyle (0,0)} .
All rational arithmetical functions are multiplicative. A multiplicative function f is a rational arithmetical function of order ( r , s ) {\displaystyle (r,s)} if and only if its Bell series is of the form f p ( x ) = ∑ n = 0 ∞ f ( p n ) x n = ( 1 − h 1 ( p ) x ) ( 1 − h 2 ( p ) x ) ⋯ ( 1 − h s ( p ) x ) ( 1 − g 1 ( p ) x ) ( 1 − g 2 ( p ) x ) ⋯ ( 1 − g r ( p ) x ) {\displaystyle {\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}={\frac {(1-h_{1}(p)x)(1-h_{2}(p)x)\cdots (1-h_{s}(p)x)}{(1-g_{1}(p)x)(1-g_{2}(p)x)\cdots (1-g_{r}(p)x)}}}} for all prime numbers p {\displaystyle p} .
The concept of a rational arithmetical function originates from R. Vaidyanathaswamy (1931).
A multiplicative function f {\displaystyle f} is said to be specially multiplicative
if there is a completely multiplicative function f A {\displaystyle f_{A}} such that
for all positive integers m {\displaystyle m} and n {\displaystyle n} , or equivalently
for all positive integers m {\displaystyle m} and n {\displaystyle n} , where μ {\displaystyle \mu } is the Möbius function.
These are known as Busche-Ramanujan identities.
In 1906, E. Busche stated the identity
and, in 1915, S. Ramanujan gave the inverse form
for k = 0 {\displaystyle k=0} . S. Chowla gave the inverse form for general k {\displaystyle k} in 1929, see P. J. McCarthy (1986). The study of Busche-Ramanujan identities begun from an attempt to better understand the special cases given by Busche and Ramanujan.
It is known that quadratic functions f = g 1 ∗ g 2 {\displaystyle f=g_{1}\ast g_{2}} satisfy the Busche-Ramanujan identities with f A = g 1 g 2 {\displaystyle f_{A}=g_{1}g_{2}} . Quadratic functions are exactly the same as specially multiplicative functions. Totients satisfy a restricted Busche-Ramanujan identity. For further details, see R. Vaidyanathaswamy (1931).
Let A = F q [ X ] , the polynomial ring over the finite field with q elements. A is a principal ideal domain and therefore A is a unique factorization domain .
A complex-valued function λ {\displaystyle \lambda } on A is called multiplicative if λ ( f g ) = λ ( f ) λ ( g ) {\displaystyle \lambda (fg)=\lambda (f)\lambda (g)} whenever f and g are relatively prime .
Let h be a polynomial arithmetic function (i.e. a function on set of monic polynomials over A ). Its corresponding Dirichlet series is defined to be
where for g ∈ A , {\displaystyle g\in A,} set | g | = q deg ( g ) {\displaystyle |g|=q^{\deg(g)}} if g ≠ 0 , {\displaystyle g\neq 0,} and | g | = 0 {\displaystyle |g|=0} otherwise.
The polynomial zeta function is then
Similar to the situation in N , every Dirichlet series of a multiplicative function h has a product representation ( Euler product ):
where the product runs over all monic irreducible polynomials P . For example, the product representation of the zeta function is as for the integers:
Unlike the classical zeta function , ζ A ( s ) {\displaystyle \zeta _{A}(s)} is a simple rational function:
In a similar way, If f and g are two polynomial arithmetic functions, one defines f * g , the Dirichlet convolution of f and g , by
where the sum is over all monic divisors d of m , or equivalently over all pairs ( a , b ) of monic polynomials whose product is m . The identity D h D g = D h ∗ g {\displaystyle D_{h}D_{g}=D_{h*g}} still holds.
Multivariate functions can be constructed using multiplicative model estimators. Where a matrix function of A is defined as D N = N 2 × N ( N + 1 ) / 2 {\displaystyle D_{N}=N^{2}\times N(N+1)/2}
a sum can be distributed across the product y t = ∑ ( t / T ) 1 / 2 u t = ∑ ( t / T ) 1 / 2 G t 1 / 2 ϵ t {\displaystyle y_{t}=\sum (t/T)^{1/2}u_{t}=\sum (t/T)^{1/2}G_{t}^{1/2}\epsilon _{t}}
For the efficient estimation of Σ(.) , the following two nonparametric regressions can be considered: y ~ t 2 = y t 2 g t = σ 2 ( t / T ) + σ 2 ( t / T ) ( ϵ t 2 − 1 ) , {\displaystyle {\tilde {y}}_{t}^{2}={\frac {y_{t}^{2}}{g_{t}}}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(\epsilon _{t}^{2}-1),}
and y t 2 = σ 2 ( t / T ) + σ 2 ( t / T ) ( g t ϵ t 2 − 1 ) . {\displaystyle y_{t}^{2}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(g_{t}\epsilon _{t}^{2}-1).}
Thus it gives an estimate value of L t ( τ ; u ) = ∑ t = 1 T K h ( u − t / T ) [ l n τ + y t 2 g t τ ] {\displaystyle L_{t}(\tau ;u)=\sum _{t=1}^{T}K_{h}(u-t/T){\begin{bmatrix}ln\tau +{\frac {y_{t}^{2}}{g_{t}\tau }}\end{bmatrix}}}
with a local likelihood function for y t 2 {\displaystyle y_{t}^{2}} with known g t {\displaystyle g_{t}} and unknown σ 2 ( t / T ) {\displaystyle \sigma ^{2}(t/T)} .
An arithmetical function f {\displaystyle f} is
quasimultiplicative if there exists a nonzero constant c {\displaystyle c} such that c f ( m n ) = f ( m ) f ( n ) {\displaystyle c\,f(mn)=f(m)f(n)} for all positive integers m , n {\displaystyle m,n} with ( m , n ) = 1 {\displaystyle (m,n)=1} . This concept originates by Lahiri (1972).
An arithmetical function f {\displaystyle f} is semimultiplicative
if there exists a nonzero constant c {\displaystyle c} , a positive integer a {\displaystyle a} and
a multiplicative function f m {\displaystyle f_{m}} such that f ( n ) = c f m ( n / a ) {\displaystyle f(n)=cf_{m}(n/a)} for all positive integers n {\displaystyle n} (under the convention that f m ( x ) = 0 {\displaystyle f_{m}(x)=0} if x {\displaystyle x} is not a positive integer.) This concept is due to David Rearick (1966).
An arithmetical function f {\displaystyle f} is Selberg multiplicative if
for each prime p {\displaystyle p} there exists a function f p {\displaystyle f_{p}} on nonnegative integers with f p ( 0 ) = 1 {\displaystyle f_{p}(0)=1} for
all but finitely many primes p {\displaystyle p} such that f ( n ) = ∏ p f p ( ν p ( n ) ) {\displaystyle f(n)=\prod _{p}f_{p}(\nu _{p}(n))} for all positive integers n {\displaystyle n} , where ν p ( n ) {\displaystyle \nu _{p}(n)} is the exponent of p {\displaystyle p} in the canonical factorization of n {\displaystyle n} .
See Selberg (1977).
It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide. They both satisfy the arithmetical identity f ( m ) f ( n ) = f ( ( m , n ) ) f ( [ m , n ] ) {\displaystyle f(m)f(n)=f((m,n))f([m,n])} for all positive integers m , n {\displaystyle m,n} . See Haukkanen (2012).
It is well known and easy to see that multiplicative functions are quasimultiplicative functions with c = 1 {\displaystyle c=1} and quasimultiplicative functions are semimultiplicative functions with a = 1 {\displaystyle a=1} . | https://en.wikipedia.org/wiki/Multiplicative_function |
In number theory , two positive integers a and b are said to be multiplicatively independent [ 1 ] if their only common integer power is 1. That is, for integers n and m , a n = b m {\displaystyle a^{n}=b^{m}} implies n = m = 0 {\displaystyle n=m=0} . Two integers which are not multiplicatively independent are said to be multiplicatively dependent.
As examples, 36 and 216 are multiplicatively dependent since 36 3 = ( 6 2 ) 3 = ( 6 3 ) 2 = 216 2 {\displaystyle 36^{3}=(6^{2})^{3}=(6^{3})^{2}=216^{2}} , whereas 2 and 3 are multiplicatively independent.
Being multiplicatively independent admits some other characterizations. a and b are multiplicatively independent if and only if log ( a ) / log ( b ) {\displaystyle \log(a)/\log(b)} is irrational. This property holds independently of the base of the logarithm .
Let a = p 1 α 1 p 2 α 2 ⋯ p k α k {\displaystyle a=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}} and b = q 1 β 1 q 2 β 2 ⋯ q l β l {\displaystyle b=q_{1}^{\beta _{1}}q_{2}^{\beta _{2}}\cdots q_{l}^{\beta _{l}}} be the canonical representations of a and b . The integers a and b are multiplicatively dependent if and only if k = l , p i = q i {\displaystyle p_{i}=q_{i}} and α i β i = α j β j {\displaystyle {\frac {\alpha _{i}}{\beta _{i}}}={\frac {\alpha _{j}}{\beta _{j}}}} for all i and j .
Büchi arithmetic in base a and b define the same sets if and only if a and b are multiplicatively dependent.
Let a and b be multiplicatively dependent integers, that is, there exists n,m>1 such that a n = b m {\displaystyle a^{n}=b^{m}} . The integers c such that the length of its expansion in base a is at most m are exactly the integers such that the length of their expansion in base b is at most n . It implies that computing the base b expansion of a number, given its base a expansion, can be done by transforming consecutive sequences of m base a digits into consecutive sequence of n base b digits.
[ 2 ] | https://en.wikipedia.org/wiki/Multiplicative_independence |
In quantum field theory , multiplicative quantum numbers are conserved quantum numbers of a special kind. A given quantum number q is said to be additive if in a particle reaction the sum of the q -values of the interacting particles is the same before and after the reaction. Most conserved quantum numbers are additive in this sense; the electric charge is one example. A multiplicative quantum number q is one for which the corresponding product, rather than the sum, is preserved.
Any conserved quantum number is a symmetry of the Hamiltonian of the system (see Noether's theorem ). Symmetry groups which are examples of the abstract group called Z 2 give rise to multiplicative quantum numbers. This group consists of an operation, P , whose square is the identity, P 2 = 1 . Thus, all symmetries which are mathematically similar to parity (physics) give rise to multiplicative quantum numbers.
In principle, multiplicative quantum numbers can be defined for any abelian group . An example would be to trade the electric charge , Q , (related to the abelian group U(1) of electromagnetism ), for the new quantum number exp(2 i π Q ) . Then this becomes a multiplicative quantum number by virtue of the charge being an additive quantum number. However, this route is usually followed only for discrete subgroups of U(1), of which Z 2 finds the widest possible use.
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multiplicative_quantum_number |
In mathematics , a multiplicative sequence or m -sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology .
Let K n be polynomials over a ring A in indeterminates p 1 , ... weighted so that p i has weight i (with p 0 = 1) and all the terms in K n have weight n (in particular K n is a polynomial in p 1 , ..., p n ). The sequence K n is multiplicative if the map
is an endomorphism of the multiplicative monoid ( A [ x 1 , x 2 , ⋯ ] [ [ z ] ] , ⋅ ) {\displaystyle (A[x_{1},x_{2},\cdots ][[z]],\cdot )} , where q n ∈ A [ x 1 , x 2 , ⋯ ] {\displaystyle q_{n}\in A[x_{1},x_{2},\cdots ]} .
The power series
is the characteristic power series of the K n . A multiplicative sequence is determined by its characteristic power series Q ( z ), and every power series with constant term 1 gives rise to a multiplicative sequence.
To recover a multiplicative sequence from a characteristic power series Q ( z ) we consider the coefficient of z j in the product
for any m > j . This is symmetric in the β i and homogeneous of weight j : so can be expressed as a polynomial K j ( p 1 , ..., p j ) in the elementary symmetric functions p of the β . Then K j defines a multiplicative sequence.
As an example, the sequence K n = p n is multiplicative and has characteristic power series 1 + z .
Consider the power series
where B k is the k -th Bernoulli number . The multiplicative sequence with Q as characteristic power series is denoted L j ( p 1 , ..., p j ).
The multiplicative sequence with characteristic power series
is denoted A j ( p 1 ,..., p j ).
The multiplicative sequence with characteristic power series
is denoted T j ( p 1 ,..., p j ): these are the Todd polynomials .
The genus of a multiplicative sequence is a ring homomorphism , from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers .
For example, the Todd genus is associated to the Todd polynomials with characteristic power series z 1 − exp ( − z ) {\displaystyle {\frac {z}{1-\exp(-z)}}} . | https://en.wikipedia.org/wiki/Multiplicative_sequence |
In the mathematical theory of automorphic representations , a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group . The multiplicity in question is the number of times a given abstract group representation is realised in a certain space, of square-integrable functions , given in a concrete way.
A multiplicity one theorem may also refer to a result about the restriction of a representation of a group G to a subgroup H . In that context, the pair ( G , H ) is called a strong Gelfand pair .
Let G be a reductive algebraic group over a number field K and let A denote the adeles of K . Let Z denote the centre of G and let ω be a continuous unitary character from Z ( K )\Z( A ) × to C × . Let L 2 0 ( G ( K )/ G ( A ), ω ) denote the space of cusp forms with central character ω on G ( A ). This space decomposes into a direct sum of Hilbert spaces
where the sum is over irreducible subrepresentations and m π are non-negative integers .
The group of adelic points of G , G ( A ), is said to satisfy the multiplicity-one property if any smooth irreducible admissible representation of G ( A ) occurs with multiplicity at most one in the space of cusp forms of central character ω , i.e. m π is 0 or 1 for all such π .
The fact that the general linear group , GL ( n ), has the multiplicity-one property was proved by Jacquet & Langlands (1970) for n = 2 and independently by Piatetski-Shapiro (1979) and Shalika ( 1974 ) for n > 2 using the uniqueness of the Whittaker model . Multiplicity-one also holds for SL (2) , but not for SL ( n ) for n > 2 ( Blasius 1994 ).
The strong multiplicity one theorem of Piatetski-Shapiro (1979) and Jacquet and Shalika ( 1981a , 1981b ) states that two cuspidal automorphic representations of the general linear group are isomorphic if their local components are isomorphic for all but a finite number of places. | https://en.wikipedia.org/wiki/Multiplicity-one_theorem |
In spectroscopy and quantum chemistry , the multiplicity of an energy level is defined as 2S+1 , where S is the total spin angular momentum . [ 1 ] [ 2 ] [ 3 ] States with multiplicity 1, 2, 3, 4, 5 are respectively called singlets , doublets , triplets , quartets and quintets. [ 2 ]
In the ground state of an atom or molecule, the unpaired electrons usually all have parallel spin. In this case the multiplicity is also equal to the number of unpaired electrons plus one. [ 4 ]
The multiplicity is often equal to the number of possible orientations of the total spin [ 3 ] relative to the total orbital angular momentum L , and therefore to the number of near– degenerate levels that differ only in their spin–orbit interaction energy.
For example, the ground state of a carbon atom is 3 P ( Term symbol ). The superscript three (read as triplet ) indicates that the multiplicity 2S+1 = 3, so that the total spin S = 1. This spin is due to two unpaired electrons , as a result of Hund's rule which favors the single filling of degenerate orbitals. The triplet consists of three states with spin components +1, 0 and −1 along the direction of the total orbital angular momentum, which is also 1 as indicated by the letter P. The total angular momentum quantum number J can vary from L + S = 2 to L − S = 0 in integer steps, so that J = 2, 1 or 0. [ 1 ] [ 2 ]
However the multiplicity equals the number of spin orientations only if S ≤ L. When S > L there are only 2L+1 orientations of total angular momentum possible, ranging from S + L to S − L. [ 2 ] [ 3 ] The ground state of the nitrogen atom is a 4 S state, for which 2S + 1 = 4 in a quartet state, S = 3/2 due to three unpaired electrons. For an S state, L = 0 so that J can only be 3/2 and there is only one level even though the multiplicity is 4.
Most stable organic molecules have complete electron shells with no unpaired electrons and therefore have singlet ground states. This is true also for inorganic molecules containing only main-group elements . Important exceptions are dioxygen (O 2 ) as well as methylene (CH 2 ) and other carbenes .
However, higher spin ground states are very common in coordination complexes of transition metals . A simple explanation of the spin states of such complexes is provided by crystal field theory .
The highest occupied orbital energy level of dioxygen is a pair of antibonding π* orbitals. In the ground state of dioxygen, this energy level is occupied by two electrons of the same spin, as shown in the molecular orbital diagram . The molecule, therefore, has two unpaired electrons and is in a triplet state.
In contrast, the first and second excited states of dioxygen are both states of singlet oxygen . Each has two electrons of opposite spin in the π* level so that S = 0 and the multiplicity is 2S + 1 = 1 in consequence.
In the first excited state, the two π* electrons are paired in the same orbital, so that there are no unpaired electrons. In the second excited state, however, the two π* electrons occupy different orbitals with opposite spin. Each is therefore an unpaired electron, but the total spin is zero and the multiplicity is 2S + 1 = 1 despite the two unpaired electrons. The multiplicity of the second excited state is therefore not equal to the number of its unpaired electrons plus one, and the rule which is usually true for ground states is invalid for this excited state.
In organic chemistry , carbenes are molecules which have carbon atoms with only six electrons in their valence shells and therefore disobey the octet rule . [ 5 ] Carbenes generally split into singlet carbenes and triplet carbenes, named for their spin multiplicities. Both have two non-bonding electrons; in singlet carbenes these exist as a lone pair and have opposite spins so that there is no net spin, while in triplet carbenes these electrons have parallel spins. [ 6 ] | https://en.wikipedia.org/wiki/Multiplicity_(chemistry) |
In statistical mechanics , multiplicity (also called statistical weight ) refers to the number of microstates corresponding to a particular macrostate of a thermodynamic system . [ 1 ] Commonly denoted Ω {\displaystyle \Omega } , it is related to the configuration entropy of an isolated system [ 2 ] via Boltzmann's entropy formula S = k B log Ω , {\displaystyle S=k_{\text{B}}\log \Omega ,} where S {\displaystyle S} is the entropy and k B {\displaystyle k_{\text{B}}} is the Boltzmann constant .
A simplified model of the two-state paramagnet provides an example of the process of calculating the multiplicity of particular macrostate. [ 1 ] This model consists of a system of N microscopic dipoles μ which may either be aligned or anti-aligned with an externally applied magnetic field B . Let N ↑ {\displaystyle N_{\uparrow }} represent the number of dipoles that are aligned with the external field and N ↓ {\displaystyle N_{\downarrow }} represent the number of anti-aligned dipoles. The energy of a single aligned dipole is U ↑ = − μ B , {\displaystyle U_{\uparrow }=-\mu B,} while the energy of an anti-aligned dipole is U ↓ = μ B ; {\displaystyle U_{\downarrow }=\mu B;} thus the overall energy of the system is U = ( N ↓ − N ↑ ) μ B . {\displaystyle U=(N_{\downarrow }-N_{\uparrow })\mu B.}
The goal is to determine the multiplicity as a function of U ; from there, the entropy and other thermodynamic properties of the system can be determined. However, it is useful as an intermediate step to calculate multiplicity as a function of N ↑ {\displaystyle N_{\uparrow }} and N ↓ . {\displaystyle N_{\downarrow }.} This approach shows that the number of available macrostates is N + 1 . For example, in a very small system with N = 2 dipoles, there are three macrostates, corresponding to N ↑ = 0 , 1 , 2. {\displaystyle N_{\uparrow }=0,1,2.} Since the N ↑ = 0 {\displaystyle N_{\uparrow }=0} and N ↑ = 2 {\displaystyle N_{\uparrow }=2} macrostates require both dipoles to be either anti-aligned or aligned, respectively, the multiplicity of either of these states is 1. However, in the N ↑ = 1 , {\displaystyle N_{\uparrow }=1,} either dipole can be chosen for the aligned dipole, so the multiplicity is 2. In the general case, the multiplicity of a state, or the number of microstates, with N ↑ {\displaystyle N_{\uparrow }} aligned dipoles follows from combinatorics , resulting in Ω = N ! N ↑ ! ( N − N ↑ ) ! = N ! N ↑ ! N ↓ ! , {\displaystyle \Omega ={\frac {N!}{N_{\uparrow }!(N-N_{\uparrow })!}}={\frac {N!}{N_{\uparrow }!N_{\downarrow }!}},} where the second step follows from the fact that N ↑ + N ↓ = N . {\displaystyle N_{\uparrow }+N_{\downarrow }=N.}
Since N ↑ − N ↓ = − U μ B , {\displaystyle N_{\uparrow }-N_{\downarrow }=-{\tfrac {U}{\mu B}},} the energy U can be related to N ↑ {\displaystyle N_{\uparrow }} and N ↓ {\displaystyle N_{\downarrow }} as follows: N ↑ = N 2 − U 2 μ B N ↓ = N 2 + U 2 μ B . {\displaystyle {\begin{aligned}N_{\uparrow }&={\frac {N}{2}}-{\frac {U}{2\mu B}}\\[4pt]N_{\downarrow }&={\frac {N}{2}}+{\frac {U}{2\mu B}}.\end{aligned}}}
Thus the final expression for multiplicity as a function of internal energy is Ω = N ! ( N 2 − U 2 μ B ) ! ( N 2 + U 2 μ B ) ! . {\displaystyle \Omega ={\frac {N!}{\left({\frac {N}{2}}-{\frac {U}{2\mu B}}\right)!\left({\frac {N}{2}}+{\frac {U}{2\mu B}}\right)!}}.}
This can be used to calculate entropy in accordance with Boltzmann's entropy formula; from there one can calculate other useful properties such as temperature and heat capacity .
This thermodynamics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multiplicity_(statistical_mechanics) |
The multiplicity function for a two state paramagnet, W(n,N), is the number of spin states such that n of the N spins point in the z-direction. This function is given by the combinatoric function C(N,n) . That is:
W ( n , N ) = ( N n ) = N ! n ! ( N − n ) ! {\displaystyle W(n,N)={N \choose n}={{N!} \over {n!(N-n)!}}}
It is primarily used in introductory statistical mechanics and thermodynamics textbooks to explain the microscopic definition of entropy to students. If the spins are non-interacting, then the multiplicity function counts the number of states which have the same energy in an external magnetic field. By definition, the entropy S is then given by the natural logarithm of this number:
S = k ln W {\displaystyle S=k\ln {W}\,} [ 1 ] Where k is the Boltzmann constant
This thermodynamics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multiplicity_function_for_N_noninteracting_spins |
In microbiology , the multiplicity of infection or MOI is the ratio of agents (e.g. phage or more generally virus , bacteria ) to infection targets (e.g. cell ). For example, when referring to a group of cells inoculated with virus particles, the MOI is the ratio of the number of virus particles to the number of target cells present in a defined space. [ 1 ]
The actual number of viruses or bacteria that will enter any given cell is a stochastic process : some cells may absorb more than one infectious agent, while others may not absorb any. Before determining the multiplicity of infection, it's absolutely necessary to have a well-isolated agent, as crude agents may not produce reliable and reproducible results. The probability that a cell will absorb n {\displaystyle n} virus particles or bacteria when inoculated with an MOI of m {\displaystyle m} can be calculated for a given population using a Poisson distribution . This application of Poisson's distribution was applied and described by Ellis and Delbrück. [ 2 ]
where m {\displaystyle m} is the multiplicity of infection or MOI, n {\displaystyle n} is the number of infectious agents that enter the infection target, and P ( n ) {\displaystyle P(n)} is the probability that an infection target (a cell) will get infected by n {\displaystyle n} infectious agents.
In fact, the infectivity of the virus or bacteria in question will alter this relationship. One way around this is to use a functional definition of infectious particles rather than a strict count, such as a plaque forming unit for viruses. [ 3 ]
For example, when an MOI of 1 (1 infectious viral particle per cell) is used to infect a population of cells, the probability that a cell will not get infected is P ( 0 ) = 36.79 % {\displaystyle P(0)=36.79\%} , and the probability that it be infected by a single particle is P ( 1 ) = 36.79 % {\displaystyle P(1)=36.79\%} , by two particles is P ( 2 ) = 18.39 % {\displaystyle P(2)=18.39\%} , by three particles is P ( 3 ) = 6.13 % {\displaystyle P(3)=6.13\%} , and so on.
The average percentage of cells that will become infected as a result of inoculation with a given MOI can be obtained by realizing that it is simply P ( n > 0 ) = 1 − P ( 0 ) {\displaystyle P(n>0)=1-P(0)} . Hence, the average fraction of cells that will become infected following an inoculation with an MOI of m {\displaystyle m} is given by:
which is approximately equal to m {\displaystyle m} for small values of m ≪ 1 {\displaystyle m\ll 1} .
As the MOI increases, the percentages of cells infected with at least one viral particle ( n > 0 {\displaystyle n>0} ) also increases. [ 4 ] | https://en.wikipedia.org/wiki/Multiplicity_of_infection |
In computing , especially digital signal processing , the multiply–accumulate ( MAC ) or multiply–add ( MAD ) operation is a common step that computes the product of two numbers and adds that product to an accumulator . The hardware unit that performs the operation is known as a multiplier–accumulator ( MAC unit ); the operation itself is also often called a MAC or a MAD operation. The MAC operation modifies an accumulator a : a ← a + ( b × c ) {\displaystyle a\gets a+(b\times c)}
When done with floating-point numbers, it might be performed with two roundings (typical in many DSPs ), or with a single rounding. When performed with a single rounding, it is called a fused multiply–add ( FMA ) or fused multiply–accumulate ( FMAC ).
Modern computers may contain a dedicated MAC, consisting of a multiplier implemented in combinational logic followed by an adder and an accumulator register that stores the result. The output of the register is fed back to one input of the adder, so that on each clock cycle, the output of the multiplier is added to the register. Combinational multipliers require a large amount of logic, but can compute a product much more quickly than the method of shifting and adding typical of earlier computers. Percy Ludgate was the first to conceive a MAC in his Analytical Machine of 1909, [ 1 ] and the first to exploit a MAC for division (using multiplication seeded by reciprocal, via the convergent series (1+ x ) −1 ). The first modern processors to be equipped with MAC units were digital signal processors , but the technique is now also common in general-purpose processors. [ 2 ] [ 3 ] [ 4 ] [ 5 ]
When done with integers , the operation is typically exact (computed modulo some power of two ). However, floating-point numbers have only a certain amount of mathematical precision . That is, digital floating-point arithmetic is generally not associative or distributive . (See Floating-point arithmetic § Accuracy problems .)
Therefore, it makes a difference to the result whether the multiply–add is performed with two roundings, or in one operation with a single rounding (a fused multiply–add). IEEE 754-2008 specifies that it must be performed with one rounding, yielding a more accurate result. [ 6 ]
A fused multiply–add ( FMA or fmadd ) [ 7 ] is a floating-point multiply–add operation performed in one step ( fused operation ), with a single rounding. That is, where an unfused multiply–add would compute the product b × c , round it to N significant bits, add the result to a , and round back to N significant bits, a fused multiply–add would compute the entire expression a + ( b × c ) to its full precision before rounding the final result down to N significant bits.
A fast FMA can speed up and improve the accuracy of many computations that involve the accumulation of products:
Fused multiply–add can usually be relied on to give more accurate results. However, William Kahan has pointed out that it can give problems if used unthinkingly. [ 8 ] If x 2 − y 2 is evaluated as (( x × x ) − y × y ) (following Kahan's suggested notation in which redundant parentheses direct the compiler to round the ( x × x ) term first) using fused multiply–add, then the result may be negative even when x = y due to the first multiplication discarding low significance bits. This could then lead to an error if, for instance, the square root of the result is then evaluated.
When implemented inside a microprocessor , an FMA can be faster than a multiply operation followed by an add. However, standard industrial implementations based on the original IBM RS/6000 design require a 2 N -bit adder to compute the sum properly. [ 9 ]
Another benefit of including this instruction is that it allows an efficient software implementation of division (see division algorithm ) and square root (see methods of computing square roots ) operations, thus eliminating the need for dedicated hardware for those operations. [ 10 ]
Some machines combine multiple fused multiply add operations into a single step, e.g. performing a four-element dot-product on two 128-bit SIMD registers a0×b0 + a1×b1 + a2×b2 + a3×b3 with single cycle throughput.
The FMA operation is included in IEEE 754-2008 .
The 1999 standard of the C programming language supports the FMA operation through the fma() standard math library function and the automatic transformation of a multiplication followed by an addition (contraction of floating-point expressions), which can be explicitly enabled or disabled with standard pragmas ( #pragma STDC FP_CONTRACT ). The GCC and Clang C compilers do such transformations by default for processor architectures that support FMA instructions. With GCC, which does not support the aforementioned pragma, [ 11 ] this can be globally controlled by the -ffp-contract command line option. [ 12 ]
The fused multiply–add operation was introduced as "multiply–add fused" in the IBM POWER1 (1990) processor, [ 13 ] but has been added to numerous processors: | https://en.wikipedia.org/wiki/Multiply–accumulate_operation |
Magnetic materials with strong spin-orbit interaction , such as: LaFeAsO, [ 1 ] [ 2 ] PrFe 4 P 12 , [ 3 ] [ 4 ] YbRu 2 Ge 2 , [ 5 ] UO 2 , [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] NpO 2 , [ 11 ] [ 12 ] [ 13 ] Ce 1−x La x B 6 , [ 14 ] URu 2 Si 2 [ 15 ] [ 16 ] [ 17 ] [ 18 ] [ 19 ] and many other compounds, are found to have magnetic ordering constituted by high rank multipoles, e.g. quadruple, octople, etc. [ 20 ] Due to the strong spin-orbit coupling, multipoles are automatically introduced to the systems when the total angular momentum quantum number J is larger than 1/2. If those multipoles are coupled by some exchange mechanisms, those multipoles could tend to have some ordering as conventional spin 1/2 Heisenberg problem. Except the multipolar ordering, many hidden order phenomena are believed closely related to the multipolar interactions [ 11 ] [ 14 ] [ 15 ]
Consider a quantum mechanical system with Hilbert space spanned by | j , m j ⟩ {\displaystyle |j,m_{j}\rangle } , where j {\displaystyle j} is the total angular momentum and m j {\displaystyle m_{j}} is its projection on the quantization axis. Then any quantum operators can be represented using the basis set { | j , m j ⟩ } {\displaystyle \lbrace |j,m_{j}\rangle \rbrace } as a matrix with dimension ( 2 j + 1 ) {\displaystyle (2j+1)} . Therefore, one can define ( 2 j + 1 ) 2 {\displaystyle (2j+1)^{2}} matrices to completely expand any quantum operator in this Hilbert space. Taking J=1/2 as an example, a quantum operator A can be expanded as
Obviously, the matrices: L i j = | i ⟩ ⟨ j | {\displaystyle L_{ij}=|i\rangle \langle j|} form a basis set in the operator space. Any quantum operator defined in this Hilbert can be expended by { L i j } {\displaystyle \lbrace L_{ij}\rbrace } operators. In the following, let's call these matrices as a super basis to distinguish the eigen basis of quantum states. More specifically the above super basis { L i j } {\displaystyle \lbrace L_{ij}\rbrace } can be called a transition super basis because it describes the transition between states | i ⟩ {\displaystyle |i\rangle } and | j ⟩ {\displaystyle |j\rangle } . In fact, this is not the only super basis that does the trick. We can also use Pauli matrices and the identity matrix to form a super basis
Since the rotation properties of σ x , σ y , σ z {\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}} follow the same rules as the rank 1 tensor of cubic harmonics T x , T y , T z {\displaystyle T_{x},T_{y},T_{z}} and the identity matrix I {\displaystyle I} follows the same rules as the rank 0 tensor T s {\displaystyle T_{s}} , the basis set { I , σ x , σ y , σ z } {\displaystyle \lbrace I,\sigma _{x},\sigma _{y},\sigma _{z}\rbrace } can be called cubic super basis. Another commonly used super basis is spherical harmonic super basis which is built by replacing the σ x , σ y {\displaystyle \sigma _{x},\ \sigma _{y}} to the raising and lowering operators { I , σ − 1 , σ 0 , σ + 1 } {\displaystyle \lbrace I,\sigma _{-1},\sigma _{0},\sigma _{+1}\rbrace }
Again, σ − 1 , σ 0 , σ + 1 {\displaystyle \sigma _{-1},\sigma _{0},\sigma _{+1}} share the same rotational properties as rank 1 spherical harmonic tensors Y − 1 1 , Y 0 1 , Y − 1 1 {\displaystyle Y_{-1}^{1},Y_{0}^{1},Y_{-1}^{1}} , so it is called spherical super basis.
Because atomic orbitals s , p , d , f {\displaystyle s,p,d,f} are also described by spherical or cubic harmonic functions, one can imagine or visualize these operators using the wave functions of atomic orbitals although they are essentially matrices not spatial functions.
If we extend the problem to J = 1 {\displaystyle J=1} , we will need 9 matrices to form a super basis. For transition super basis, we have { L i j ; i , j = 1 ∼ 3 } {\displaystyle \lbrace L_{ij};i,j=1\sim 3\rbrace } . For cubic super basis, we have { T s , T x , T y , T z , T x y , T y z , T z x , T x 2 − y 2 , T 3 z 2 − r 2 } {\displaystyle \lbrace T_{s},T_{x},T_{y},T_{z},T_{xy},T_{yz},T_{zx},T_{x^{2}-y^{2}},T_{3z^{2}-r^{2}}\rbrace } . For spherical super basis, we have { Y 0 0 , Y − 1 1 , Y 0 1 , Y − 1 1 , Y − 2 2 , Y − 1 2 , Y 0 2 , Y 1 2 , Y 2 2 } {\displaystyle \lbrace Y_{0}^{0},Y_{-1}^{1},Y_{0}^{1},Y_{-1}^{1},Y_{-2}^{2},Y_{-1}^{2},Y_{0}^{2},Y_{1}^{2},Y_{2}^{2}\rbrace } . In group theory, T s / Y 0 0 {\displaystyle T_{s}/Y_{0}^{0}} are called scalar or rank 0 tensor, T x , y z , / Y − 1 , 0 , + 1 1 {\displaystyle T_{x,yz,}/Y_{-1,0,+1}^{1}} are called dipole or rank 1 tensors, T x y , y z , z x , x 2 − y 2 , 3 z 2 − r 2 / Y − 2 , − 1 , 0 , + 1 , + 2 2 {\displaystyle T_{xy,yz,zx,x^{2}-y^{2},3z^{2}-r^{2}}/Y_{-2,-1,0,+1,+2}^{2}} are called quadrupole or rank 2 tensors. [ 20 ]
The example tells us, for a J {\displaystyle J} -multiplet problem, one will need all rank 0 ∼ 2 J {\displaystyle 0\sim 2J} tensor operators to form a complete super basis. Therefore, for a J = 1 {\displaystyle J=1} system, its density matrix must have quadrupole components. This is the reason why a J > 1 / 2 {\displaystyle J>1/2} problem will automatically introduce high-rank multipoles to the system [ 21 ] [ 22 ]
A general definition of spherical harmonic super basis of a J {\displaystyle J} -multiplet problem can be expressed as [ 20 ]
where the parentheses denote a 3-j symbol ; K is the rank which ranges 0 ∼ 2 J {\displaystyle 0\sim 2J} ; Q is the
projection index of rank K which ranges from −K to +K. A cubic harmonic super basis where all the tensor operators are hermitian can be defined as
Then, any quantum operator A {\displaystyle A} defined in the J {\displaystyle J} -multiplet Hilbert space can be expanded as
where the expansion coefficients can be obtained by taking the trace inner product, e.g. α K Q = T r [ A Y K Q † ] {\displaystyle \alpha _{K}^{Q}=Tr[AY_{K}^{Q\dagger }]} .
Apparently, one can make linear combination of these operators to form a new super basis that have different symmetries.
Using the addition theorem of tensor operators, the product of a rank n tensor and a rank m tensor can generate a new tensor with rank n+m ~ |n-m|. Therefore, a high rank tensor can be expressed as the product of low rank tensors. This convention is useful to interpret the high rank multipolar exchange terms as a "multi-exchange" process of dipoles (or pseudospins). For example, for the spherical harmonic tensor operators of J = 1 {\displaystyle J=1} case, we have
If so, a quadrupole-quadrupole interaction (see next section) can be considered as a two steps dipole-dipole interaction. For example, Y 2 i + 2 i Y 2 j − 2 j = 4 Y 1 i + 1 i Y 1 i + 1 i Y 1 j − 1 j Y 1 j − 1 j {\displaystyle Y_{2_{i}}^{+2_{i}}Y_{2_{j}}^{-2_{j}}=4Y_{1_{i}}^{+1_{i}}Y_{1_{i}}^{+1_{i}}Y_{1_{j}}^{-1_{j}}Y_{1_{j}}^{-1_{j}}} , so the one step quadrupole transition Y 2 i + 2 i {\displaystyle Y_{2_{i}}^{+2_{i}}} on site i {\displaystyle i} now becomes a two steps of dipole transition Y 1 i + 1 i Y 1 i + 1 i {\displaystyle Y_{1_{i}}^{+1_{i}}Y_{1_{i}}^{+1_{i}}} . Hence not only inter-site-exchange but also intra-site-exchange terms appear (so called multi-exchange). If J {\displaystyle J} is even larger, one can expect more complicated intra-site-exchange terms would appear. However, one has to note that it is not a perturbation expansion but just a mathematical technique. The high rank terms are not necessarily smaller than low rank terms. In many systems, high rank terms are more important than low rank terms. [ 20 ]
There are four major mechanisms to induce exchange interactions between two magnetic moments in a system: [ 20 ] 1). Direct exchange 2). RKKY 3). Superexchange 4). Spin-Lattice. No matter which one is dominated, a general form of the exchange interaction can be written as [ 21 ]
where i , j {\displaystyle i,j} are the site indexes and C K i K j Q i Q j {\displaystyle C_{K_{i}K_{j}}^{Q{i}Q_{j}}} is the coupling constant that couples two multipole moments T K i Q i {\displaystyle T_{K_{i}}^{Q_{i}}} and T K j Q j {\displaystyle T_{K_{j}}^{Q_{j}}} . One can immediately find if K {\displaystyle K} is restricted to 1 only, the Hamiltonian reduces to conventional Heisenberg model.
An important feature of the multipolar exchange Hamiltonian is its anisotropy. [ 21 ] The value of coupling constant C K i K j Q i Q j {\displaystyle C_{K_{i}K_{j}}^{Q{i}Q_{j}}} is usually very sensitive to the relative angle between two multipoles. Unlike conventional spin only exchange Hamiltonian where the coupling constants are isotropic in a homogeneous system, the highly anisotropic atomic orbitals (recall the shape of the s , p , d , f {\displaystyle s,p,d,f} wave functions) coupling to the system's magnetic moments will inevitably introduce huge anisotropy even in a homogeneous system. This is one of the main reasons that most multipolar orderings tend to be non-colinear.
Unlike magnetic spin ordering where the antiferromagnetism can be defined by flipping the magnetization axis of two neighbor sites from a ferromagnetic configuration, flipping of the magnetization axis of a multipole is usually meaningless. Taking a T y z {\displaystyle T_{yz}} moment as an example, if one flips the z-axis by making a π {\displaystyle \pi } rotation toward the y-axis, it just changes nothing. Therefore, a suggested definition [ 21 ] of antiferromagnetic multipolar ordering is to flip their phases by π {\displaystyle \pi } , i.e. T y z → e i π T y z = − T y z {\displaystyle T_{yz}\rightarrow e^{i\pi }T_{yz}=-T_{yz}} . In this regard, the antiferromagnetic spin ordering is just a special case of this definition, i.e. flipping the phase of a dipole moment is equivalent to flipping its magnetization axis. As for high rank multipoles, e.g. T y z {\displaystyle T_{yz}} , it actually becomes a π / 2 {\displaystyle \pi /2} rotation and for T 3 z 2 − r 2 {\displaystyle T_{3z^{2}-r^{2}}} it is even not any kind of rotation.
Calculation of multipolar exchange interactions remains a challenging issue in many aspects. Although there were many works based on fitting the model Hamiltonians with experiments, predictions of the coupling constants based on first-principle schemes remain lacking. Currently there are two studies implemented first-principles approach to explore multipolar exchange interactions. An early study was developed in 80's. It is based on a mean field approach that can greatly reduce the complexity of coupling constants induced by RKKY mechanism, so the multipolar exchange Hamiltonian can be described by just a few unknown parameters and can be obtained by fitting with experiment data. [ 23 ] Later on, a first-principles approach to estimate the unknown parameters was further developed and got good agreements with a few selected compounds, e.g. cerium momnpnictides. [ 24 ] Another first-principle approach was also proposed recently. [ 21 ] It maps all the coupling constants induced by all static exchange mechanisms to a series of DFT+U total energy calculations and got agreement with uranium dioxide. | https://en.wikipedia.org/wiki/Multipolar_exchange_interaction |
Transitions between excited states (or excited states and the ground state) of a nuclide lead to the emission of gamma quanta . These can be classified by their multipolarity . [ 1 ] There are two kinds: electric and magnetic multipole radiation. Each of these, being electromagnetic radiation, consists of an electric and a magnetic field.
Electric dipole, quadrupole, octupole… radiation (generally: 2 ℓ {\displaystyle \ell } pole radiation) is also designated as E1, E2, E3,… radiation (generally: E ℓ {\displaystyle \ell } radiation). [ note 1 ]
Similarly, magnetic dipole , quadrupole, octupole… radiation (generally: 2 ℓ {\displaystyle \ell } pole radiation) is designated as M1, M2, M3,… radiation (generally: M ℓ {\displaystyle \ell } radiation).
There is no monopole radiation ( ℓ = 0 {\displaystyle \ell =0} ). [ 1 ]
In quantum mechanics , angular momentum is quantized. The various multipole fields have particular values of angular momentum: E ℓ {\displaystyle \ell } radiation carries an angular momentum ℓ {\displaystyle \ell } in units of ℏ {\displaystyle \hbar } ; likewise, M ℓ {\displaystyle \ell } radiation carries an angular momentum ℓ {\displaystyle \ell } in units of ℏ {\displaystyle \hbar } . The conservation of angular momentum leads to selection rules , i.e., rules defining which multipoles may or may not be emitted in particular transitions.
To make a simple classical comparison, consider the figure of the oscillating dipole. It produces electric field lines travelling outwards, intertwined with magnetic field lines, according to Maxwell's equations . This system of field lines then corresponds to that of E1 radiation. Similar considerations hold for oscillating electric or magnetic multipoles of higher order.
Conversely, it is plausible that the multipolarity of radiation can be determined from the angular distribution of the emitted radiation.
A state of a nuclide is described by its energy above the ground state, by its angular momentum J (in units of ℏ {\displaystyle \hbar } ), and by its parity , i.e., its behaviour under reflection ( positive + or negative − ). Since the spin of nucleons is ½ (in units of ℏ {\displaystyle \hbar } ), and since orbital angular momentum has integer values, J may be an integer or a half integer number.
Electric and magnetic multipole radiations of the same order ℓ {\displaystyle \ell } (i.e., dipole, or quadrupole...) carry the same angular momentum ℓ {\displaystyle \ell } (in units of ℏ {\displaystyle \hbar } ), but differ in parity. The following relations hold for ℓ > 0 {\displaystyle \ell >0} : [ 1 ]
The designation " electric multipole radiation" seems appropriate since the major part of that radiation is produced by the charge density in the source; [ 1 ] conversely, the " magnetic multipole radiation" is mainly due to the current density of the source. [ 1 ]
In electric multipole radiation, the electric field has a radial component; in magnetic multipole radiation, the magnetic field has a radial component. [ 1 ]
An example: in the simplified decay scheme of 60 Co above, the angular momenta and the parities of the various states are shown (A plus sign means even parity, a minus sign means odd parity). Consider the 1.33 MeV transition to the ground state. Clearly, this must carry away an angular momentum of 2, without change of parity. It is therefore an E2 transition. The case of the 1.17 MeV transition is a bit more complex: going from J = 4 to J = 2, all values of angular momentum from 2 to 6 could be emitted. But in practice, the smallest values are most likely, so it is also a quadrupole transition, and it is E2 since there is no parity change. | https://en.wikipedia.org/wiki/Multipolarity_of_gamma_radiation |
The Multipole Density Formalism (also referred to as Hansen-Coppens Formalism ) is an X-ray crystallography method of electron density modelling proposed by Niels K. Hansen and Philip Coppens in 1978. Unlike the commonly used Independent Atom Model , the Hansen-Coppens Formalism presents an aspherical approach, allowing one to model the electron distribution around a nucleus separately in different directions and therefore describe numerous chemical features of a molecule inside the unit cell of an examined crystal in detail.
The Independent Atom Model (abbreviated to IAM), upon which the Multipole Model is based, is a method of charge density modelling. It relies on an assumption that electron distribution around the atom is isotropic , and that therefore charge density is dependent only on the distance from a nucleus. The choice of the radial function used to describe this electron density is arbitrary, granted that its value at the origin is finite. In practice either Gaussian- or Slater-type 1s-orbital functions are used. [ 1 ]
Due to its simplistic approach, this method provides a straightforward model that requires no additional parameters (other than positional and Debye–Waller factors ) to be refined. This allows the IAM to perform satisfactorily while a relatively low amount of data from the diffraction experiment is available. However, the fixed shape of the singular basis function prevents any detailed description of aspherical atomic features.
In order to adjust some valence shell parameters, the Kappa formalism was proposed. [ 2 ] It introduces two additional refineable parameters: an outer shell population (denoted as P v a l {\displaystyle P_{val}} ) and its expansion/contraction ( κ {\displaystyle \kappa } ). Therefore, the electron density is formulated as:
While P v a l {\displaystyle P_{val}} , being responsible for the charge flow part, is linearly coupled with partial charge , the normalised κ {\displaystyle \kappa } parameter scales radial coordinate r {\displaystyle r} . Therefore, lowering the κ {\displaystyle \kappa } parameter results in expansion of the outer shell and, conversely, raising it results in contraction. [ 3 ] Although the Kappa formalism is still, strictly speaking, a spherical method, it is an important step towards understanding modern approaches as it allows one to distinguish chemically different atoms of the same element .
In the multipole model description, the charge density around a nucleus is given by the following equation:
The spherical part remains almost indistinguishable from the Kappa formalism, the only difference being one parameter corresponding to the population of the inner shell . The real strength of the Hansen-Coppens formalism lies in the right, deformational part of the equation. Here κ ′ {\displaystyle \kappa '} fulfils a role similar to κ {\displaystyle \kappa } in the Kappa formalism (expansion/contraction of the aspherical part), whereas individual R l {\displaystyle R_{l}} are fixed spherical functions, analogous to ρ {\displaystyle \rho } . Spherical harmonics Y l m {\displaystyle Y_{lm}} (each with its populational parameter P l m {\displaystyle P_{lm}} ) are, however, introduced to simulate the electrically anisotropic charge distribution. [ 4 ]
In this approach, a fixed coordinate system for each atom needs to be applied. Although at first glance it seems practical to arbitrarily and indiscriminately make it contingent on the unit cell for all atoms present, it is far more beneficial to assign each atom its own local coordinates , which allows for focusing on hybridisation -specific interactions . While the singular sigma bond of the hydrogen can be described well using certain z-parallel pseudoorbitals , xy-plane oriented multipoles with a 3-fold rotational symmetry will prove more beneficial for flat aromatic structures. [ 5 ]
The primary advantage of the Hansen-Coppens formalism is its ability to free the model from spherical restraints and describe the surroundings of a nucleus far more accurately. In this way it becomes possible to examine some molecular features which would normally be only roughly approximated or completely ignored.
X-ray crystallography allows the researcher to precisely determine the position of peak electron density and to reason about the placement of nuclei based on this information. This approach works without any problems for heavy (non-hydrogen) atoms, whose inner shell electrons contribute to the density function to a far greater degree then outer shell electrons.
However, hydrogen atoms possess a feature unique among all the elements - they possess exactly one electron, which additionally is located on their valence shell and therefore is involved in creating strong covalent bonds with atoms of various other elements. While a bond is forming, the maximum of the electron density function moves significantly away from the nucleus and towards the other atom. This prevents any spherical approach from determining hydrogen position correctly by itself. Therefore, usually the hydrogen position is estimated basing on neutron crystallography data for similar molecules, [ 6 ] or it is not modelled at all in the case of low-quality diffraction data.
It is possible (albeit disputable) to freely refine hydrogen atoms' positions using the Hansen-Coppens formalism, after releasing the bond lengths from any restraints derived from neutron measurements. [ 7 ] The bonding orbital simulated with adequate multipoles describes the density distribution neatly while preserving believable bond lengths. It may be worth approximating hydrogen atoms' anisotropic displacement parameters , e.g. using SHADE , before introducing the formalism and, possibly, discarding bond distance constraints. [ 8 ]
In order to analyse the length and strength of various interactions within the molecule, Richard Bader's " Atoms in molecules " theorem may be applied. Due to the complex description of the electron field provided by this aspherical model, it becomes possible to establish realistic bond paths between interacting atoms as well as to find and characterise their critical points . Deeper insight into this data yields useful information about bond strength , type , polarity or ellipticity, and when compared with other molecules brings greater understanding about the actual electron structure of the examined compound. [ 9 ]
Due to the fact that for each multipole of every atom its population is being refined independently, individual charges will rarely be integers . In real cases, electron density flows freely through the molecule and is not bound by any restrictions resulting from the outdated Bohr atom model and found in IAM. Therefore, through e.g. an accurate Bader analysis, net atomic charges may be estimated, which again is beneficial for deepening the understanding of systems under investigation.
Although the Multipole Formalism is a simple and straightforward alternative means of structure refinement, it is definitely not flawless. While usually for each atom either three or nine parameters are to be refined, depending on whether an anisotropic displacement is being taken into account or not, a full multipole description of heavy atoms belonging to the fourth and subsequent periods (such as chlorine , iron or bromine ) requires refinement of up to 37 parameters. [ 10 ] This proves problematic for any crystals possessing large asymmetric units (especially macromolecular compounds) and renders a refinement using the Hansen-Coppens Formalism unachievable for low-quality data with an unsatisfactory ratio of independent reflections to refined parameters.
Caution should be taken while refining some of the parameters simultaneously (i.e. κ {\displaystyle \kappa } or κ ′ {\displaystyle \kappa '} , multipole populations and thermal parameters), as they may correlate strongly, resulting in an unstable refinement or unphysical parameter values. Applying additional constraints resulting from local symmetry for each atom in a molecule (which decreases the number of refined multipoles) [ 1 ] or importing populational parameters from existing databases [ 11 ] [ 12 ] may also be necessary to achieve a passable model. On the other hand, the aforementioned approaches significantly reduce the amount of information required from experiments, while preserving some level of detail concerning aspherical charge distribution. [ 5 ] Therefore, even macromolecular structures with satisfactory X-ray diffraction data can be modelled aspherically in a similar fashion. [ 13 ]
Despite their similarity, individual multipoles do not correspond to atomic projections of molecular orbitals of a wavefuntion as resulting from quantum calculations . Nevertheless, as brilliantly summarized by Stewart, "The structure of the model crystal density, as a superposition of pseudoatoms [...] does have quantitative features which are close to many results based on quantum chemical calculations". [ 14 ] If the overlap between the atomic wavefunctions is small enough, as it occurs for example in transition metal complexes, the atomic multipoles may be correlated with the atomic valence orbitals and multipolar coefficients may be correlated with populations of metal d-orbitals. [ 15 ] A stronger correlation between the X-ray measured diffracted intensities and quantum mechanical wavefunctions is possible using the wavefunction based methods [ 16 ] of Quantum Crystallography , as for example the X-ray atomic orbital model, [ 17 ] the so-called experimental wavefunction [ 18 ] or the Hirshfeld Atom Refinement . [ 19 ] | https://en.wikipedia.org/wiki/Multipole_density_formalism |
Multiprocessing ( MP ) is the use of two or more central processing units (CPUs) within a single computer system . [ 1 ] [ 2 ] The term also refers to the ability of a system to support more than one processor or the ability to allocate tasks between them. There are many variations on this basic theme, and the definition of multiprocessing can vary with context, mostly as a function of how CPUs are defined ( multiple cores on one die , multiple dies in one package , multiple packages in one system unit , etc.).
A multiprocessor is a computer system having two or more processing units (multiple processors) each sharing main memory and peripherals, in order to simultaneously process programs. [ 3 ] [ 4 ] A 2009 textbook defined multiprocessor system similarly, but noted that the processors may share "some or all of the system’s memory and I/O facilities"; it also gave tightly coupled system as a synonymous term. [ 5 ]
At the operating system level, multiprocessing is sometimes used to refer to the execution of multiple concurrent processes in a system, with each process running on a separate CPU or core, as opposed to a single process at any one instant. [ 6 ] [ 7 ] When used with this definition, multiprocessing is sometimes contrasted with multitasking , which may use just a single processor but switch it in time slices between tasks (i.e. a time-sharing system ). Multiprocessing however means true parallel execution of multiple processes using more than one processor. [ 7 ] Multiprocessing doesn't necessarily mean that a single process or task uses more than one processor simultaneously; the term parallel processing is generally used to denote that scenario. [ 6 ] Other authors prefer to refer to the operating system techniques as multiprogramming and reserve the term multiprocessing for the hardware aspect of having more than one processor. [ 2 ] [ 8 ] The remainder of this article discusses multiprocessing only in this hardware sense.
In Flynn's taxonomy , multiprocessors as defined above are MIMD machines. [ 9 ] [ 10 ] As the term "multiprocessor" normally refers to tightly coupled systems in which all processors share memory, multiprocessors are not the entire class of MIMD machines, which also contains message passing multicomputer systems. [ 9 ]
In a multiprocessing system, all CPUs may be equal, or some may be reserved for special purposes. A combination of hardware and operating system software design considerations determine the symmetry (or lack thereof) in a given system. For example, hardware or software considerations may require that only one particular CPU respond to all hardware interrupts, whereas all other work in the system may be distributed equally among CPUs; or execution of kernel-mode code may be restricted to only one particular CPU, whereas user-mode code may be executed in any combination of processors. Multiprocessing systems are often easier to design if such restrictions are imposed, but they tend to be less efficient than systems in which all CPUs are utilized.
Systems that treat all CPUs equally are called symmetric multiprocessing (SMP) systems. In systems where all CPUs are not equal, system resources may be divided in a number of ways, including asymmetric multiprocessing (ASMP), non-uniform memory access (NUMA) multiprocessing, and clustered multiprocessing.
In a master/slave multiprocessor system, the master CPU is in control of the computer and the slave CPU(s) performs assigned tasks. The CPUs can be completely different in terms of speed and architecture. Some (or all) of the CPUs can share a common bus, each can also have a private bus (for private resources), or they may be isolated except for a common communications pathway. Likewise, the CPUs can share common RAM and/or have private RAM that the other processor(s) cannot access. The roles of master and slave can change from one CPU to another.
Two early examples of a mainframe master/slave multiprocessor are the Bull Gamma 60 and the Burroughs B5000 . [ 11 ]
An early example of a master/slave multiprocessor system of microprocessors is the Tandy/Radio Shack TRS-80 Model 16 desktop computer which came out in February 1982 and ran the multi-user/multi-tasking Xenix operating system, Microsoft's version of UNIX (called TRS-XENIX). The Model 16 has two microprocessors: an 8-bit Zilog Z80 CPU running at 4 MHz, and a 16-bit Motorola 68000 CPU running at 6 MHz. When the system is booted, the Z-80 is the master and the Xenix boot process initializes the slave 68000, and then transfers control to the 68000, whereupon the CPUs change roles and the Z-80 becomes a slave processor responsible for all I/O operations including disk, communications, printer and network, as well as the keyboard and integrated monitor, while the operating system and applications run on the 68000 CPU. The Z-80 can be used to do other tasks.
The earlier TRS-80 Model II , which was released in 1979, could also be considered a multiprocessor system as it had both a Z-80 CPU and an Intel 8021 [ 12 ] microcontroller in the keyboard. The 8021 made the Model II the first desktop computer system with a separate detachable lightweight keyboard connected with by a single thin flexible wire, and likely the first keyboard to use a dedicated microcontroller, both attributes that would later be copied years later by Apple and IBM.
In multiprocessing, the processors can be used to execute a single sequence of instructions in multiple contexts ( single instruction, multiple data or SIMD, often used in vector processing ), multiple sequences of instructions in a single context ( multiple instruction, single data or MISD, used for redundancy in fail-safe systems and sometimes applied to describe pipelined processors or hyper-threading ), or multiple sequences of instructions in multiple contexts ( multiple instruction, multiple data or MIMD).
Tightly coupled multiprocessor systems contain multiple CPUs that are connected at the bus level. These CPUs may have access to a central shared memory (SMP or UMA ), or may participate in a memory hierarchy with both local and shared memory (SM)( NUMA ). The IBM p690 Regatta is an example of a high end SMP system. Intel Xeon processors dominated the multiprocessor market for business PCs and were the only major x86 option until the release of AMD 's Opteron range of processors in 2004. Both ranges of processors had their own onboard cache but provided access to shared memory; the Xeon processors via a common pipe and the Opteron processors via independent pathways to the system RAM .
Chip multiprocessors, also known as multi-core computing, involves more than one processor placed on a single chip and can be thought of the most extreme form of tightly coupled multiprocessing. Mainframe systems with multiple processors are often tightly coupled.
Loosely coupled multiprocessor systems (often referred to as clusters ) are based on multiple standalone relatively low processor count commodity computers interconnected via a high speed communication system ( Gigabit Ethernet is common). A Linux Beowulf cluster is an example of a loosely coupled system.
Tightly coupled systems perform better and are physically smaller than loosely coupled systems, but have historically required greater initial investments and may depreciate rapidly; nodes in a loosely coupled system are usually inexpensive commodity computers and can be recycled as independent machines upon retirement from the cluster.
Power consumption is also a consideration. Tightly coupled systems tend to be much more energy-efficient than clusters. This is because a considerable reduction in power consumption can be realized by designing components to work together from the beginning in tightly coupled systems, whereas loosely coupled systems use components that were not necessarily intended specifically for use in such systems.
Loosely coupled systems have the ability to run different operating systems or OS versions on different systems.
Merging data from multiple threads or processes may incur significant overhead due to conflict resolution , data consistency , versioning, and synchronization. [ 13 ] | https://en.wikipedia.org/wiki/Multiprocessing |
Multiprotocol Extensions for BGP ( MBGP or MP-BGP ), sometimes referred to as Multiprotocol BGP or Multicast BGP and defined in IETF RFC 4760, [ 1 ] is an extension to Border Gateway Protocol (BGP) that allows different types of addresses (known as address families) to be distributed in parallel. Whereas standard BGP supports only IPv4 unicast addresses, Multiprotocol BGP supports IPv4 and IPv6 addresses and it supports unicast and multicast variants of each. Multiprotocol BGP allows information about the topology of IP multicast -capable routers to be exchanged separately from the topology of normal IPv4 unicast routers. Thus, it allows a multicast routing topology different from the unicast routing topology. Although MBGP enables the exchange of inter-domain multicast routing information, other protocols such as the Protocol Independent Multicast family are needed to build trees and forward multicast traffic.
As an enhancement of BGP-4, MP-BGP provides routing information for various protocols, such as IPv6 (BGP4+) and multicast:
Multiprotocol BGP is also widely deployed in case of MPLS L3 VPN , to exchange VPN labels learned for the routes from the customer sites over the MPLS network, in order to distinguish between different customer sites when the traffic from the other customer sites comes to the provider edge router (PE router) for routing.
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multiprotocol_BGP |
In quantum chemistry , the multireference configuration interaction ( MRCI ) method consists of a configuration interaction expansion of the eigenstates of the electronic molecular Hamiltonian in a set of Slater determinants which correspond to excitations of the ground state electronic configuration but also of some excited states . The Slater determinants from which the excitations are performed are called reference determinants . The higher excited determinants (also called configuration state functions (CSFs) or shortly configurations) are then chosen either by the program according to some perturbation theoretical ansatz according to a threshold provided by the user or simply by truncating excitations from these references to singly, doubly, ... excitations resulting in MRCIS, MRCISD, etc.
For the ground state using more than one reference configuration means a better correlation and so a lower energy. The problem of size inconsistency of truncated CI-methods is not solved by taking more references.
As a result of a MRCI calculation one gets a more balanced correlation of the ground and excited states . For quantitative good energy differences (excitation energies) one has to be careful in selecting the references. Taking only the dominant configuration of an excited state into the reference space leads to a correlated (lower) energy of the excited state. The generally too-high excitation energies of CIS or CISD are lowered. But usually excited states have more than one dominant configuration and so the ground state is more correlated due to: a) now including some configurations with higher excitations (triply and quadruply in MRCISD); b) the neglect of other dominant configurations of the excited states which are still uncorrelated.
Selecting the references can be done manually ( Φ 1 , Φ 2 , Φ 5 , . . . {\displaystyle \Phi _{1},\Phi _{2},\Phi _{5},...} ), automatically (all possible configurations within an active space of some orbitals) or semiautomatically (taking all configurations as references that have been shown to be important in a previous CI or MRCI calculation)
This method has been implemented first by Robert Buenker and Sigrid D. Peyerimhoff in the seventies under the name Multi-Reference Single and Double Configuration Interaction ( MRSDCI ). [ 1 ] [ 2 ] MRCI was further streamlined in 1988 by Hans-Joachim Werner and Peter Knowles, which made previous MRCI procedures more generalizable. [ 3 ]
The MRCI method can also be implemented in semi-empirical methods. An example for this is the OM2/MRCI method developed by Walter Thiel's group.
This quantum chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multireference_configuration_interaction |
The multiregional hypothesis , multiregional evolution ( MRE ), or polycentric hypothesis , is a scientific model that provides an alternative explanation to the more widely accepted "Out of Africa" model of monogenesis for the pattern of human evolution .
Multiregional evolution holds that the human species first arose around two million years ago and subsequent human evolution has been within a single, continuous human species. This species encompasses all archaic human forms such as Homo erectus , Denisovans , and Neanderthals as well as modern forms, and evolved worldwide to the diverse populations of anatomically modern humans ( Homo sapiens ).
The hypothesis contends that the mechanism of clinal variation through a model of "centre and edge" allowed for the necessary balance between genetic drift , gene flow , and selection throughout the Pleistocene , as well as overall evolution as a global species, but while retaining regional differences in certain morphological features. [ 1 ] Proponents of multiregionalism point to fossil and genomic data and continuity of archaeological cultures as support for their hypothesis.
The multiregional hypothesis was first proposed in 1984, and then revised in 2003. In its revised form, it is similar to the assimilation model, which holds that modern humans originated in Africa and today share a predominant recent African origin, but have also absorbed small, geographically variable, degrees of admixture from other regional ( archaic ) hominin species. [ 2 ]
The multiregional hypothesis is not currently the most accepted theory of modern human origin among scientists. "The African replacement model has gained the widest acceptance owing mainly to genetic data (particularly mitochondrial DNA) from existing populations. This model is consistent with the realization that modern humans cannot be classified into subspecies or races, and it recognizes that all populations of present-day humans share the same potential." [ 3 ] The African replacement model is also known as the "out of Africa" theory, which is currently the most widely accepted model. It proposes that Homo sapiens evolved in Africa before migrating across the world." [ 4 ] And: "The primary competing scientific hypothesis is currently recent African origin of modern humans, which proposes that modern humans arose as a new species in Africa around 100-200,000 years ago, moving out of Africa around 50-60,000 years ago to replace existing human species such as Homo erectus and the Neanderthals without interbreeding. [ 5 ] [ 6 ] [ 7 ] [ 8 ] This differs from the multiregional hypothesis in that the multiregional model predicts interbreeding with preexisting local human populations in any such migration." [ 8 ] [ 9 ]
The Multiregional hypothesis was proposed in 1984 by Milford H. Wolpoff , Alan Thorne and Xinzhi Wu . [ 10 ] [ 11 ] [ 1 ] Wolpoff credits Franz Weidenreich 's "Polycentric" hypothesis of human origins as a major influence, but cautions that this should not be confused with polygenism , or Carleton Coon 's model that minimized gene flow. [ 11 ] [ 12 ] [ 13 ] According to Wolpoff, multiregionalism was misinterpreted by William W. Howells , who confused Weidenreich's hypothesis with a polygenic "candelabra model" in his publications spanning five decades:
How did Multiregional evolution get stigmatized as polygeny? We believe it comes from the confusion of Weidenreich's ideas, and ultimately of our own, with Coon's. The historic reason for linking Coon's and Weidenreich's ideas came from the mischaracterizations of Weidenreich's Polycentric model as a candelabra (Howells, 1942, 1944, 1959, 1993), that made his Polycentric model appear much more similar to Coon's than it actually was. [ 14 ]
Through the influence of Howells, many other anthropologists and biologists have confused multiregionalism with polygenism i.e. separate or multiple origins for different populations. Alan Templeton for example notes that this confusion has led to the error that gene flow between different populations was added to the Multiregional hypothesis as a "special pleading in response to recent difficulties", despite the fact: "parallel evolution was never part of the multiregional model, much less its core, whereas gene flow was not a recent addition , but rather was present in the model from the very beginning" [ 15 ] (emphasis in original). Despite this, multiregionalism is still confused with polygenism, or Coon's model of racial origins, from which Wolpoff and his colleagues have distanced themselves. [ 16 ] [ 17 ] Wolpoff has also defended Wiedenreich's Polycentric hypothesis from being labeled polyphyletic. Weidenreich himself in 1949 wrote: "I may run the risk of being misunderstood, namely that I believe in polyphyletic evolution of man". [ 18 ]
In 1998, Wu founded a China-specific Multiregional model called "Continuity with [Incidental] Hybridization". [ 19 ] [ 20 ] Wu's variant only applies the Multiregional hypothesis to the East Asian fossil record, and is popular among Chinese scientists. [ 21 ] However, James Leibold, a political historian of modern China, has argued the support for Wu's model is largely rooted in Chinese nationalism . [ 22 ] Outside of China, the Multiregional hypothesis has limited support, held only by a small number of paleoanthropologists. [ 23 ]
Chris Stringer , a leading proponent of the more mainstream recent African origin theory , debated Multiregionalists such as Wolpoff and Thorne in a series of publications throughout the late 1980s and 1990s. [ 24 ] [ 25 ] [ 26 ] [ 27 ] Stringer describes how he considers the original Multiregional hypothesis to have been modified over time into a weaker variant that now allows a much greater role for Africa in human evolution, including anatomical modernity (and subsequently less regional continuity than was first proposed). [ 28 ]
Stringer distinguishes the original or "classic" Multiregional model as having existed from 1984 (its formulation) until 2003, to a "weak" post-2003 variant that has "shifted close to that of the Assimilation Model". [ 29 ] [ 30 ]
The finding that " Mitochondrial Eve " was relatively recent and African seemed to give the upper hand to the proponents of the Out of Africa hypothesis. But in 2002, Alan Templeton published a genetic analysis involving other loci in the genome as well, and this showed that some variants that are present in modern populations existed already in Asia hundreds of thousands of years ago. [ 31 ] This meant that even if our male line ( Y chromosome ) and our female line ( mitochondrial DNA ) came out of Africa in the last 100,000 years or so, we have inherited other genes from populations that were already outside of Africa. Since this study other studies have been done using much more data (see Phylogeography ).
Proponents of the multiregional hypothesis see regional continuity of certain morphological traits spanning the Pleistocene in different regions across the globe as evidence against a single replacement model from Africa. In general, three major regions are recognized: Europe , China , and Indonesia (often including Australia ). [ 32 ] [ 33 ] [ 34 ] Wolpoff cautions that the continuity in certain skeletal features in these regions should not be seen in a racial context, instead calling them morphological clades ; defined as sets of traits that "uniquely characterise a geographic region". [ 35 ] According to Wolpoff and Thorne (1981): "We do not regard a morphological clade as a unique lineage, nor do we believe it necessary to imply a particular
taxonomic status for it". [ 36 ] Critics of multiregionalism have pointed out that no single human trait is unique to a geographical region (i.e. confined to one population and not found in any other) but Wolpoff et al. (2000) note that regional continuity only recognizes combinations of features, not traits if individually accessed, a point they elsewhere compare to the forensic identification of a human skeleton:
Regional continuity ... is not the claim that such features do not appear elsewhere; the genetic structure of the human species makes such a possibility unlikely to the extreme. There may be uniqueness in combinations of traits, but no single trait is likely to have been unique in a particular part of the world although it might appear to be so because of the incomplete sampling provided by the spotty human fossil record.
Combinations of features are "unique" in the sense of being found in only one region, or more weakly limited to one region at high frequency (very rarely in another). Wolpoff stresses that regional continuity works in conjunction with genetic exchanges between populations. Long-term regional continuity in certain morphological traits is explained by Alan Thorne 's "centre and edge" [ 37 ] population genetics model which resolves Weidenreich's paradox of "how did populations retain geographical distinctions and yet evolve together?". For example, in 2001 Wolpoff and colleagues published an analysis of character traits of the skulls of early modern human fossils in Australia and central Europe. They concluded that the diversity of these recent humans could not "result exclusively from a single late Pleistocene dispersal", and implied dual ancestry for each region, involving interbreeding with Africans. [ 38 ]
Thorne held that there was regional continuity in Indonesia and Australia for a morphological clade. [ 39 ] [ 40 ] This sequence is said to consist of the earliest fossils from Sangiran , Java, that can be traced through Ngandong and found in prehistoric and recent Aboriginal Australians . In 1991, Andrew Kramer tested 17 proposed morphological clade features. He found that: "a plurality (eight) of the seventeen non-metric features link Sangiran to modern Australians" and that these "are suggestive of morphological continuity, which implies the presence of a genetic continuum in Australasia dating back at least one million years" [ 41 ] but Colin Groves has criticized Kramer's methodology, pointing out that the polarity of characters was not tested and that the study is actually inconclusive. [ 42 ] Phillip Habgood discovered that the characters said to be unique to the Australasian region by Thorne are plesiomorphic :
...it is evident that all of the characters proposed... to be 'clade features' linking Indonesian Homo erectus material with Australian Aboriginal crania are retained primitive features present on Homo erectus and archaic Homo sapiens crania in general. Many are also commonly found on the crania and mandibles of anatomically-modern Homo sapiens from other geographical locations, being especially prevalent on the robust Mesolithic skeletal material from North Africa." [ 43 ]
Yet, regardless of these criticisms Habgood (2003) allows for limited regional continuity in Indonesia and Australia, recognizing four plesiomorphic features which do not appear in such a unique combination on fossils in any other region: a sagittally flat frontal bone, with a posterior position of minimum frontal breadth, great facial prognathism, and zygomaxillary tuberosities. [ 44 ] This combination, Habgood says, has a "certain Australianness about it".
Wolpoff, initially skeptical of Thorne's claims, became convinced when reconstructing the Sangiran 17 Homo erectus skull from Indonesia, when he was surprised that the skull's face to vault angle matched that of the Australian modern human Kow Swamp 1 skull in excessive prognathism. Durband (2007) in contrast states that "features cited as showing continuity between Sangiran 17 and the Kow Swamp sample disappeared in the new, more orthognathic reconstruction of that fossil that was recently completed". [ 45 ] Baba et al. who newly restored the face of Sangiran 17 concluded: "regional continuity in Australasia is far less evident than Thorne and Wolpoff argued". [ 46 ]
Xinzhi Wu has argued for a morphological clade in China spanning the Pleistocene, characterized by a combination of 10 features. [ 47 ] [ 48 ] The sequence is said to start with Lantian and Peking Man , traced to Dali , to Late Pleistocene specimens (e.g. Liujiang) and recent Chinese. Habgood in 1992 criticized Wu's list, pointing out that most of the 10 features in combination appear regularly on fossils outside China. [ 49 ] He did though note that three combined: a non-depressed nasal root, non-projecting perpendicularly oriented nasal bones and facial flatness are unique to the Chinese region in the fossil record and may be evidence for limited regional continuity. However, according to Chris Stringer , Habgood's study suffered from not including enough fossil samples from North Africa, many of which exhibit the small combination he considered to be region-specific to China. [ 27 ]
Facial flatness as a morphological clade feature has been rejected by many anthropologists since it is found on many early African Homo erectus fossils, and is therefore considered plesiomorphic, [ 50 ] but Wu has responded that the form of facial flatness in the Chinese fossil record appears distinct to other (i.e. primitive) forms. Toetik Koesbardiati in her PhD thesis "On the Relevance of the Regional Continuity Features of the Face in East Asia" also found that a form of facial flatness is unique to China (i.e. only appears there at high frequency, very rarely elsewhere) but cautions that this is the only available evidence for regional continuity: "Only two features appear to show a tendency as suggested by the Multiregional model: flatness at the upper face expressed by an obtuse nasio-frontal angle and flatness at the middle part of the face expressed by an obtuse zygomaxillay angle".
Shovel-shaped incisors are commonly cited as evidence for regional continuity in China. [ 51 ] [ 52 ] Stringer (1992) however found that shovel-shaped incisors are present on >70% of the early Holocene Wadi Halfa fossil sample from North Africa, and common elsewhere. [ 53 ] Frayer, et al. (1993) have criticized Stringer's method of scoring shovel-shaped incisor teeth. They discuss the fact that there are different degrees of "shovelled" e.g. trace (+), semi (++), and marked (+++), but that Stringer misleadingly lumped all these together: "...combining shoveling categories in this manner is biologically meaningless and misleading, as the statistic cannot be validly compared with the very high frequencies for the marked shoveling category reported for East Asians." [ 33 ] Palaeoanthropologist Fred H. Smith (2009) also emphasizes that: "It is the pattern of shoveling that identities as an East Asian regional feature, not just the occurrence of shoveling of any sort". [ 2 ] Multiregionalists argue that marked (+++) shovel-shaped incisors only appear in China at a high frequency, and have <10% occurrence elsewhere.
Since the early 1990s, David W. Frayer has described what he regards as a morphological clade in Europe. [ 54 ] [ 55 ] [ 56 ] The sequence starts with the earliest dated Neanderthal specimens ( Krapina and Saccopastore skulls ) traced through the mid- Late Pleistocene (e.g. La Ferrassie 1 ) to Vindija Cave , and late Upper Palaeolithic Cro-Magnons or recent Europeans. Although many anthropologists consider Neanderthals and Cro Magnons morphologically distinct, [ 57 ] [ 58 ] Frayer maintains quite the opposite and points to their similarities, which he argues is evidence for regional continuity:
"Contrary to Brauer's recent pronouncement that there is a large and generally recognized morphological gap between the Neanderthals and the early moderns, the actual evidence provided by the extensive fossil record of late Pleistocene Europe shows considerable continuity between Neanderthals and subsequent Europeans." [ 33 ]
Frayer et al. (1993) consider there to be at least four features in combination that are unique to the European fossil record: a horizontal-oval shaped mandibular foramen , anterior mastoid tubercle, suprainiac fossa , and narrowing of the nasal breadth associated with tooth-size reduction. Regarding the latter, Frayer observes a sequence of nasal narrowing in Neanderthals, following through to late Upper Palaeolithic and Holocene (Mesolithic) crania. His claims are disputed by others, [ 59 ] but have received support from Wolpoff, who regards late Neanderthal specimens to be "transitional" in nasal form between earlier Neanderthals and later Cro Magnons. [ 60 ] Based on other cranial similarities, Wolpoff et al. (2004) argue for a sizable Neanderthal contribution to modern Europeans. [ 61 ]
More recent claims regarding continuity in skeletal morphology in Europe focus on fossils with both Neanderthal and modern anatomical traits, to provide evidence of interbreeding rather than replacement. [ 62 ] [ 63 ] [ 64 ] Examples include the Lapedo child found in Portugal [ 65 ] and the Oase 1 mandible from Peștera cu Oase , Romania, [ 66 ] though the "Lapedo child" is disputed by some. [ 67 ]
A 1987 analysis of mitochondrial DNA from 147 people by Cann et al. from around the world indicated that their mitochondrial lineages all coalesced in a common ancestor from Africa between 140,000 and 290,000 years ago. [ 68 ] The analysis suggested that this reflected the worldwide expansion of modern humans as a new species, replacing, rather than mixing with, local archaic humans outside of Africa. Such a recent replacement scenario is not compatible with the Multiregional hypothesis and the mtDNA results led to increased popularity for the alternative single replacement theory . [ 69 ] [ 70 ] [ 71 ] According to Wolpoff and colleagues: [ 72 ]
When they were first published, the Mitochondrial Eve results were clearly incongruous with Multiregional evolution, and we wondered how the two could be reconciled.
Multiregionalists have responded to what they see as flaws in the Eve theory, [ 73 ] and have offered contrary genetic evidences. [ 74 ] [ 75 ] [ 76 ] Wu and Thorne have questioned the reliability of the molecular clock used to date Eve. [ 77 ] [ 78 ] Multiregionalists point out that Mitochondrial DNA alone can not rule out interbreeding between early modern and archaic humans, since archaic human mitochondrial strains from such interbreeding could have been lost due to genetic drift or a selective sweep . [ 79 ] [ 80 ] Wolpoff for example states that Eve is "not the most recent common ancestor of all living people" since "Mitochondrial history is not population history". [ 81 ]
Neanderthal mitochondrial DNA ( mtDNA ) sequences from Feldhofer and Vindija Cave are substantially different from modern human mtDNA. [ 82 ] [ 83 ] [ 84 ] Multiregionalists however have discussed the fact that the average difference between the Feldhofer sequence and living humans is less than that found between chimpanzee subspecies, [ 85 ] [ 86 ] and therefore that while Neanderthals were different subspecies , they were still human and part of the same lineage.
Initial analysis of Y chromosome DNA, which like mitochondrial DNA, is inherited from only one parent, was consistent with a recent African replacement model. However, the mitochondrial and Y chromosome data could not be explained by the same modern human expansion out of Africa; the Y chromosome expansion would have involved genetic mixing that retained regionally local mitochondrial lines. In addition, the Y chromosome data indicated a later expansion back into Africa from Asia, demonstrating that gene flow between regions was not unidirectional. [ 87 ]
An early analysis of 15 noncoding sites on the X chromosome found additional inconsistencies with the recent African replacement hypothesis. The analysis found a multimodal distribution of coalescence times to the most recent common ancestor for those sites, contrary to the predictions for recent African replacement; in particular, there were more coalescence times near 2 million years ago ( mya ) than expected, suggesting an ancient population split around the time humans first emerged from Africa as Homo erectus , rather than more recently as suggested by the mitochondrial data. While most of these X chromosome sites showed greater diversity in Africa, consistent with African origins, a few of the sites showed greater diversity in Asia rather than Africa. For four of the 15 gene sites that did show greater diversity in Africa, the sites' varying diversity by region could not be explained by simple expansion from Africa, as would be required by the recent African replacement hypothesis. [ 88 ]
Later analyses of X chromosome and autosomal DNA continued to find sites with deep coalescence times inconsistent with a single origin of modern humans, [ 89 ] [ 90 ] [ 91 ] [ 92 ] [ 93 ] diversity patterns inconsistent with a recent expansion from Africa, [ 94 ] or both. [ 95 ] [ 96 ] For example, analyses of a region of RRM2P4 ( ribonucleotide reductase M2 subunit pseudogene 4) showed a coalescence time of about 2 Mya, with a clear root in Asia, [ 97 ] [ 98 ] while the MAPT locus at 17q 21.31 is split into two deep genetic lineages, one of which is common in and largely confined to the present European population, suggesting inheritance from Neanderthals. [ 99 ] [ 100 ] [ 101 ] [ 102 ] In the case of the Microcephalin D allele, evidence for rapid recent expansion indicated introgression from an archaic population. [ 103 ] [ 104 ] [ 105 ] [ 106 ] However, later analysis, including of the genomes of Neanderthals, did not find the Microcephalin D allele (in the proposed archaic species), nor evidence that it had introgressed from an archaic lineage as previously suggested. [ 107 ] [ 108 ] [ 109 ]
In 2001, a DNA study of more than 12,000 men from 163 East Asian regions showed that all of them carry a mutation that originated in Africa about 35,000 to 89,000 years ago and these "data do not support even a minimal in situ hominid contribution in the origin of anatomically modern humans in East Asia". [ 110 ]
In a 2005 review and analysis of the genetic lineages of 25 chromosomal regions, Alan Templeton found evidence of more than 34 occurrences of gene flow between Africa and Eurasia. Of these occurrences, 19 were associated with continuous restricted gene exchange through at least 1.46 million years ago; only 5 were associated with a recent expansion from Africa to Eurasia. Three were associated with the original expansion of Homo erectus out of Africa around 2 million years ago, 7 with an intermediate expansion out of Africa at a date consistent with the expansion of Acheulean tool technology, and a few others with other gene flows such as an expansion out of Eurasia and back into Africa subsequent to the most recent expansion out of Africa. Templeton rejected a hypothesis of complete recent African replacement with greater than 99% certainty ( p < 10 −17 ). [ 111 ]
Recent analyses of DNA taken directly from Neanderthal specimens indicates that they or their ancestors contributed to the genome of all humans outside of Africa, indicating there was some degree of interbreeding with Neanderthals before their replacement. [ 112 ] It has also been shown that Denisova hominins contributed to the DNA of Melanesians and Australians through interbreeding. [ 113 ]
By 2006, extraction of DNA directly from some archaic human samples was becoming possible. The earliest analyses were of Neanderthal DNA, and indicated that the Neanderthal contribution to modern human genetic diversity was no more than 20%, with a most likely value of 0%. [ 114 ] By 2010, however, detailed DNA sequencing of the Neanderthal specimens from Europe indicated that the contribution was nonzero, with Neanderthals sharing 1-4% more genetic variants with living non-Africans than with living humans in sub-Saharan Africa. [ 115 ] [ 116 ] In late 2010, a recently discovered non-Neanderthal archaic human, the Denisova hominin from south-western Siberia, was found to share 4–6% more of its genome with living Melanesian humans than with any other living group, supporting admixture between two regions outside of Africa. [ 117 ] [ 118 ] In August 2011, human leukocyte antigen (HLA) alleles from the archaic Denisovan and Neanderthal genomes were found to show patterns in the modern human population demonstrating origins from these non-African populations; the ancestry from these archaic alleles at the HLA-A site was more than 50% for modern Europeans, 70% for Asians, and 95% for Papua New Guineans. [ 119 ] Proponents of the multiregional hypothesis believe the combination of regional continuity inside and outside of Africa and lateral gene transfer between various regions around the world supports the multiregional hypothesis. However, "Out of Africa" Theory proponents also explain this with the fact that genetic changes occur on a regional basis rather than a continental basis, and populations close to each other are likely to share certain specific regional SNPs while sharing most other genes in common. [ 120 ] [ 121 ] Migration Matrix theory (A=Mt) indicates that dependent upon the potential contribution of Neanderthal ancestry, we would be able to calculate the percentage of Neanderthal mtDNA contribution to the human species. As we do not know the specific migration matrix, we are unable to input the exact data, which would answer these questions irrefutably. [ 85 ] | https://en.wikipedia.org/wiki/Multiregional_origin_of_modern_humans |
Multiscale Electrophysiology Format ( MEF ) was developed to handle the large amounts of data produced by large-scale electrophysiology in human and animal subjects. MEF can store any time series data up to 24 bits in length, and employs lossless range encoded difference compression . Subject identifying information in the file header can be encrypted using 128-bit AES encryption in order to comply with HIPAA requirements for patient privacy when transmitting data across an open network.
Compressed data is stored in independent blocks to allow direct access to the data, facilitate parallel processing and limit the effects of potential damage to files. Data fidelity is ensured by a 32-bit cyclic redundancy check in each compressed data block using the Koopman polynomial (0xEB31D82E), which has a Hamming distance of from 4 to 114 kbits.
A formal specification [ 1 ] and source code [ 2 ] are available online. MEF_import [ 3 ] is an EEGLAB plugin to import MEF data into EEGLAB. | https://en.wikipedia.org/wiki/Multiscale_Electrophysiology_Format |
Multiscale decision-making , also referred to as multiscale decision theory ( MSDT ), is an approach in operations research that combines game theory , multi-agent influence diagrams , in particular dependency graphs , and Markov decision processes to solve multiscale challenges [ 1 ] in sociotechnical systems . MSDT considers interdependencies within and between the following scales: system level, time and information.
Multiscale decision theory builds upon decision theory and multiscale mathematics . Multiscale decision theory can model and analyze complex decision-making networks that exhibit multiscale phenomena. The theory's results can be used by mechanism designers and decision-makers in organizations and complex systems to improve system performance and decision quality.
Multiscale decision theory has been applied to manufacturing enterprise enterprises, [ 2 ] [ 3 ] service systems , [ 4 ] supply chain management , [ 5 ] healthcare , [ 6 ] systems engineering , [ 7 ] among others. In healthcare , for example, MSDT has been used to identify multi-level incentives that can improve healthcare value (quality of outcomes per dollar spent). The Multiscale Decision Making Laboratory at Virginia Tech directed by Dr. Christian Wernz is working at the forefront of MSDT theory and applications.
Multiscale decision theory is related to: | https://en.wikipedia.org/wiki/Multiscale_decision-making |
Multiscale modeling or multiscale mathematics is the field of solving problems that have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, [ 1 ] [ 2 ] [ 3 ] solids, [ 2 ] [ 4 ] polymers, [ 5 ] [ 6 ] proteins, [ 7 ] [ 8 ] [ 9 ] [ 10 ] nucleic acids [ 11 ] as well as various physical and chemical phenomena (like adsorption, chemical reactions, diffusion ). [ 9 ] [ 12 ] [ 13 ] [ 14 ]
An example of such problems involve the Navier–Stokes equations for incompressible fluid flow.
ρ 0 ( ∂ t u + ( u ⋅ ∇ ) u ) = ∇ ⋅ τ , ∇ ⋅ u = 0. {\displaystyle {\begin{array}{lcl}\rho _{0}(\partial _{t}\mathbf {u} +(\mathbf {u} \cdot \nabla )\mathbf {u} )=\nabla \cdot \tau ,\\\nabla \cdot \mathbf {u} =0.\end{array}}}
In a wide variety of applications, the stress tensor τ {\displaystyle \tau } is given as a linear function of the gradient ∇ u {\displaystyle \nabla u} . Such a choice for τ {\displaystyle \tau } has been proven to be sufficient for describing the dynamics of a broad range of fluids. However, its use for more complex fluids such as polymers is dubious. In such a case, it may be necessary to use multiscale modeling to accurately model the system such that the stress tensor can be extracted without requiring the computational cost of a full microscale simulation. [ 15 ]
Horstemeyer 2009, [ 16 ] 2012 [ 17 ] presented a historical review of the different disciplines (mathematics, physics, and materials science) for solid materials related to multiscale materials modeling.
The recent surge of multiscale modeling from the smallest scale (atoms) to full system level (e.g., autos) related to solid mechanics that has now grown into an international multidisciplinary activity was birthed from an unlikely source. Since the US Department of Energy (DOE) national labs started to reduce nuclear underground tests in the mid-1980s, with the last one in 1992, the idea of simulation-based design and analysis concepts were birthed. Multiscale modeling was a key in garnering more precise and accurate predictive tools. In essence, the number of large-scale systems level tests that were previously used to validate a design was reduced to nothing, thus warranting the increase in simulation results of the complex systems for design verification and validation purposes.
Essentially, the idea of filling the space of system-level “tests” was then proposed to be filled by simulation results. After the Comprehensive Test Ban Treaty of 1996 in which many countries pledged to discontinue all systems-level nuclear testing, programs like the Advanced Strategic Computing Initiative (ASCI) were birthed within the Department of Energy (DOE) and managed by the national labs within the US. Within ASCI, the basic recognized premise was to provide more accurate and precise simulation-based design and analysis tools. Because of the requirements for greater complexity in the simulations, parallel computing and multiscale modeling became the major challenges that needed to be addressed. With this perspective, the idea of experiments shifted from the large-scale complex tests to multiscale experiments that provided material models with validation at different length scales. If the modeling and simulations were physically based and less empirical, then a predictive capability could be realized for other conditions. As such, various multiscale modeling methodologies were independently being created at the DOE national labs: Los Alamos National Lab (LANL), Lawrence Livermore National Laboratory (LLNL), Sandia National Laboratories (SNL), and Oak Ridge National Laboratory (ORNL). In addition, personnel from these national labs encouraged, funded, and managed academic research related to multiscale modeling. Hence, the creation of different methodologies and computational algorithms for parallel environments gave rise to different emphases regarding multiscale modeling and the associated multiscale experiments.
The advent of parallel computing also contributed to the development of multiscale modeling. Since more degrees of freedom could be resolved by parallel computing environments, more accurate and precise algorithmic formulations could be admitted. This thought also drove the political leaders to encourage the simulation-based design concepts.
At LANL, LLNL, and ORNL, the multiscale modeling efforts were driven from the materials science and physics communities with a bottom-up approach. Each had different programs that tried to unify computational efforts, materials science information, and applied mechanics algorithms with different levels of success. Multiple scientific articles were written, and the multiscale activities took different lives of their own. At SNL, the multiscale modeling effort was an engineering top-down approach starting from continuum mechanics perspective, which was already rich with a computational paradigm. SNL tried to merge the materials science community into the continuum mechanics community to address the lower-length scale issues that could help solve engineering problems in practice.
Once this management infrastructure and associated funding was in place at the various DOE institutions, different academic research projects started, initiating various satellite networks of multiscale modeling research. Technological transfer also arose into other labs within the Department of Defense and industrial research communities.
The growth of multiscale modeling in the industrial sector was primarily due to financial motivations. From the DOE national labs perspective, the shift from large-scale systems experiments mentality occurred because of the 1996 Nuclear Ban Treaty. Once industry realized that the notions of multiscale modeling and simulation-based design were invariant to the type of product and that effective multiscale simulations could in fact lead to design optimization, a paradigm shift began to occur, in various measures within different industries, as cost savings and accuracy in product warranty estimates were rationalized.
The aforementioned DOE multiscale modeling efforts were hierarchical in nature. The first concurrent multiscale model occurred when Michael Ortiz (Caltech) took the molecular dynamics code Dynamo, developed by Mike Baskes at Sandia National Labs, and with his students embedded it into a finite element code for the first time. [ 18 ] Martin Karplus , Michael Levitt , and Arieh Warshel received the Nobel Prize in Chemistry in 2013 for the development of a multiscale model method using both classical and quantum mechanical theory which were used to model large complex chemical systems and reactions. [ 8 ] [ 9 ] [ 10 ]
In physics and chemistry, multiscale modeling is aimed at the calculation of material properties or system behavior on one level using information or models from different levels. On each level, particular approaches are used for the description of a system. The following levels are usually distinguished: level of quantum mechanical models (information about electrons is included), level of molecular dynamics models (information about individual atoms is included), coarse-grained models (information about atoms and/or groups of atoms is included), mesoscale or nano-level (information about large groups of atoms and/or molecule positions is included), level of continuum models, level of device models. Each level addresses a phenomenon over a specific window of length and time. Multiscale modeling is particularly important in integrated computational materials engineering since it allows the prediction of material properties or system behavior based on knowledge of the process-structure-property relationships. [ citation needed ]
In operations research , multiscale modeling addresses challenges for decision-makers that come from multiscale phenomena across organizational, temporal, and spatial scales. This theory fuses decision theory and multiscale mathematics and is referred to as multiscale decision-making . Multiscale decision-making draws upon the analogies between physical systems and complex man-made systems. [ citation needed ]
In meteorology, multiscale modeling is the modeling of the interaction between weather systems of different spatial and temporal scales that produces the weather that we experience. The most challenging task is to model the way through which the weather systems interact as models cannot see beyond the limit of the model grid size. In other words, to run an atmospheric model that is having a grid size (very small ~ 500 m ) which can see each possible cloud structure for the whole globe is computationally very expensive. On the other hand, a computationally feasible Global climate model (GCM), with grid size ~ 100 km , cannot see the smaller cloud systems. So we need to come to a balance point so that the model becomes computationally feasible and at the same time we do not lose much information, with the help of making some rational guesses, a process called parametrization. [ citation needed ]
Besides the many specific applications, one area of research is methods for the accurate and efficient solution of multiscale modeling problems. The primary areas of mathematical and algorithmic development include: | https://en.wikipedia.org/wiki/Multiscale_modeling |
Multiscale tomography (or multi-length scale tomography) is a form of tomography spanning large orders of magnitude in resolution, often utilizing many different forms of tomography together to do so. [ 1 ] The forms of tomography combined in the process depend on what is being studied and the details needed. Each form of tomography has an optimal range of optical resolution that it can function across, but many modern materials and applications need information beyond the range of a single form of tomography. Combining this information using many forms of tomography can help provide a holistic view of the system being looked at, and is important for computer simulations .
This article related to medical imaging is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multiscale_tomography |
Multiscale turbulence is a class of turbulent flows in which the chaotic motion of the fluid is forced at different length and/or time scales. [ 1 ] [ 2 ] This is usually achieved by immersing in a moving fluid a body with a multiscale, often fractal -like, arrangement of length scales. This arrangement of scales can be either passive [ 3 ] [ 4 ] or active [ 5 ]
As turbulent flows contain eddies with a wide range of scales, exciting the turbulence at particular scales (or range of scales) allows one to fine-tune the properties of that flow. Multiscale turbulent flows have been successfully applied in different fields., [ 6 ] such as:
Multiscale turbulence has also played an important role into probing the internal structure of turbulence . [ 15 ] This sort of turbulence allowed researchers to unveil a novel dissipation law in which the parameter C ϵ {\displaystyle C_{\epsilon }} in
is not constant, as required by the Richardson - Kolmogorov energy cascade . This new law [ 15 ] can be expressed as C ϵ ∝ R e I m R e L n {\displaystyle C_{\epsilon }\propto {\frac {Re_{I}^{m}}{Re_{L}^{n}}}} , with m ≈ 1 ≈ n {\displaystyle m\approx 1\approx n} , where R e I {\displaystyle Re_{I}} and R e L {\displaystyle Re_{L}} are Reynolds numbers based, respectively, on initial/global conditions (such as free-stream velocity and the object's length scale ) and local conditions (such as the rms velocity and integral length scale). This new dissipation law characterises non-equilibrium turbulence apparently universally in various flows (not just multiscale turbulence) and results from non-equilibrium unsteady energy cascade . This imbalance implies that new mean flow scalings exist for free shear turbulent flows, as already observed in axisymmetric wakes [ 15 ] [ 16 ] | https://en.wikipedia.org/wiki/Multiscale_turbulence |
A multiseat , multi-station or multiterminal system is a single computer which supports multiple independent local users at the same time.
A "seat" consists of all hardware devices assigned to a specific workplace at which one user sits at and interacts with the computer. It consists of at least one graphics device (graphics card or just an output (e.g. HDMI / VGA / DisplayPort port) and the attached monitor/video projector) for the output and a keyboard and a mouse for the input. It can also include video cameras, sound cards and more.
Since the 1960s computers have been shared between users. Especially in the early days of computing when computers were extremely expensive the usual paradigm was a central mainframe computer connected to numerous terminals. With the advent of personal computing this paradigm has been largely replaced by personal computers (or one computer per user).
Multiseat setups are a return to this multiuser paradigm but based around a PC which supports a number of zero-clients usually consisting of a terminal per user (screen, keyboard, mouse).
In some situations a multiseat setup is more cost-effective because it is not necessary to buy separate motherboards , microprocessors, RAM, hard disks and other components for each user. For example, buying one high speed CPU , usually costs less than buying several slower CPUs.
In the 1970s, it was very commonplace to connect multiple computer terminals to a single mainframe computer , even graphical terminals. Early terminals were connected with RS-232 type serial connections , either directly, or through modems . With the advent of Internet Protocol based networking , it became possible for multiple users to log into a host using telnet or – for a graphic environment – an X Window System "server". These systems would retain a physically secure " root console " for system administration and direct access to the host machine.
Support for multiple consoles in a PC running the X interface was implemented in 2001 by Miguel Freitas, using the Linux operating system and the X11 graphical system (at the time maintained by XFree86 ). [ 1 ] This was done using a patch in the display server to execute several instances of X at the same time such that each one captures specific mouse and keyboard events and the graphical content. This method received the name of multiseat or multiterminal.
In 2001, Thinsoft BeTwin offered a multiseat solution for Windows, utilizing multiple graphics cards and peripherals attached to a single host PC. [ 2 ]
In 2002 a Canadian company, Userful Corporation, released Userful Multiplier, a multiseat Linux software solution that enables up to 10 users to simultaneously share one computer. [ 3 ] Earlier they worked on a kernel-based approach to a multi-station platform computer, but abandoned the idea due to a problem with multiple video card support.
Other solutions appeared in 2003, such Svetoslav Slavtchev, Aivils Stoss and James Simmons worked, with the evdev and Faketty [ 4 ] [ 5 ] approach modifying the Linux kernel and letting more than one user independently use the same machine. In that time, the Linux Console Project [ 6 ] also proposed an idea to use multiple independent consoles and then multiple independent keyboards and mice in a project called "Backstreet Ruby". [ 7 ] Backstreet Ruby is a kernel patch for the Linux kernel. It is a back port to Linux-2.4 of the Ruby kernel tree. The aim of the Linux Console developers is to enhance and reorganize the input, the console and the framebuffer subsystems in the Linux kernel, so they can work independent from each other and to allow multi-desktop operation. The Backstreet Ruby idea was never finished.
In 2005, the C3SL team (Center for Scientific Computing and Free Software), [ 8 ] from the Federal University of Parana in Brazil, created a solution based on nested display servers, such as Xnest and Xephyr . [ 9 ] With this solution, each nested display server runs in each screen of a host display server (e.g. Xorg ) and a modification to the nested servers let each one exclusively acquire its mouse and keyboard. In 2008, the C3SL group released the Multiseat Display Manager (MDM) [ 10 ] to ease the process of installation and configuration of a multiseat box. This group, also in 2008, conceived a live-CD [ 11 ] for test purposes.
In 2007, NComputing entered the market with a Windows-based multiseat product, the X-series [ 12 ] or Xtenda system, which uses a PCI add-in card to connect terminal units containing video, keyboard, mouse, and audio jacks, allowing 3 to 6 additional user seats to be added to a PC. [ 13 ] The X-series also offered Linux compatibility. [ 14 ]
In 2010, Microsoft began offering Windows MultiPoint Server , allowing one machine to host multiple users utilizing separate graphics cards and peripherals.
Automatic multiseat with USB docking stations is a feature of Fedora 17 . [ 15 ] [ 16 ]
Each user will require a monitor , keyboard and mouse connected to the host machine. For example, to make a four-head (four users) system would require four monitors, four keyboards , four mice and two dual-output, or one quad-output video card. USB keyboards and mice are typically recommended instead of PS/2 connections, as they can be connected to a USB hub . Additional devices and peripherals such as cameras, flash storage drives, card readers and touch screens could also be assigned to each seat. An alternative to multiple physical video cards and connections is DisplayLink over USB.
Multiseat on modern Linux systems is provided by systemd-logind [ 19 ] and configured through the loginctl command [ 20 ] or through ID_SEAT or ID_AUTOSEAT udev variables. [ 21 ]
Certain specialized USB hubs, when connected, automatically results in a seat without any configuration required. [ 22 ]
For Windows 2000 , XP and Vista operating systems, there are several commercial products to implement multiseat configurations for two or more seats.
An operating system designed specifically for multiseat setups entitled Windows MultiPoint Server was announced on February 24, 2010. It uses Remote Desktop (Terminal Services) technologies in Windows Server 2008 R2 to provide multiseat functionality. This functionality was incorporated into Windows Server proper as of Windows Server 2016 in a new server role entitled MultiPoint Services, but this server role was removed in Windows Server 2019 owing to Microsoft ceasing development of the service in 2018.
Instead of relying on operating system support for multiseat configuration, a hypervisor can be configured to run multiple virtual machines, each configured to interface one connected seat by I/O virtualization methods. Input devices can be attached to the virtual machines through USB Redirection, and entire GPUs can be attached through Intel VT-d.
The virtualization-based 2-seat [ 23 ] and 7-seat [ 24 ] systems with Unraid as the host operating system has been demonstrated. Each seat has exclusive control of one of the Windows guest operating systems running on the host. There is a dedicated high-end graphics card for each guest, which it takes full advantage of via the use of VT-d, making the system capable of hosting demanding video game sessions at full quality simultaneously on all seats.
In February, 2009, The Brazil Ministry of Education committed to deploy 350,000 Linux-based multiseat computing stations in more than 45,000 rural and urban schools across the country. The chosen companies to implement this project were the Canadian multiseat Linux software company Userful Corporation, and its Brazilian IT partner ThinNetworks. [ 25 ]
One of multiterminal's successful cases is happening at Paraná Digital project. It is creating multiterminal laboratories on 2000 public schools of the state of Paraná ( Brazil ). More than 1.5 million users will benefit from the 40,000 terminals when the project is finished. The laboratories have four-head multiterminals running Debian . The cost of all the hardware is 50% less than the normal price, and there is absolutely no cost with software . This project developer is C3SL ( Center for Scientific Computing and Free Software ).
Since 2008, electrical and computer engineering students from Michigan State University have installed multiterminal systems with internet access in three schools in Mto wa Mbu, Tanzania . The purpose of the project is to study the impact of having computer systems with internet access in an education system that cannot afford other educational resources such as books. The computer systems run Ubuntu 8.04 32-bit and utilize the open source Multiseat Display Manager created by C3SL . The research will eventually be used to present to government officials of third world countries in effort to showcase the positive impact of having cost-effective computing systems in schools. The project is sponsored by George and Vickie Rock and the Dow Chemical Company . [ 26 ] [ 27 ] [ 28 ] | https://en.wikipedia.org/wiki/Multiseat_configuration |
The multislice algorithm [ 1 ] is a method for the simulation of the elastic scattering of an electron beam with matter, including all multiple scattering effects. The method is reviewed in the book by John M. Cowley , [ 2 ] and also the work by Ishizuka. [ 3 ] The algorithm is used in the simulation of high resolution transmission electron microscopy (HREM) micrographs, and serves as a useful tool for analyzing experimental images. [ 4 ] This article describes some relevant background information, the theoretical basis of the technique, approximations used, and several software packages that implement this technique. Some of the advantages and limitations of the technique and important considerations that need to be taken into account are described.
The multislice method has found wide application in electron microscopy and crystallography . The mapping from a crystal structure to its image or electron diffraction pattern is relatively well understood and documented. However, the reverse mapping from electron micrograph images to the crystal structure is generally more complicated. The fact that the images are two-dimensional projections of three-dimensional crystal structure makes it tedious to compare these projections to all plausible crystal structures. Hence, the use of numerical techniques in simulating results for different crystal structure is integral to the field of electron microscopy and crystallography. Several software packages exist to simulate electron micrographs.
There are two widely used simulation techniques that exist in literature: the Bloch wave method, [ 5 ] derived from Hans Bethe 's original theoretical treatment, [ 6 ] and the multislice method. This article focuses on the multislice method for simulation of dynamical diffraction, including multiple elastic scattering effects. Most of the packages that exist implement the multislice algorithm along with Fourier analysis to incorporate electron lens aberration effects to determine electron microscope image and address aspects such as phase contrast and diffraction contrast. For electron microscope samples in the form of a thin crystalline slab in the transmission geometry, the aim of these software packages is to provide a map of the crystal potential, however this inversion process is greatly complicated by the presence of multiple elastic scattering.
The first description of what is now known as the multislice theory was given in the classic paper by Cowley and Moodie. [ 1 ] In this work, the authors describe scattering of electrons using a physical optics approach without invoking quantum mechanical arguments. Many other derivations of these iterative equations have since been given using alternative methods, such as Greens functions, differential equations, scattering matrices or path integral methods, see for instance the book by Lianmao Peng , Sergei Dudarev and Michael Whelan . [ 7 ]
A summary of the development of a computer algorithm from the multislice theory of Cowley and Moodie for numerical computation was reported by Goodman and Moodie. [ 8 ] They also discussed in detail the relationship of the multislice to the other formulations. Specifically, using Zassenhaus's theorem, this paper gives the mathematical path from multislice to 1. Schrödinger equation , 2. Darwin's differential equations, widely used for diffraction contrast Transmission electron microscopy (TEM) image simulations - the Howie -Whelan equations, [ 9 ] 3. Sturkey's scattering matrix method. [ 10 ] 4. the free-space propagation case, 5. The phase grating approximation, 6. A new "thick-phase grating" approximation, which has never been used, 7. Moodie's polynomial expression for multiple scattering, 8. The Feynman path-integral formulation, and 9. relationship of multislice to the Born series . The relationship between algorithms is summarized in Section 5.11 of Spence (2013), [ 11 ] (see Figure 5.9).
The form of multislice algorithm presented here has been adapted from Peng, Dudarev and Whelan 2003. [ 7 ] The multislice algorithm is an approach to solving the Schrödinger equation :
− ℏ 2 2 m ∂ 2 Ψ ( x , t ) ∂ x 2 + V ( x , t ) Ψ ( x , t ) = E Ψ ( x , t ) {\displaystyle {\begin{aligned}-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi (x,t)}{\partial x^{2}}}+V(x,t)\Psi (x,t)&=E\Psi (x,t)\end{aligned}}}
In 1957, Cowley and Moodie showed that the Schrödinger equation can be solved analytically to evaluate the amplitudes of diffracted beams. [ 1 ] Subsequently, the effects of dynamical diffraction can be calculated and the resulting simulated image will exhibit good similarities with the actual image taken from a microscope under dynamical conditions. Furthermore, the multislice algorithm does not make any assumption about the periodicity of the structure and can thus be used to simulate HREM images of aperiodic systems as well.
The following section will include a mathematical formulation of the multislice algorithm. The Schrödinger equation can also be represented in the form of incident and scattered wave as:
Ψ ( r ) = Ψ 0 ( r ) + ∫ G ( r , r ′ ) V ( r ′ ) Ψ ( r ′ ) d r ′ {\displaystyle {\begin{aligned}\Psi ({\mathbf {r} })&=\Psi _{0}({\mathbf {r} })+\int {G({\mathbf {r,r'} })V({\mathbf {r'} })\Psi ({\mathbf {r'} })d{\mathbf {r'} }}\end{aligned}}}
where G ( r , r ′ ) {\displaystyle G(\mathbf {r,r'} )} is the Green's function that represents the amplitude of the electron wave function at a point r {\displaystyle \mathbf {r} } due to a source at point r ′ {\displaystyle \mathbf {r'} } .
Hence for an incident plane wave of the form Ψ ( r ) = exp ( i k ⋅ r ) {\displaystyle \Psi (r)=\exp(i\mathbf {k\cdot r} )} the Schrödinger equation can be written as
We then choose the coordinate axis in such a way that the incident beam hits the sample at (0,0,0) in the z ^ {\displaystyle {\hat {z}}} -direction, i.e., k = ( 0 , 0 , k ) {\textstyle \mathbf {k} =(0,0,k)} . Now we consider a wave-function Ψ ( r ) = ϕ ( r ) exp ( i k ⋅ r ) {\displaystyle \Psi (r)=\phi (\mathbf {r} )\exp(i\mathbf {k\cdot r} )} with a modulation function ϕ ( r ) {\displaystyle \phi ({\mathbf {r} })} for the amplitude. Equation ( 1 ) becomes then an equation for the modulation function, i.e.,
ϕ ( r ) = 1 − m 2 π ℏ 2 ∫ exp [ i k | r − r ′ | − i k ⋅ ( r − r ′ ) ] | r − r ′ | V ( r ′ ) ϕ ( r ′ ) d r ′ {\displaystyle {\begin{aligned}\phi ({\mathbf {r} })&=1-{\frac {m}{2\pi \hbar ^{2}}}\int {{\frac {\exp[ik|{\mathbf {r-r'} }|-i{\mathbf {k} }\cdot ({\mathbf {r-r'} })]}{|{\mathbf {r-r'} }|}}V({\mathbf {r'} )\phi ({\mathbf {r'} })}dr'}\end{aligned}}} .
Now we make substitutions with regards to the coordinate system we have adhered, i.e.,
k ⋅ ( r − r ′ ) = k ( z − z ′ ) | r − r ′ | ≈ ( z − z ′ ) + ( X − X ′ ) 2 / 2 ( z − z ′ ) {\displaystyle {\begin{aligned}{\mathbf {k} }\cdot ({\mathbf {r-r'} })&=k(z-z')\\|{\mathbf {r-r'} }|&\approx (z-z')+({\mathbf {X-X'} })^{2}/{2(z-z')}\end{aligned}}}
where X = ( x y ) {\displaystyle {\boldsymbol {X}}={\begin{pmatrix}x\\y\end{pmatrix}}} .
Thus
ϕ ( r ) = 1 − i π E λ ∫ ∫ z ′ = − ∞ z ′ = z V ( X ′ , z ′ ) ϕ ( X ′ , z ′ ) 1 i λ ( z − z ′ ) exp ( i k | X − X ′ | 2 2 ( z − z ′ ) ) d X ′ d z ′ {\displaystyle {\begin{aligned}\phi ({\mathbf {r} })=1-i{\frac {\pi }{E\lambda }}\int \int \limits _{z'=-\infty }^{z'=z}V({\mathbf {X'} },z')\phi ({\mathbf {X'} },z'){\frac {1}{i\lambda (z-z')}}\exp \left(ik{\frac {|{\mathbf {X-X'} }|^{2}}{2(z-z')}}\right)d{\mathbf {X'} }dz'\end{aligned}}} ,
where λ = 2 π / k {\displaystyle \lambda =2\pi /k} is the wavelength of the electrons with energy E = ℏ 2 k 2 / 2 m {\displaystyle E=\hbar ^{2}k^{2}/{2m}} and σ = π / E λ {\displaystyle {\begin{aligned}\sigma =\pi /E\lambda \end{aligned}}} is the interaction constant. So far we have set up the mathematical formulation of wave mechanics without addressing the scattering in the material. Further we need to address the transverse spread, which is done in terms of the Fresnel propagation function
p ( X , z ) = 1 i z λ exp ( i k X 2 2 z ) {\displaystyle {\begin{aligned}p({\mathbf {X} },z)={\frac {1}{iz\lambda }}\exp \left(ik{\frac {{\mathbf {X} }^{2}}{2z}}\right)\end{aligned}}} .
The thickness of each slice over which the iteration is performed is usually small and as a result within a slice the potential field can be approximated to be constant V ( X ′ , z ) {\displaystyle V({\mathbf {X'} },z)} . Subsequently, the modulation function can be represented as:
ϕ ( X , z n + 1 ) = ∫ p ( X − X ′ , z n + 1 − z n ) ϕ ( X , z n ) exp ( − i σ ∫ z n z n + 1 V ( X ′ , z ′ ) d z ′ ) d X ′ {\displaystyle {\begin{aligned}\phi ({\mathbf {X} },z_{n+1})=\int p({\mathbf {X} }-{\mathbf {X'} },z_{n+1}-z_{n})\phi ({\mathbf {X} },z_{n})\exp \left(-i\sigma \int \limits _{z_{n}}^{z_{n+1}}V({\mathbf {X'} },z')dz'\right)dX'\end{aligned}}}
We can therefore represent the modulation function in the next slice
ϕ n + 1 = ϕ ( X , z n + 1 ) = [ q n ϕ n ] ∗ p n {\displaystyle {\begin{aligned}\phi _{n+1}=\phi ({\mathbf {X} },z_{n+1})=[q_{n}\phi _{n}]*p_{n}\end{aligned}}}
where, * represents convolution, p n = p ( X , z n + 1 − z n ) {\displaystyle p_{n}=p({\mathbf {X} },z_{n+1}-z_{n})} and q n ( X ) {\displaystyle q_{n}({\mathbf {X} })} defines the transmission function of the slice.
q n ( X ) = exp { − i σ ∫ z n z n + 1 V ( X , z ′ ) d z ′ } {\displaystyle {\begin{aligned}q_{n}({\mathbf {X} })=\exp\{-i\sigma \int \limits _{z_{n}}^{z_{n+1}}V({\mathbf {X} },z')dz'\}\end{aligned}}}
Hence, the iterative application of the aforementioned procedure will provide a full interpretation of the sample in context. Further, it should be reiterated that no assumptions have been made on the periodicity of the sample apart from assuming that the potential V ( X , z ) {\displaystyle V(\mathbf {X} ,z)} is uniform within the slice. As a result, it is evident that this method in principle will work for any system. However, for aperiodic systems in which the potential will vary rapidly along the beam direction, the slice thickness has to be significantly small and hence will result in higher computational expense.
The basic premise is to calculate diffraction from each layer of atoms using fast Fourier transforms (FFT) and multiplying each by a phase grating term. The wave is then multiplied by a propagator, inverse Fourier transformed, multiplied by a phase grating term yet again, and the process is repeated. The use of FFTs allows a significant computational advantage over the Bloch Wave method in particular, since the FFT algorithm involves N log N {\displaystyle N\log N} steps compared to the diagonalization problem of the Bloch wave solution which scales as N 2 {\displaystyle N^{2}} where N {\displaystyle N} is the number of atoms in the system. (See Table 1 for comparison of computational time).
The most important step in performing a multislice calculation is setting up the unit cell and determining an appropriate slice thickness. In general, the unit cell used for simulating images will be different from the unit cell that defines the crystal structure of a particular material. The primary reason for this due to aliasing effects which occur due to wraparound errors in FFT calculations. The requirement is to add additional “padding” to the unit cell has earned the nomenclature “super cell” and the requirement to add these additional pixels to the basic unit cell comes at a computational price.
To illustrate the effect of choosing a slice thickness that is too thin, consider a simple example. The Fresnel propagator describes the propagation of electron waves in the z direction (the direction of the incident beam) in a solid:
ϕ ~ ( u , z ) = ϕ ~ ( u , z = 0 ) exp ( π i λ u 2 z ) {\displaystyle {\tilde {\phi }}(\mathbf {u} ,z)={\tilde {\phi }}(\mathbf {u} ,z=0)\exp(\pi i\lambda \mathbf {u} ^{2}z)}
Where u {\displaystyle \mathbf {u} } is the reciprocal lattice coordinate, z is the depth in the sample, and λ {\displaystyle \lambda } is the wavelength of the electron wave (related to the wave vector by the relation k = 2 π / λ {\displaystyle k=2\pi /\lambda } ). In the case of the small-angle approximation ( θ ∼ {\displaystyle \theta \sim } 100 mRad) we can approximate the phase shift as Δ z {\displaystyle \Delta z} . For 100 mRad the error d − S {\displaystyle d-S} is on the order of 0.5% since cos ( 0.1 ) = 0.995 {\displaystyle \cos(0.1)=0.995} . For small angles this approximation holds regardless of how many slices there are, although choosing a Δ z {\displaystyle \Delta z} greater than the lattice parameter (or half the lattice parameter in the case of perovskites ) for a multislice simulation would result in missing atoms that should be in the crystal potential.
Additional practical concerns are how to effectively include effects such as inelastic and diffuse scattering, quantized excitations (e.g. plasmons, phonons, excitons), etc. There was one code that took these things into consideration through a coherence function approach [ 12 ] called Yet Another Multislice (YAMS), but the code is no longer available either for download or purchase.
There are several software packages available to perform multislice simulations of images. Among these is NCEMSS, NUMIS, MacTempas, and Kirkland . Other programs exist but unfortunately many have not been maintained (e.g. SHRLI81 by Mike O’Keefe of Lawrence Berkeley National Lab and Cerius2 of Accerlys). A brief chronology of multislice codes is given in Table 2, although this is by no means exhaustive.
This software is developed by Earl Kirkland of Cornell University. This code is freely available as an interactive Java applet and as standalone code written in C/C++. The Java applet is ideal for a quick introduction and simulations under a basic incoherent linear imaging approximation. The ACEM code accompanies an excellent text of the same name by Kirkland which describes the background theory and computational techniques for simulating electron micrographs (including multislice) in detail. The main C/C++ routines use a command line interface (CLI) for automated batching of many simulation. The ACEM package also includes a graphical user interface that is more appropriate for beginners. The atomic scattering factors in ACEM are accurately characterized by a 12-parameter fit of Gaussians and Lorentzians to relativistic Hartree–Fock calculations.
This package was released from the National Center for High Resolution Electron Microscopy. This program uses a mouse-drive graphical user interface and is written by Roar Kilaas and Mike O’Keefe of Lawrence Berkeley National Laboratory. While the code is no longer developed, the program is available through the Electron Direct Methods (EDM) package written by Laurence D. Marks of Northwestern University. Debye-Waller factors can be included in as a parameter to account for diffuse scattering, although the accuracy is unclear (i.e. a good guess of the Debye-Waller factor is needed).
The Northwestern University Multislice and Imaging System ( NUMIS ) is a package is written by Laurence Marks of Northwestern University. It uses a command line interface (CLI) and is based on UNIX. A structure file must be provided as input in order to run use this code, which makes it ideal for advanced users. The NUMIS multislice programs use the conventional multislice algorithm by calculating the wavefunction of electrons at the bottom of a crystal and simulating the image taking into account various instrument-specific parameters including C s {\displaystyle C_{s}} and convergence. This program is good to use if one already has structure files for a material that have been used in other calculations (for example, Density Functional Theory). These structure files can be used to general X-Ray structure factors which are then used as input for the PTBV routine in NUMIS. Microscope parameters can be changed through the MICROVB routine.
This software is specifically developed to run in Mac OS X by Roar Kilaas of Lawrence Berkeley National Laboratory. It is designed to have a user-friendly user interface and has been well-maintained relative to many other codes (last update May 2013). It is available (for a fee) from here .
This is a software for multislice simulation was written in FORTRAN 77 by J. M. Zuo, while he was a postdoc research fellow at Arizona State University under the guidance of John C. H. Spence . The source code was published in the book of Electron Microdiffraction. [ 13 ] A comparison between multislice and Bloch wave simulations for ZnTe was also published in the book. A separate comparison between several multislice algorithms at the year of 2000 was reported. [ 14 ]
The Quantitative TEM/STEM (QSTEM) simulations software package was written by Christopher Koch of Humboldt University of Berlin in Germany. Allows simulation of HAADF, ADF, ABF-STEM, as well as conventional TEM and CBED. The executable and source code are available as a free download on the Koch group website .
This is a code written by Vincenzo Grillo of the Institute for Nanoscience (CNR) in Italy. This code is essentially a graphical frontend to the multislice code written by Kirkland, with more additional features. These include tools to generate complex crystalline structures, simulate HAADF images and model the STEM probe, as well as modeling of strain in materials. Tools for image analysis (e.g. GPA) and filtering are also available.
The code is updated quite often with new features and a user mailing list is maintained. Freely available on their website .
Multi-slice image simulations for high-resolution scanning and coherent imaging transmission electron microscopy written by Juri Barthel from the Ernst Ruska-Centre at the Jülich Research Centre . The software comprises a graphical user interface version for direct visualization of STEM image calculations, as well as a bundle of command-line modules for more comprehensive calculation tasks. The programs have been written using Visual C++, Fortran 90, and Perl. Executable binaries for Microsoft Windows 32-bit and 64-bit operating systems are available for free from the website .
OpenCL accelerated multislice software written by Adam Dyson and Jonathan Peters from University of Warwick . clTEM is under development as of October 2019.
The code cudaEM is a multi-GPU enabled code based on CUDA for multislice simulations developed by the group of Stephen Pennycook.
An extension of the multislice algorithm is the magnetic multislice method, also referred to as the Pauli multislice method, which enables the simulation of electron microscopy of magnetic materials. [ 15 ] [ 16 ] This method is based on the Pauli equation , incorporating the effects of spin and orbital angular momentum of electrons in elastic scattering. Unlike the conventional Schrödinger equation-based multislice approach, the magnetic multislice method explicitly accounts for the interaction of electron beams with the magnetic fields in a solid.
The multislice method has been extended to model inelastic scattering mechanisms involving phonons [ 17 ] and magnons . [ 18 ]
The frozen phonon multislice method (FPMS) was developed to simulate the impact of thermal vibrations on electron diffraction. [ 17 ] First introduced in the early 1990s, FPMS approximates phonon-induced distortions by averaging over static snapshots of atomic displacements sampled from thermal distributions. [ 17 ] [ 19 ]
Inspired by FPMS, the frozen magnon multislice method (FMMS) was introduced to model the diffuse scattering of electrons due to spin-wave excitations (magnons) . [ 18 ] FMMS follows an analogous approach to FPMS but is implemented within the Pauli multislice framework, allowing for the simulation of inelastic magnetic scattering processes.
Building on FPMS and FMMS, additional theoretical methods have been developed to compute electron energy loss spectroscopy (EELS) and electron energy gain spectroscopy (EEGS) in TEM / STEM : | https://en.wikipedia.org/wiki/Multislice |
Multispecies Coalescent Process is a stochastic process model that describes the genealogical relationships for a sample of DNA sequences taken from several species. [ 1 ] [ 2 ] It represents the application of coalescent theory to the case of multiple species. The multispecies coalescent results in cases where the relationships among species for an individual gene (the gene tree ) can differ from the broader history of the species (the species tree ). It has important implications for the theory and practice of phylogenetics [ 3 ] [ 4 ] and for understanding genome evolution.
A gene tree is a binary graph that describes the evolutionary relationships between a sample of sequences for a non-recombining locus. A species tree describes the evolutionary relationships between a set of species, assuming tree-like evolution. However, several processes can lead to discordance between gene trees and species trees . The Multispecies Coalescent model provides a framework for inferring species phylogenies while accounting for ancestral polymorphism and gene tree-species tree conflict. The process is also called the Censored Coalescent . [ 1 ]
Besides species tree estimation, the multispecies coalescent model also provides a framework for using genomic data to address a number of biological problems, such as estimation of species divergence times, population sizes of ancestral species, species delimitation, and inference of cross-species gene flow. [ 5 ] [ 6 ]
If we consider a rooted three-taxon tree, the simplest non-trivial phylogenetic tree, there are three different tree topologies [ 7 ] but four possible gene trees. [ 8 ] The existence of four distinct gene trees despite the smaller number of topologies reflects the fact that there are topologically identical gene tree that differ in their coalescent times. In the type 1 tree the alleles in species A and B coalesce after the speciation event that separated the A-B lineage from the C lineage. In the type 2 tree the alleles in species A and B coalesce before the speciation event that separated the A-B lineage from the C lineage (in other words, the type 2 tree is a deep coalescence tree). The type 1 and type 2 gene trees are both congruent with the species tree. The other two gene trees differ from the species tree; the two discordant gene trees are also deep coalescence trees.
The distribution of times to coalescence is actually continuous for all of these trees. In other words, the exact coalescent time for any two loci with the same gene tree may differ. However, it is convenient to break up the trees based on whether the coalescence occurred before or after the earliest speciation event.
Given the internal branch length in coalescent units it is straightforward to calculate the probability of each gene tree. [ 9 ] For diploid organisms the branch length in coalescent units is the number of generations between the speciation events divided by twice the effective population size. Since all three of the deep coalescence tree are equiprobable and two of those deep coalescence tree are discordant it is easy to see that the probability that a rooted three-taxon gene tree will be congruent with the species tree is:
P ( c o n g r u e n c e ) = 1 − 2 3 exp ( − T ) = 1 − 2 3 exp ( − t 2 N e ) {\displaystyle {\begin{aligned}P(congruence)&=1-{\frac {2}{3}}\exp(-T)=1-{\frac {2}{3}}\exp(-{\frac {t}{2N_{e}}})\end{aligned}}}
Where the branch length in coalescent units ( T ) is also written in an alternative form: the number of generations ( t ) divided by twice the effective population size ( N e ). Pamilo and Nei [ 9 ] also derived the probability of congruence for rooted trees of four and five taxa as well as a general upper bound on the probability of congruence for larger trees. Rosenberg [ 10 ] followed up with equations used for the complete set of topologies (although the large number of distinct phylogenetic trees that becomes possible as the number of taxa increases [ 7 ] makes these equations impractical unless the number of taxa is very limited).
The phenomenon of hemiplasy is a natural extension of the basic idea underlying gene tree-species tree discordance. If we consider the distribution of some character that disagrees with the species tree it might reflect homoplasy (multiple independent origins of the character or a single origin followed by multiple losses) or it could reflect hemiplasy (a single origin of the trait that is associated with a gene tree that disagrees with the species tree).
The phenomenon called incomplete lineage sorting (often abbreviated ILS in the scientific literatures [ 11 ] ) is linked to the phenomenon. If we examine the illustration of hemiplasy with using a rooted four-taxon tree (see image to the right) the lineage between the common ancestor of taxa A, B, and C and the common ancestor of taxa A and B must be polymorphic for the allele with the derived trait (e.g., a transposable element insertion [ 12 ] ) and the allele with the ancestral trait. The concept of incomplete lineage sorting ultimately reflects on persistence of polymorphisms across one or more speciation events.
The probability density of the gene trees under the multispecies coalescent model is discussed along with its use for parameter estimation using multi-locus sequence data.
In the basic multispecies coalescent model, the species phylogeny is assumed to be known. Complete isolation after species divergence, with no migration, hybridization, or introgression is also assumed. We assume no recombination so that all the sites within the locus share the same gene tree (topology and coalescent times). However, the basic model can be extended in different ways to accommodate migration or introgression, population size changes, recombination. [ 13 ] [ 14 ]
The model and implementation of this method can be applied to any species tree. As an example, the species tree of the great apes : humans (H), chimpanzees (C), gorillas (G) and orangutans (O) is considered. The topology of the species tree, (((HC)G)O)), is assumed known and fixed in the analysis (Figure 1). [ 1 ] Let D = { D i } {\displaystyle D=\{D_{i}\}} be the entire data set, where D i {\displaystyle {D_{i}}} represent the sequence alignment at locus i {\displaystyle i} , with i = 1 , 2 , … , L {\displaystyle i=1,2,\ldots ,L} for a total of L {\displaystyle L} loci.
The population size of a current species is considered only if more than one individual is sampled from that species at some loci.
The parameters in the model for the example of Figure 1 include the three divergence times τ H C {\displaystyle \tau _{HC}} , τ H C G {\displaystyle \tau _{HCG}} and τ H C G O {\displaystyle \tau _{HCGO}} and population size parameters θ H {\displaystyle \theta _{H}} for humans; θ C {\displaystyle \theta _{C}} for chimpanzees; and θ H C {\displaystyle \theta _{HC}} , θ H C G {\displaystyle \theta _{HCG}} and θ H C G O {\displaystyle \theta _{HCGO}} for the three ancestral species.
The divergence times ( τ {\displaystyle \tau } 's) are measured by the expected number of mutations per site from the ancestral node in the species tree to the present time (Figure 1 of Rannala and Yang, 2003).
Therefore, the parameters are Θ = { θ H , θ C , θ H C , θ H C G , θ H C G O , τ H C , τ H C G , τ H C G O } {\displaystyle \Theta =\{\theta _{H},\theta _{C},\theta _{HC},\theta _{HCG},\theta _{HCGO},\tau _{HC},\tau _{HCG},\tau _{HCGO}\}} .
The joint distribution of f ( T i , t i ∣ Θ ) {\displaystyle f(T_{i},t_{i}\mid \Theta )} is derived directly in this section. [ 1 ] Two sequences from different species can coalesce only in one populations that are ancestral to the two species. For example, sequences H and G can coalesce in populations HCG or HCGO, but not in populations H or HC. The coalescent processes in different populations are different.
For each population, the genealogy is traced backward in time, until the end of the population at time τ {\displaystyle \tau } , and the number of lineages ( m ) {\displaystyle (m)} entering the population and the number of lineages leaving it ( n ) {\displaystyle (n)} are recorded. For example, m = 3 , n = 2 , {\displaystyle m=3,n=2,} and τ = τ H C {\displaystyle \tau =\tau _{HC}} , for population H (Table 1). [ 1 ] This process is called a censored coalescent process because the coalescent process for one population may be terminated before all lineages that entered the population have coalesced. If n ≥ 1 {\displaystyle n\geq 1} the population consists of n {\displaystyle n} disconnected subtrees or lineages.
With one time unit defined as the time taken to accumulate one mutation per site, any two lineages coalesce at the rate 2 θ {\displaystyle {\frac {2}{\theta }}} . The waiting time t j {\displaystyle t_{j}} until the next coalescent event, which reduces the number of lineages from j {\displaystyle j} to j − 1 {\displaystyle j-1} has exponential density
If n ≥ 1 {\displaystyle n\geq 1} , the probability that no coalescent event occurs between the last one and the end of the population at time τ {\displaystyle \tau } ; i.e. during the time interval τ − ( t m + t m − 1 + … + t n + 1 ) {\displaystyle \tau -(t_{m}+t_{m-1}+\ldots +t_{n+1})} . This probability is exp { − n ( n − 1 ) θ [ τ − ( t m + t m − 1 + … + t n + 1 ) ] {\displaystyle \exp\{-{\frac {n(n-1)}{\theta }}[\tau -(t_{m}+t_{m-1}+\ldots +t_{n+1})]} and is 1 if n = 1 {\displaystyle n=1} .
(Note: One should recall that the probability of no events over time interval t {\displaystyle t} for a Poisson process with rate λ {\displaystyle \lambda } is e − λ t {\displaystyle e^{-\lambda t}} . Here the coalescent rate when there are n {\displaystyle n} lineages is λ = n ( n − 1 ) θ {\displaystyle \lambda ={\frac {n(n-1)}{\theta }}} .)
In addition, to derive the probability of a particular gene tree topology in the population, if a coalescent event occurs in a sample of j {\displaystyle j} lineages, the probability that a particular pair of lineages coalesce is 1 / ( j 2 ) = 2 / j ( j − 1 ) , j = m , m − 1 , … , n + 1 {\displaystyle 1/{\binom {j}{2}}=2/j(j-1),\quad j=m,m-1,\ldots ,n+1} .
Multiplying these probabilities together, the joint probability distribution of the gene tree topology in the population and its coalescent times t m , t m + 1 , … , t n + 1 {\displaystyle t_{m},t_{m+1},\ldots ,t_{n+1}} as
The probability of the gene tree and coalescent times for the locus is the product of such probabilities across all the populations. Therefore, the gene genealogy of Figure 1, [ 1 ] [ 15 ] we have
f ( G i ∣ Θ ) = [ 2 / θ H exp { − 6 t 3 ( H ) / θ H } exp { − 2 ( τ H C − t 3 ( H ) ) / θ H } ] × [ 2 / θ C exp { − 2 t 2 ( C ) / θ C } ] × [ 2 / θ H C exp { − 6 t 3 H C / θ H C } ] × [ 2 / θ H C exp { − 2 t 2 H C / θ H C } ] × [ exp { − 2 ( τ H C G − τ H G − ( t 3 H C + t 2 H C ) ) / θ H C G } ] × [ 2 / θ H C G O exp { − 6 t 3 H C G O / θ H C G O } ] × [ 2 / θ H C G O exp { − 2 t 2 H C G O / θ H C G O } ] {\displaystyle {\begin{aligned}f(G_{i}\mid \Theta )&=[2/\theta _{H}\exp\{-6t_{3}^{(H)}/\theta _{H}\}\exp\{-2(\tau _{HC}-t_{3}^{(H)})/\theta _{H}\}]\\&{}\times [2/\theta _{C}\exp\{-2t_{2}^{(C)}/\theta _{C}\}]\\&{}\times [2/\theta _{HC}\exp\{-6t_{3}^{HC}/\theta _{HC}\}]\times [2/\theta _{HC}\exp\{-2t_{2}^{HC}/\theta _{HC}\}]\\&{}\times [\exp\{-2(\tau _{HCG}-\tau _{HG}-(t_{3}^{HC}+t_{2}^{HC}))/\theta _{HCG}\}]\\&{}\times [2/\theta _{HCGO}\exp\{-6t_{3}^{HCGO}/\theta _{HCGO}\}]\times [2/\theta _{HCGO}\exp\{-2t_{2}^{HCGO}/\theta _{HCGO}\}]\end{aligned}}}
The gene genealogy G i {\displaystyle G_{i}} at each locus i {\displaystyle i} is represented by the tree topology T i {\displaystyle T_{i}} and the coalescent times t i {\displaystyle t_{i}} . Given the species tree and the parameters Θ {\displaystyle \Theta } on it, the probability distribution of G i = { T i , t i } {\displaystyle G_{i}=\{T_{i},t_{i}\}} is specified by the coalescent process as
where f ( G i ∣ Θ ) = f ( T i , t i ∣ Θ ) {\displaystyle f(G_{i}\mid \Theta )=f(T_{i},t_{i}\mid \Theta )} is the probability density for the gene tree at locus locus i {\displaystyle i} , [ 1 ] and the product is because we assume that the gene trees are independent given the parameters.
The probability of data D i {\displaystyle D_{i}} given the gene tree and coalescent times (and thus branch lengths) at the locus, f ( D i ∣ G i ) {\displaystyle f(D_{i}\mid G_{i})} , is Felsenstein's phylogenetic likelihood. [ 16 ] Due to the assumption of independent evolution across the loci,
The likelihood function or the probability of the sequence data given the parameters Θ {\displaystyle \Theta } is then an average over the unobserved gene trees
where the integration represents summation over all possible gene tree topologies ( T i {\displaystyle T_{i}} ) and, for each possible topology at each locus, integration over the coalescent times t i {\displaystyle t_{i}} . [ 17 ] This is in general intractable except for very small species trees.
In Bayesian inference , we assign a prior on the parameters, f ( Θ ) {\displaystyle f(\Theta )} , and then the posterior is given as
where again the integration represents summation over all possible gene tree topologies ( T i {\displaystyle T_{i}} ) and integration over the coalescent times t i {\displaystyle t_{i}} . In practice this integration over the gene trees is achieved through a Markov chain Monte Carlo algorithm, which samples from the joint conditional distribution of the parameters and the gene trees
The above assumes that the species tree is fixed. In species-tree estimation, the species tree ( S {\displaystyle S} ) changes as well, so that the joint conditional distribution (from which the MCMC samples) is
where f ( S ) {\displaystyle f(S)} is the prior on species trees.
As a major departure from two-step summary methods, full-likelihood methods average over the gene trees. This means that they make use of information in the branch lengths (coalescent times) on the gene trees and accommodate their uncertainties (due to limited sequence length in the alignments) at the same time. It also explains why full-likelihood methods are computationally much more demanding than two-step summary methods.
The integration or summation over the gene trees in the definition of the likelihood function above is virtually impossible to compute except for very small species trees with only two or three species. [ 18 ] Full-likelihood or full-data methods, based on calculation of the likelihood function on sequence alignments, have thus mostly relied on Markov chain Monte Carlo algorithms. MCMC algorithms under the multispecies coalescent model are similar to those used in Bayesian phylogenetics but are distinctly more complex, mainly due to the fact that the gene trees at multiple loci and the species tree have to be compatible: sequence divergence has to be older than species divergence. As a result, changing the species tree while the gene trees are fixed (or changing a gene tree while the species tree is fixed) leads to inefficient algorithms with poor mixing properties. Considerable efforts have been taken to design smart algorithms that change the species tree and gene trees in a coordinated manner, as in the rubber-band algorithm for changing species divergence times, [ 1 ] the coordinated NNI, SPR and NodeSlider moves. [ 19 ] [ 20 ]
Consider for example the case of two species ( A and B ) and two sequences at each locus, with a sequence divergence time t i {\displaystyle t_{i}} at locus i {\displaystyle i} . We have t i < τ {\displaystyle t_{i}<\tau } for all i {\displaystyle i} . When we want to change the species divergence time τ {\displaystyle \tau } within the constraint of the current t i {\displaystyle t_{i}} , we may have very little room for change, as τ {\displaystyle \tau } may be virtually identical to the smallest of the t i {\displaystyle t_{i}} . The rubber-band algorithm [ 1 ] changes τ {\displaystyle \tau } without consideration of the t i {\displaystyle t_{i}} , and then modifies the t i {\displaystyle t_{i}} deterministically in the same way that marks on a rubber band move when the rubber band is held from a fixed point pulled towards one end. In general, the rubber-band move guarantees that the ages of nodes in the gene trees are modified so that they remain compatible with the modified species divergence time.
Full likelihood methods tend to reach their limit when the data consist of a few hundred loci, even though more than 10,000 loci have been analyzed in a few published studies. [ 21 ] [ 22 ]
The basic multispecies coalescent model can be extended in a number of ways to accommodate major factors of the biological process of reproduction and drift. [ 13 ] [ 14 ] For example, incorporating continuous-time migration leads to the MSC+M (for MSC with migration) model, also known as the isolation-with-migration or IM models. [ 23 ] [ 24 ] Incorporating episodic hybridization/introgression leads to the MSC with introgression (MSci) [ 25 ] or multispecies-network-coalescent (MSNC) model. [ 26 ] [ 27 ]
The multispecies coalescent has profound implications for the theory and practice of molecular phylogenetics. [ 3 ] [ 4 ] Since individual gene trees can differ from the species tree one cannot estimate the tree for a single locus and assume that the gene tree correspond the species tree. In fact, one can be virtually certain that any individual gene tree will differ from the species tree for at least some relationships when any reasonable number of taxa are considered. However, gene tree-species tree discordance has an impact on the theory and practice of species tree estimation that goes beyond the simple observation that one cannot use a single gene tree to estimate the species tree because there is a part of parameter space where the most frequent gene tree is incongruent with the species tree. This part of parameter space is called the anomaly zone [ 28 ] and any discordant gene trees that are more expected to arise more often than the gene tree. that matches the species tree are called anomalous gene trees .
The existence of the anomaly zone implies that one cannot simply estimate a large number of gene trees and assume the gene tree recovered the largest number of times is the species tree. Of course, estimating the species tree by a "democratic vote" of gene trees would only work for a limited number of taxa outside of the anomaly zone given the extremely large number of phylogenetic trees that are possible. [ 7 ] However, the existence of the anomalous gene trees also means that simple methods for combining gene trees, like the majority rule extended ("greedy") consensus method or the matrix representation with parsimony (MRP) supertree [ 29 ] [ 30 ] approach, will not be consistent estimators of the species tree [ 31 ] [ 32 ] (i.e., they will be misleading). Simply generating the majority-rule consensus tree for the gene trees, where groups that are present in at least 50% of gene trees are retained, will not be misleading as long as a sufficient number of gene trees are used. [ 31 ] However, this ability of the majority-rule consensus tree for a set of gene trees to avoid incorrect clades comes at the cost of having unresolved groups.
Simulations have shown that there are parts of species tree parameter space where maximum likelihood estimates of phylogeny are incorrect trees with increasing probability as the amount of data analyzed increases. [ 33 ] This is important because the "concatenation approach," where multiple sequence alignments from different loci are concatenated to form a single large supermatrix alignment that is then used for maximum likelihood (or Bayesian MCMC ) analysis, is both easy to implement and commonly used in empirical studies. This represents a case of model misspecification because the concatenation approach implicitly assumes that all gene trees have the same topology. [ 34 ] Indeed, it has now been proven that analyses of data generated under the multispecies coalescent using maximum likelihood analysis of a concatenated data are not guaranteed to converge on the true species tree as the number of loci used for the analysis increases [ 35 ] [ 36 ] [ 37 ] (i.e., maximum likelihood concatenation is statistically inconsistent).
There are two basic approaches for phylogenetic estimation in the multispecies coalescent framework: 1) full-likelihood or full-data methods which operate on multilocus sequence alignments directly, including both maximum likelihood and Bayesian methods, and 2) summary methods, which use a summary of the original sequence data, including the two-step methods that use estimated gene trees as summary input and SVDQuartets, which use site pattern counts pooled over loci as summary input. | https://en.wikipedia.org/wiki/Multispecies_coalescent_process |
The Multispectral Scanner (MSS) is one of the Earth 's observing sensors introduced in the Landsat program . A Multispectral Scanner was placed aboard each of the first five Landsat satellites . [ 1 ]
The scanner was designed at Hughes Aerospace by Virginia Norwood . Her design called for a six band scanner, but the first one launched had only four bands. For her work on the design Norwood is called "The Mother of Landsat." [ 2 ]
This astronomy -related article is a stub . You can help Wikipedia by expanding it .
This article about one or more spacecraft of the United States is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multispectral_Scanner |
Multispectral imaging captures image data within specific wavelength ranges across the electromagnetic spectrum . The wavelengths may be separated by filters or detected with the use of instruments that are sensitive to particular wavelengths, including light from frequencies beyond the visible light range (i.e. infrared and ultraviolet ). It can allow extraction of additional information the human eye fails to capture with its visible receptors for red, green and blue . It was originally developed for military target identification and reconnaissance. Early space-based imaging platforms incorporated multispectral imaging technology [ 1 ] to map details of the Earth related to coastal boundaries, vegetation, and landforms. [ 2 ] Multispectral imaging has also found use in document and painting analysis. [ 3 ] [ 4 ]
Multispectral imaging measures light in a small number (typically 3 to 15) of spectral bands . Hyperspectral imaging is a special case of spectral imaging where often hundreds of contiguous spectral bands are available. [ 5 ]
For different purposes, different combinations of spectral bands can be used. They are usually represented with red, green, and blue channels. Mapping of bands to colors depends on the purpose of the image and the personal preferences of the analysts. Thermal infrared is often omitted from consideration due to poor spatial resolution, except for special purposes.
Many other combinations are in use. NIR is often shown as red, causing vegetation-covered areas to appear red.
The wavelengths are approximate; exact values depend on the particular instruments (e.g. characteristics of satellite's sensors for Earth observation, characteristics of illumination and sensors for document analysis):
Unlike other aerial photographic and satellite image interpretation work, these multispectral images do not make it easy to identify directly the feature type by visual inspection. Hence the remote sensing data has to be classified first, followed by processing by various data enhancement techniques so as to help the user to understand the features that are present in the image.
Such classification is a complex task which involves rigorous validation of the training samples depending on the classification algorithm used. The techniques can be grouped mainly into two types.
Supervised classification makes use of training samples. Training samples are areas on the ground for which there is ground truth , that is, what is there is known. The spectral signatures of the training areas are used to search for similar signatures in the remaining pixels of the image, and we will classify accordingly. This use of training samples for classification is called supervised classification. Expert knowledge is very important in this method since the selection of the training samples and a biased selection can badly affect the accuracy of classification. Popular techniques include the maximum likelihood principle and convolutional neural network . The maximum likelihood principle calculates the probability of a pixel belonging to a class (i.e. feature) and allots the pixel to its most probable class. Newer convolutional neural network based methods [ 6 ] account for both spatial proximity and entire spectra to determine the most likely class.
In case of unsupervised classification no prior knowledge is required for classifying the features of the image. The natural clustering or grouping of the pixel values (i.e. the gray levels of the pixels) are observed. Then a threshold is defined for adopting the number of classes in the image. The finer the threshold value, the more classes there will be. However, beyond a certain limit the same class will be represented in different classes in the sense that variation in the class is represented. After forming the clusters, ground truth validation is done to identify the class the image pixel belongs to. Thus in this unsupervised classification a priori information about the classes is not required. One of the popular methods in unsupervised classification is k-means clustering .
Multispectral imaging measures light emission and is often used in detecting or tracking military targets. In 2003, researchers at the United States Army Research Laboratory and the Federal Laboratory Collaborative Technology Alliance reported a dual band multispectral imaging focal plane array (FPA). This FPA allowed researchers to look at two infrared (IR) planes at the same time. [ 9 ] Because mid-wave infrared (MWIR) and long wave infrared (LWIR) technologies measure radiation inherent to the object and require no external light source, they also are referred to as thermal imaging methods.
The brightness of the image produced by a thermal imager depends on the objects emissivity and temperature. [ 10 ] Every material has an infrared signature that aids in the identification of the object. [ 11 ] These signatures are less pronounced in hyperspectral systems (which image in many more bands than multispectral systems) and when exposed to wind and, more dramatically, to rain. [ 11 ] Sometimes the surface of the target may reflect infrared energy. This reflection may misconstrue the true reading of the objects’ inherent radiation. [ 12 ] Imaging systems that use MWIR technology function better with solar reflections on the target's surface and produce more definitive images of hot objects, such as engines, compared to LWIR technology. [ 13 ] However, LWIR operates better in hazy environments like smoke or fog because less scattering occurs in the longer wavelengths. [ 10 ] Researchers claim that dual-band technologies combine these advantages to provide more information from an image, particularly in the realm of target tracking. [ 9 ]
For nighttime target detection, thermal imaging outperformed single-band multispectral imaging. Dual band MWIR and LWIR technology resulted in better visualization during the nighttime than MWIR alone. Citation Citation. The US Army reports that its dual band LWIR/MWIR FPA demonstrated better visualizing of tactical vehicles than MWIR alone after tracking them through both day and night. [ citation needed ]
By analyzing the emissivity of ground surfaces, multispectral imaging can detect the presence of underground missiles. Surface and sub-surface soil possess different physical and chemical properties that appear in spectral analysis. [ 11 ] Disturbed soil has increased emissivity in the wavelength range of 8.5 to 9.5 micrometers while demonstrating no change in wavelengths greater than 10 micrometers. [ 9 ] The US Army Research Laboratory's dual MWIR/LWIR FPA used "red" and "blue" detectors to search for areas with enhanced emissivity. The red detector acts as a backdrop, verifying realms of undisturbed soil areas, as it is sensitive to the 10.4 micrometer wavelength. The blue detector is sensitive to wavelengths of 9.3 micrometers. If the intensity of the blue image changes when scanning, that region is likely disturbed . The scientists reported that fusing these two images increased detection capabilities. [ 9 ]
Intercepting an intercontinental ballistic missile (ICBM) in its boost phase requires imaging of the hard body as well as the rocket plumes. MWIR presents a strong signal from highly heated objects including rocket plumes, while LWIR produces emissions from the missile's body material. The US Army Research Laboratory reported that with their dual-band MWIR/LWIR technology, tracking of the Atlas 5 Evolved Expendable Launch Vehicles, similar in design to ICBMs, picked up both the missile body and plumage. [ 9 ]
Most radiometers for remote sensing (RS) acquire multispectral images. Dividing the spectrum into many bands, multispectral is the opposite of panchromatic , which records only the total intensity of radiation falling on each pixel . [ 14 ] Usually, Earth observation satellites have three or more radiometers . Each acquires one digital image (in remote sensing, called a 'scene') in a small spectral band. The bands are grouped into wavelength regions based on the origin of the light and the interests of the researchers.
Modern weather satellites produce imagery in a variety of spectra. [ 15 ]
Multispectral imaging combines two to five spectral imaging bands of relatively large bandwidth into a single optical system. A multispectral system usually provides a combination of visible (0.4 to 0.7 µm), near infrared (NIR; 0.7 to 1 µm), short-wave infrared (SWIR; 1 to 1.7 µm), mid-wave infrared (MWIR; 3.5 to 5 µm) or long-wave infrared (LWIR; 8 to 12 µm) bands into a single system. — Valerie C. Coffey [ 16 ]
In the case of Landsat satellites, several different band designations have been used, with as many as 11 bands ( Landsat 8 ) comprising a multispectral image. [ 17 ] [ 18 ] [ 19 ] Spectral imaging with a higher radiometric resolution
(involving hundreds or thousands of bands), finer spectral resolution (involving smaller bands), or wider spectral coverage may be called hyperspectral or ultraspectral. [ 19 ]
Multispectral imaging can be employed for investigation of paintings and other works of art. [ 3 ] The painting is irradiated by ultraviolet , visible and infrared rays and the reflected radiation is recorded in a camera sensitive in this region of the spectrum. The image can also be registered using the transmitted instead of reflected radiation. In special cases the painting can be irradiated by UV , VIS or IR rays and the fluorescence of pigments or varnishes can be registered. [ 20 ]
Multispectral analysis has assisted in the interpretation of ancient papyri , such as those found at Herculaneum , by imaging the fragments in the infrared range (1000 nm). Often, the text on the documents appears to the naked eye as black ink on black paper. At 1000 nm, the difference in how paper and ink reflect infrared light makes the text clearly readable. It has also been used to image the Archimedes palimpsest by imaging the parchment leaves in bandwidths from 365–870 nm, and then using advanced digital image processing techniques to reveal the undertext with Archimedes' work. [ 21 ] Multispectral imaging has been used in a Mellon Foundation project at Yale University to compare inks in medieval English manuscripts. [ 4 ]
Multispectral imaging has also been used to examine discolorations and stains on old books and manuscripts. Comparing the "spectral fingerprint" of a stain to the characteristics of known chemical substances can make it possible to identify the stain. This technique has been used to examine medical and alchemical texts, seeking hints about the activities of early chemists and the possible chemical substances they may have used in their experiments. Like a cook spilling flour or vinegar on a cookbook, an early chemist might have left tangible evidence on the pages of the ingredients used to make medicines. [ 22 ] | https://en.wikipedia.org/wiki/Multispectral_imaging |
A multistage rocket or step rocket [ 1 ] is a launch vehicle that uses two or more rocket stages , each of which contains its own engines and propellant . A tandem or serial stage is mounted on top of another stage; a parallel stage is attached alongside another stage. The result is effectively two or more rockets stacked on top of or attached next to each other. Two-stage rockets are quite common, but rockets with as many as five separate stages have been successfully launched.
By jettisoning stages when they run out of propellant, the mass of the remaining rocket is decreased. Each successive stage can also be optimized for its specific operating conditions, such as decreased atmospheric pressure at higher altitudes. This staging allows the thrust of the remaining stages to more easily accelerate the rocket to its final velocity and height.
In serial or tandem staging schemes, the first stage is at the bottom and is usually the largest, the second stage and subsequent upper stages are above it, usually decreasing in size. In parallel staging schemes solid or liquid rocket boosters are used to assist with launch. These are sometimes referred to as "stage 0". In the typical case, the first-stage and booster engines fire to propel the entire rocket upwards. When the boosters run out of fuel, they are detached from the rest of the rocket (usually with some kind of small explosive charge or explosive bolts ) and fall away. The first stage then burns to completion and falls off. This leaves a smaller rocket, with the second stage on the bottom, which then fires. Known in rocketry circles as staging , this process is repeated until the desired final velocity is achieved. In some cases with serial staging, the upper stage ignites before the separation—the interstage ring is designed with this in mind, and the thrust is used to help positively separate the two vehicles.
Only multistage rockets have reached orbital speed . Single-stage-to-orbit designs are sought, but have not yet been demonstrated.
Multi-stage rockets overcome a limitation imposed by the laws of physics on the velocity change achievable by a rocket stage. The limit depends on the fueled-to-dry mass ratio and on the effective exhaust velocity of the engine. This relation is given by the classical rocket equation :
where:
The delta v required to reach low Earth orbit (or the required velocity of a sufficiently heavy suborbital payload) requires a wet to dry mass ratio larger than has been achieved in a single rocket stage. The multistage rocket overcomes this limit by splitting the delta-v into fractions. As each lower stage drops off and the succeeding stage fires, the rest of the rocket is still traveling near the burnout speed. Each lower stage's dry mass includes the propellant in the upper stages, and each succeeding upper stage has reduced its dry mass by discarding the useless dry mass of the spent lower stages. [ 2 ]
A further advantage is that each stage can use a different type of rocket engine, each tuned for its particular operating conditions. Thus the lower-stage engines are designed for use at atmospheric pressure, while the upper stages can use engines suited to near vacuum conditions. Lower stages tend to require more structure than upper as they need to bear their own weight plus that of the stages above them. Optimizing the structure of each stage decreases the weight of the total vehicle and provides further advantage.
The advantage of staging comes at the cost of the lower stages lifting engines which are not yet being used, as well as making the entire rocket more complex and harder to build than a single stage. In addition, each staging event is a possible point of launch failure, due to separation failure, ignition failure, or stage collision. Nevertheless, the savings are so great that every rocket ever used to deliver a payload into orbit has had staging of some sort.
One of the most common measures of rocket efficiency is its specific impulse, which is defined as the thrust per flow rate (per second) of propellant consumption: [ 3 ]
When rearranging the equation such that thrust is calculated as a result of the other factors, we have:
These equations show that a higher specific impulse means a more efficient rocket engine, capable of burning for longer periods of time. In terms of staging, the initial rocket stages usually have a lower specific impulse rating, trading efficiency for superior thrust in order to quickly push the rocket into higher altitudes. Later stages of the rocket usually have a higher specific impulse rating because the vehicle is further outside the atmosphere and the exhaust gas does not need to expand against as much atmospheric pressure.
When selecting the ideal rocket engine to use as an initial stage for a launch vehicle, a useful performance metric to examine is the thrust-to-weight ratio, and is calculated by the equation:
The common thrust-to-weight ratio of a launch vehicle is within the range of 1.3 to 2.0. [ 3 ] Another performance metric to keep in mind when designing each rocket stage in a mission is the burn time, which is the amount of time the rocket engine will last before it has exhausted all of its propellant. For most non-final stages, thrust and specific impulse can be assumed constant, which allows the equation for burn time to be written as:
Where m 0 {\displaystyle m_{\mathrm {0} }} and m f {\displaystyle m_{\mathrm {f} }} are the initial and final masses of the rocket stage respectively. In conjunction with the burnout time, the burnout height and velocity are obtained using the same values, and are found by these two equations:
When dealing with the problem of calculating the total burnout velocity or time for the entire rocket system, the general procedure for doing so is as follows: [ 3 ]
The burnout time does not define the end of the rocket stage's motion, as the vehicle will still have a velocity that will allow it to coast upward for a brief amount of time until the acceleration of the planet's gravity gradually changes it to a downward direction. The velocity and altitude of the rocket after burnout can be easily modeled using the basic physics equations of motion.
When comparing one rocket with another, it is impractical to directly compare the rocket's certain trait with the same trait of another because their individual attributes are often not independent of one another. For this reason, dimensionless ratios have been designed to enable a more meaningful comparison between rockets. The first is the initial to final mass ratio, which is the ratio between the rocket stage's full initial mass and the rocket stage's final mass once all of its fuel has been consumed. The equation for this ratio is:
Where m E {\displaystyle m_{\mathrm {E} }} is the empty mass of the stage, m p {\displaystyle m_{\mathrm {p} }} is the mass of the propellant, and m P L {\displaystyle m_{\mathrm {PL} }} is the mass of the payload. [ 4 ] The second dimensionless performance quantity is the structural ratio, which is the ratio between the empty mass of the stage, and the combined empty mass and propellant mass as shown in this equation: [ 4 ]
The last major dimensionless performance quantity is the payload ratio, which is the ratio between the payload mass and the combined mass of the empty rocket stage and the propellant:
After comparing the three equations for the dimensionless quantities, it is easy to see that they are not independent of each other, and in fact, the initial to final mass ratio can be rewritten in terms of structural ratio and payload ratio: [ 4 ]
These performance ratios can also be used as references for how efficient a rocket system will be when performing optimizations and comparing varying configurations for a mission.
For initial sizing, the rocket equations can be used to derive the amount of propellant needed for the rocket based on the specific impulse of the engine and the total impulse required in N·s. The equation is:
where g is the gravity constant of Earth. [ 3 ] This also enables the volume of storage required for the fuel to be calculated if the density of the fuel is known, which is almost always the case when designing the rocket stage. The volume is yielded when dividing the mass of the propellant by its density. Asides from the fuel required, the mass of the rocket structure itself must also be determined, which requires taking into account the mass of the required thrusters, electronics, instruments, power equipment, etc. [ 3 ] These are known quantities for typical off the shelf hardware that should be considered in the mid to late stages of the design, but for preliminary and conceptual design, a simpler approach can be taken. Assuming one engine for a rocket stage provides all of the total impulse for that particular segment, a mass fraction can be used to determine the mass of the system. The mass of the stage transfer hardware such as initiators and safe-and-arm devices are very small by comparison and can be considered negligible.
For modern day solid rocket motors, it is a safe and reasonable assumption to say that 91 to 94 percent of the total mass is fuel. [ 3 ] It is also important to note there is a small percentage of "residual" propellant that will be left stuck and unusable inside the tank, and should also be taken into consideration when determining amount of fuel for the rocket. A common initial estimate for this residual propellant is five percent. With this ratio and the mass of the propellant calculated, the mass of the empty rocket weight can be determined.
Sizing rockets using a liquid bipropellant requires a slightly more involved approach because there are two separate tanks that are required: one for the fuel, and one for the oxidizer. The ratio of these two quantities is known as the mixture ratio, and is defined by the equation:
Where m o x {\displaystyle m_{\mathrm {ox} }} is the mass of the oxidizer and m f u e l {\displaystyle m_{\mathrm {fuel} }} is the mass of the fuel. This mixture ratio not only governs the size of each tank, but also the specific impulse of the rocket. Determining the ideal mixture ratio is a balance of compromises between various aspects of the rocket being designed, and can vary depending on the type of fuel and oxidizer combination being used. For example, a mixture ratio of a bipropellant could be adjusted such that it may not have the optimal specific impulse, but will result in fuel tanks of equal size. This would yield simpler and cheaper manufacturing, packing, configuring, and integrating of the fuel systems with the rest of the rocket, [ 3 ] and can become a benefit that could outweigh the drawbacks of a less efficient specific impulse rating. But suppose the defining constraint for the launch system is volume, and a low density fuel is required such as hydrogen. This example would be solved by using an oxidizer-rich mixture ratio, reducing efficiency and specific impulse rating, but will meet a smaller tank volume requirement.
The ultimate goal of optimal staging is to maximize the payload ratio (see ratios under performance), meaning the largest amount of payload is carried up to the required burnout velocity using the least amount of non-payload mass, which comprises everything else. This goal assumes that the cost of a rocket launch is proportional to the total liftoff mass of the rocket, which is a rule of thumb in rocket engineering. Here are a few quick rules and guidelines to follow in order to reach optimal staging: [ 3 ]
The payload ratio can be calculated for each individual stage, and when multiplied together in sequence, will yield the overall payload ratio of the entire system. It is important to note that when computing payload ratio for individual stages, the payload includes the mass of all the stages after the current one. The overall payload ratio is:
Where n is the number of stages the rocket system comprises. Similar stages yielding the same payload ratio simplify this equation, however that is seldom the ideal solution for maximizing payload ratio, and ΔV requirements may have to be partitioned unevenly as suggested in guideline tips 1 and 2 from above. Two common methods of determining this perfect ΔV partition between stages are either a technical algorithm that generates an analytical solution that can be implemented by a program, or simple trial and error. [ 3 ] For the trial and error approach, it is best to begin with the final stage, calculating the initial mass which becomes the payload for the previous stage. From there it is easy to progress all the way down to the initial stage in the same manner, sizing all the stages of the rocket system.
Restricted rocket staging is based on the simplified assumption that each of the stages of the rocket system have the same specific impulse, structural ratio, and payload ratio, the only difference being the total mass of each increasing stage is less than that of the previous stage. Although this assumption may not be the ideal approach to yielding an efficient or optimal system, it greatly simplifies the equations for determining the burnout velocities, burnout times, burnout altitudes, and mass of each stage. This would make for a better approach to a conceptual design in a situation where a basic understanding of the system behavior is preferential to a detailed, accurate design.
One important concept to understand when undergoing restricted rocket staging, is how the burnout velocity is affected by the number of stages that split up the rocket system. Increasing the number of stages for a rocket while keeping the specific impulse, payload ratios and structural ratios constant will always yield a higher burnout velocity than the same systems that use fewer stages. However, the law of diminishing returns is evident in that each increment in number of stages gives less of an improvement in burnout velocity than the previous increment. The burnout velocity gradually converges towards an asymptotic value as the number of stages increases towards a very high number. [ 4 ] In addition to diminishing returns in burnout velocity improvement, the main reason why real world rockets seldom use more than three stages is because of increase of weight and complexity in the system for each added stage, ultimately yielding a higher cost for deployment.
Hot-staging is a type of rocket staging in which the next stage fires its engines before separation instead of after. [ 5 ] During hot-staging, the earlier stage throttles down its engines. [ 5 ] Hot-staging may reduce the complexity of stage separation, and gives a small extra payload capacity to the booster. [ 5 ] It also eliminates the need for ullage motors , as the acceleration from the nearly spent stage keeps the propellants settled at the bottom of the tanks. Hot-staging is used on Soviet-era Russian rockets such as Soyuz [ 6 ] [ 7 ] and Proton-M . [ 8 ] The N1 rocket was designed to use hot staging, but none of the test flights lasted long enough for this to occur. Starting with the Titan II, the Titan family of rockets used hot staging. SpaceX retrofitted their Starship rocket to use hot staging after its first flight , making it the largest rocket ever to do so, as well as the first reusable vehicle to utilize hot staging. [ 9 ]
A rocket system that implements tandem staging means that each individual stage runs in order one after the other. The rocket breaks free from the previous stage, then begins burning through the next stage in straight succession. On the other hand, a rocket that implements parallel staging has two or more different stages that are active at the same time. For example, the Space Shuttle has two Solid Rocket Boosters that burn simultaneously. Upon launch, the boosters ignite, and at the end of the stage, the two boosters are discarded while the external fuel tank is kept for another stage. [ 3 ] Most quantitative approaches to the design of the rocket system's performance are focused on tandem staging, but the approach can be easily modified to include parallel staging. To begin with, the different stages of the rocket should be clearly defined. Continuing with the previous example, the end of the first stage which is sometimes referred to as 'stage 0', can be defined as when the side boosters separate from the main rocket. From there, the final mass of stage one can be considered the sum of the empty mass of stage one, the mass of stage two (the main rocket and the remaining unburned fuel) and the mass of the payload. [ original research? ]
High-altitude and space-bound upper stages are designed to operate with little or no atmospheric pressure. This allows the use of lower pressure combustion chambers and engine nozzles with optimal vacuum expansion ratios . Some upper stages, especially those using hypergolic propellants like Delta-K or Ariane 5 ES second stage, are pressure fed , which eliminates the need for complex turbopumps . Other upper stages, such as the Centaur or DCSS , use liquid hydrogen expander cycle engines, or gas generator cycle engines like the Ariane 5 ECA's HM7B or the S-IVB 's J-2 . These stages are usually tasked with completing orbital injection and accelerating payloads into higher energy orbits such as GTO or to escape velocity . Upper stages, such as Fregat , used primarily to bring payloads from low Earth orbit to GTO or beyond are sometimes referred to as space tugs . [ 10 ]
Each individual stage is generally assembled at its manufacturing site and shipped to the launch site; the term vehicle assembly refers to the mating of all rocket stage(s) and the spacecraft payload into a single assembly known as a space vehicle . Single-stage vehicles ( suborbital ), and multistage vehicles on the smaller end of the size range, can usually be assembled directly on the launch pad by lifting the stage(s) and spacecraft vertically in place by means of a crane.
This is generally not practical for larger space vehicles, which are assembled off the pad and moved into place on the launch site by various methods. NASA's Apollo / Saturn V crewed Moon landing vehicle, and Space Shuttle , were assembled vertically onto mobile launcher platforms with attached launch umbilical towers, in a Vehicle Assembly Building , and then a special crawler-transporter moved the entire vehicle stack to the launch pad in an upright position. In contrast, vehicles such as the Russian Soyuz rocket and the SpaceX Falcon 9 are assembled horizontally in a processing hangar, transported horizontally, and then brought upright at the pad.
Spent upper stages of launch vehicles are a significant source of space debris remaining in orbit in a non-operational state for many years after use, and occasionally, large debris fields created from the breakup of a single upper stage while in orbit. [ 11 ]
After the 1990s, spent upper stages are generally passivated after their use as a launch vehicle is complete in order to minimize risks while the stage remains derelict in orbit . [ 12 ] Passivation means removing any sources of stored energy remaining on the vehicle, as by dumping fuel or discharging batteries.
Many early upper stages, in both the Soviet and U.S. space programs, were not passivated after mission completion. During the initial attempts to characterize the space debris problem, it became evident that a good proportion of all debris was due to the breaking up of rocket upper stages, particularly unpassivated upper-stage propulsion units. [ 11 ]
An illustration and description in the 14th century Chinese Huolongjing by Jiao Yu and Liu Bowen shows the oldest known multistage rocket; this was the " fire-dragon issuing from the water " (火龙出水, huǒ lóng chū shuǐ), which was used mostly by the Chinese navy. [ 13 ] [ 14 ] It was a two-stage rocket that had booster rockets that would eventually burn out, yet, before they did so, automatically ignited a number of smaller rocket arrows that were shot out of the front end of the missile, which was shaped like a dragon's head with an open mouth. [ 14 ] The British scientist and historian Joseph Needham points out that the written material and depicted illustration of this rocket come from the oldest stratum of the Huolongjing , which can be dated roughly 1300–1350 AD (from the book's part 1, chapter 3, page 23). [ 14 ]
Another example of an early multistaged rocket is the Juhwa (走火) of Korean development. It was proposed by medieval Korean engineer, scientist and inventor Ch'oe Mu-sŏn and developed by the Firearms Bureau (火㷁道監) during the 14th century. [ 15 ] [ 16 ] The rocket had the length of 15 cm and 13 cm; the diameter was 2.2 cm. It was attached to an arrow 110 cm long; experimental records show that the first results were around 200m in range. [ 17 ] There are records that show Korea kept developing this technology until it came to produce the Singijeon , or 'magical machine arrows' in the 16th century.
The earliest experiments with multistage rockets in Europe were made in 1551 by Austrian Conrad Haas (1509–1576), the arsenal master of the town of Hermannstadt , Transylvania (now Sibiu/Hermannstadt, Romania). This concept was developed independently by at least five individuals:
The first high-speed multistage rockets were the RTV-G-4 Bumper rockets tested at the White Sands Proving Ground and later at Cape Canaveral from 1948 to 1950. These consisted of a V-2 rocket and a WAC Corporal sounding rocket. The greatest altitude ever reached was 393 km, attained on February 24, 1949, at White Sands.
In 1947, the Soviet rocket engineer and scientist Mikhail Tikhonravov developed a theory of parallel stages, which he called "packet rockets". In his scheme, three parallel stages were fired from liftoff , but all three engines were fueled from the outer two stages, until they are empty and could be ejected. This is more efficient than sequential staging, because the second-stage engine is never just dead weight. In 1951, Soviet engineer and scientist Dmitry Okhotsimsky carried out a pioneering engineering study of general sequential and parallel staging, with and without the pumping of fuel between stages. The design of the R-7 Semyorka emerged from that study. The trio of rocket engines used in the first stage of the American Atlas I and Atlas II launch vehicles, arranged in a row, used parallel staging in a similar way: the outer pair of booster engines existed as a jettisonable pair which would, after they shut down, drop away with the lowermost outer skirt structure, leaving the central sustainer engine to complete the first stage's engine burn towards apogee or orbit.
Separation of each portion of a multistage rocket introduces additional risk into the success of the launch mission. Reducing the number of separation events results in a reduction in complexity . [ 23 ] Separation events occur when stages or strap-on boosters separate after use, when the payload fairing separates prior to orbital insertion, or when used, a launch escape system which separates after the early phase of a launch. Pyrotechnic fasteners , or in some cases pneumatic systems like on the Falcon 9 Full Thrust , are typically used to separate rocket stages.
A two-stage-to-orbit ( TSTO ) or two-stage rocket launch vehicle is a spacecraft in which two distinct stages provide propulsion consecutively in order to achieve orbital velocity. It is intermediate between a three-stage-to-orbit launcher and a hypothetical single-stage-to-orbit (SSTO) launcher. [ citation needed ]
The three-stage-to-orbit launch system is a commonly used rocket system to attain Earth orbit. The spacecraft uses three distinct stages to provide propulsion consecutively in order to achieve orbital velocity. It is intermediate between a four-stage-to-orbit launcher and a two-stage-to-orbit launcher.
Other designs (in fact, most modern medium- to heavy-lift designs) do not have all three stages inline on the main stack, instead having strap-on boosters for the "stage-0" with two core stages. In these designs, the boosters and first stage fire simultaneously instead of consecutively, providing extra initial thrust to lift the full launcher weight and overcome gravity losses and atmospheric drag. The boosters are jettisoned a few minutes into flight to reduce weight.
The four-stage-to-orbit launch system is a rocket system used to attain Earth orbit. The spacecraft uses four distinct stages to provide propulsion consecutively in order to achieve orbital velocity. It is intermediate between a five-stage-to-orbit launcher and a three-stage-to-orbit launcher, most often used with solid-propellant launch systems.
Other designs do not have all four stages inline on the main stack, instead having strap-on boosters for the "stage-0" with three core stages. In these designs, the boosters and first stage fire simultaneously instead of consecutively, providing extra initial thrust to lift the full launcher weight and overcome gravity losses and atmospheric drag. The boosters are jettisoned a few minutes into flight to reduce weight. | https://en.wikipedia.org/wiki/Multistage_rocket |
A multistorey car park [ 1 ] [ 2 ] ( Commonwealth English ) or parking garage ( American English ), [ 1 ] also called a multistorey , [ 3 ] parking building , parking structure , parkade ( Canadian ), parking ramp , parking deck , or indoor parking , is a building designed for car, motorcycle, and bicycle parking in which parking takes place on more than one floor or level. The first known multistorey facility was built in London in 1901 and the first underground parking was built in Barcelona in 1904 (see history ). [ 1 ] The term multistorey (or multistory) is almost never used in the United States , because almost all parking structures have multiple parking levels. Parking structures may be heated if they are enclosed.
Design of parking structures can add considerable cost for planning new developments, with costs in the United States around $28,000 per space and $56,000 per space for underground (excluding the cost of land), and can be required by cities in parking mandates for new buildings. [ 4 ] Some cities such as London have abolished previously enacted minimum parking requirements. [ 5 ] Minimum parking requirements are a hallmark of zoning and planning codes for municipalities in the US. (States do not prescribe parking requirements, while counties and cities can). [ 6 ]
The earliest known multi-storey car park was opened in May 1901 by City & Suburban Electric Carriage Company at 6 Denman Street, central London. The location had space for 100 vehicles over seven floors, totaling 19,000 square feet. The same company opened a second location in 1902 for 230 vehicles. The company specialized in the sale, storage, valeting, and on-demand delivery of electric vehicles that could travel about 40 miles and had a top speed of 20 miles per hour.
The earliest known parking garage in the United States was built in 1918 for the Hotel La Salle at 215 West Washington Street in the West Loop area of downtown Chicago, Illinois. It was designed by Holabird and Roche. [ 7 ] The Hotel La Salle was demolished in 1976, but the parking structure remained because it had been designated as preliminary landmark status [ 8 ] and the structure was several blocks from the hotel. It was demolished in 2005 after failing to receive landmark status from the city of Chicago. [ 9 ] A 49-storey apartment tower, 215 West, has taken its place, also featuring a parking garage. [ 10 ] When the Capital Garage in Washington, D.C. was built in 1927, it was reportedly the largest parking structure of its kind in the country. It was imploded in 1974. [ 11 ]
The movement of vehicles between floors can take place by means of:
Where the car park is built on sloping land, it may be split-level or have sloped parking.
Many parking structures are independent buildings dedicated exclusively to that use. The design loads for car parks are often less than the office building they serve (50 psf versus 80 [100] psf), leading to long floor spans of 55–65 feet that permit cars to park in rows without supporting columns in between [called long span]. Podium parking below high-rise and mid-rise buildings are often short-span 25–30 feet clear between columns, since office/residential/retail floors above require more support [100 psf per International Building Code]. Columns in short -span structures obstruct row based parking spaces and will be less efficient than long-span designs; parking efficiency is measured in cars per level square footage [car count/level area]. Common structural systems in the United States for long-span structures are prestressed concrete double-tee floor systems, post-tensioned cast-in-place concrete floor systems or short-span podium parking with post-tensioned slabs and drop panels [drop heads. Steel embeds or thicker slabs can eliminate the need for drop panels, providing higher clearances for higher profile vehicles.]
In recent times, parking structures built to serve residential and some business properties have been built as part of a larger building, often underground as part of the basement, such as the parking lot at the Atlantic Station redevelopment in Atlanta . This saves land for other uses (as opposed surface parking), is cheaper and more practical in most cases than a separate structure, and is hidden from view. It protects customers and their cars from weather such as rain, snow, or hot summer sunshine that raises a vehicle's interior temperature to extremely high levels. Underground parking of only two levels was considered an innovative concept in 1964, when developer Louis Lesser developed a two-level underground parking structure under six 10-storey high-rise residential halls at California State University, Los Angeles , which lacked space for horizontal expansion in the 176-acre (0.71 km 2 ) university. The simple two-level parking structure was considered unusual enough in 1964 that a separate newspaper section entitled "Parking Underground" described the parking lot as an innovative "concept" and as "subterranean spaces". [ 12 ] [ 13 ] In Toronto , a 2,400 space underground parking structure below Nathan Phillips Square is one of the world's largest.
Parking which serve shopping centers can be built adjacent to the center for easier access at each floor between shops and parking. One example is Mall of America in Bloomington, Minnesota , USA, which has two large parking lots attached to the building, at the eastern and western ends. A common position for parking within shopping centers in the UK is on the roof, around the various utility systems, enabling customers to take lifts straight down into the center. Examples of such are The Oracle in Reading and Festival Place in Basingstoke . Parking garages without mixed use can provide excellent uses for the Roof area: The Grove Parking Garage is the site for movies on its 8th level roof, [ 14 ] The Grand Prix of Long Beach, CA can be viewed from the Roof level of The Aquarium of the Pacific Parking Garage and The Pike Parking Garage (opposite the Queensway Structure) were built with a thickened post-tensioned roof slab to accommodate crowds of people.
These parking structures often have low ceiling clearances [7'-2" and 8'-4" for accessible parking], which restrict access by full-size vans and other large vehicles. On 15 December 2013, a man was killed during a robbery in the parking garage at The Mall at Short Hills in Millburn, New Jersey . The paramedics responding to the shooting were delayed because their ambulance was too large to enter the structure. [ 15 ]
In the United States, costs for parking garages are estimated to cost between $25,000 per space, with underground parking costing around $35,000 per space. [ 5 ]
Parking structures are subjected to the heavy and shifting loads of moving vehicles, and must bear the associated physical stresses. Expansion joints are used between sections not only for thermal expansion but to accommodate the flexing of the structure's sections due to vehicle traffic. Parking structures are generally not subject to building inspections after being checked for their initial occupancy permit. Seismic retrofits can be applied where earthquakes are an issue.
Some parking structures have partly collapsed, either during construction or years later. In July 2009 a fourth-floor section failed at the Centergy building in midtown Atlanta , pancaking down and destroying more than 30 vehicles but injuring no-one. In December 2007, a car crashed into the wall of the deck at the SouthPark Mall in Charlotte, North Carolina , weakening it and causing a small collapse which destroyed two cars below. On the same day, one under construction in Jacksonville, Florida collapsed as concrete was being poured on the sixth floor. [ 16 ]
In November 2008, the sudden collapse of the middle level of a deck in Montreal was preceded by warning signs some weeks before, including cracks and water leaks. [ 17 ]
In June 2012, the Algo Centre Mall 's rooftop parking deck collapsed into the building, crashing through the upper level lottery kiosk adjacent to the food court and escalators to the ground floor below, killing two people. [ 18 ]
In October 2012 four people were killed and nine more injured when a parking structure under construction at a campus of Miami-Dade College in Florida collapsed, [ 19 ] purportedly due to an unfinished column. [ 20 ]
The Surfside condominium's main building's collapse that killed ninety-eight people was likely caused by the failure of the long-term degradation of reinforced concrete structural support in the basement-level parking garage. [ 21 ]
As multi-storey car parks have become more common since the middle of the twentieth century, many constructions of such structures have been using precast concrete to reduce the construction time. The design involves putting parking structure parts together. The parts of precast concrete include multi-storey structural wall panels, interior and exterior columns, structural floors, girders , wall panels, stairs, and slabs. The precast concrete parts are transported using flatbed semi-trailers to the sites. The structural floor modules may need to be laid tilted during the transportation in order to cover as large floor area as possible while they can be easily transported on the roadways. The modules are lifted using precast concrete lifting anchor systems at the sites for assembly. Decorations may include using of covers to close the holes in the precast concrete that contains the lifting anchors, and installing facades to the exterior of the structures.
In modern construction of the precast modules, there are other features to improve the strength of the structure. An example is to use prestressed strands on post-tensioned concrete for the construction of the shear walls . Another example is the use of carbon-fiber-reinforced polymer to replace steel wire mesh to lighten the load and yield more corrosion resistance especially for the cold-climate areas which use salt for melting snow. [ 22 ]
These structures are not usually known for their architectural value. As Architectural Record has noted, "In the Pantheon of Building Types, the parking garage lurks somewhere in the vicinity of prisons and toll plazas." [ 23 ] The New York Times has labeled parking structures as "the grim afterthought of American design". [ 24 ]
A handful of structures have received considerable praise for their design, including
The term multistorey car park (often abbreviated to multistorey or multistory ) [ 3 ] is used in the United Kingdom, Hong Kong, and many Commonwealth of Nations countries, and it is nowadays most commonly spelled without a hyphen. [ 1 ] [ 2 ] In the United States, the term parking structure is used, [ 26 ] especially when it is necessary to distinguish such a structure from the "garage" connected with a house. In some places in North America, "parking garage" refers only to an indoor, often underground, structure. Outdoor, multi-level parking facilities are referred to by a number of regional terms:
Architects and structural engineers in the USA are likely to call it a parking structure since their work is all about structures and since that term is the vernacular in the United States. [ citation needed ] When constructed as the base of a high-rise, it is sometimes called a parking podium . [ 30 ] United States building codes use the term open parking garage to refer to a structure designed for car storage that has openings along at least 40% of the perimeter, as opposed to an enclosed parking garage that requires mechanical ventilation. [ 31 ] Natural or mechanical ventilation provides fresh air flow to disperse car exhaust in normal conditions, or hot gas and smoke in case of fire.
Typically parking consultants in the UK describe the number of car park floors in terms of "G+x". G stands for ground and x for the number of floors above ground. For example, G+5 is a multi-storey car park structure with a ground floor and 5 floors above that, i.e. a total of 6 floors. The preceding does not apply to the United States where B+x refers to basement levels ascending in number x while descending in elevation, L1 or ground level [unlike European standards where ground level is below level 1] with added levels as L2, L3, L4, etc. [ 26 ]
Structure car parks are car parks made of structural steel components connected to each other to carry the loads and provide full structural rigidity.
Steel is a high-strength material requiring less material than other types of structures like concrete and timber. Steel construction features:
The ceiling slab of the steel structure car park is typically made of composite material such as corrugated steel sheets and concrete. The surface of the first-floor parking can be left bare or covered with epoxy or tarmac.
Demand, steel features, and innovation have led to the development of a foundationless, modular, removable steel car park structure.
Parking demand often grows quickly, significantly and sometimes unexpectedly. Modular steel car parks could be the proper solution if the surface area available is not sufficient and can be expanded upward, or whenever it is not feasible to build up a multi-storey parking.
The development of the building concept of modular car parks came about by using the modular assembling method of vertical and horizontal elements (such as columns and beams)
Modular car park structures are versatile and can be built in phases or in different sizes and shape.
The solution makes it possible to develop a parking structure even in case of particular conditions or constraints, such as archaeological sites or city centres, because it allows:
These parking structures are generally demountable and can be relocated to avoid making the choice of converting a surface to parking area irrevocably. They could be used as permanent structures or are conceived as temporary parking facilities for temporary parking demand needs. A number of parking decks have been demounted after a few years – to make room for the development of a permanent structure – and relocated to respond to local parking demand.
The earliest use of an automated parking system (APS) was in Paris in 1905 at the Garage Rue de Ponthieu. [ 32 ] The APS consisted of a groundbreaking [ 32 ] multi-storey concrete structure with an internal elevator to transport cars to upper levels where attendants parked the cars. [ 33 ] A 1931 Popular Mechanics article speculated about design for an underground garage where the car is taken to a parking area by a conveyor and then an elevator to shuttles mounted on rails. [ 34 ]
The total cost of ownership of automated parking needs to be carefully considered. The actual cost of construction of automated car parks is typically higher than conventional car park structures, however, this can be offset by the higher space efficiency including reduced excavation waste from minimized footprints. The cost of the mechanical equipment needed to transport the cars needs to be added to the building cost. In addition, operation and maintenance costs of the mechanical equipment need to be added in order to determine the total cost of ownership. Other costs could be saved, for example, there is no need for an energy-intensive ventilating system , since cars are not driven inside and human cashiers or security personnel may not be needed. For naturally ventilated car parks structures, the ventilation equipment is not needed. [ 35 ]
Automated car parks rely on similar technology to that used for mechanical handling and document retrieval. The driver leaves the car in an entrance module, and it is then transported to a parking slot by a robotic trolley . For the driver, the process of parking is reduced to leaving the car inside an entrance module.
At peak periods a wait may occur before entering or leaving, because loading passengers and luggage occurs at the entrance and exit rather than at the parking stall. This loading blocks the entrance or exit from being available to others. It is generally not recommended to use automated car parks for high peak hour volume facilities.
Additional factors that need to be taken into consideration are:
Automotive factories and car dealerships often use automated car parks to store inventory, which makes best use of space if they operate in urban areas, plus the car park may be decorated to promote the brand. [ 36 ] For instance at the Autostadt there are two 60 meter/200 ft tall glass silos (AutoTürme) used as storage for new Volkswagens. The two towers are connected to the Volkswagen factory by a 700-metre tunnel. When cars arrive at the towers they are carried up at a speed of 1.5 metres per second. The render for the Autostadt shows 6 towers. When purchasing a car from Volkswagen (the main brand only, not the sub-brands) in select European countries, it is optional if the customer wants it delivered to the dealership where it was bought or if the customer wants to travel to Autostadt to pick it up. If the latter is chosen, the Autostadt supplies the customer with free entrance, meal tickets and a variety of events building up to the point where the customer can follow on screen as the automatic elevator picks up the selected car in one of the silos. The car is then transported out to the customer without having driven a single meter, and the odometer is thus on "0". [ 37 ]
Automated car parks have been popular for multistorey residential buildings in New York City and Paris . In Toronto , automated car parks are gradually catching in the downtown core condominium developments since the 2010s, due to developers having to meet city-mandated minimum parking space requirements while building on increasingly smaller lots. [ 38 ] [ 39 ]
Modern car parks utilize a variety of technologies to help motorists find unoccupied parking spaces, car location when returning to the vehicle and improve their experience. These include adaptive lighting , sensors and parking space LED indicators (red for occupied, green for available and blue is reserved for the disabled; above every parking space), indoor positioning system (IPS), including QR code , and mobile payment options. The Santa Monica Place shopping mall in California has cameras on each stall that can help count the lot occupancy and find lost cars. [ 41 ]
Online booking technology service providers have been created to help drivers find long-term parking in an automated manner, while also providing significant savings for those who book parking spaces ahead of time. They use real-time inventory management checking technology to display car parks with availability, sorted by price and distance from the airport.
Other recent developments in technology include Vehicles Detection and Count Systems, Point of Sale & Revenue Control Systems, Traffic & Capacity Monitoring Systems, Valet Parking Point of Sale, Management & Revenue Control Systems. These systems help in way finding for parking clients with space availability shown at every turn, space monitoring using retrofit wifi transmitters in each space to update the space availability signs and to alert parking management of bottle necks and intervention measures. Revenue Control, Capacity Management, and Valet Point of Sale is a major issue for Office and Retail parking management and is also a means of parking management intervention, where website update the status of all of these issues for exclusive use by management. Irvine Spectrum Center, Irvine CA, with 3 parking structures, uses all of these systems [ 42 ] The City of Santa Monica uses Traffic and Capacity Monitoring with its 30 parking structures. Disneyland, in Anaheim CA uses most of these hi-tech solutions on its 8 garages.
In 1991, a 1975 marine vessel was transformed into a floating pontoon multi-storey car parking facility. The ship was given the new name P-Arken (a pun on the words park and ark ) and it is permanently towed in Gothenburg 's harbour Lilla Bommen near Skeppsbron . [ 43 ] [ 44 ] [ 45 ] [ 46 ]
In November 2019, a fully-clad parking barge for automobiles was patented in the United States. Its angular sides are designed to protect against driving wind, rain, and debris. [ 47 ] [ 48 ]
In October 2009, the National Building Museum opened an exhibition solely devoted to the study of parking garages and their impact on the built environment. This exhibition, titled House of Cars: Innovation and the Parking Garage , [ 49 ] was on view until 11 July 2010. | https://en.wikipedia.org/wiki/Multistorey_car_park |
Multiuser detection deals with demodulation of the mutually interfering digital streams of information that occur in areas such as wireless communications, high-speed data transmission, DSL , satellite communication, digital television, and magnetic recording. [ 1 ] It is also being currently investigated for demodulation in low-power inter-chip and intra-chip communication. Multiuser detection encompasses both receiver technologies devoted to joint detection of all the interfering signals [ 2 ] or to single-user receivers which are interested in recovering only one user but are robustified against multiuser interference and not just background noise.
Mutual interference is unavoidable in modern spectrally efficient wireless systems: even when using orthogonal multiplexing systems such as TDMA, synchronous CDMA or OFDMA , multiuser interference originates from channel distortion and from out-of-cell interference. In addition, in multi-antenna ( MIMO ) systems, the digitally modulated streams emanating from different antennas interfere at the receiver, and the MIMO receiver uses multiuser detection techniques to separate them. [ 3 ] By exploiting the structure of the interfering signals, multiuser detection can increase spectral efficiency, receiver sensitivity, and the number of users the system can sustain.
Because of the mistaken belief in some quarters of the spread spectrum community that little could be gained from receivers more sophisticated than the single-user matched filter, multiuser detection did not start developing until the early 1980s. [ 4 ] Verdu [ 5 ] showed that the near-far problem suffered by CDMA was not inherent to this multiplexing technology and could be overcome by an optimum receiver that demodulates all users simultaneously. Verdu's receiver consisted of a bank of matched filters followed by a Viterbi algorithm . In the context of the capacity of the narrowband Gaussian two-user multiple-access channel, Cover [ 6 ] showed the achievability of the capacity region by means of a successive cancellation receiver, which decodes one user treating the other as noise, re-encodes its signal and subtracts it from the received signal. The same near-far resistance of the optimum receiver can be achieved with the decorrelating receiver proposed in. [ 7 ] Adaptive multiuser detectors that do not require prior knowledge of the interfering waveforms have also been proposed. [ 8 ] [ 9 ] | https://en.wikipedia.org/wiki/Multiuser_detection |
In statistics, in particular in the design of experiments , a multi-valued treatment is a treatment that can take on more than two values. It is related to the dose-response model in the medical literature.
Generally speaking, treatment levels may be finite or infinite as well as ordinal or cardinal, which leads to a large collection of possible treatment effects to be studied in applications. [ 1 ] One example is the effect of different levels of program participation (e.g. full-time and part-time) in a job training program. [ 2 ]
Assume there exists a finite collection of multi-valued treatment status T = { 0 , 1 , 2 , … , J , } {\displaystyle T=\{0,1,2,\ldots ,J,\}} with J some fixed integer. As in the potential outcomes framework, denote Y ( j ) ⊂ R {\displaystyle Y(j)\subset R} the collection of potential outcomes under the treatment J , and Y = ∑ j = 0 J D j Y ( j ) {\displaystyle Y=\textstyle \sum _{j=0}^{J}\displaystyle D_{j}Y(j)} denotes the observed outcome and D j {\displaystyle D_{j}} is an indicator that equals 1 when the treatment equals j and 0 when it does not equal j , leading to a fundamental problem of causal inference . [ 3 ] A general framework that analyzes ordered choice models in terms of marginal treatment effects and average treatment effects has been extensively discussed by Heckman and Vytlacil. [ 4 ]
Recent work in the econometrics and statistics literature has focused on estimation and inference for multivalued treatments and ignorability conditions for identifying the treatment effects. In the context of program evaluation, the propensity score has been generalized to allow for multi-valued treatments, [ 5 ] while other work has also focused on the role of the conditional mean independence assumption. [ 6 ] Other recent work has focused more on the large sample properties of an estimator of the marginal mean treatment effect conditional on a treatment level in the context of a difference-in-differences model, [ 7 ] and on the efficient estimation of multi-valued treatment effects in a semiparametric framework. [ 8 ] | https://en.wikipedia.org/wiki/Multivalued_treatment |
Multivariable calculus (also known as multivariate calculus ) is the extension of calculus in one variable to calculus with functions of several variables : the differentiation and integration of functions involving multiple variables ( multivariate ), rather than just one. [ 1 ]
Multivariable calculus may be thought of as an elementary part of calculus on Euclidean space . The special case of calculus in three dimensional space is often called vector calculus .
In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the domain is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces:
The consequence of the first difference is the difference in the definition of the limit and differentiation. Directional limits and derivatives define the limit and differential along a 1D parametrized curve, reducing the problem to the 1D case. Further higher-dimensional objects can be constructed from these operators.
The consequence of the second difference is the existence of multiple types of integration, including line integrals , surface integrals and volume integrals . Due to the non-uniqueness of these integrals, an antiderivative or indefinite integral cannot be properly defined.
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions.
A limit along a path may be defined by considering a parametrised path s ( t ) : R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}} in n-dimensional Euclidean space. Any function f ( x → ) : R n → R m {\displaystyle f({\overrightarrow {x}}):\mathbb {R} ^{n}\to \mathbb {R} ^{m}} can then be projected on the path as a 1D function f ( s ( t ) ) {\displaystyle f(s(t))} . The limit of f {\displaystyle f} to the point s ( t 0 ) {\displaystyle s(t_{0})} along the path s ( t ) {\displaystyle s(t)} can hence be defined as
Note that the value of this limit can be dependent on the form of s ( t ) {\displaystyle s(t)} , i.e. the path chosen, not just the point which the limit approaches. [ 1 ] : 19–22 For example, consider the function
If the point ( 0 , 0 ) {\displaystyle (0,0)} is approached through the line y = k x {\displaystyle y=kx} , or in parametric form:
Then the limit along the path will be:
On the other hand, if the path y = ± x 2 {\displaystyle y=\pm x^{2}} (or parametrically, x ( t ) = t , y ( t ) = ± t 2 {\displaystyle x(t)=t,\,y(t)=\pm t^{2}} ) is chosen, then the limit becomes:
Since taking different paths towards the same point yields different values, a general limit at the point ( 0 , 0 ) {\displaystyle (0,0)} cannot be defined for the function.
A general limit can be defined if the limits to a point along all possible paths converge to the same value, i.e. we say for a function f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} that the limit of f {\displaystyle f} to some point x 0 ∈ R n {\displaystyle x_{0}\in \mathbb {R} ^{n}} is L, if and only if
for all continuous functions s ( t ) : R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}} such that s ( t 0 ) = x 0 {\displaystyle s(t_{0})=x_{0}} .
From the concept of limit along a path, we can then derive the definition for multivariate continuity in the same manner, that is: we say for a function f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} that f {\displaystyle f} is continuous at the point x 0 {\displaystyle x_{0}} , if and only if
for all continuous functions s ( t ) : R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}} such that s ( t 0 ) = x 0 {\displaystyle s(t_{0})=x_{0}} .
As with limits, being continuous along one path s ( t ) {\displaystyle s(t)} does not imply multivariate continuity.
Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example. [ 1 ] : 17–19 For example, for a real-valued function f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } with two real-valued parameters, f ( x , y ) {\displaystyle f(x,y)} , continuity of f {\displaystyle f} in x {\displaystyle x} for fixed y {\displaystyle y} and continuity of f {\displaystyle f} in y {\displaystyle y} for fixed x {\displaystyle x} does not imply continuity of f {\displaystyle f} .
Consider
It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle ( 0 , 1 ) × ( 0 , 1 ) {\displaystyle (0,1)\times (0,1)} . Furthermore, the functions defined for constant x {\displaystyle x} and y {\displaystyle y} and 0 ≤ a ≤ 1 {\displaystyle 0\leq a\leq 1} by
are continuous. Specifically,
However, consider the parametric path x ( t ) = t , y ( t ) = t {\displaystyle x(t)=t,\,y(t)=t} . The parametric function becomes
Therefore,
It is hence clear that the function is not multivariate continuous, despite being continuous in both coordinates.
From the Lipschitz continuity condition for f {\displaystyle f} we have
where K {\displaystyle K} is the Lipschitz constant. Note also that, as s ( t ) {\displaystyle s(t)} is continuous at t 0 {\displaystyle t_{0}} , for every δ > 0 {\displaystyle \delta >0} there exists a ϵ > 0 {\displaystyle \epsilon >0} such that | s ( t ) − s ( t 0 ) | < δ {\displaystyle |s(t)-s(t_{0})|<\delta } ∀ | t − t 0 | < ϵ {\displaystyle \forall |t-t_{0}|<\epsilon } .
Hence, for every α > 0 {\displaystyle \alpha >0} , choose δ = α K {\displaystyle \delta ={\frac {\alpha }{K}}} ; there exists an ϵ > 0 {\displaystyle \epsilon >0} such that for all t {\displaystyle t} satisfying | t − t 0 | < ϵ {\displaystyle |t-t_{0}|<\epsilon } , | s ( t ) − s ( t 0 ) | < δ {\displaystyle |s(t)-s(t_{0})|<\delta } , and | f ( s ( t ) ) − f ( s ( t 0 ) ) | ≤ K | s ( t ) − s ( t 0 ) | < K δ = α {\displaystyle |f(s(t))-f(s(t_{0}))|\leq K|s(t)-s(t_{0})|<K\delta =\alpha } . Hence lim t → t 0 f ( s ( t ) ) {\displaystyle \lim _{t\to t_{0}}f(s(t))} converges to f ( s ( t 0 ) ) {\displaystyle f(s(t_{0}))} regardless of the precise form of s ( t ) {\displaystyle s(t)} .
The derivative of a single-variable function is defined as
Using the extension of limits discussed above, one can then extend the definition of the derivative to a scalar-valued function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } along some path s ( t ) : R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}} :
Unlike limits, for which the value depends on the exact form of the path s ( t ) {\displaystyle s(t)} , it can be shown that the derivative along the path depends only on the tangent vector of the path at s ( t 0 ) {\displaystyle s(t_{0})} , i.e. s ′ ( t 0 ) {\displaystyle s'(t_{0})} , provided that f {\displaystyle f} is Lipschitz continuous at s ( t 0 ) {\displaystyle s(t_{0})} , and that the limit exits for at least one such path.
For s ( t ) {\displaystyle s(t)} continuous up to the first derivative (this statement is well defined as s {\displaystyle s} is a function of one variable), we can write the Taylor expansion of s {\displaystyle s} around t 0 {\displaystyle t_{0}} using Taylor's theorem to construct the remainder:
where τ ∈ [ t 0 , t ] {\displaystyle \tau \in [t_{0},t]} .
Substituting this into 10 ,
where τ ( h ) ∈ [ t 0 , t 0 + h ] {\displaystyle \tau (h)\in [t_{0},t_{0}+h]} .
Lipschitz continuity gives us | f ( x ) − f ( y ) | ≤ K | x − y | {\displaystyle |f(x)-f(y)|\leq K|x-y|} for some finite K {\displaystyle K} , ∀ x , y ∈ R n {\displaystyle \forall x,y\in \mathbb {R} ^{n}} . It follows that | f ( x + O ( h ) ) − f ( x ) | ∼ O ( h ) {\displaystyle |f(x+O(h))-f(x)|\sim O(h)} .
Note also that given the continuity of s ′ ( t ) {\displaystyle s'(t)} , s ′ ( τ ) = s ′ ( t 0 ) + O ( h ) {\displaystyle s'(\tau )=s'(t_{0})+O(h)} as h → 0 {\displaystyle h\to 0} .
Substituting these two conditions into 12 ,
whose limit depends only on s ′ ( t 0 ) {\displaystyle s'(t_{0})} as the dominant term.
It is therefore possible to generate the definition of the directional derivative as follows: The directional derivative of a scalar-valued function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } along the unit vector u ^ {\displaystyle {\hat {\mathbf {u}}}} at some point x 0 ∈ R n {\displaystyle x_{0}\in \mathbb {R} ^{n}} is
or, when expressed in terms of ordinary differentiation,
which is a well defined expression because f ( x 0 + u ^ t ) {\displaystyle f(x_{0}+{\hat {\mathbf {u}}}t)} is a scalar function with one variable in t {\displaystyle t} .
It is not possible to define a unique scalar derivative without a direction; it is clear for example that ∇ u ^ f ( x 0 ) = − ∇ − u ^ f ( x 0 ) {\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=-\nabla _{-{\hat {\mathbf {u}}}}f(x_{0})} . It is also possible for directional derivatives to exist for some directions but not for others.
The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. [ 1 ] : 26ff
A partial derivative may be thought of as the directional derivative of the function along a coordinate axis.
Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus , the del operator ( ∇ {\displaystyle \nabla } ) is used to define the concepts of gradient , divergence , and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as a linear transformation which directly varies from point to point in the domain of the function.
Differential equations containing partial derivatives are called partial differential equations or PDEs. These equations are generally more difficult to solve than ordinary differential equations , which contain derivatives with respect to only one variable. [ 1 ] : 654ff
The multiple integral extends the concept of the integral to functions of any number of variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubini's theorem guarantees that a multiple integral may be evaluated as a repeated integral or iterated integral as long as the integrand is continuous throughout the domain of integration. [ 1 ] : 367ff
The surface integral and the line integral are used to integrate over curved manifolds such as surfaces and curves .
In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus: [ 1 ] : 543ff
In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem , which applies to the integration of differential forms over manifolds . [ 2 ]
Techniques of multivariable calculus are used to study many objects of interest in the material world. In particular,
Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom . Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics .
Multivariate calculus is used in the optimal control of continuous time dynamic systems . It is used in regression analysis to derive formulas for estimating relationships among various sets of empirical data .
Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior. In economics , for example, consumer choice over a variety of goods, and producer choice over various inputs to use and outputs to produce, are modeled with multivariate calculus.
Non-deterministic, or stochastic systems can be studied using a different kind of mathematics, such as stochastic calculus . | https://en.wikipedia.org/wiki/Multivariable_calculus |
Multivariate optical computing , also known as molecular factor computing, is an approach to the development of compressed sensing spectroscopic instruments, particularly for industrial applications such as process analytical support. "Conventional" spectroscopic methods often employ multivariate and chemometric methods, such as multivariate calibration , pattern recognition , and classification , to extract analytical information (including concentration) from data collected at many different wavelengths. Multivariate optical computing uses an optical computer to analyze the data as it is collected. The goal of this approach is to produce instruments which are simple and rugged, yet retain the benefits of multivariate techniques for the accuracy and precision of the result.
An instrument which implements this approach may be described as a multivariate optical computer . Since it describes an approach, rather than any specific wavelength range, multivariate optical computers may be built using a variety of different instruments (including Fourier Transform Infrared ( FTIR ) [ 1 ] and Raman [ 2 ] ).
The "software" in multivariate optical computing is encoded directly into an optical element spectral calculation engine such as an interference filter based multivariate optical element (MOE), holographic grating , liquid crystal tunable filter , spatial light modulator (SLM), or digital micromirror device (DMD) and is specific to the particular application. The optical pattern for the spectral calculation engine is designed for the specific purpose of measuring the magnitude of that multi-wavelength pattern in the spectrum of a sample, without actually measuring a spectrum. [ 3 ]
Multivariate optical computing allows instruments to be made with the mathematics of pattern recognition designed directly into an optical computer, which extracts information from light without recording a spectrum. This makes it possible to achieve the speed, dependability, and ruggedness necessary for real time, in-line process control instruments.
Multivariate optical computing encodes an analog optical regression vector of a transmission function for an optical element. Light which emanates from a sample contains the spectral information of that sample, whether the spectrum is discovered or not. As light passes from a sample through the element, the normalized intensity, which is detected by a broad band detector, is proportional to the dot product of the regression vector with that spectrum, i.e. is proportional to the concentration of the analyte for which the regression vector was designed. The quality of the analysis is then equal to the quality of the regression vector which is encoded. If the resolution of the regression vector is encoded to the resolution of the laboratory instrument from which that regression vector was designed and the resolution of the detector is equivalent, then the measurement made by Multivariate Optical Computing will be equivalent to that laboratory instrument by conventional means. The technique is making headway in a niche market for harsh environment detection. Specifically the technique has been adopted for use in the oil industry for detection of hydrocarbon composition in oil wells and pipeline monitoring. In such situations, laboratory quality measurements are necessary, but in harsh environments. [ 4 ]
Although the concept of using a single optical element for analyte regression and detection was suggested in 1986, [ 5 ] the first full MOC concept device was published in 1997 from the Myrick group at the University of South Carolina , [ 6 ] with a subsequent demonstration in 2001. [ 7 ] The technique has received much recognition in the optics industry as a new method to perform optical analysis with advantages for harsh environment sensing. [ 4 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] The technique has been applied to Raman spectroscopy, [ 2 ] [ 11 ] [ 12 ] fluorescence spectroscopy , [ 12 ] [ 13 ] [ 14 ] [ 15 ] [ 16 ] [ 17 ] [ 18 ] [ 19 ] absorbance spectroscopy in the UV-Vis , [ 7 ] [ 20 ] NIR [ 21 ] [ 22 ] [ 23 ] and MIR , [ 24 ] [ 25 ] microscopy , [ 26 ] reflectance spectroscopy [ 27 ] and hyperspectral imaging . [ 11 ] [ 20 ] [ 22 ] [ 23 ] [ 27 ] [ 28 ] [ 29 ] In the years since first demonstration, applications have been demonstrated for defence, [ 30 ] forensics, [ 31 ] monitoring of chemical reactions, [ 6 ] [ 32 ] environmental monitoring , [ 8 ] [ 33 ] [ 34 ] recycling, [ 21 ] [ 35 ] food and drug, [ 28 ] [ 36 ] medical and life sciences, [ 14 ] [ 15 ] [ 16 ] [ 17 ] [ 18 ] [ 19 ] and the petroleum industry. [ 4 ] [ 10 ] [ 25 ] [ 32 ] [ 37 ] [ 38 ] [ 39 ] [ 40 ] [ 41 ] [ 42 ] The first published demonstration for use of MOC in the harsh environments, was 2012 with a laboratory study with temperatures from 150F to 350F and pressures from 3000psi to 20,000psi, [ 10 ] followed in 2013 with field trials in oil wells. [ 42 ] | https://en.wikipedia.org/wiki/Multivariate_optical_computing |
A multivariate optical element (MOE), is the key part of a multivariate optical computer ; an alternative to conventional spectrometry for the chemical analysis of materials.
It is helpful to understand how light is processed in a multivariate optical computer, as compared to how it is processed in a spectrometer. For example, if we are studying the composition of a powder mixture using diffuse reflectance, a suitable light source is directed at the powder mixture and light is collected, usually with a lens, after it has scattered from the powder surface. Light entering a spectrometer first strikes a device (either a grating or interferometer ) that separates light of different wavelengths to be measured. A series of independent measurements is used to estimate the full spectrum of the mixture, and the spectrometer renders a measurement of the spectral intensity at many wavelengths. Multivariate statistics can then be applied to the spectrum produced.
In contrast, when using multivariate optical computing, the light entering the instrument strikes an application specific multivariate optical element, which is uniquely tuned to the pattern that needs to be measured using multivariate analysis.
This system can produce the same result that multivariate analysis of a spectrum would produce. Thus, it can generally produce the same accuracy as laboratory grade spectroscopic systems, but with the fast speed inherent with a pure, passive, optical computer. The multivariate optical computer makes use of optical computing to realize the performance of a full spectroscopic system using traditional multivariate analysis. A side benefit is that the throughput and efficiency of the system is higher than conventional spectrometers, which increases the speed of analysis by orders of magnitude.
While each chemical problem presents its own unique challenges and opportunities, the design of a system for a specific analysis is complex and requires the assembly of several pieces of a spectroscopic puzzle. The data necessary for a successful design are spectral characteristics of light sources, detectors and a variety of optics to be used in the final assemblage, dispersion characteristics of the materials used in the wavelength range of interest, and a set of calibrated sample spectra for pattern-recognition-based analysis. With these pieces assembled, suitable application specific multivariate optical computer designs can be generated and the performance accurately modeled and predicted. | https://en.wikipedia.org/wiki/Multivariate_optical_element |
In multilinear algebra , a multivector , sometimes called Clifford number or multor , [ 1 ] is an element of the exterior algebra Λ( V ) of a vector space V . This algebra is graded , associative and alternating , and consists of linear combinations of simple k -vectors [ 2 ] (also known as decomposable k -vectors [ 3 ] or k -blades ) of the form
where v 1 , … , v k {\displaystyle v_{1},\ldots ,v_{k}} are in V .
A k -vector is such a linear combination that is homogeneous of degree k (all terms are k -blades for the same k ). Depending on the authors, a "multivector" may be either a k -vector or any element of the exterior algebra (any linear combination of k -blades with potentially differing values of k ). [ 4 ]
In differential geometry , a k -vector is usually a vector in the exterior algebra of the tangent vector space of a smooth manifold ; that is, it is an antisymmetric tensor obtained by taking linear combinations of the exterior product of k tangent vectors , for some integer k ≥ 0 . A differential k -form is a k -vector in the exterior algebra of the dual of the tangent space, which is also the dual of the exterior algebra of the tangent space.
For k = 0, 1, 2 and 3 , k -vectors are often called respectively scalars , vectors , bivectors and trivectors ; they are respectively dual to 0-forms, 1-forms, 2-forms and 3-forms . [ 5 ] [ 6 ]
The exterior product (also called the wedge product) used to construct multivectors is multilinear (linear in each input), associative and alternating. This means for vectors u , v and w in a vector space V and for scalars α , β , the exterior product has the properties:
The exterior product of k vectors or a sum of such products (for a single k ) is called a grade k multivector, or a k -vector. The maximum grade of a multivector is the dimension of the vector space V .
Linearity in either input together with the alternating property implies linearity in the other input. The multilinearity of the exterior product allows a multivector to be expressed as a linear combination of exterior products of basis vectors of V . The exterior product of k basis vectors of V is the standard way of constructing each basis element for the space of k -vectors, which has dimension ( n k ) in the exterior algebra of an n -dimensional vector space. [ 2 ]
The k -vector obtained from the exterior product of k separate vectors in an n -dimensional space has components that define the projected ( k − 1) -volumes of the k - parallelotope spanned by the vectors. The square root of the sum of the squares of these components defines the volume of the k -parallelotope. [ 2 ] [ 7 ]
The following examples show that a bivector in two dimensions measures the area of a parallelogram, and the magnitude of a bivector in three dimensions also measures the area of a parallelogram. Similarly, a three-vector in three dimensions measures the volume of a parallelepiped.
It is easy to check that the magnitude of a three-vector in four dimensions measures the volume of the parallelepiped spanned by these vectors.
Properties of multivectors can be seen by considering the two-dimensional vector space V = R 2 . Let the basis vectors be e 1 and e 2 , so u and v are given by
and the multivector u ∧ v , also called a bivector, is computed to be
The vertical bars denote the determinant of the matrix, which is the area of the parallelogram spanned by the vectors u and v . The magnitude of u ∧ v is the area of this parallelogram. Notice that because V has dimension two the basis bivector e 1 ∧ e 2 is the only multivector in Λ V .
The relationship between the magnitude of a multivector and the area or volume spanned by the vectors is an important feature in all dimensions. Furthermore, the linear functional version of a multivector that computes this volume is known as a differential form.
More features of multivectors can be seen by considering the three-dimensional vector space V = R 3 . In this case, let the basis vectors be e 1 , e 2 , and e 3 , so u , v and w are given by
and the bivector u ∧ v is computed to be
The components of this bivector are the same as the components of the cross product. The magnitude of this bivector is the square root of the sum of the squares of its components.
This shows that the magnitude of the bivector u ∧ v is the area of the parallelogram spanned by the vectors u and v as it lies in the three-dimensional space V . The components of the bivector are the projected areas of the parallelogram on each of the three coordinate planes.
Notice that because V has dimension three, there is one basis three-vector in Λ V . Compute the three-vector
u ∧ v ∧ w = ( u ∧ v ) ∧ w = ( | u 2 v 2 u 3 v 3 | ( e 2 ∧ e 3 ) + | u 1 v 1 u 3 v 3 | ( e 1 ∧ e 3 ) + | u 1 v 1 u 2 v 2 | ( e 1 ∧ e 2 ) ) ∧ ( w 1 e 1 + w 2 e 2 + w 3 e 3 ) = | u 2 v 2 u 3 v 3 | ( e 2 ∧ e 3 ) ∧ ( w 1 e 1 + w 2 e 2 + w 3 e 3 ) + | u 1 v 1 u 3 v 3 | ( e 1 ∧ e 3 ) ∧ ( w 1 e 1 + w 2 e 2 + w 3 e 3 ) + | u 1 v 1 u 2 v 2 | ( e 1 ∧ e 2 ) ∧ ( w 1 e 1 + w 2 e 2 + w 3 e 3 ) = | u 2 v 2 u 3 v 3 | ( e 2 ∧ e 3 ) ∧ w 1 e 1 + | u 2 v 2 u 3 v 3 | ( e 2 ∧ e 3 ) ∧ w 2 e 2 + | u 2 v 2 u 3 v 3 | ( e 2 ∧ e 3 ) ∧ w 3 e 3 e 2 ∧ e 2 = 0 ; e 3 ∧ e 3 = 0 + | u 1 v 1 u 3 v 3 | ( e 1 ∧ e 3 ) ∧ w 1 e 1 + | u 1 v 1 u 3 v 3 | ( e 1 ∧ e 3 ) ∧ w 2 e 2 + | u 1 v 1 u 3 v 3 | ( e 1 ∧ e 3 ) ∧ w 3 e 3 e 1 ∧ e 1 = 0 ; e 3 ∧ e 3 = 0 + | u 1 v 1 u 2 v 2 | ( e 1 ∧ e 2 ) ∧ w 1 e 1 + | u 1 v 1 u 2 v 2 | ( e 1 ∧ e 2 ) ∧ w 2 e 2 + | u 1 v 1 u 2 v 2 | ( e 1 ∧ e 2 ) ∧ w 3 e 3 e 1 ∧ e 1 = 0 ; e 2 ∧ e 2 = 0 = | u 2 v 2 u 3 v 3 | ( e 2 ∧ e 3 ) ∧ w 1 e 1 + | u 1 v 1 u 3 v 3 | ( e 1 ∧ e 3 ) ∧ w 2 e 2 + | u 1 v 1 u 2 v 2 | ( e 1 ∧ e 2 ) ∧ w 3 e 3 = − w 1 | u 2 v 2 u 3 v 3 | ( e 2 ∧ e 1 ∧ e 3 ) − w 2 | u 1 v 1 u 3 v 3 | ( e 1 ∧ e 2 ∧ e 3 ) + w 3 | u 1 v 1 u 2 v 2 | ( e 1 ∧ e 2 ∧ e 3 ) = w 1 | u 2 v 2 u 3 v 3 | ( e 1 ∧ e 2 ∧ e 3 ) − w 2 | u 1 v 1 u 3 v 3 | ( e 1 ∧ e 2 ∧ e 3 ) + w 3 | u 1 v 1 u 2 v 2 | ( e 1 ∧ e 2 ∧ e 3 ) = ( w 1 | u 2 v 2 u 3 v 3 | − w 2 | u 1 v 1 u 3 v 3 | + w 3 | u 1 v 1 u 2 v 2 | ) ( e 1 ∧ e 2 ∧ e 3 ) = | u 1 v 1 w 1 u 2 v 2 w 2 u 3 v 3 w 3 | ( e 1 ∧ e 2 ∧ e 3 ) {\displaystyle {\begin{aligned}&\mathbf {u} \wedge \mathbf {v} \wedge \mathbf {w} =(\mathbf {u} \wedge \mathbf {v} )\wedge \mathbf {w} \\{}={}&\left({\begin{vmatrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)+{\begin{vmatrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{3}\right)+{\begin{vmatrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\right)\right)\wedge \left(w_{1}\mathbf {e} _{1}+w_{2}\mathbf {e} _{2}+w_{3}\mathbf {e} _{3}\right)\\{}={}&{\begin{vmatrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)\wedge \left(w_{1}\mathbf {e} _{1}+w_{2}\mathbf {e} _{2}+w_{3}\mathbf {e} _{3}\right)\\&{}+{\begin{vmatrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{3}\right)\wedge \left(w_{1}\mathbf {e} _{1}+w_{2}\mathbf {e} _{2}+w_{3}\mathbf {e} _{3}\right)\\&{}+{\begin{vmatrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\right)\wedge \left(w_{1}\mathbf {e} _{1}+w_{2}\mathbf {e} _{2}+w_{3}\mathbf {e} _{3}\right)\\{}={}&{\begin{vmatrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)\wedge w_{1}\mathbf {e} _{1}+{\cancel {{\begin{vmatrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)\wedge w_{2}\mathbf {e} _{2}}}+{\cancel {{\begin{vmatrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)\wedge w_{3}\mathbf {e} _{3}}}&&\mathbf {e} _{2}\wedge \mathbf {e} _{2}=0;\mathbf {e} _{3}\wedge \mathbf {e} _{3}=0\\&{}+{\cancel {{\begin{vmatrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{3}\right)\wedge w_{1}\mathbf {e} _{1}}}+{\begin{vmatrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{3}\right)\wedge w_{2}\mathbf {e} _{2}+{\cancel {{\begin{vmatrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{3}\right)\wedge w_{3}\mathbf {e} _{3}}}&&\mathbf {e} _{1}\wedge \mathbf {e} _{1}=0;\mathbf {e} _{3}\wedge \mathbf {e} _{3}=0\\&{}+{\cancel {{\begin{vmatrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\right)\wedge w_{1}\mathbf {e} _{1}}}+{\cancel {{\begin{vmatrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\right)\wedge w_{2}\mathbf {e} _{2}}}+{\begin{vmatrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\right)\wedge w_{3}\mathbf {e} _{3}&&\mathbf {e} _{1}\wedge \mathbf {e} _{1}=0;\mathbf {e} _{2}\wedge \mathbf {e} _{2}=0\\{}={}&{\begin{vmatrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)\wedge w_{1}\mathbf {e} _{1}+{\begin{vmatrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{3}\right)\wedge w_{2}\mathbf {e} _{2}+{\begin{vmatrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\right)\wedge w_{3}\mathbf {e} _{3}\\{}={}&-w_{1}{\begin{vmatrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{2}\wedge \mathbf {e} _{1}\wedge \mathbf {e} _{3}\right)-w_{2}{\begin{vmatrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)+w_{3}{\begin{vmatrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)\\{}={}&w_{1}{\begin{vmatrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)-w_{2}{\begin{vmatrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)+w_{3}{\begin{vmatrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)\\{}={}&\left(w_{1}{\begin{vmatrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{vmatrix}}-w_{2}{\begin{vmatrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{vmatrix}}+w_{3}{\begin{vmatrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{vmatrix}}\right)\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)\\{}={}&{\begin{vmatrix}u_{1}&v_{1}&w_{1}\\u_{2}&v_{2}&w_{2}\\u_{3}&v_{3}&w_{3}\end{vmatrix}}\left(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3}\right)\\\end{aligned}}}
This shows that the magnitude of the three-vector u ∧ v ∧ w is the volume of the parallelepiped spanned by the three vectors u , v and w .
In higher-dimensional spaces, the component three-vectors are projections of the volume of a parallelepiped onto the coordinate three-spaces, and the magnitude of the three-vector is the volume of the parallelepiped as it sits in the higher-dimensional space.
In this section, we consider multivectors on a projective space P n , which provide a convenient set of coordinates for lines, planes and hyperplanes that have properties similar to the homogeneous coordinates of points, called Grassmann coordinates . [ 8 ]
Points in a real projective space P n are defined to be lines through the origin of the vector space R n +1 . For example, the projective plane P 2 is the set of lines through the origin of R 3 . Thus, multivectors defined on R n +1 can be viewed as multivectors on P n .
A convenient way to view a multivector on P n is to examine it in an affine component of P n , which is the intersection of the lines through the origin of R n +1 with a selected hyperplane, such as H: x n +1 = 1 . Lines through the origin of R 3 intersect the plane E: z = 1 to define an affine version of the projective plane that only lacks the points for which z = 0 , called the points at infinity.
Points in the affine component E: z = 1 of the projective plane P 2 have coordinates x = ( x , y , 1) . A linear combination of two points p = ( p 1 , p 2 , 1) and q = ( q 1 , q 2 , 1) defines a plane in R 3 that intersects E in the line joining p and q . The multivector p ∧ q defines a parallelogram in R 3 given by
Notice that substitution of α p + β q for p multiplies this multivector by a constant. Therefore, the components of p ∧ q are homogeneous coordinates for the plane through the origin of R 3 .
The set of points x = ( x , y , 1) on the line through p and q is the intersection of the plane defined by p ∧ q with the plane E: z = 1 . These points satisfy x ∧ p ∧ q = 0 , that is,
which simplifies to the equation of a line
This equation is satisfied by points x = α p + β q for real values of α and β.
The three components of p ∧ q that define the line λ are called the Grassmann coordinates of the line. Because three homogeneous coordinates define both a point and a line, the geometry of points is said to be dual to the geometry of lines in the projective plane. This is called the principle of duality .
Three-dimensional projective space P 3 consists of all lines through the origin of R 4 . Let the three-dimensional hyperplane, H: w = 1 , be the affine component of projective space defined by the points x = ( x , y , z , 1) . The multivector p ∧ q ∧ r defines a parallelepiped in R 4 given by
Notice that substitution of α p + β q + γ r for p multiplies this multivector by a constant. Therefore, the components of p ∧ q ∧ r are homogeneous coordinates for the 3-space through the origin of R 4 .
A plane in the affine component H: w = 1 is the set of points x = ( x , y , z , 1) in the intersection of H with the 3-space defined by p ∧ q ∧ r . These points satisfy x ∧ p ∧ q ∧ r = 0 , that is,
which simplifies to the equation of a plane
This equation is satisfied by points x = α p + β q + γ r for real values of α , β and γ .
The four components of p ∧ q ∧ r that define the plane λ are called the Grassmann coordinates of the plane. Because four homogeneous coordinates define both a point and a plane in projective space, the geometry of points is dual to the geometry of planes.
A line as the join of two points: In projective space the line λ through two points p and q can be viewed as the intersection of the affine space H: w = 1 with the plane x = α p + β q in R 4 . The multivector p ∧ q provides homogeneous coordinates for the line
These are known as the Plücker coordinates of the line, though they are also an example of Grassmann coordinates.
A line as the intersection of two planes: A line μ in projective space can also be defined as the set of points x that form the intersection of two planes π and ρ defined by grade three multivectors, so the points x are the solutions to the linear equations
In order to obtain the Plucker coordinates of the line μ , map the multivectors π and ρ to their dual point coordinates using the right complement, denoted by an overline, as in [ 9 ]
then
So, the Plücker coordinates of the line μ are given by
where the underline denotes the left complement. The left complement of the wedge product of right complements is called the antiwedge product, denoted by a downward pointing wedge, allowing us to write μ = π ∨ ρ . {\displaystyle \mu =\pi \vee \rho .}
W. K. Clifford combined multivectors with the inner product defined on the vector space, in order to obtain a general construction for hypercomplex numbers that includes the usual complex numbers and Hamilton's quaternions . [ 10 ] [ 11 ]
The Clifford product between two vectors u and v is bilinear and associative like the exterior product, and has the additional property that the multivector uv is coupled to the inner product u ⋅ v by Clifford's relation,
Clifford's relation retains the anticommuting property for vectors that are perpendicular. This can be seen from the mutually orthogonal unit vectors e i , i = 1, ..., n in R n : Clifford's relation yields
which shows that the basis vectors mutually anticommute,
In contrast to the exterior product, the Clifford product of a vector with itself is not zero. To see this, compute the product
which yields
The set of multivectors constructed using Clifford's product yields an associative algebra known as a Clifford algebra . Inner products with different properties can be used to construct different Clifford algebras. [ 12 ] [ 13 ]
The term k-blade was used in Clifford Algebra to Geometric Calculus (1984) [ 14 ]
Multivectors play a central role in the mathematical formulation of physics known as geometric algebra. According to David Hestenes ,
In 2003 the term blade for a multivector that can be written as the exterior product of [a scalar and] a set of vectors was used by C. Doran and A. Lasenby. Here, by the statement "Any multivector can be expressed as the sum of blades", scalars are implicitly defined as 0-blades. [ 16 ]
In geometric algebra , a multivector is defined to be the sum of different-grade k -blades , such as the summation of a scalar , a vector , and a 2-vector. [ 17 ] A sum of only k -grade components is called a k -vector, [ 18 ] or a homogeneous multivector. [ 19 ]
The highest grade element in a space is called a pseudoscalar .
If a given element is homogeneous of a grade k , then it is a k -vector, but not necessarily a k -blade. Such an element is a k -blade when it can be expressed as the exterior product of k vectors. A geometric algebra generated by a four-dimensional vector space illustrates the point with an example: The sum of any two blades with one taken from the XY-plane and the other taken from the ZW-plane will form a 2-vector that is not a 2-blade. In a geometric algebra generated by a vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.
In the presence of a volume form (such as given an inner product and an orientation), pseudovectors and pseudoscalars can be identified with vectors and scalars, which is routine in vector calculus , but without a volume form this cannot be done without making an arbitrary choice.
In the algebra of physical space (the geometric algebra of Euclidean 3-space, used as a model of (3+1)-spacetime), a sum of a scalar and a vector is called a paravector , and represents a point in spacetime (the vector the space, the scalar the time).
A bivector is an element of the antisymmetric tensor product of a tangent space with itself.
In geometric algebra , also, a bivector is a grade 2 element (a 2-vector) resulting from the wedge product of two vectors, and so it is geometrically an oriented area , in the same way a vector is an oriented line segment. If a and b are two vectors, the bivector a ∧ b has
Bivectors are connected to pseudovectors , and are used to represent rotations in geometric algebra.
As bivectors are elements of a vector space Λ 2 V (where V is a finite-dimensional vector space with dim V = n ), it makes sense to define an inner product on this vector space as follows. First, write any element F ∈ Λ 2 V in terms of a basis ( e i ∧ e j ) 1 ≤ i < j ≤ n of Λ 2 V as
where the Einstein summation convention is being used.
Now define a map G : Λ 2 V × Λ 2 V → R by insisting that
where G a b c d {\displaystyle G_{abcd}} are a set of numbers.
Bivectors play many important roles in physics, for example, in the classification of electromagnetic fields . | https://en.wikipedia.org/wiki/Multivector |
In mathematical set theory , the multiverse view is that there are many models of set theory, but no "absolute", "canonical" or "true" model. The various models are all equally valid or true, though some may be more useful or attractive than others. The opposite view is the "universe" view of set theory in which all sets are contained in some single ultimate model.
The collection of countable transitive models of ZFC (in some universe) is called the hyperverse and is very similar to the "multiverse".
A typical difference between the universe and multiverse views is the attitude to the continuum hypothesis . In the universe view the continuum hypothesis is a meaningful question that is either true or false though we have not yet been able to decide which. In the multiverse view it is meaningless to ask whether the continuum hypothesis is true or false before selecting a model of set theory. Another difference is that the statement "For every transitive model of ZFC there is a larger model of ZFC in which it is countable" is true in some versions of the multiverse view of mathematics but is false in the universe view.
This mathematical logic -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Multiverse_(set_theory) |
Multiverse Computing is a Spanish quantum computing software company headquartered in San Sebastián , Spain, with offices in Paris, Munich, London, Toronto and Sherbrooke , Canada. The company applies artificial intelligence (AI), quantum and quantum-inspired algorithms to problems in energy, logistics, manufacturing, mobility, life sciences, finance, cybersecurity, chemistry, materials science and aerospace. [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ]
Its quantum and quantum-inspired software platform Singularity is designed to allow users without prior knowledge of quantum computing to use quantum algorithms via tools such as Microsoft Excel . [ 7 ]
Multiverse was co-founded in 2019 by Enrique Lizaso, Román Orús , Alfonso Rubio and Sam Mugel. [ 8 ] Lizaso, treasurer and member of the governing board of the European Quantum Industry Consortium, and Orús, Ikerbasque Research Professor at Donostia International Physics Center and former Marie Curie Fellow at the Max Planck Institute of Quantum Optics, were chatting on WhatsApp when the idea for a quantum computing company for finance was born. [ 1 ]
In 2021, the company announced €12.5 million in funding from the European_Innovation_Council (EIC) Accelerator program. [ 8 ] This was followed by a €25 million funding round in 2024, valuing the startup at €100 million. [ 9 ] [ 10 ] Later that year, the startup was selected by the EIC’s Scaling Club – with a budget of $10 billion – as one of 48 companies to receive support with funding, mentoring and partnership matchmaking. [ 11 ]
In April 2022, the company partnered with the Bank of Canada to explore how quantum computing can simulate cryptocurrency adoption. This research made Canada the first G7 country to explore the model of complex networks and cryptocurrencies through quantum computing. [ 5 ] That July, Multiverse partnered with Bosch to integrate quantum algorithms into digital twin simulation workflow to scale simulations more efficiently and improve the accuracy of defect detection. [ 12 ] Later that year, BASF used Multiverse’s Singularity to develop models for foreign exchange trading optimization between U.S. and EU currency. [ 13 ]
Additional organizations exploring Multiverse’s quantum software include BBVA , Crédit Agricole CIB , and Repsol . [ 1 ] [ 14 ] [ 15 ] In 2022, Gartner recognized Multiverse Computing as a “Gartner Cool Vendor” for offering “quantum software technologies and services that enable integration of quantum solutions exploration in the financial services industry.” [ 5 ] Today, Multiverse is one of the largest quantum software companies in the world and has raised over €127.3 million. [ 1 ] [ 8 ] [ 16 ] [ 17 ]
Multiverse Computing’s algorithms have been implemented across verticals such as energy, manufacturing, logistics, finance, chemistry, space, and cybersecurity. [ 1 ] In addition to quantum machine learning and optimization algorithms, the company uses quantum-inspired tensor networks to improve efficiency in solving industrial challenges. [ 18 ] Tensor networks are used to model quantum systems, specifically quantum systems of many particles, and more recently are being applied to model artificial intelligence systems. [ 19 ] [ 20 ]
In 2023, the company launched CompactifAI, software that utilizes these networks to reduce the computational costs and energy requirements of training and operating large language models (LLMs). By minimizing size, memory, and storage needs of the models, the tensor networks can enhance efficiency and portability of these models. [ 21 ]
Earlier that year, Credit Agricole CIB released the results of multiple experiments with Multiverse that explored the use of quantum and quantum-inspired computing for the valuation of derivatives as well as solving the anticipation of credit rating downgrades. [ 14 ]
In 2024, the company secured a $1.4M contract with the German Aerospace Center to develop single photon detectors with applications ranging from quantum computing to deep-space communication and bio-imaging. [ 22 ] [ 23 ]
Multiverse Computing won the 2024 Future Unicorn award from DIGITALEUROPE, recognizing the company’s potential for digital transformation across cybersecurity, health and AI. [ 17 ] The company was also selected as a top 100 most promising AI company in the world in quantum AI software by CB Insights in 2023. [ 24 ] | https://en.wikipedia.org/wiki/Multiverse_Computing |
Communication between neurons happens primarily through chemical neurotransmission at the synapse . Neurotransmitters are packaged into synaptic vesicles for release from the presynaptic cell into the synapse, from where they diffuse and can bind to postsynaptic receptors. While most presynaptic cells are historically thought to release one vesicle at a time per active site, more recent research has pointed towards the possibility of multiple vesicles being released from the same active site (multivesicular release; MVR) in response to an action potential .
In the nervous system there are primarily two ways of propagating signals. By far the most common method of intracellular signal propagation is the action potential. [ 1 ] The dendrites of neurons contain ionotropic (aka ligand-gated ion channel ) and metabotropic neurotransmitter receptors that bind chemical neurotransmitters. At ionotropic receptors, these chemical neurotransmitters cause quick changes in ion flux into or out of the cell. The resulting internal voltage change in the dendrites is propagated towards the cell body and axon hillock , where a large concentration of voltage-gated ion channels typically exists. If some voltage threshold is met, voltage gated sodium channels open up, letting in a critical charge of sodium, and the positive current propagates down the axon towards the presynaptic axon terminal . This action potential leads to neurotransmitter vesicular release at in this terminal.
While action potentials are the typical means of signal propagation in the nervous system, some sensory neurons use graded potentials to trigger vesicular release. These cells are typically short enough that regenerative action potentials aren't needed to cause a large enough voltage change at the presynaptic terminal. For example, photoreceptor cells in the eye produce graded potentials in response to light, and these graded potentials can directly lead to neurotransmitter release. [ 2 ]
Most presynaptic terminals release small numbers of neurotransmitter containing vesicles even when action potentials are not present. This is stochastic and the probability of release (Pr) can be modified by numerous factors including the presence and speed of an action potential. [ 3 ] These vesicles are released at synaptic active zones, areas of the axon terminal that have all of the machinery and conditions necessary to specialize in vesicle fusion with the plasma membrane. Until relatively recently, the prevailing hypothesis was that only one vesicle at a time is released from these active zones. [ 4 ] However, research over the past several decades has added support for an additional mechanism of neurotransmitter vesicle release.
While univesicular release is still believed to make up a substantial portion of transmitter release events, large fluctuations in postsynaptic cell current measurements and high transmitter concentration in some synapses have led to the hypothesis that multiple vesicles can be released per active zone with each action potential. Since vesicular release happens on the scale of microseconds, it has been difficult to capture direct electron microscopic evidence of this phenomenon. There is however considerable functional data that supports MVR throughout much of the brain and sensory neuron synapses. [ 5 ]
MVR is thought to affect signal strength in postsynaptic neurons that typically have low receptor occupancy; this number can vary widely throughout the nervous system. This means that for however many receptors are found on a postsynaptic cell in the area of presynaptic cell vesicle release, only a small number of them would typically be occupied by neurotransmitter released from one vesicle (each vesicle can contain up to approximately 10,000 molecules of neurotransmitter). [ 6 ] MVR increases the likelihood that an action potential in a presynaptic cell will result in a postsynaptic cell chance in action potential likelihood. This could be either more or fewer action potentials, depending upon if the neurotransmitter / receptor combo is excitatory or inhibitory . [ 7 ]
MVR likely plays a substantial role in hippocampal signaling and memory . MVR exists in both near-synchronous (mulitiple vesicles released within tens of microseconds) and desynchronous (multiple vesicles released over a somewhat longer timescale). These different types of MVR can have differential effects on the post-synaptic neuron. Near-synchronous MVR leads to a fast onset and decline of synapse neurotransmitter levels as the molecules diffuse away or are taken up by neurotransmitter reuptake transporters (e.g. synapse glutamate transporters in neurons and glia. In the hippocampus, AMPA receptors that bind glutamate released in this manner produce a relatively large positive change in postsynaptic dendrite current (also called the excitatory postsynaptic current; EPSC). When the same receptors are exposed to de-synchronous MVR, which can lead to the same level of synaptic neurotransmitter but over a much longer timescale, the AMPA receptors can become desensitized, which leads to a smaller EPSC. This desensitization can reduce AMPA receptor availability and contribute to short-term depression , one of the fundamental mechanisms of learning and memory in hippocampal circuits. [ 5 ]
NMDA receptors are one of the other main types of glutamatergic receptors in the hippocampus. NMDA receptors allow calcium into the postsynaptic cell when they bind glutamate. Previous work in the hippocampus has shown that the response to stimulation of a single axon can be highly variable. Unlike earlier electrophysiology studies that studied postsynaptic currents due to calcium influx at multiple synapses, improved experimental techniques have allowed the isolation and study of individual synapses. This revolution in experimental technique has helped answer the question if the variable response to stimulation is due to UVR at a varying number of synapses, or to MVR at a single synapse. Individual synapse studies have found that after presynaptic axon stimulation, the postsynaptic NMDA-mediated calcium influx is highly variable, supporting the hypothesis that MVR can play a role at these synapses as well. [ 5 ]
One of the main benefits of MVR is thought to be the maintenance of information fidelity. Many sensory synapses take advantage of MVR to regulate firing duration and frequency, and to ensure reliable and sustained postsynaptic firing at high frequencies. Small changes in postsynaptic current are less likely to affect action potential generation than large ones are, hence the "information" held within presynaptic cell activity is less likely to make it farther downstream in the circuit. Since MVR results in release of more neurotransmitter, the changes in postsynaptic current tend to be larger, and information is more likely to be reliably propagated downstream. [ 8 ]
MVR is also thought to participate in vesicular release at ribbon synapses that are prominent in the sensory nervous system. [ 5 ] Presynaptic cells at primary synapses in the sensory nervous system typically are smaller than many other neurons, and can rely on graded potentials in the cells to reliably trigger neurotransmitter release at their synaptic terminals. [ 9 ] These graded potentials can last for several seconds and result in the release of thousands of neurotransmitter vesicles from a neuron. These synapses have specialized ribbon-like protein structures that can bind thousands of vesicles at a time, and MVR is capable of modulating a high level of vesicular release to transmit meaningful changes in sensation to the next sensory neuron downstream.
The balance between UVR and MVR is not necessarily static at any given synapse, as it can change over time. During development, many synapses undergo activity dependent pruning . [ 10 ] In general, those that are less active are most likely to be removed from the nervous system compared to those that are more active. This phenomenon has been observed in several brain areas, including the cerebellum. In this structure, many climbing fibers synapse onto individual Purkinje neurons . The climbing fiber axon terminals with the highest level of MVR versus UVR tend to cause the largest change in Purkinje neuron calcium influx, and these synapses are typically retained in the developing circuitry compared to those that have a lower level of calcium influx. [ 5 ] In the developing auditory system, MVR becomes less prominent as the Calyx of Held synapses mature and change shape. At this location, these shape changes increase action potential speed and decrease Pr. The decrease in Pr and MVR reduces synaptic glutamate and AMPA receptor desensitization, leading to a higher frequency of action potentials in the postsynaptic neuron. These maturation induced changes in MVR are also likely relevant at other locations in the developing nervous system. | https://en.wikipedia.org/wiki/Multivesicular_release |
The Mumm rearrangement is an organic reaction and a rearrangement reaction . [ 1 ] [ 2 ] It describes a 1,3(O-N) acyl transfer of an acyl imidate or isoimide group to an imide .
The reaction is of relevance as part of the Ugi reaction . | https://en.wikipedia.org/wiki/Mumm_rearrangement |
The Munching Square is a display hack dating back to the PDP-1 (ca. 1962, reportedly discovered by Jackson Wright ), which employs a trivial computation (repeatedly plotting the graph Y = X XOR T for successive values of T [ 1 ] ) to produce an impressive display of moving and growing squares that devour the screen. The initial value of T is treated as a parameter, which, when well-chosen, can produce intricate effects. Some of these, later (re)discovered on the LISP machine , have been christened munching triangles (using bitwise AND instead of XOR, and toggling points instead of plotting them), munching w's, and munching mazes. More generally, suppose a graphics program produces an ever-changing display of some basic form, foo , on a display terminal, and does it using a relatively simple program; then the program (or the resulting display) is likely to be referred to as munching foos . [ 2 ]
This article is based in part on the Jargon File , version 5.0.1, which is in the public domain.
This software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Munching_square |
Mundus subterraneus, quo universae denique naturae divitiae (very roughly "The subterranean world, all its riches" [ 1 ] ) is a scientific textbook written by Athanasius Kircher , and published in 1665. The work depicts Earth's geography through textual description, as well as lavish illustrations. [ 2 ]
Kircher had already advocated in favor of rationalism and natural laws in the explanation of phenomena , rejecting miraculous explanations. He expanded on the concept with this book.
Diatribe de Progidiosis Crucibus ("Diatribe of Prodigious Crosses") is Kircher's most succinct and explicit statement in favour of seeking rational causes for phenomena through an understanding of natural laws, derived from observation, rather than seeking miraculous explanations. [ 3 ] : 233–4 He pursued this in greater detail in Mundus Subterraneus (1665). [ 4 ] : 154
This article about a book on physical or human geography is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Mundus_Subterraneus |
Municipal or urban engineering applies the tools of science, art and engineering in an urban environment.
Municipal engineering is concerned with municipal infrastructure . This involves specifying, designing, constructing, and maintaining streets , sidewalks , water supply networks , sewers , street lighting , municipal solid waste management and disposal, storage depots for various bulk materials used for maintenance and public works (salt, sand, etc.), public parks and cycling infrastructure .
In the case of underground utility networks, it may also include the civil portion (conduits and access chambers) of the local distribution networks of electrical and telecommunications services. It can also include the optimizing of garbage collection and bus service networks. Some of these disciplines overlap with other civil engineering specialties, however municipal engineering focuses on the coordination of these infrastructure networks and services, as they are often built simultaneously (for a given street or development project), and managed by the same municipal authority.
Modern municipal engineering finds its origins in the 19th-century United Kingdom , following the Industrial Revolution and the growth of large industrial cities. The threat to urban populations from epidemics of waterborne diseases such as cholera and typhus led to the development of a profession devoted to "sanitary science" that later became "municipal engineering". [ citation needed ]
A key figure of the so-called "public health movement" was Edwin Chadwick , author of the parliamentary report, [ 1 ] published in 1842. [ citation needed ]
Early British legislation included:
This legislation provided local authorities with powers to undertake municipal engineering projects and to appoint borough surveyors (later known as "municipal engineers"). [ citation needed ]
In the U.K, the Association of Municipal Engineers, (subsequently named Institution of Municipal Engineers [ 2 ] ), was established in 1874 under the encouragement of the Institution of Civil Engineers, to address the issue of the application of sanitary science. By the early 20th century, Municipal Engineering had become a broad discipline embracing many of the responsibilities undertaken by local authorities, including roads, drainage, flood control, coastal engineering, public health, waste management, street cleaning, water supply, sewers, waste water treatment, crematoria, public baths, slum clearance , town planning, public housing, energy supply, parks, leisure facilities, libraries, town halls and other municipal buildings. [ citation needed ]
In the UK, the development of different strands of knowledge necessary for the management of municipal infrastructure led to the emergence of separate specialised institutions, including:
In 1984 the Institution of Municipal Engineers merged with the Institution of Civil Engineers. [ citation needed ]
Since the 1970s, there has been a global trend toward increasing privatisation and outsourcing of municipal engineering services. [ citation needed ]
In the UK in the 1990s a change in management philosophy brought the demise of the traditional organisational structure of boroughs where the three functions of town clerk, borough treasurer and borough engineer were replaced by an administrative structure with a larger number of specialised departments. [ citation needed ]
In the late 1990s and early 21st century there was increasing dissatisfaction over what was perceived to be fractured and dysfunctional public services designed along narrow specialties. A more holistic approach to urban engineering began to emerge as an alternative concept. Critics of the specialised approach included the Commission for Architecture and the Built Environment that complained that the specialised approach to management of the public realm focussed too much on the on efficient movement of vehicles rather than the more general interests of local communities. [ citation needed ]
In the United Kingdom there is no longer any formal professional qualification in municipal engineering although there are degree courses available in urban engineering. [ citation needed ]
A professional certificate in Urban Engineering is available from through the Institution of Incorporated Engineers via the Public Realm Information and Advice Network. [ citation needed ]
The British Institution of Civil Engineers (ICE) caters to practitioners employed in the public sector, private consultancy and academia through its Proceedings Journal Municipal Engineer. The journal, first published in 1873, has a global scope and covers the whole life cycle of municipal services addressing technical, political and community issues. [ 3 ] In addition an Expert Panel responds on behalf of ICE to Government consultations and is represented on the International Federation of Municipal Engineering. [ citation needed ]
The International Federation of Municipal Engineering (IFME) is an organisation comprising professional municipal engineers from all round the world. IFME's mission is to connect municipal engineers, public works professionals, public agencies, institutions and businesses around the world in order that they can share a global pool of knowledge and experience. The aim is to foster continued improvement in the quality of public works and wider community services. [ citation needed ]
The inaugural meeting was held in 1960 at the UNESCO headquarters in Paris. Membership has grown steadily and in 2009 [ 4 ] comprised representatives from national associations in: Australia, Canada, Denmark, Estonia, Finland, Italy, Israel, The Netherlands, New Zealand, Norway, Southern Africa (South Africa, Botawana, Namibia & Zimbabwe), Sweden, UK (England, Scotland, Wales & Northern Ireland) and USA. Belgium and San Marino are presently Corresponding Members. [ citation needed ]
Municipal or urban engineering combines elements of environmental engineering , water resources engineering and transport engineering . [ citation needed ]
Today, municipal engineering may be confused with urban design or urban planning . Whereas the urbanist or urban planner may design the general layout of streets and public places, the municipal engineer is concerned with the detailed design. For example, in the case of the design of a new street, the urbanist may specify the general layout of the street, including landscaping, surface finishings and urban accessories, but the municipal engineer will prepare the detailed plans and specifications for the roads, sidewalks, municipal services and street lighting. [ citation needed ]
In the case of large buildings or plants, facilities or campuses, site civil works may be required that are similar in scope or type as municipal infrastructure, namely, access roads, parking lots, potable water supply (including fire hydrants), on-site waste water treatment plants, site drainage including sedimentation and retention ponds or basins, etc. In most engineering consulting firms, Structural Engineering and Municipal Infrastructure are typically separate departments. On a large construction project, the civil engineering design will typically be divided into a structural portion, designed by structural engineers and typically focused on the buildings, and "civil" portion, designed by municipal engineers and focused on the site. [ citation needed ] | https://en.wikipedia.org/wiki/Municipal_or_urban_engineering |
Muometric navigation is positioning, navigation and timing using cosmic ray muons and other cosmic particles. [ 1 ]
It is possible to determine locations with GNSS satellites with well-known positions and time. GNSS is often used by critically important governmental organizations for navigating ships and planes, but the signals can be easily jammed and spoofed. [ 2 ] In 2020 Hiroyuki K.M. Tanaka created an entirely new approach from GNSS that locates the receiver's position with cosmic-ray muons. [ 3 ] [ 4 ] [ 5 ]
Muometric techniques include the muometric positioning system (muPS), [ 6 ] the muometric wireless navigation system (MuWNS) [ 7 ] or muPS Wireless Navigation System (muWNS), [ 1 ] cosmic time synchronizer (CTS) [ 8 ] and cosmic time calibrator (CTC). [ 9 ]
The muometric positioning and navigation techniques are based on the time-of-flight of relativistic cosmic-ray muons between reference detectors and the receiver detector usually located indoor, underground, or underwater. [ 10 ] Instead of receiving a GNSS signal, they detect cosmic-ray muons. Three or more reference detectors are deployed with known positions and time-references. [ 11 ] Like GNSS, clocks between the reference receivers and the receiver must be well-synchronized. [ 1 ] Unlike GNSS, this technology enables navigation in Arctic areas [ 12 ] where GNSS satellite access is limited due to orbital constraints of these satellites. [ 2 ]
The initial prototype required wiring between the receiver and each reference detector for accurate time synchronization. However, this configuration restricted the range of applicability of the system. [ 4 ] Efforts to find a way to navigate without wires, growing out of the success of this initial system replaced wires with a clock. muWNS is expected to be applied to rescue teams, for example, to guide robots underwater and underground [ 13 ] by positioning inside tunnels, [ 14 ] in a building or mine collapse. [ 15 ]
The indoor muometric positioning accuracy is 3.9 cm as of 2023. [ 16 ]
Precise timekeeping generally requires GNSS and atomic clock systems, but these are expensive and unavailable in indoor, underground or underwater areas. [ 17 ] Also, GNSS is vulnerable to cyber-attack and disruption. [ 18 ] A method using cosmic particles was proposed to precisely track time to solve this problem. [ 17 ] Cosmic rays collide in the atmosphere, generating particle showers . [ 19 ] The muons in these showers travel close to the speed of light and spread out as they travel through the atmosphere. They reach the ground at almost the same time, so by sharing the information provided by these muons, clocks can be synchronized. [ 20 ] A recent demonstration showed its synchronization precision of tens of nanoseconds over a distance of 60 m. [ 21 ]
Random timestamps generated in this cosmic time synchronizing scheme can be used in turn for generating random numbers for secure data transfer. [ 22 ] The sender and receiver use the same muons to create truly random cryptographic keys from the timestamp. Based on the precise time delay between the sender and the receiver calculated from the distance between the detectors within 10 meters each other, [ 23 ] the receiver knows the private key without having to directly exchange it between the sender and the receiver. [ 22 ] Applications potentially range from secure cloud storage, communications, to virtual currency generation. [ 23 ] | https://en.wikipedia.org/wiki/Muometric_navigation |
Muon-catalyzed fusion (abbreviated as μCF or MCF ) is a process that allows nuclear fusion to take place at temperatures that are significantly lower than those required for thermonuclear fusion , even at room temperature or lower. It is one of the few known ways of catalyzing nuclear fusion reactions.
Muons are unstable subatomic particles that are similar to electrons but 207 times more massive. If a muon replaces one of the electrons in a hydrogen molecule , the nuclei are consequently drawn 186 [ 1 ] [ 2 ] times closer than in a normal molecule, due to the reduced mass being 186 times the mass of an electron. When the nuclei move closer together, the fusion probability increases, to the point where a significant number of fusion events can happen at room temperature.
Methods for obtaining muons, however, require far more energy than can be produced by the resulting fusion reactions. Muons have a mean lifetime of 2.2 μs , much longer than that of many other subatomic particles but nevertheless far too brief to allow their useful storage. [ 3 ]
To create useful room-temperature muon-catalyzed fusion, reactors would need a cheap, efficient muon source and/or a way for each individual muon to catalyze many more fusion reactions.
Andrei Sakharov and F.C. Frank [ 4 ] predicted the phenomenon of muon-catalyzed fusion on theoretical grounds before 1950. Yakov Borisovich Zel'dovich [ 5 ] also wrote about the phenomenon of muon-catalyzed fusion in 1954. Luis W. Alvarez et al. , [ 6 ] when analyzing the outcome of some experiments with muons incident on a hydrogen bubble chamber at Berkeley in 1956, observed muon-catalysis of exothermic p–d, proton and deuteron, nuclear fusion , which results in a helion , a gamma ray , and a release of about 5.5 MeV of energy. The Alvarez experimental results, in particular, spurred John David Jackson to publish one of the first comprehensive theoretical studies of muon-catalyzed fusion in his ground-breaking 1957 paper. [ 7 ] This paper contained the first serious speculations on useful energy release from muon-catalyzed fusion. Jackson concluded that it would be impractical as an energy source, unless the "alpha-sticking problem" (see below) could be solved, leading potentially to an energetically cheaper and more efficient way of utilizing the catalyzing muons. [ 7 ]
If muon-catalyzed d–t nuclear fusion is realized practically, it will be a much more attractive way of generating power than conventional nuclear fission reactors because muon-catalyzed d–t nuclear fusion (like most other types of nuclear fusion ), produces far fewer harmful (and far less long-lived) radioactive wastes.
The large number of neutrons produced in muon-catalyzed d–t nuclear fusions may be used to breed fissile fuels from fertile material – for example, thorium -232 could breed uranium -233 in this way. [ note 1 ] The fissile fuels that have been bred can then be "burned," either in a conventional critical nuclear fission reactor or in an unconventional subcritical fission reactor , for example, a reactor using nuclear transmutation to process nuclear waste , or a reactor using the energy amplifier concept devised by Carlo Rubbia and others.
Another benefit of muon-catalyzed fusion is that the fusion process can start with pure deuterium gas without tritium. Plasma fusion reactors like ITER or Wendelstein X7 need tritium to initiate and also need a tritium factory. Muon-catalyzed fusion generates tritium under operation and increases operating efficiency up to an optimum point when the deuterium–tritium ratio reaches about 1:1. Muon-catalyzed fusion can operate as a tritium factory and deliver tritium for material and plasma fusion research.
Except for some refinements, little has changed since Jackson's 1957 assessment of the feasibility of muon-catalyzed fusion other than Vesman's 1967 prediction of the hyperfine resonant formation of the muonic (d–μ–t) + molecular ion which was subsequently experimentally observed. This helped spark renewed interest in the whole field of muon-catalyzed fusion, which remains an active area of research worldwide. However, as Jackson observed in his paper, muon-catalyzed fusion is "unlikely" to provide "useful power production ... unless an energetically cheaper way of producing μ − -mesons [ note 2 ] can be found." [ 7 ]
One practical problem with the muon-catalyzed fusion process is that muons are unstable, decaying in 2.2 μs (in their rest frame ). [ 8 ] Hence, there needs to be some cheap means of producing muons, and the muons must be arranged to catalyze as many nuclear fusion reactions as possible before decaying.
Another, and in many ways more serious, problem is the "alpha-sticking" problem, which was recognized by Jackson in his 1957 paper. [ 7 ] [ note 3 ] The α-sticking problem is the approximately 1% probability of the muon "sticking" to the alpha particle that results from deuteron–triton nuclear fusion , thereby effectively removing the muon from the muon-catalysis process altogether. Even if muons were absolutely stable, each muon could catalyze, on average, only about 100 d–t fusions before sticking to an alpha particle, which is only about one-fifth the number of muon catalyzed d–t fusions needed for break-even , where as much thermal energy is generated as electrical energy is consumed to produce the muons in the first place, according to Jackson's rough estimate. [ 7 ]
More recent measurements seem to point to more encouraging values for the α-sticking probability, finding the α-sticking probability to be around 0.3% to 0.5%, which could mean as many as about 200 (even up to 350) muon-catalyzed d–t fusions per muon. [ 9 ] Indeed, the team led by Steven E. Jones achieved 150 d–t fusions per muon (average) at the Los Alamos Meson Physics Facility . [ 10 ] The results were promising and almost enough to reach theoretical break-even. Unfortunately, these measurements for the number of muon-catalyzed d–t fusions per muon are still not enough to reach industrial break-even. Even with break-even, the conversion efficiency from thermal energy to electrical energy is only about 40% or so, further limiting viability. The best recent estimates of the electrical "energy cost" per muon is about 6 GeV with accelerators that are (coincidentally) about 40% efficient at transforming electrical energy from the power grid into acceleration of the deuterons.
As of 2012, no practical method of producing energy through this means has been published, although some discoveries using the Hall effect show promise. [ 11 ] [ failed verification ]
According to Gordon Pusch, a physicist at Argonne National Laboratory , various breakeven calculations on muon-catalyzed fusion omit the heat energy the muon beam itself deposits in the target. [ 12 ] By taking this factor into account, muon-catalyzed fusion can already exceed breakeven; however, the recirculated power is usually very large compared to power out to the electrical grid (about 3–5 times as large, according to estimates). Despite this rather high recirculated power, the overall cycle efficiency is comparable to conventional fission reactors; however the need for 4–6 MW of electrical generating capacity for each megawatt out to the grid probably represents an unacceptably large capital investment. Pusch suggested using Bogdan Maglich's " migma " self-colliding beam concept to significantly increase the muon production efficiency, by eliminating target losses, and using tritium nuclei as the driver beam, to optimize the number of negative muons.
In 2021, Kelly, Hart and Rose [ 13 ] produced a μCF model whereby the ratio, Q , of thermal energy produced to the kinetic energy of the accelerated deuterons used to create negative pions (and thus negative muons through pion decay) was optimized. In this model, the heat energy of the incoming deuterons as well as that of the particles produced due to the deuteron beam impacting a tungsten target was recaptured to the extent possible, as suggested by Gordon Pusch in the previous paragraph. Additionally, heat energy due to tritium breeding in a lithium-lead shell was recaptured, as suggested by Jändel, Danos and Rafelski in 1988. [ 14 ] The best Q value was found to be about 130% assuming that 50% of the muons produced were actually utilized for fusion catalysis. Furthermore, assuming that the accelerator was 18% efficient at transforming electrical energy into deuteron kinetic energy and conversion efficiency of heat energy into electrical energy of 60%, they estimate that, currently, the amount of electrical energy that could be produced by a μCF reactor would be 14% of the electrical energy consumed. In order for this to improve, they suggest that some combination of a) increasing accelerator efficiency and b) increasing the number of fusion reactions per negative muon above the assumed level of 150 would be needed.
To create this effect, a stream of negative muons, most often created by decaying pions , is sent to a block that may be made up of all three hydrogen isotopes (protium, deuterium, and/or tritium), where the block is usually frozen, and the block may be at temperatures of about 3 kelvin (−270 °C) or so. The muon may bump the electron from one of the hydrogen isotopes. The muon, 207 times more massive than the electron, effectively shields and reduces the electromagnetic repulsion between two nuclei and draws them much closer into a covalent bond than an electron can. Because the nuclei are so close, the strong nuclear force is able to bind both nuclei together. They fuse, release the catalytic muon (most of the time), and part of the original mass of both nuclei is released as energetic particles, as with any other type of nuclear fusion . The release of the catalytic muon is critical to continue the reactions. The majority of the muons continue to bond with other hydrogen isotopes and continue fusing nuclei together. However, not all of the muons are recycled: some bond with other debris emitted following the fusion of the nuclei (such as alpha particles and helions ), removing the muons from the catalytic process. This gradually reduces the rate of the reactions, as muons with which the nuclei may bond become depleted. The average number of reactions achieved in the lab can be as high as 150 d–t fusions per muon.
In the muon-catalyzed fusion of most interest, a positively charged deuteron (d), a positively charged triton (t), and a muon essentially form a positively charged muonic molecular heavy hydrogen ion (d–μ–t) + . The muon, with a rest mass 207 times greater than the rest mass of an electron, [ 8 ] is able to drag the more massive triton and deuteron 207 times closer together to each other [ 1 ] [ 2 ] in the muonic (d–μ–t) + molecular ion than can an electron in the corresponding electronic (d–e–t) + molecular ion. The average separation between the triton and the deuteron in the electronic molecular ion is about one angstrom (100 pm ), [ 7 ] [ note 4 ] so the average separation between the triton and the deuteron in the muonic molecular ion is 207 times smaller than that. [ note 5 ] Due to the strong nuclear force , whenever the triton and the deuteron in the muonic molecular ion happen to get even closer to each other during their periodic vibrational motions, the probability is very greatly enhanced that the positively charged triton and the positively charged deuteron would undergo quantum tunnelling through the repulsive Coulomb barrier that acts to keep them apart. Indeed, the quantum mechanical tunnelling probability depends roughly exponentially on the average separation between the triton and the deuteron, allowing a single muon to catalyze the d–t nuclear fusion in less than about half a picosecond , once the muonic molecular ion is formed. [ 7 ]
The formation time of the muonic molecular ion is one of the "rate-limiting steps" in muon-catalyzed fusion that can easily take up to ten thousand or more picoseconds in a liquid molecular deuterium and tritium mixture (D 2 , DT, T 2 ), for example. [ 7 ] Each catalyzing muon thus spends most of its ephemeral existence of 2.2 microseconds, [ 8 ] as measured in its rest frame , wandering around looking for suitable deuterons and tritons with which to bind.
Another way of looking at muon-catalyzed fusion is to try to visualize the ground state orbit of a muon around either a deuteron or a triton. Suppose the muon happens to have fallen into an orbit around a deuteron initially, which it has about a 50% chance of doing if there are approximately equal numbers of deuterons and tritons present, forming an electrically neutral muonic deuterium atom (d–μ) 0 that acts somewhat like a "fat, heavy neutron" due both to its relatively small size (again, 207 times smaller than an electrically neutral electronic deuterium atom (d–e) 0 ) and to the very effective "shielding" by the muon of the positive charge of the proton in the deuteron. Even so, the muon still has a much greater chance of being transferred to any triton that comes near enough to the muonic deuterium than it does of forming a muonic molecular ion. The electrically neutral muonic tritium atom (t–μ) 0 thus formed will act somewhat like an even "fatter, heavier neutron," but it will most likely hang on to its muon, eventually forming a muonic molecular ion, most likely due to the resonant formation of a hyperfine molecular state within an entire deuterium molecule D 2 (d=e 2 =d), with the muonic molecular ion acting as a "fatter, heavier nucleus" of the "fatter, heavier" neutral "muonic/electronic" deuterium molecule ([d–μ–t]=e 2 =d), as predicted by Vesman, an Estonian graduate student, in 1967. [ 15 ]
Once the muonic molecular ion state is formed, the shielding by the muon of the positive charges of the proton of the triton and the proton of the deuteron from each other allows the triton and the deuteron to tunnel through the Coulomb barrier in time span of order of a nanosecond [ 16 ] The muon survives the d–t muon-catalyzed nuclear fusion reaction and remains available (usually) to catalyze further d–t muon-catalyzed nuclear fusions. Each exothermic d–t nuclear fusion releases about 17.6 MeV of energy in the form of a "very fast" neutron having a kinetic energy of about 14.1 MeV and an alpha particle α (a helium -4 nucleus) with a kinetic energy of about 3.5 MeV. [ 7 ] An additional 4.8 MeV can be gleaned by having the fast neutrons moderated in a suitable "blanket" surrounding the reaction chamber, with the blanket containing lithium -6, whose nuclei, known by some as "lithions," readily and exothermically absorb thermal neutrons , the lithium-6 being transmuted thereby into an alpha particle and a triton. [ note 6 ]
The first kind of muon–catalyzed fusion to be observed experimentally, by L.W. Alvarez et al. , [ 6 ] was protium (H or 1 H 1 ) and deuterium (D or 1 H 2 ) muon-catalyzed fusion. The fusion rate for p–d (or pd) muon-catalyzed fusion has been estimated to be about a million times slower than the fusion rate for d–t muon-catalyzed fusion . [ 7 ] [ note 7 ]
Of more practical interest, deuterium–deuterium muon-catalyzed fusion has been frequently observed and extensively studied experimentally, in large part because deuterium already exists in relative abundance and, like protium, deuterium is not at all radioactive. (Tritium rarely occurs naturally, and is radioactive with a half-life of about 12.5 years. [ 8 ] )
The fusion rate for d–d muon-catalyzed fusion has been estimated to be only about 1% of the fusion rate for d–t muon-catalyzed fusion, but this still gives about one d–d nuclear fusion every 10 to 100 picoseconds or so. [ 7 ] However, the energy released with every d–d muon-catalyzed fusion reaction is only about 20% or so of the energy released with every d–t muon-catalyzed fusion reaction. [ 7 ] Moreover, the catalyzing muon has a probability of sticking to at least one of the d–d muon-catalyzed fusion reaction products that Jackson in this 1957 paper [ 7 ] estimated to be at least 10 times greater than the corresponding probability of the catalyzing muon sticking to at least one of the d–t muon-catalyzed fusion reaction products, thereby preventing the muon from catalyzing any more nuclear fusions. Effectively, this means that each muon catalyzing d–d muon-catalyzed fusion reactions in pure deuterium is only able to catalyze about one-tenth of the number of d–t muon-catalyzed fusion reactions that each muon is able to catalyze in a mixture of equal amounts of deuterium and tritium, and each d–d fusion only yields about one-fifth of the yield of each d–t fusion, thereby making the prospects for useful energy release from d–d muon-catalyzed fusion at least 50 times worse than the already dim prospects for useful energy release from d–t muon-catalyzed fusion.
Potential "aneutronic" (or substantially aneutronic) nuclear fusion possibilities, which result in essentially no neutrons among the nuclear fusion products, are almost certainly not very amenable to muon-catalyzed fusion. [ 7 ] One such essentially aneutronic nuclear fusion reaction involves a deuteron from deuterium fusing with a helion (He +2 ) from helium-3 , which yields an energetic alpha particle and a much more energetic proton , both positively charged (with a few neutrons coming from inevitable d–d nuclear fusion side reactions). However, one muon with only one negative electric charge is incapable of shielding both positive charges of a helion from the one positive charge of a deuteron. The chances of the requisite two muons being present simultaneously are exceptionally remote.
The term "cold fusion" was coined to refer to muon-catalyzed fusion in a 1956 New York Times article about Luis W. Alvarez 's paper. [ 17 ]
In 1957 Theodore Sturgeon wrote a novelette, " The Pod in the Barrier ", in which humanity has ubiquitous cold fusion reactors that work with muons. The reaction is "When hydrogen one and hydrogen two are in the presence of Mu mesons, they fuse into helium three, with an energy yield in electron volts of 5.4 times ten to the fifth power". Unlike the thermonuclear bomb contained in the Pod (which is used to destroy the Barrier) they can become temporarily disabled by "concentrated disbelief" that muon fusion works. [ 18 ]
In Sir Arthur C. Clarke 's third novel in the Space Odyssey series, 2061: Odyssey Three , muon-catalyzed fusion is the technology that allows mankind to achieve easy interplanetary travel. The main character, Heywood Floyd, compares Luis Alvarez to Lord Rutherford for underestimating the future potential of their discoveries. | https://en.wikipedia.org/wiki/Muon-catalyzed_fusion |
Muon capture is the capture of a negative muon by a proton , usually resulting in production of a neutron and a neutrino , and sometimes a gamma photon .
Muon capture by heavy nuclei often leads to emission of particles; most often neutrons , but charged particles can be emitted as well.
Ordinary muon capture ( OMC ) involves capture of a negative muon from the atomic orbital without emission of a gamma photon:
Radiative muon capture ( RMC ) is a radiative version of OMC, where a gamma photon is emitted:
Theoretical motivation for the study of muon capture on the proton is its connection to the proton's induced pseudoscalar form factor g p .
Muon capture is being investigated for practical application in radioactive waste disposal, for example in the artificial transmutation of large quantities of long-lived radioactive waste that have been produced globally by fission reactors . Radioactive waste can be transmuted to stable isotopes following irradiation by an incident muon ( μ − ) beam from a compact proton accelerator source.
This particle physics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Muon_capture |
Muon spin spectroscopy , also known as μSR , is an experimental technique based on the implantation of spin-polarized muons in matter and on the detection of the influence of the atomic, molecular or crystalline surroundings on their spin motion. The motion of the muon spin is due to the magnetic field experienced by the particle and may provide information on its local environment in a very similar way to other magnetic resonance [ a ] techniques, such as electron spin resonance (ESR or EPR) and, more closely, nuclear magnetic resonance (NMR).
Muon spin spectroscopy is an atomic, molecular and condensed matter experimental technique that exploits nuclear detection methods. In analogy with the acronyms for the previously established spectroscopies NMR and ESR , muon spin spectroscopy is also known as μSR. The acronym stands for muon spin rotation, relaxation, or resonance, depending respectively on whether the muon spin motion is predominantly a rotation (more precisely a precession around a still magnetic field ), a relaxation towards an equilibrium direction, or a more complex dynamic dictated by the addition of short radio frequency pulses. μSR does not require any radio-frequency technique to align the probing spin.
More generally speaking, muon spin spectroscopy includes any study of the interactions of the muon's magnetic moment with its surroundings when implanted into any kind of matter. Its two most notable features are its ability to study local environments, due to the short effective range of muon interactions with matter, and the characteristic time-window (10 −13 – 10 −5 s) of the dynamical processes in atomic, molecular and condensed media. The closest parallel to μSR is "pulsed NMR", in which one observes time-dependent transverse nuclear polarization or the so-called " free induction decay " of the nuclear polarization. However, a key difference is that in μSR one uses a specifically implanted spin (the muon's) and does not rely on internal nuclear spins.
Although particles are used as a probe, μSR is not a diffraction technique. A clear distinction between the μSR technique and those involving neutrons or X-rays is that scattering is not involved. Neutron diffraction techniques, for example, use the change in energy and/or momentum of a scattered neutron to deduce the sample properties. In contrast, the implanted muons are not diffracted but remain in a sample until they decay. Only a careful analysis of the decay product (i.e. a positron ) provides information about the interaction between the implanted muon and its environment in the sample.
As with many of the other nuclear methods, μSR relies on discoveries and developments made in the field of particle physics. Following the discovery of the muon by Seth Neddermeyer and Carl D. Anderson in 1936, pioneer experiments on its properties were performed with cosmic rays . Indeed, with one muon hitting each square centimeter of the earth's surface every minute, the muons constitute the foremost constituent of cosmic rays arriving at ground level. However, μSR experiments require muon fluxes of the order of 10 4 − 10 7 {\displaystyle 10^{4}-10^{7}} muons per second per square centimeter. Such fluxes can only be obtained in high-energy particle accelerators which have been developed during the last 50 years.
The collision of an accelerated proton beam (typical energy 600 MeV) with the nuclei of a production target produces positive pions ( π + {\displaystyle \pi ^{+}} ) via the possible reactions:
From the subsequent weak decay of the pions (MEAN lifetime τ π + {\displaystyle \tau _{\pi ^{+}}} = 26.03 ns) positive muons ( μ + {\displaystyle \mu ^{+}} ) are formed via the two body decay :
Parity violation in the weak interactions implies that only left-handed neutrinos exist, with their spin antiparallel to their linear momentum (likewise only right-handed anti-neutrino are found in nature). Since the pion is spinless both the neutrino and the μ + {\displaystyle \mu ^{+}} are ejected with spin antiparallel to their momentum in the pion rest frame. This is the key to provide spin-polarised muon beams. According to the value of the pion momentum different types of μ + {\displaystyle \mu ^{+}} -beams are available for μSR measurements.
Muon beams are classified into three types based on the energy of the muons being produced: high-energy, surface or "Arizona", and ultra-slow muon beams.
High-energy muon beams are formed by the pions escaping the production target at high energies. They are collected over a certain solid angle by quadrupole magnets and directed onto a decay section consisting of a long superconducting solenoid with a field of several tesla. If the pion momentum is not too high, a large fraction of the pions will have decayed before they reach the end of the solenoid. In the laboratory frame the polarization of a high-energy muon beam is limited to about 80% and its energy is of the order of ~40-50MeV. Although such a high energy beam requires the use of suitable moderators and samples with sufficient thickness, it guarantees a homogeneous implantation of the muons in the sample volume. Such beams are also used to study specimens inside of recipients, e.g. samples inside pressure cells. Such muon beams are available at PSI , TRIUMF , J-PARC and RIKEN-RAL .
The second type of muon beam is often called the surface or Arizona beam (recalling the pioneering work of Pifer et al. [ 1 ] [ 2 ] from the University of Arizona ). In these beams, muons arise from pions decaying at rest inside but near the surface of the production target. Such muons are 100% polarized, ideally monochromatic, and have a very low momentum of 29.8 MeV/c (corresponding to a kinetic energy of 4.1 MeV). They have a range width in matter of the order of 180 mg/cm 2 . The paramount advantage of this type of beam is the ability to use relatively thin samples. Beams of this type are available at PSI (Swiss Muon Source SμS), TRIUMF, J-PARC, ISIS Neutron and Muon Source and RIKEN-RAL.
Positive muon beams of even lower energy ( ultra-slow muons with energy down to the eV-keV range) can be obtained by further reducing the energy of an Arizona beam by utilizing the energy-loss characteristics of large band gap solid moderators. This technique was pioneered by researchers at the TRIUMF cyclotron facility in Vancouver, B.C. , Canada . It was christened with the acronym μSOL (muon separator on-line) and initially employed LiF as the moderating solid. [ 3 ] The same 1986 paper also reported the observation of negative muonium ions (i.e., Mu − or μ + e − e − ) in vacuum. In 1987, the slow μ + production rate was increased 100-fold using thin-film rare-gas solid moderators, producing a usable flux of low-energy positive muons. [ 4 ] This production technique was subsequently adopted by PSI for their low-energy positive muon beam facility. The tunable energy range of such muon beams corresponds to implantation depths in solids of less than a nanometer up to several hundred nanometers. Therefore, the study of magnetic properties as a function of the distance from the surface of the sample is possible. At the present time, PSI is the only facility where such a low-energy muon beam is available on a regular basis. Technical developments have been also conducted at RIKEN-RAL, but with a strongly reduced low-energy muon rate. J-PARC is projecting the development of a high-intensity low-energy muon beam. [ when? ]
In addition to the above-mentioned classification based on energy, muon beams are also divided according to the time structure of the particle accelerator, i.e. continuous or pulsed.
For continuous muon sources no dominating time structure is present. By selecting an appropriate incoming muon rate, muons are implanted into the sample one-by-one. The main advantage is that the time resolution is solely determined by the detector construction and the read-out electronics. There are two main limitations for this type of source, however: (i) unrejected charged particles accidentally hitting the detectors produce non-negligible random background counts; this compromises measurements after a few muon lifetimes, when the random background exceeds the true decay events; and (ii) the requirement to detect muons one at a time sets a maximum event rate. The background problem can be reduced by the use of electrostatic deflectors to ensure that no muons enter the sample before the decay of the previous muon. PSI and TRIUMF host the two continuous muon sources available for μSR experiments.
At pulsed muon sources protons hitting the production target are bunched into short, intense, and widely separated pulses that provide a similar time structure in the secondary muon beam. An advantage of pulsed muon sources is that the event rate is only limited by detector construction. Furthermore, detectors are active only after the incoming muon pulse, strongly reducing the accidental background counts. The virtual absence of background allows the extension of the time window for measurements up to about ten times the muon mean lifetime. The principal downside is that the width of the muon pulse limits the time resolution. ISIS Neutron and Muon Source and J-PARC are the two pulsed muon sources available for μSR experiments.
The muons are implanted into the sample of interest where they lose energy very quickly. Fortunately, this deceleration process occurs in such a way that it does not jeopardize a μSR measurement. On one side it is very fast (much faster than 100 ps), which is much shorter than a typical μSR time window (up to 20 μs), and on the other side, all the processes involved during the deceleration are Coulombic ( ionization of atoms, electron scattering , electron capture ) in origin and do not interact with the muon spin, so that the muon is thermalized without any significant loss of polarization.
The positive muons usually adopt interstitial sites of the crystallographic lattice , markedly distinguished by their electronic (charge) state. The spectroscopy of a muon chemically bound to an unpaired electron is remarkably different from that of all other muon states, which motivates the historical distinction in paramagnetic and diamagnetic states. Note that many diamagnetic muon states really behave like paramagnetic centers, according to the standard definition of a paramagnet . For example, in most metallic samples, which are Pauli paramagnets , the muon's positive charge is collectively screened by a cloud of conduction electrons. Thus, in metals, the muon is not bound to a single electron, hence it is in the so-called diamagnetic state and behaves like a free muon. In insulators or semiconductors a collective screening cannot take place and the muon will usually pick up one electron and form a so-called muonium (Mu=μ + +e − ), which has similar size ( Bohr radius ), reduced mass , and ionization energy to the hydrogen atom. This is the prototype of the so-called paramagnetic state.
The decay of the positive muon into a positron and two neutrinos occurs via the weak interaction process after a mean lifetime of
τ μ = 2.197034(21) μs:
Parity violation in the weak interaction leads in this more complicated case ( three body decay ) to an anisotropic distribution of the positron emission with respect to the spin direction of the μ + at the decay time. The positron emission probability is given by
where θ {\displaystyle \theta } is the angle between the positron trajectory and the μ + -spin, and a {\displaystyle a} is an intrinsic asymmetry parameter determined by the weak decay mechanism. This anisotropic emission constitutes in fact the basics for the μSR technique.
The average asymmetry A {\displaystyle A} is measured over a statistical ensemble of implanted muons and it depends on further experimental parameters, such as the beam spin polarization P μ {\displaystyle P_{\mu }} , close to one, as already mentioned . Theoretically A {\displaystyle A} =1/3 is obtained if all emitted positrons are detected with the same efficiency, irrespective of their energy. Practically, values of A {\displaystyle A} ≈ 0.25 are routinely obtained.
The muon spin motion may be measured over a time scale dictated by the muon decay , i.e. a few times τ μ , roughly 10 μs. The asymmetry in the muon decay correlates the positron emission and the muon spin directions. The simplest example is when the spin direction of all muons remains constant in time after implantation (no motion). In this case the asymmetry shows up as an imbalance between the positron counts in two equivalent detectors placed in front and behind the sample, along the beam axis. Each of them records an exponentially decaying rate as a function of the time t elapsed from implantation, according to
with α = ± 1 {\displaystyle \alpha =\pm 1} for the detector looking towards and away from the spin arrow, respectively. Considering that the huge muon spin polarization is completely outside thermal equilibrium, a dynamical relaxation towards the equilibrium unpolarized state typically shows up in the count rate, as an additional decay factor in front of the experimental asymmetry parameter, A . A magnetic field parallel to the initial muon spin direction probes the dynamical relaxation rate as a function of the additional muon Zeeman energy , without introducing additional coherent spin dynamics. This experimental arrangement is called Longitudinal Field (LF) μSR.
A special case of LF μSR is Zero Field (ZF) μSR, when the external magnetic field is zero. This experimental condition is particularly important since it allows to probe any internal quasi-static (i.e. static on the muon time-scale) magnetic field of field distribution at the muon site. Internal quasi-static fields may appear spontaneously, not induced by the magnetic response of the sample to an external field They are produced by disordered nuclear magnetic moments or, more importantly, by ordered electron magnetic moments and orbital currents.
Another simple type of μSR experiment is when implanted all muon spins precess coherently around the external magnetic field of modulus B {\displaystyle B} , perpendicular to the beam axis, causing the count unbalance to oscillate at the corresponding Larmor frequency ω {\displaystyle \omega } between the same two detectors, according to
Since the Larmor frequency is ω = γ μ B {\displaystyle \omega =\gamma _{\mu }B} , with a gyromagnetic ratio γ μ = 851.616 {\displaystyle \gamma _{\mu }=851.616} Mrad(sT) −1 , the frequency spectrum obtained by means of this experimental arrangement provides a direct measure of the internal magnetic field intensity distribution. The distribution produces an additional decay factor of the experimental asymmetry A . This method is usually referred to as Transverse Field (TF) μSR.
A more general case is when the initial muon spin direction (coinciding with the detector axis) forms an angle θ {\displaystyle \theta } with the field direction. In this case the muon spin precession describes a cone which results in both a longitudinal component, A cos 2 θ {\displaystyle A\cos ^{2}\theta } , and a transverse precessing component, A sin 2 θ cos ω t {\displaystyle A\sin ^{2}\theta \cos \omega t} , of the total asymmetry. ZF μSR experiments in the presence of a spontaneous internal field fall into this category as well.
Muon spin rotation and relaxation are mostly performed with positive muons. They are well suited to the study of magnetic fields at the atomic scale inside matter, such as those produced by various kinds of magnetism and/or superconductivity encountered in compounds occurring in nature or artificially produced by modern material science .
The London penetration depth is one of the most important parameters characterizing a superconductor because its inverse square provides a measure of the density n s of Cooper pairs . The dependence of n s on temperature and magnetic field directly indicates the symmetry of the superconducting gap. Muon spin spectroscopy provides a way to measure the penetration depth, and so has been used to study high-temperature cuprate superconductors since their discovery in 1986.
Other important fields of application of μSR exploit the fact that positive muons capture electrons to form muonium atoms which behave chemically as light isotopes of the hydrogen atom. This allows investigation of the largest known kinetic isotope effect in some of the simplest types of chemical reactions, as well as the early stages of formation of radicals in organic chemicals. Muonium is also studied as an analogue of hydrogen in semiconductors , where hydrogen is one of the most ubiquitous impurities.
μSR requires a particle accelerator for the production of a muon beam. This is presently achieved at few large scale facilities in the world: the CMMS continuous source at TRIUMF in Vancouver, Canada; the SμS continuous source at the Paul Scherrer Institut (PSI) in Villigen, Switzerland; the ISIS Neutron and Muon Source and RIKEN-RAL pulsed sources at the Rutherford Appleton Laboratory in Chilton, United Kingdom; and the J-PARC facility in Tokai, Japan, where a new pulsed source is being built to replace that at KEK in Tsukuba, Japan.
Muon beams are also available at the Laboratory of Nuclear Problems, Joint Institute for Nuclear Research (JINR) in Dubna, Russia.
The International Society for μSR Spectroscopy (ISMS) exists to promote the worldwide advancement of μSR. Membership in the society is open free of charge to all individuals in academia, government laboratories and industry who have an interest in the society's goals. | https://en.wikipedia.org/wiki/Muon_spin_spectroscopy |
Mura ( 斑 ) is a Japanese word meaning "unevenness; irregularity; lack of uniformity; nonuniformity; inequality", [ 1 ] and is a key concept in the Toyota Production System (TPS) as one of the three types of waste ( muda , mura , muri ). [ 2 ] Waste in this context refers to the wasting of time or resources rather than wasteful by-products and should not be confused with waste reduction . Toyota adopted these three Japanese words as part of their product improvement program, due to their familiarity in common usage.
Mura , in terms of business/process improvement, is avoided through just-in-time manufacturing systems, which are based on keeping little or no inventory. These systems supply the production process with the right part, at the right time, in the right amount, using first-in, first-out (FIFO) component flow. Just-in-time systems create a "pull system" in which each sub-process withdraws its needs from the preceding sub-processes, and ultimately from an outside supplier. When a preceding process does not receive a request or withdrawal it does not make more parts. This type of system is designed to maximize productivity by minimizing storage overhead.
For example:
If parts or material defects are found in one process, the just-in-time approach requires that the problem be quickly identified and corrected.
Production leveling , also called heijunka , and frequent deliveries to customer are key to identifying and eliminating mura . The use of different types of kanban to control inventory at different stages in the process are key to ensuring that "pull" is happening between sub-processes. Leveling production, even when different products are produced in the same system, will aid in scheduling work in a standard way that encourages lower costs.
It is also possible to smooth the workflow by having one operator work across several machines in a process rather than having different operators; in a sense merging several sub-processes under one operator. The fact that there is one operator will force a smoothness across the operations because the workpiece flows with the operator. There is no reason why the several operators cannot all work across these several machines following each other and carrying their workpiece with them. [ 3 ] This multiple machine handling is called "multi-process handling" in the Toyota Production System.
Another means of detecting and reducing mura is increasing the process' standardization—ensuring that all workers understand and can handle each type of request that they come across along a clear, step-by-step protocol. Working to simplify the process as much as possible will also help to drive down the unevenness-generating complexities. You can also aid variability detection by performance monitoring through histograms and statistical control charts .
Some processes have considerable lead time . Some processes have unusually high costs for waiting or downtime. When this is the case, it is often desirable to try to predict the upcoming demand from a sub-process before pull occurs or a card is generated. The smoother the process, the more accurately this can be done from analysis of previous historical experience.
Some processes have asymmetric cost. In such situations, it may be better to err away from the higher cost error. In this case, there appears to be waste and higher average error, but the waste or errors are smaller ones and in aggregate leads to lower costs and more customer value.
For example, consider running a call center. It may be more effective to have low cost call center operators wait for high value clients rather than risk losing high value clients by making them wait. Given the asymmetric cost of these errors—particularly if the processes are not smooth—it may be prudent to have what seems like a surplus of call center operators that appear to be "wasting" call center operator time, rather than commit the higher-cost error of losing the occasional high value client. | https://en.wikipedia.org/wiki/Mura_(Japanese_term) |
The Murahashi Coupling is a cross coupling reaction . The coupling partners are organolithiums and organic halides. Transition metal catalysts are required. [ 1 ] The reaction was first reported by Shun-Ichi Murahashi in 1974. [ 2 ] This reaction is notable for using organolithiums as opposed to other cross-coupling reactions which utilize various metal-carbon compounds (metal = tin, magnesium, boron, silicon, zinc). Since the production of these other coupling reagents relies heavily upon organolithiums (especially in the case of organozinc and organomagnesium compounds), in bypassing these intermediates, this process is much more efficient. It has further been shown that the Murahashi reaction proceeds with greater selectivity, faster reaction times, and lower reaction temperatures than other similar coupling reactions while maintaining similar or higher yields. [ 3 ] [ 4 ] [ 5 ]
Murahashi first reported C(sp 2 )—C(sp 3 ) coupling in 1974. These reactions were not metal catalyzed but were promoted by solvent, THF . The following year, Murahashi showed that coupling can occur between organolithium and alkenyl halides in the presence of tetrakis(triphenylphosphine)palladium catalyst. In 1979, Murahashi published the catalytic version of this reaction. [ 6 ]
The catalytic cycle of the Murahashi coupling is comparable to that of other extensively studied palladium-catalyzed cross-coupling reactions. The cycle proposed by Murahashi involves four intermediates and two oxidation states of palladium, palladium(0) and palladium(II). At the start of the cycle, oxidative addition occurs between the Pd(0) catalyst (1) and an organo-halide to form an organo-Pd(II) complex (2) . A transmetallation then occurs between (2) and an organo-lithium to yield the trans-hetero-organometallic complex (3) and the lithium-halide. In order to perform the reductive elimination that ultimately yields the final coupled product, a trans-cis-isomerization of (3) must occur to bring the ligands cis to each other, resulting in the cis-hetero-organometallic complex (4) . Finally, reductive elimination of (4) regenerates the Pd(0) catalyst and creates the C(sp 2 )--C(sp 3 ) bond in the coupled product. [ 2 ]
Organolithium reagents pose some danger because they are highly pyrophoric. [ 7 ] The reaction is typically carried out in toluene as a solvent . The mixture is often carried out between 0–80 °C and quenched with acid to deactivate any remaining organolithium. [ 6 ]
While Pd(0)-catalyzed reactions are highly efficient for the cross-coupling of organo-halides and organolithium reagents, ruthenium catalysts have also been demonstrated. [ 2 ] | https://en.wikipedia.org/wiki/Murahashi_coupling |
In organic chemistry , the Murai reaction is an organic reaction that uses C-H activation to create a new C-C bond between a terminal or strained internal alkene and an aromatic compound using a ruthenium catalyst. [ 1 ] The reaction, named after Shinji Murai, was first reported in 1993. While not the first example of C-H activation, the Murai reaction is notable for its high efficiency and scope. [ 2 ] [ 3 ] Previous examples of such hydroarylations required more forcing conditions and narrow scope. [ 4 ] [ 1 ]
The reaction was initially demonstrated using a ketone as the directing group , but other functional groups have been reported, including esters , imines, nitriles, and imidates. [ 5 ] Murai reactions have also been reported with disubstituted alkynes . [ 6 ] Bidentate directing groups allow ortho alkylation of aromatic rings with α,β-unsaturated ketones, which typically are unreactive in Murai reactions. [ 7 ]
Early examples of the reaction suffered from side products of alkylation at both ortho positions. This problem can be partially solved using an ortho methyl blocking group. Unfortunately, with ortho methyl groups both the rate and generality of the reaction are reduced. [ 3 ] Substituents at the meta position influence regioselectivity. [ 8 ] The reaction preferentially adds at the least sterically hindered ortho position, except when there is a meta group capable of coordinating with the Ru catalyst. Methoxyacetophenones show preferential reaction at the more hindered position.
A variety of Ru catalysts catalyze the Murai reaction, including RuH 2 (CO)(PPh 3 ) 3 , RuH 2 (PPh 3 ) 4 , Ru(CO) 2 (PPh 3 ) 3 , and Ru 3 (CO) 12 . [ 9 ]
A detailed mechanism for the Murai reaction has not been elucidated. Experimental and computational studies give evidence for at least two different mechanisms, depending on the catalyst . [ 10 ] [ 11 ] For catalysts such as [Ru(H) 2 (CO)(PR 3 ) 3 ] which are active as Ru 0 , a combination of computational density functional studies and experimental evidence has resulted in the following proposed mechanism: [ 3 ] [ 11 ] [ 12 ]
It is proposed that at high temperatures RuH 2 (CO)(PPh 3 ) 3 converts to an unsaturated Ru(CO)(PPh 3 ) n species. [ 9 ] The catalytic cycle is proposed to begin with coordination of the ketone followed by oxidative addition of a C-H bond. The resulting five-coordinated metallocycle is stabilized by an agostic interaction . The C-C bond formation is the rate limiting step.
The complex [Ru( o -C 6 H 4 PPh 2 )(H)(CO)(PPh 3 ) 2 ] catalyzes the Murai reaction at room temperature. [ 13 ] [ 14 ] [ 15 ] For [Ru(H) 2 (H 2 ) 2 (PR 3 ) 2 ], the active complex is [Ru(H) 2 (PR 3 ) 2 ].
After the active form of the ruthenium catalyst complex is generated from 1 , acetophenone coordinates to the complex via its carbonyl oxygen and agostically via its ortho C-H bond ( 2 ). As in the Ru 0 proposed mechanism, this agostic interaction leads to the oxidative addition of the ortho C-H. Reductive elimination releases H 2 , which remains coordinated, giving complex 3 . Coordination of ethylene and decoordination of the ketone results in complex 4 which then undergoes migratory insertion of ethylene into the hydride to give 5 . Following oxidative addition of H 2 ( 6 ), the complex reductively eliminates the product to give the product agostically bound to the complex. Coordination of another acetophenone molecule regenerates complex 2 . | https://en.wikipedia.org/wiki/Murai_reaction |
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