Misplaced Pages

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112-639: Palazzo Valmarana is a palace in Vicenza . It was built by Italian Renaissance architect Andrea Palladio in 1565 for the noblewoman Isabella Nogarola Valmarana. Since 1994 it is part of the UNESCO World Heritage Site " City of Vicenza and the Palladian Villas of the Veneto ". The foundation medal of this building bears engraved the date 1566 as well as the bust of Isabella Nogarola Valmarana , and it

224-727: A Roman aqueduct exist outside the Porta Santa Croce . During the decline of the Western Roman Empire , Heruls , Vandals , Alaric and his Visigoths , as well as the Huns laid waste to the area, but the city recovered after the Ostrogoth conquest in 489 AD, before being conquered by the Byzantine Empire soon after. It was also an important Lombard city and then a Frankish center. Numerous Benedictine monasteries were built in

336-467: A bas-relief of a warrior who bears the Valmarana arms . Rather than abstract geometrical constructions, the compositional logic of these civil and religious façades derived from Palladio’s familiarity with the techniques of draughting, in particular the orthogonal representations by which he visualised projects and reconstructed antique buildings, and which moreover allowed him a punctilious control over

448-455: A Hilbert space that is not orthogonal is called an oblique projection . The eigenvalues of a projection matrix must be 0 or 1. For example, the function which maps the point ( x , y , z ) {\displaystyle (x,y,z)} in three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} to the point ( x , y , 0 ) {\displaystyle (x,y,0)}

560-469: A ball B P x {\displaystyle B_{P\mathbf {x} }} (with positive radius) centered on P x {\displaystyle P\mathbf {x} } that is wholly contained in the image P ( B x ) {\displaystyle P(B_{\mathbf {x} })} . Let W {\displaystyle W} be a finite-dimensional vector space and P {\displaystyle P} be

672-562: A basis for the orthogonal complement of the kernel of the projection, and assemble these vectors in the matrix B {\displaystyle B} . Then the projection P {\displaystyle P} (with the condition k ≥ 1 {\displaystyle k\geq 1} ) is given by P = A ( B T A ) − 1 B T . {\displaystyle P=A\left(B^{\mathsf {T}}A\right)^{-1}B^{\mathsf {T}}.} This expression generalizes

784-417: A basis in which P {\displaystyle P} has the form where r {\displaystyle r} is the rank of P {\displaystyle P} . Here I r {\displaystyle I_{r}} is the identity matrix of size r {\displaystyle r} , 0 d − r {\displaystyle 0_{d-r}}

896-1146: A closed subspace follows from the Hilbert projection theorem . An orthogonal projection is a bounded operator . This is because for every v {\displaystyle \mathbf {v} } in the vector space we have, by the Cauchy–Schwarz inequality : ‖ P v ‖ 2 = ⟨ P v , P v ⟩ = ⟨ P v , v ⟩ ≤ ‖ P v ‖ ⋅ ‖ v ‖ {\displaystyle \left\|P\mathbf {v} \right\|^{2}=\langle P\mathbf {v} ,P\mathbf {v} \rangle =\langle P\mathbf {v} ,\mathbf {v} \rangle \leq \left\|P\mathbf {v} \right\|\cdot \left\|\mathbf {v} \right\|} Thus ‖ P v ‖ ≤ ‖ v ‖ {\displaystyle \left\|P\mathbf {v} \right\|\leq \left\|\mathbf {v} \right\|} . For finite-dimensional complex or real vector spaces,

1008-727: A closed subspace. Furthermore, the kernel of a continuous projection (in fact, a continuous linear operator in general) is closed. Thus a continuous projection P {\displaystyle P} gives a decomposition of X {\displaystyle X} into two complementary closed subspaces: X = rg ⁡ ( P ) ⊕ ker ⁡ ( P ) = ker ⁡ ( 1 − P ) ⊕ ker ⁡ ( P ) {\displaystyle X=\operatorname {rg} (P)\oplus \ker(P)=\ker(1-P)\oplus \ker(P)} . The converse holds also, with an additional assumption. Suppose U {\displaystyle U}

1120-441: A façade surviving with its original plaster and marmorino . In 1960, the ruined palace was sold by the Valmarana family to Vittor Luigi Braga Rosa, who led an extended restoration, rebuilding the parts demolished by war. He also enriched the palace with many decorations and artworks, coming from other destroyed palaces, in particular a collection of Seicento paintings by Giulio Carpioni with mythological themes. The façade of

1232-602: A projection on W {\displaystyle W} . Suppose the subspaces U {\displaystyle U} and V {\displaystyle V} are the image and kernel of P {\displaystyle P} respectively. Then P {\displaystyle P} has the following properties: The image and kernel of a projection are complementary , as are P {\displaystyle P} and Q = I − P {\displaystyle Q=I-P} . The operator Q {\displaystyle Q}

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1344-496: A projection, even if they are orthogonal. If two projections commute then their product is a projection, but the converse is false: the product of two non-commuting projections may be a projection. If two orthogonal projections commute then their product is an orthogonal projection. If the product of two orthogonal projections is an orthogonal projection, then the two orthogonal projections commute (more generally: two self-adjoint endomorphisms commute if and only if their product

1456-498: A projection. The relation P 2 = P {\displaystyle P^{2}=P} implies 1 = P + ( 1 − P ) {\displaystyle 1=P+(1-P)} and X {\displaystyle X} is the direct sum rg ⁡ ( P ) ⊕ rg ⁡ ( 1 − P ) {\displaystyle \operatorname {rg} (P)\oplus \operatorname {rg} (1-P)} . However, in contrast to

1568-443: A projection. This implies that an orthogonal projection P {\displaystyle P} is always a positive semi-definite matrix . In general, the corresponding eigenspaces are (respectively) the kernel and range of the projection. Decomposition of a vector space into direct sums is not unique. Therefore, given a subspace V {\displaystyle V} , there may be many projections whose range (or kernel)

1680-588: A roster of 35+ athletes. Vicenza is home to football club L.R. Vicenza Virtus , formerly Lanerossi Vicenza and Vicenza Calcio, which currently compete in Serie C . Their home venue is the Stadio Romeo Menti . Vicenza is home to Rangers Rugby Vicenza, a rugby union team who compete in Serie A2. Vicenza's cuisine reflects its humble, agricultural past: simple, hearty meals made with fresh local ingredients that reflect

1792-990: A vector. One can define a projection of y {\displaystyle \mathbf {y} } onto V {\displaystyle V} as proj V ⁡ y = y ⋅ u i u i ⋅ u i u i {\displaystyle \operatorname {proj} _{V}\mathbf {y} ={\frac {\mathbf {y} \cdot \mathbf {u} ^{i}}{\mathbf {u} ^{i}\cdot \mathbf {u} ^{i}}}\mathbf {u} ^{i}} where repeated indices are summed over ( Einstein sum notation ). The vector y {\displaystyle \mathbf {y} } can be written as an orthogonal sum such that y = proj V ⁡ y + z {\displaystyle \mathbf {y} =\operatorname {proj} _{V}\mathbf {y} +\mathbf {z} } . proj V ⁡ y {\displaystyle \operatorname {proj} _{V}\mathbf {y} }

1904-410: Is V {\displaystyle V} . If a projection is nontrivial it has minimal polynomial x 2 − x = x ( x − 1 ) {\displaystyle x^{2}-x=x(x-1)} , which factors into distinct linear factors, and thus P {\displaystyle P} is diagonalizable . The product of projections is not in general

2016-731: Is complete , i.e. when V {\displaystyle V} is a Hilbert space , the concept of orthogonality can be used. A projection P {\displaystyle P} on a Hilbert space V {\displaystyle V} is called an orthogonal projection if it satisfies ⟨ P x , y ⟩ = ⟨ x , P y ⟩ {\displaystyle \langle P\mathbf {x} ,\mathbf {y} \rangle =\langle \mathbf {x} ,P\mathbf {y} \rangle } for all x , y ∈ V {\displaystyle \mathbf {x} ,\mathbf {y} \in V} . A projection on

2128-426: Is twinned with: Ex comuni Orthogonal projection A projection on a vector space V {\displaystyle V} is a linear operator P : V → V {\displaystyle P\colon V\to V} such that P 2 = P {\displaystyle P^{2}=P} . When V {\displaystyle V} has an inner product and

2240-600: Is a closed subspace of X {\displaystyle X} . If there exists a closed subspace V {\displaystyle V} such that X = U ⊕ V , then the projection P {\displaystyle P} with range U {\displaystyle U} and kernel V {\displaystyle V} is continuous. This follows from the closed graph theorem . Suppose x n → x and Px n → y . One needs to show that P x = y {\displaystyle Px=y} . Since U {\displaystyle U}

2352-630: Is a projection along U {\displaystyle U} onto V {\displaystyle V} . In infinite-dimensional vector spaces, the spectrum of a projection is contained in { 0 , 1 } {\displaystyle \{0,1\}} as ( λ I − P ) − 1 = 1 λ I + 1 λ ( λ − 1 ) P . {\displaystyle (\lambda I-P)^{-1}={\frac {1}{\lambda }}I+{\frac {1}{\lambda (\lambda -1)}}P.} Only 0 or 1 can be an eigenvalue of

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2464-469: Is a projection. Boundedness of φ {\displaystyle \varphi } implies continuity of P {\displaystyle P} and therefore ker ⁡ ( P ) = rg ⁡ ( I − P ) {\displaystyle \ker(P)=\operatorname {rg} (I-P)} is a closed complementary subspace of U {\displaystyle U} . Projections (orthogonal and otherwise) play

2576-415: Is also a projection as the image and kernel of P {\displaystyle P} become the kernel and image of Q {\displaystyle Q} and vice versa. We say P {\displaystyle P} is a projection along V {\displaystyle V} onto U {\displaystyle U} (kernel/image) and Q {\displaystyle Q}

2688-501: Is also an oblique projection. The singular values of P {\displaystyle P} and I − P {\displaystyle I-P} can be computed by an orthonormal basis of A {\displaystyle A} . Let Q A {\displaystyle Q_{A}} be an orthonormal basis of A {\displaystyle A} and let Q A ⊥ {\displaystyle Q_{A}^{\perp }} be

2800-540: Is also clear that A A T {\displaystyle AA^{\mathsf {T}}} is the identity operator on U {\displaystyle U} . The orthonormality condition can also be dropped. If u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} is a (not necessarily orthonormal) basis with k ≥ 1 {\displaystyle k\geq 1} , and A {\displaystyle A}

2912-416: Is an open map , meaning that it maps each open set in the domain to an open set in the subspace topology of the image . That is, for any vector x {\displaystyle \mathbf {x} } and any ball B x {\displaystyle B_{\mathbf {x} }} (with positive radius) centered on x {\displaystyle \mathbf {x} } , there exists

3024-697: Is an orthogonal projection onto the xy -plane. This function is represented by the matrix P = [ 1 0 0 0 1 0 0 0 0 ] . {\displaystyle P={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}}.} The action of this matrix on an arbitrary vector is P [ x y z ] = [ x y 0 ] . {\displaystyle P{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}.} To see that P {\displaystyle P}

3136-412: Is an orthogonal projection. When these basis vectors are not orthogonal to the kernel, the projection is an oblique projection, or just a projection. Let P : V → V {\displaystyle P\colon V\to V} be a linear operator such that P 2 = P {\displaystyle P^{2}=P} and assume that P {\displaystyle P}

3248-466: Is approximately 60 kilometres (37 mi) west of Venice and 200 kilometres (120 mi) east of Milan . Vicenza is a thriving and cosmopolitan city, with a rich history and culture, and many museums, art galleries, piazzas , villas , churches and elegant Renaissance palazzi . With the Palladian villas of the Veneto in the surrounding area, and his renowned Teatro Olimpico ("Olympic Theater"),

3360-570: Is closed and { Px n } ⊂ U , y lies in U {\displaystyle U} , i.e. Py = y . Also, x n − Px n = ( I − P ) x n → x − y . Because V {\displaystyle V} is closed and {( I − P ) x n } ⊂ V , we have x − y ∈ V {\displaystyle x-y\in V} , i.e. P ( x − y ) = P x − P y = P x − y = 0 {\displaystyle P(x-y)=Px-Py=Px-y=0} , which proves

3472-451: Is commonly used in areas such as machine learning . Any projection P = P 2 {\displaystyle P=P^{2}} on a vector space of dimension d {\displaystyle d} over a field is a diagonalizable matrix , since its minimal polynomial divides x 2 − x {\displaystyle x^{2}-x} , which splits into distinct linear factors. Thus there exists

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3584-443: Is complex-valued, the transpose in the above equation is replaced by a Hermitian transpose). This operator leaves u invariant, and it annihilates all vectors orthogonal to u {\displaystyle \mathbf {u} } , proving that it is indeed the orthogonal projection onto the line containing u . A simple way to see this is to consider an arbitrary vector x {\displaystyle \mathbf {x} } as

3696-460: Is from Morocco ). The city is predominantly Roman Catholic , but due to immigration, it now has some Orthodox Christian , Muslim and Sikh followers. In 1994 UNESCO inscribed "Vicenza, City of Palladio" on its list of World Heritage Sites . In 1996, the site was expanded to include the Palladian villas outside the core area, and accordingly renamed " City of Vicenza and the Palladian Villas of

3808-575: Is given by P A x = argmin y ∈ range ⁡ ( A ) ⁡ ‖ x − y ‖ D 2 {\textstyle P_{A}x=\operatorname {argmin} _{y\in \operatorname {range} (A)}\left\|x-y\right\|_{D}^{2}} . Then P A = A ( A T D A ) − 1 A T D . {\displaystyle P_{A}=A\left(A^{\mathsf {T}}DA\right)^{-1}A^{\mathsf {T}}D.} When

3920-561: Is in the image of P {\displaystyle P} , and vector x 2 = x − P ( x ) . {\displaystyle \mathbf {x} _{2}=\mathbf {x} -P(\mathbf {x} ).} So P ( x 2 ) = P ( x ) − P 2 ( x ) = 0 {\displaystyle P(\mathbf {x} _{2})=P(\mathbf {x} )-P^{2}(\mathbf {x} )=\mathbf {0} } , and then x 2 {\displaystyle \mathbf {x} _{2}}

4032-579: Is in the kernel of P {\displaystyle P} , which is the null space of A . {\displaystyle A.} In other words, the vector x 1 {\displaystyle \mathbf {x} _{1}} is in the column space of A , {\displaystyle A,} so x 1 = A w {\displaystyle \mathbf {x} _{1}=A\mathbf {w} } for some k {\displaystyle k} dimension vector w {\displaystyle \mathbf {w} } and

4144-724: Is indeed a projection, i.e., P = P 2 {\displaystyle P=P^{2}} , we compute P 2 [ x y z ] = P [ x y 0 ] = [ x y 0 ] = P [ x y z ] . {\displaystyle P^{2}{\begin{bmatrix}x\\y\\z\end{bmatrix}}=P{\begin{bmatrix}x\\y\\0\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}=P{\begin{bmatrix}x\\y\\z\end{bmatrix}}.} Observing that P T = P {\displaystyle P^{\mathrm {T} }=P} shows that

4256-497: Is indeed a projection. The projection P {\displaystyle P} is orthogonal if and only if α = 0 {\displaystyle \alpha =0} because only then P T = P . {\displaystyle P^{\mathrm {T} }=P.} By definition, a projection P {\displaystyle P} is idempotent (i.e. P 2 = P {\displaystyle P^{2}=P} ). Every projection

4368-654: Is not a projection if ‖ u ‖ ≠ 1. {\displaystyle \left\|\mathbf {u} \right\|\neq 1.} After dividing by u T u = ‖ u ‖ 2 , {\displaystyle \mathbf {u} ^{\mathsf {T}}\mathbf {u} =\left\|\mathbf {u} \right\|^{2},} we obtain the projection u ( u T u ) − 1 u T {\displaystyle \mathbf {u} \left(\mathbf {u} ^{\mathsf {T}}\mathbf {u} \right)^{-1}\mathbf {u} ^{\mathsf {T}}} onto

4480-612: Is not the zero operator. Let the vectors u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} form a basis for the range of P {\displaystyle P} , and assemble these vectors in the n × k {\displaystyle n\times k} matrix A {\displaystyle A} . Then k ≥ 1 {\displaystyle k\geq 1} , otherwise k = 0 {\displaystyle k=0} and P {\displaystyle P}

4592-399: Is orthogonal if and only if k = 0 {\displaystyle k=0} ) and the σ i {\displaystyle \sigma _{i}} -blocks correspond to the oblique components. When the underlying vector space X {\displaystyle X} is a (not necessarily finite-dimensional) normed vector space , analytic questions, irrelevant in

Palazzo Valmarana - Misplaced Pages Continue

4704-1087: Is orthogonal if and only if it is self-adjoint . Using the self-adjoint and idempotent properties of P {\displaystyle P} , for any x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } in W {\displaystyle W} we have P x ∈ U {\displaystyle P\mathbf {x} \in U} , y − P y ∈ V {\displaystyle \mathbf {y} -P\mathbf {y} \in V} , and ⟨ P x , y − P y ⟩ = ⟨ x , ( P − P 2 ) y ⟩ = 0 {\displaystyle \langle P\mathbf {x} ,\mathbf {y} -P\mathbf {y} \rangle =\langle \mathbf {x} ,\left(P-P^{2}\right)\mathbf {y} \rangle =0} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle }

4816-406: Is projection on X {\displaystyle X} , i.e. P 2 = P {\displaystyle P^{2}=P} , then it is easily verified that ( 1 − P ) 2 = ( 1 − P ) {\displaystyle (1-P)^{2}=(1-P)} . In other words, 1 − P {\displaystyle 1-P} is also

4928-1320: Is self-adjoint). When the vector space W {\displaystyle W} has an inner product and is complete (is a Hilbert space ) the concept of orthogonality can be used. An orthogonal projection is a projection for which the range U {\displaystyle U} and the kernel V {\displaystyle V} are orthogonal subspaces . Thus, for every x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } in W {\displaystyle W} , ⟨ P x , ( y − P y ) ⟩ = ⟨ ( x − P x ) , P y ⟩ = 0 {\displaystyle \langle P\mathbf {x} ,(\mathbf {y} -P\mathbf {y} )\rangle =\langle (\mathbf {x} -P\mathbf {x} ),P\mathbf {y} \rangle =0} . Equivalently: ⟨ x , P y ⟩ = ⟨ P x , P y ⟩ = ⟨ P x , y ⟩ . {\displaystyle \langle \mathbf {x} ,P\mathbf {y} \rangle =\langle P\mathbf {x} ,P\mathbf {y} \rangle =\langle P\mathbf {x} ,\mathbf {y} \rangle .} A projection

5040-413: Is sometimes denoted as y ^ {\displaystyle {\hat {\mathbf {y} }}} . There is a theorem in linear algebra that states that this z {\displaystyle \mathbf {z} } is the smallest distance (the orthogonal distance ) from y {\displaystyle \mathbf {y} } to V {\displaystyle V} and

5152-438: Is the partial isometry that vanishes on the orthogonal complement of U {\displaystyle U} , and A {\displaystyle A} is the isometry that embeds U {\displaystyle U} into the underlying vector space. The range of P A {\displaystyle P_{A}} is therefore the final space of A {\displaystyle A} . It

5264-667: Is the zero matrix of size d − r {\displaystyle d-r} , and ⊕ {\displaystyle \oplus } is the direct sum operator. If the vector space is complex and equipped with an inner product , then there is an orthonormal basis in which the matrix of P is where σ 1 ≥ σ 2 ≥ ⋯ ≥ σ k > 0 {\displaystyle \sigma _{1}\geq \sigma _{2}\geq \dots \geq \sigma _{k}>0} . The integers k , s , m {\displaystyle k,s,m} and

5376-507: Is the direct sum X = U ⊕ V {\displaystyle X=U\oplus V} , then the operator defined by P ( u + v ) = u {\displaystyle P(u+v)=u} is still a projection with range U {\displaystyle U} and kernel V {\displaystyle V} . It is also clear that P 2 = P {\displaystyle P^{2}=P} . Conversely, if P {\displaystyle P}

5488-1519: Is the inner product associated with W {\displaystyle W} . Therefore, P {\displaystyle P} and I − P {\displaystyle I-P} are orthogonal projections. The other direction, namely that if P {\displaystyle P} is orthogonal then it is self-adjoint, follows from the implication from ⟨ ( x − P x ) , P y ⟩ = ⟨ P x , ( y − P y ) ⟩ = 0 {\displaystyle \langle (\mathbf {x} -P\mathbf {x} ),P\mathbf {y} \rangle =\langle P\mathbf {x} ,(\mathbf {y} -P\mathbf {y} )\rangle =0} to ⟨ x , P y ⟩ = ⟨ P x , P y ⟩ = ⟨ P x , y ⟩ = ⟨ x , P ∗ y ⟩ {\displaystyle \langle \mathbf {x} ,P\mathbf {y} \rangle =\langle P\mathbf {x} ,P\mathbf {y} \rangle =\langle P\mathbf {x} ,\mathbf {y} \rangle =\langle \mathbf {x} ,P^{*}\mathbf {y} \rangle } for every x {\displaystyle x} and y {\displaystyle y} in W {\displaystyle W} ; thus P = P ∗ {\displaystyle P=P^{*}} . The existence of an orthogonal projection onto

5600-459: Is the latter who signed the construction contracts with the builders in December 1565. Nevertheless, no doubts can remain about the role her deceased husband, Giovanni Alvise (died 1558), played in choosing Palladio as designer of his family palace. In 1549, along with Girolamo Chiericati and Giangiorgio Trissino , Giovanni Alvise Valmarana had publicly supported Palladio’s project for the porticoes of

5712-415: Is the matrix with these vectors as columns, then the projection is: P A = A ( A T A ) − 1 A T . {\displaystyle P_{A}=A\left(A^{\mathsf {T}}A\right)^{-1}A^{\mathsf {T}}.} The matrix A {\displaystyle A} still embeds U {\displaystyle U} into

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5824-443: Is the zero operator. The range and the kernel are complementary spaces, so the kernel has dimension n − k {\displaystyle n-k} . It follows that the orthogonal complement of the kernel has dimension k {\displaystyle k} . Let v 1 , … , v k {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{k}} form

5936-403: Is therefore not necessarily equal. Let V {\displaystyle V} be a vector space (in this case a plane) spanned by orthogonal vectors u 1 , u 2 , … , u p {\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\dots ,\mathbf {u} _{p}} . Let y {\displaystyle y} be

6048-1218: The k × k {\displaystyle k\times k} -matrix B T A {\displaystyle B^{\mathsf {T}}A} is invertible. So the equation B T ( x − A w ) = 0 {\displaystyle B^{\mathsf {T}}(\mathbf {x} -A\mathbf {w} )=\mathbf {0} } gives the vector w = ( B T A ) − 1 B T x . {\displaystyle \mathbf {w} =(B^{\mathsf {T}}A)^{-1}B^{\mathsf {T}}\mathbf {x} .} In this way, P x = x 1 = A w = A ( B T A ) − 1 B T x {\displaystyle P\mathbf {x} =\mathbf {x} _{1}=A\mathbf {w} =A(B^{\mathsf {T}}A)^{-1}B^{\mathsf {T}}\mathbf {x} } for any vector x ∈ V {\displaystyle \mathbf {x} \in V} and hence P = A ( B T A ) − 1 B T {\displaystyle P=A(B^{\mathsf {T}}A)^{-1}B^{\mathsf {T}}} . In

6160-578: The Fall of the Venetian Republic in 1797, under Napoleonic rule, it was made a duché grand-fief (not a grand duchy, but a hereditary (extinguished in 1896), nominal duchy, a rare honor reserved for French officials) within Napoleon's personal Kingdom of Italy for general Caulaincourt , also imperial Grand-Écuyer. One of the consequences of the city's occupation was the destruction of a prized silver model of

6272-554: The Third War of Italian independence . Vicenza's area was a location of major combat in both World War I (on the Asiago plateau) and World War II (a focal center of the Italian resistance), and it was the most damaged city in Veneto by Allied bombings , including many of its monuments; the civil victims were over 2,000. The end of World War II was followed by a period of depression, caused by

6384-569: The conjugate transpose is used instead of the transpose. Further details on sums of projectors can be found in Banerjee and Roy (2014). Also see Banerjee (2004) for application of sums of projectors in basic spherical trigonometry . The term oblique projections is sometimes used to refer to non-orthogonal projections. These projections are also used to represent spatial figures in two-dimensional drawings (see oblique projection ), though not as frequently as orthogonal projections. Whereas calculating

6496-544: The nave and aisles are projected onto the same plane in the Venetian church, so too on the façade of the Palazzo Valmarana the stratification of two systems becomes evident: the giant order of the six Composite pilasters seems to be superimposed on the minor order of Corinthian pilasters, in a far more evident way at the edges where the absence of the final pilaster serves to reveal the underlying order which supports

6608-620: The orthogonal complement of Q A {\displaystyle Q_{A}} . Denote the singular values of the matrix Q A T A ( B T A ) − 1 B T Q A ⊥ {\displaystyle Q_{A}^{T}A(B^{T}A)^{-1}B^{T}Q_{A}^{\perp }} by the positive values γ 1 ≥ γ 2 ≥ … ≥ γ k {\displaystyle \gamma _{1}\geq \gamma _{2}\geq \ldots \geq \gamma _{k}} . With this,

6720-664: The orthogonal complement . For Banach spaces, a one-dimensional subspace always has a closed complementary subspace. This is an immediate consequence of Hahn–Banach theorem . Let U {\displaystyle U} be the linear span of u {\displaystyle u} . By Hahn–Banach, there exists a bounded linear functional φ {\displaystyle \varphi } such that φ ( u ) = 1 . The operator P ( x ) = φ ( x ) u {\displaystyle P(x)=\varphi (x)u} satisfies P 2 = P {\displaystyle P^{2}=P} , i.e. it

6832-615: The standard inner product can be substituted for ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } . A simple case occurs when the orthogonal projection is onto a line. If u {\displaystyle \mathbf {u} } is a unit vector on the line, then the projection is given by the outer product P u = u u T . {\displaystyle P_{\mathbf {u} }=\mathbf {u} \mathbf {u} ^{\mathsf {T}}.} (If u {\displaystyle \mathbf {u} }

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6944-456: The "city of Palladio" has been listed as a UNESCO World Heritage Site since 1994. Vicenza had an estimated population of 115,927 and a metropolitan area of 270,000 in 2008. Vicenza is the third-largest Italian industrial centre as measured by the value of its exports, and is one of the country's wealthiest cities, in large part due to its textile and steel industries, which employ tens of thousands of people. Additionally, about one fifth of

7056-535: The Basilica Palladiana , evidently on the basis of an opinion formed six years prior, when Giovanni Alvise supervised the execution of ephemeral structures, conceived by Palladio under Trissino’s direction, to honour the entrance into Vicenza of Bishop Niccolò Ridolfi (1543). Furthermore, it was a space designed by Palladio – the Valmarana Chapel in the church of Santa Corona – which would eventually host

7168-488: The Italian average of 9.45 births. In 2010, 83.5% of the population was Italian . From 1876 to 1976 it has been calculated that over 1,000,000 people from the province of Vicenza have emigrated, with more than 3,000,000 people of Vicentino descent living around the world (most common migrational currents included Brazil , the United States , Canada , Australia , Germany , France , Belgium and Switzerland ) escaping

7280-457: The Palazzo Valmarana is both one of Palladio's most extraordinary and most individual realizations. For the first time in a palace, a giant order embraces the entire vertical expanse of the building: evidently this was a solution which found its origins in Palladio's experimentation with the façades of religious buildings, such as the almost contemporary façade of San Francesco della Vigna . Just as

7392-590: The Paleo-Veneti in their fight against the Celtic tribes that populated north-western Italy. The Roman presence in the area grew exponentially over time and the Paleo-Veneti (whose culture mirrored Etruscan and Greek values more so than Celtic ones) were gradually assimilated. In 157 BC, the city was a de facto Roman centre and was given the name of Vicetia or Vincentia , meaning "victorious". The citizens of Vicetia received Roman citizenship and were inscribed into

7504-568: The Roman tribe Romilia in 49 BC. The city was known for its agriculture, brickworks, marble quarry, and wool industry and had some importance as a way-station on the important road from Mediolanum (Milan) to Aquileia , near Tergeste (Trieste), but it was overshadowed by its neighbor Patavium ( Padua ). Little survives of the Roman city, but three of the bridges across the Bacchiglione and Retrone rivers are of Roman origin, and isolated arches of

7616-483: The Second Lombard League against Emperor Frederick II who sacked the city in 1237, after which it was annexed to Ezzelino's dominions. On his death the old oligarchic republic political structure was restored – a consiglio maggiore ("grand council") of four hundred members and a consiglio minore ("small council") of forty members – and it formed a league with Padua, Treviso and Verona . Three years later

7728-516: The Veneto ". Vicenza is home to twenty-three buildings designed by Palladio. Famous examples include: Some of the main historical churches: The surrounding country is predominantly agricultural. Major products are wine, wheat, corn, olive oil (in the Barbarano area) and cherries and asparagus are a particularity of Bassano. There are also quarries of marble, sulphur, copper, and silver mines, and beds of lignite and kaolin; mineral springs also abound,

7840-618: The Vicentines entrusted the protection of the city to Padua , so as to safeguard republican liberty; but this protectorate ( custodia ) quickly became dominion, and for that reason Vicenza in 1311 submitted to the Scaligeri lords of Verona , who fortified it against the Visconti of Milan . Vicenza came under the rule of the Republic of Venice in 1404, and its subsequent history is that of Venice. It

7952-666: The Vicenza area, beginning in the 6th century. In 899, Vicenza was destroyed by Magyar raiders. In 1001, Otto III handed over the government of the city to the bishop , and its communal organization had an opportunity to develop, separating soon from the episcopal authority. It took an active part in the League with Verona and, most of all, in the Lombard League (1164–1167) against Emperor Frederick I Barbarossa compelling Padua and Treviso to join: its podestà , Ezzelino II il Balbo ,

8064-651: The assumption that the integer k ≥ 1 {\displaystyle k\geq 1} , and let A {\displaystyle A} denote the n × k {\displaystyle n\times k} matrix whose columns are u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} , i.e., A = [ u 1 ⋯ u k ] {\displaystyle A={\begin{bmatrix}\mathbf {u} _{1}&\cdots &\mathbf {u} _{k}\end{bmatrix}}} . Then

8176-534: The case that P {\displaystyle P} is an orthogonal projection, we can take A = B {\displaystyle A=B} , and it follows that P = A ( A T A ) − 1 A T {\displaystyle P=A\left(A^{\mathsf {T}}A\right)^{-1}A^{\mathsf {T}}} . By using this formula, one can easily check that P = P T {\displaystyle P=P^{\mathsf {T}}} . In general, if

8288-587: The city, massive and disorganized urbanization and employment of foreign immigrants increased. Vicenza is home to the US Army post Caserma Ederle (Camp Ederle), also known as the U.S. Army Garrison Vicenza . In 1965, Caserma Ederle became the headquarters of the Southern European Task Force, which includes the 173d Airborne Brigade . In January 2006, the European Gendarmerie Force

8400-575: The city, the Jewel of Vicenza . After 1814, Vicenza passed to the Austrian Empire . In 1848, however, the populace rose against Austria, more violently than in any other Italian centre apart from Milan and Brescia (the city would receive the highest award for military valour for the courage displayed by revolutionaries in this period). As a part of the Kingdom of Lombardy–Venetia , it was annexed to Italy after

8512-424: The claim. The above argument makes use of the assumption that both U {\displaystyle U} and V {\displaystyle V} are closed. In general, given a closed subspace U {\displaystyle U} , there need not exist a complementary closed subspace V {\displaystyle V} , although for Hilbert spaces this can always be done by taking

8624-567: The country's gold and jewelry is made in Vicenza, greatly contributing to the city's economy. Another important sector is the engineering/computer components industry ( Federico Faggin , the microprocessor's co-inventor, was born in Vicenza). Vicentia was settled by the Italic Euganei tribe and then by the Paleo- Veneti tribe in the 3rd and 2nd centuries BC. The Romans allied themselves with

8736-473: The devastation during the two world wars. In the 1960s, the whole central part of Veneto, witnessed a strong economic development caused by the emergence of small and medium family businesses, ranging in a vast array of products (that often emerged illegally) that paved the way for what would be known as the "miracolo del nord-est" ("miracle of the northeast"). In the following years, the economic development grew vertiginously. Huge industrial areas sprouted around

8848-546: The devastation left by poverty, war and sickness. Today, almost 100,000 Vicenza citizens live and work abroad and the city has morphed from a land of emigration to a land of immigration. The largest immigrant group comes from the United States (about 9,000 people, partly due to the presence of the military base ). Other ethnic minorities comes from other European nations (the largest being Serbia , Romania , and Moldova ), South Asian (the largest being Bangladesh and Pakistan ), sub-saharan Africa , and North Africa (largest

8960-412: The finite-dimensional case, need to be considered. Assume now X {\displaystyle X} is a Banach space . Many of the algebraic results discussed above survive the passage to this context. A given direct sum decomposition of X {\displaystyle X} into complementary subspaces still specifies a projection, and vice versa. If X {\displaystyle X}

9072-419: The finite-dimensional case, projections need not be continuous in general. If a subspace U {\displaystyle U} of X {\displaystyle X} is not closed in the norm topology, then the projection onto U {\displaystyle U} is not continuous. In other words, the range of a continuous projection P {\displaystyle P} must be

9184-403: The fitted value of an ordinary least squares regression requires an orthogonal projection, calculating the fitted value of an instrumental variables regression requires an oblique projection. A projection is defined by its kernel and the basis vectors used to characterize its range (which is a complement of the kernel). When these basis vectors are orthogonal to the kernel, then the projection

9296-1990: The following holds: I = [ A B ] [ A B ] − 1 [ A T B T ] − 1 [ A T B T ] = [ A B ] ( [ A T B T ] [ A B ] ) − 1 [ A T B T ] = [ A B ] [ A T A O O B T B ] − 1 [ A T B T ] = A ( A T A ) − 1 A T + B ( B T B ) − 1 B T {\displaystyle {\begin{aligned}I&={\begin{bmatrix}A&B\end{bmatrix}}{\begin{bmatrix}A&B\end{bmatrix}}^{-1}{\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}^{-1}{\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}\\&={\begin{bmatrix}A&B\end{bmatrix}}\left({\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}{\begin{bmatrix}A&B\end{bmatrix}}\right)^{-1}{\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}\\&={\begin{bmatrix}A&B\end{bmatrix}}{\begin{bmatrix}A^{\mathsf {T}}A&O\\O&B^{\mathsf {T}}B\end{bmatrix}}^{-1}{\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}\\[4pt]&=A\left(A^{\mathsf {T}}A\right)^{-1}A^{\mathsf {T}}+B\left(B^{\mathsf {T}}B\right)^{-1}B^{\mathsf {T}}\end{aligned}}} If

9408-545: The formula for orthogonal projections given above. A standard proof of this expression is the following. For any vector x {\displaystyle \mathbf {x} } in the vector space V {\displaystyle V} , we can decompose x = x 1 + x 2 {\displaystyle \mathbf {x} =\mathbf {x} _{1}+\mathbf {x} _{2}} , where vector x 1 = P ( x ) {\displaystyle \mathbf {x} _{1}=P(\mathbf {x} )}

9520-442: The jewelry and clothing factories. Important vicentino clothing firms include: Diesel , Pal Zileri , Marzotto , Bottega Veneta , Marlboro Classics etc. The Gold Exposition is world-famous and it takes place in Vicenza twice a year (January and September). Other industries worthy of mention are the woollen and silk, pottery, tanneries, and musical instruments. The headquarters of the bicycle component manufacturer Campagnolo and

9632-613: The largest singular values of P {\displaystyle P} and I − P {\displaystyle I-P} are equal, and thus that the matrix norm of the oblique projections are the same. However, the condition number satisfies the relation κ ( I − P ) = σ 1 1 ≥ σ 1 σ k = κ ( P ) {\displaystyle \kappa (I-P)={\frac {\sigma _{1}}{1}}\geq {\frac {\sigma _{1}}{\sigma _{k}}}=\kappa (P)} , and

9744-458: The matrix A {\displaystyle A} by A + = ( A ∗ A ) − 1 A ∗ {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}} since A {\displaystyle A} has full column rank, so P = A A + {\displaystyle P=AA^{+}} . I − P {\displaystyle I-P}

9856-427: The mortal remains of Giovanni Alvise and Isabella, on the commission of their son Leonardo. On the site later occupied by the new 16th century (Cinquecento) palace, the Valmarana family possessed buildings right from the end of the 15th century (Quattrocento), which were progressively combined until they became the object of Palladio’s renovation. The planimetric irregularity of the internal spaces doubtless derives from

9968-480: The most famous being those of Recoaro. Massive industrial areas surround the city and extend extensively in the western and eastern hinterland, with numerous steel and textile factories located in the Montecchio Maggiore , Chiampo and Sovizzo area in the west and Camisano Vicentino and Torri di Quartesolo in the east, areas characterised by a disorganised and extensive cementifaction. Elite sectors are

10080-453: The oblique orientation of the façade and of pre-existing walls. In this sense it becomes quite evident just how much the Olympian regularity of the palace illustrated in I quattro libri dell'architettura (1570) was the product of Palladio’s usual theoretical abstraction, especially since not only was the extension of the palace beyond the square courtyard never realised, but nor does it seem it

10192-901: The orthogonal condition is enhanced to A T W B = A T W T B = 0 {\displaystyle A^{\mathsf {T}}WB=A^{\mathsf {T}}W^{\mathsf {T}}B=0} with W {\displaystyle W} non-singular, the following holds: I = [ A B ] [ ( A T W A ) − 1 A T ( B T W B ) − 1 B T ] W . {\displaystyle I={\begin{bmatrix}A&B\end{bmatrix}}{\begin{bmatrix}\left(A^{\mathsf {T}}WA\right)^{-1}A^{\mathsf {T}}\\\left(B^{\mathsf {T}}WB\right)^{-1}B^{\mathsf {T}}\end{bmatrix}}W.} All these formulas also hold for complex inner product spaces, provided that

10304-427: The population, compared to pensioners, who number 21.60%. This compares with the Italian average of 18.06% (minors) and 19.94% (pensioners). The average age of Vicenza residents is 43 compared to the Italian average of 42. In the five years between 2002 and 2007, the population of Vicenza grew by 3.72%, while Italy as a whole grew by 3.85%. The current birth rate of Vicenza is 9.16 births per 1,000 inhabitants compared to

10416-855: The projection is an orthogonal projection. A simple example of a non-orthogonal (oblique) projection is P = [ 0 0 α 1 ] . {\displaystyle P={\begin{bmatrix}0&0\\\alpha &1\end{bmatrix}}.} Via matrix multiplication , one sees that P 2 = [ 0 0 α 1 ] [ 0 0 α 1 ] = [ 0 0 α 1 ] = P . {\displaystyle P^{2}={\begin{bmatrix}0&0\\\alpha &1\end{bmatrix}}{\begin{bmatrix}0&0\\\alpha &1\end{bmatrix}}={\begin{bmatrix}0&0\\\alpha &1\end{bmatrix}}=P.} showing that P {\displaystyle P}

10528-499: The projection is given by: P A = A A T {\displaystyle P_{A}=AA^{\mathsf {T}}} which can be rewritten as P A = ∑ i ⟨ u i , ⋅ ⟩ u i . {\displaystyle P_{A}=\sum _{i}\langle \mathbf {u} _{i},\cdot \rangle \mathbf {u} _{i}.} The matrix A T {\displaystyle A^{\mathsf {T}}}

10640-405: The projection operator. If [ A B ] {\displaystyle {\begin{bmatrix}A&B\end{bmatrix}}} is a non-singular matrix and A T B = 0 {\displaystyle A^{\mathsf {T}}B=0} (i.e., B {\displaystyle B} is the null space matrix of A {\displaystyle A} ),

10752-420: The properties of the dot product of parallel and perpendicular vectors. This formula can be generalized to orthogonal projections on a subspace of arbitrary dimension . Let u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} be an orthonormal basis of the subspace U {\displaystyle U} , with

10864-557: The protective wear for sports manufacturer Dainese are located here. Vicenza railway station , opened in 1846, forms part of the Milan–Venice railway , and is also a junction of two branch lines, to Schio and Treviso. Vicenza is home to Vicenza Hurricanes American Football team which currently plays in League 2 . Founded in 2009, the Hurricanes have a junior team and a senior team with

10976-831: The province's geographical diversity. Unlike Venetian cuisine where fish reigns supreme, game meat, cheeses and vegetables take center stage accompanied by polenta, soft from the stove or day-old sliced and grilled over the fireplace embers, better yet cooked in a pan under the spit where it lightly fries in meat drippings to create a crunchy golden outer crust. Vicenza is known for its simple dishes, and often famous cheeses, fruits, ingredients and wines, such as sopressa vicentina, Asiago cheese , Marostica cherries, Nanto truffles, Bassano del Grappa asparagus and Breganze Cabernet wine. The inhabitants of Vicenza are jokingly referred to by other Italians as mangiagatti , or "cat eaters". Purportedly, Vicentini turned to cats for sustenance during times of famine, such as during World War II . Vicenza

11088-480: The range space of the projection is generated by a frame (i.e. the number of generators is greater than its dimension), the formula for the projection takes the form: P A = A A + {\displaystyle P_{A}=AA^{+}} . Here A + {\displaystyle A^{+}} stands for the Moore–Penrose pseudoinverse . This is just one of many ways to construct

11200-481: The real numbers σ i {\displaystyle \sigma _{i}} are uniquely determined. 2 k + s + m = d {\displaystyle 2k+s+m=d} . The factor I m ⊕ 0 s {\displaystyle I_{m}\oplus 0_{s}} corresponds to the maximal invariant subspace on which P {\displaystyle P} acts as an orthogonal projection (so that P itself

11312-574: The relationships between the building’s interior and exterior. Vicenza Vicenza ( / v ɪ ˈ tʃ ɛ n t s ə / vih- CHENT -sə ; Italian: [viˈtʃɛntsa] ; Venetian : Vicensa [viˈtʃeŋ(t)sa] ) is a city in northeastern Italy. It is in the Veneto region, at the northern base of the Monte Berico , where it straddles the River Bacchiglione . Vicenza

11424-562: The singular values for I − P {\displaystyle I-P} are σ i = { 1 + γ i 2 1 ≤ i ≤ k 1 k + 1 ≤ i ≤ n − k 0 otherwise {\displaystyle \sigma _{i}={\begin{cases}{\sqrt {1+\gamma _{i}^{2}}}&1\leq i\leq k\\1&k+1\leq i\leq n-k\\0&{\text{otherwise}}\end{cases}}} This implies that

11536-420: The singular values for P {\displaystyle P} are: σ i = { 1 + γ i 2 1 ≤ i ≤ k 0 otherwise {\displaystyle \sigma _{i}={\begin{cases}{\sqrt {1+\gamma _{i}^{2}}}&1\leq i\leq k\\0&{\text{otherwise}}\end{cases}}} and

11648-465: The subspace spanned by u {\displaystyle u} . In the general case, we can have an arbitrary positive definite matrix D {\displaystyle D} defining an inner product ⟨ x , y ⟩ D = y † D x {\displaystyle \langle x,y\rangle _{D}=y^{\dagger }Dx} , and the projection P A {\displaystyle P_{A}}

11760-1138: The sum of a component on the line (i.e. the projected vector we seek) and another perpendicular to it, x = x ∥ + x ⊥ {\displaystyle \mathbf {x} =\mathbf {x} _{\parallel }+\mathbf {x} _{\perp }} . Applying projection, we get P u x = u u T x ∥ + u u T x ⊥ = u ( sgn ⁡ ( u T x ∥ ) ‖ x ∥ ‖ ) + u ⋅ 0 = x ∥ {\displaystyle P_{\mathbf {u} }\mathbf {x} =\mathbf {u} \mathbf {u} ^{\mathsf {T}}\mathbf {x} _{\parallel }+\mathbf {u} \mathbf {u} ^{\mathsf {T}}\mathbf {x} _{\perp }=\mathbf {u} \left(\operatorname {sgn} \left(\mathbf {u} ^{\mathsf {T}}\mathbf {x} _{\parallel }\right)\left\|\mathbf {x} _{\parallel }\right\|\right)+\mathbf {u} \cdot \mathbf {0} =\mathbf {x} _{\parallel }} by

11872-414: The underlying vector space but is no longer an isometry in general. The matrix ( A T A ) − 1 {\displaystyle \left(A^{\mathsf {T}}A\right)^{-1}} is a "normalizing factor" that recovers the norm. For example, the rank -1 operator u u T {\displaystyle \mathbf {u} \mathbf {u} ^{\mathsf {T}}}

11984-808: The vector x 2 {\displaystyle \mathbf {x} _{2}} satisfies B T x 2 = 0 {\displaystyle B^{\mathsf {T}}\mathbf {x} _{2}=\mathbf {0} } by the construction of B {\displaystyle B} . Put these conditions together, and we find a vector w {\displaystyle \mathbf {w} } so that B T ( x − A w ) = 0 {\displaystyle B^{\mathsf {T}}(\mathbf {x} -A\mathbf {w} )=\mathbf {0} } . Since matrices A {\displaystyle A} and B {\displaystyle B} are of full rank k {\displaystyle k} by their construction,

12096-477: The vector space is over complex number field, one then uses the Hermitian transpose A ∗ {\displaystyle A^{*}} and has the formula P = A ( A ∗ A ) − 1 A ∗ {\displaystyle P=A\left(A^{*}A\right)^{-1}A^{*}} . Recall that one can express the Moore–Penrose inverse of

12208-531: Was besieged by the Emperor Sigismund , and Maximilian I held possession of it in 1509 and 1516. Vicenza was a candidate to host the Council of Trent . The 16th century was the time of Andrea Palladio , who left many outstanding examples of his art with palaces and villas in the city's territory, which before Palladio's passage, was arguably the most downtrodden and esthetically lacking city in Veneto. After

12320-575: Was captain of the league. When peace was restored, however, the old rivalry with Padua, Bassano , and other cities was renewed, besides which there were the internal factions of the Vivaresi ( Ghibellines ) and the Maltraversi ( Guelphs ). The tyrannical Ezzelino III from Bassano drove the Guelphs out of Vicenza, and caused his brother, Alberico , to be elected podestà in 1230. The independent commune joined

12432-462: Was even intended by Leonardo Valmarana, who bought up neighbouring properties rather than continuing the construction of the family palace. The palazzo was heavily damaged during the World War II by an Allied bombing on 18 March 1945. The roof, part of the attic and most of the main hall at the piano nobile were destroyed. The façade however remained intact, and today represents a rare example of

12544-491: Was inaugurated in Vicenza. Vicenza lies in the Veneto region, at the northern base of Monte Berico , where it straddles the Bacchiglione River . Vicenza is approximately 60 kilometres (37 mi) west of Venice and 200 kilometres (120 mi) east of Milan . In 2007, there were 114,268 people residing in Vicenza of whom 47.6% were male and 52.4% were female. Minors (children ages 18 and younger) totalled 17.17% of

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