Projection (linear algebra)

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The transformation P is the orthogonal projection onto the line m.

In linear algebra and functional analysis, a projection is a linear transformation ${\displaystyle P}$ from a vector space to itself (an endomorphism) such that ${\displaystyle P\circ P=P}$. That is, whenever ${\displaystyle P}$ is applied twice to any vector, it gives the same result as if it were applied once (i.e. ${\displaystyle P}$ is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.

Definitions

A projection on a vector space ${\displaystyle V}$ is a linear operator ${\displaystyle P:V\to V}$ such that ${\displaystyle P^2=P}$.

When ${\displaystyle V}$ has an inner product and is complete, i.e. when ${\displaystyle V}$ is a Hilbert space, the concept of orthogonality can be used. A projection ${\displaystyle P}$ on a Hilbert space ${\displaystyle V}$ is called an orthogonal projection if it satisfies ${\displaystyle ⟨P\mathbf{x},\mathbf{y}⟩=⟨\mathbf{x},P\mathbf{y}⟩}$ for all ${\displaystyle \mathbf{x},\mathbf{y}\in V}$. A projection on a Hilbert space that is not orthogonal is called an oblique projection.

Projection matrix

• A square matrix ${\displaystyle P}$ is called a projection matrix if it is equal to its square, i.e. if ${\displaystyle P^2=P}$.
• A square matrix ${\displaystyle P}$ is called an orthogonal projection matrix if ${\displaystyle P^2=P=P^{\mathrm{T}}}$ for a real matrix, and respectively ${\displaystyle P^2=P=P^{\ast }}$ for a complex matrix, where ${\displaystyle P^{\mathrm{T}}}$ denotes the transpose of ${\displaystyle P}$ and ${\displaystyle P^{\ast }}$ denotes the adjoint or Hermitian transpose of ${\displaystyle P}$.
• A projection matrix that is not an orthogonal projection matrix is called an oblique projection matrix.

The eigenvalues of a projection matrix must be 0 or 1.

Examples

Orthogonal projection

For example, the function which maps the point ${\displaystyle (x,y,z)}$ in three-dimensional space ${\displaystyle {\mathbb{R}}^3}$ to the point ${\displaystyle (x,y,0)}$ is an orthogonal projection onto the xy-plane. This function is represented by the matrix

$${\displaystyle P=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right].}$$

The action of this matrix on an arbitrary vector is

$${\displaystyle P\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}x\\ y\\ 0\end{array}\right].}$$

To see that ${\displaystyle P}$ is indeed a projection, i.e., ${\displaystyle P=P^2}$, we compute

$${\displaystyle P^2\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=P\left[\begin{array}{c}x\\ y\\ 0\end{array}\right]=\left[\begin{array}{c}x\\ y\\ 0\end{array}\right]=P\left[\begin{array}{c}x\\ y\\ z\end{array}\right].}$$

Observing that ${\displaystyle P^{\mathrm{T}}=P}$ shows that the projection is an orthogonal projection.

Oblique projection

A simple example of a non-orthogonal (oblique) projection is

$${\displaystyle P=\left[\begin{array}{cc}0& 0\\ \alpha & 1\end{array}\right].}$$

Via matrix multiplication, one sees that

$${\displaystyle P^2=\left[\begin{array}{cc}0& 0\\ \alpha & 1\end{array}\right]\left[\begin{array}{cc}0& 0\\ \alpha & 1\end{array}\right]=\left[\begin{array}{cc}0& 0\\ \alpha & 1\end{array}\right]=P.}$$

showing that ${\displaystyle P}$ is indeed a projection.

The projection ${\displaystyle P}$ is orthogonal if and only if ${\displaystyle \alpha =0}$ because only then ${\displaystyle P^{\mathrm{T}}=P.}$

Properties and classification

The transformation T is the projection along k onto m. The range of T is m and the kernel is k.

Idempotence

By definition, a projection ${\displaystyle P}$ is idempotent (i.e. ${\displaystyle P^2=P}$).

Open map

Every projection 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 ${\displaystyle \mathbf{x}}$ and any ball ${\displaystyle B_{\mathbf{x}}}$ (with positive radius) centered on ${\displaystyle \mathbf{x}}$, there exists a ball ${\displaystyle B_{P\mathbf{x}}}$ (with positive radius) centered on ${\displaystyle P\mathbf{x}}$ that is wholly contained in the image ${\displaystyle P(B_{\mathbf{x}})}$.

Complementarity of image and kernel

Let ${\displaystyle W}$ be a finite-dimensional vector space and ${\displaystyle P}$ be a projection on ${\displaystyle W}$. Suppose the subspaces ${\displaystyle U}$ and ${\displaystyle V}$ are the image and kernel of ${\displaystyle P}$ respectively. Then ${\displaystyle P}$ has the following properties:

1. ${\displaystyle P}$ is the identity operator ${\displaystyle I}$ on ${\displaystyle U}$:
   $${\displaystyle \mathrm{\forall }\mathbf{x}\in U:P\mathbf{x}=\mathbf{x}.}$$

   
2. We have a direct sum ${\displaystyle W=U\oplus V}$. Every vector ${\displaystyle \mathbf{x}\in W}$ may be decomposed uniquely as ${\displaystyle \mathbf{x}=\mathbf{u}+\mathbf{v}}$ with ${\displaystyle \mathbf{u}=P\mathbf{x}}$ and ${\displaystyle \mathbf{v}=\mathbf{x}-P\mathbf{x}=\left(I-P\right)\mathbf{x}}$, and where ${\displaystyle \mathbf{u}\in U,\mathbf{v}\in V.}$

The image and kernel of a projection are complementary, as are ${\displaystyle P}$ and ${\displaystyle Q=I-P}$. The operator ${\displaystyle Q}$ is also a projection as the image and kernel of ${\displaystyle P}$ become the kernel and image of ${\displaystyle Q}$ and vice versa. We say ${\displaystyle P}$ is a projection along ${\displaystyle V}$ onto ${\displaystyle U}$ (kernel/image) and ${\displaystyle Q}$ is a projection along ${\displaystyle U}$ onto ${\displaystyle V}$.

Spectrum

In infinite-dimensional vector spaces, the spectrum of a projection is contained in ${\displaystyle \{0,1\}}$ as

$${\displaystyle (\lambda I-P{)}^{-1}=\frac{1}{\lambda }I+\frac{1}{\lambda (\lambda -1)}P.}$$

Only 0 or 1 can be an eigenvalue of a projection. This implies that an orthogonal projection ${\displaystyle P}$ 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 ${\displaystyle V}$, there may be many projections whose range (or kernel) is ${\displaystyle V}$.

If a projection is nontrivial it has minimal polynomial ${\displaystyle x^2-x=x(x-1)}$, which factors into distinct linear factors, and thus ${\displaystyle P}$ is diagonalizable.

Product of projections

The product of projections is not in general 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 is self-adjoint).

Orthogonal projections

Main articles: Hilbert projection theorem and Complemented subspace

When the vector space ${\displaystyle W}$ 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 ${\displaystyle U}$ and the kernel ${\displaystyle V}$ are orthogonal subspaces. Thus, for every ${\displaystyle \mathbf{x}}$ and ${\displaystyle \mathbf{y}}$ in ${\displaystyle W}$, ${\displaystyle ⟨P\mathbf{x},(\mathbf{y}-P\mathbf{y})⟩=⟨(\mathbf{x}-P\mathbf{x}),P\mathbf{y}⟩=0}$. Equivalently:

$${\displaystyle ⟨\mathbf{x},P\mathbf{y}⟩=⟨P\mathbf{x},P\mathbf{y}⟩=⟨P\mathbf{x},\mathbf{y}⟩.}$$

A projection is orthogonal if and only if it is self-adjoint. Using the self-adjoint and idempotent properties of ${\displaystyle P}$, for any ${\displaystyle \mathbf{x}}$ and ${\displaystyle \mathbf{y}}$ in ${\displaystyle W}$ we have ${\displaystyle P\mathbf{x}\in U}$, ${\displaystyle \mathbf{y}-P\mathbf{y}\in V}$, and

$${\displaystyle ⟨P\mathbf{x},\mathbf{y}-P\mathbf{y}⟩=⟨\mathbf{x},\left(P-P^2\right)\mathbf{y}⟩=0}$$

where ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is the inner product associated with ${\displaystyle W}$. Therefore, ${\displaystyle P}$ and ${\displaystyle I-P}$ are orthogonal projections.^[3] The other direction, namely that if ${\displaystyle P}$ is orthogonal then it is self-adjoint, follows from the implication from ${\displaystyle ⟨(\mathbf{x}-P\mathbf{x}),P\mathbf{y}⟩=⟨P\mathbf{x},(\mathbf{y}-P\mathbf{y})⟩=0}$ to

$${\displaystyle ⟨\mathbf{x},P\mathbf{y}⟩=⟨P\mathbf{x},P\mathbf{y}⟩=⟨P\mathbf{x},\mathbf{y}⟩=⟨\mathbf{x},P^{\ast }\mathbf{y}⟩}$$

for every ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle W}$; thus ${\displaystyle P=P^{\ast }}$.

The existence of an orthogonal projection onto a closed subspace follows from the Hilbert projection theorem.

Properties and special cases

An orthogonal projection is a bounded operator. This is because for every ${\displaystyle \mathbf{v}}$ in the vector space we have, by the Cauchy–Schwarz inequality:

$${\displaystyle {\Vert P\mathbf{v}\Vert }^2=⟨P\mathbf{v},P\mathbf{v}⟩=⟨P\mathbf{v},\mathbf{v}⟩\le \Vert P\mathbf{v}\Vert \cdot \Vert \mathbf{v}\Vert }$$

Thus ${\displaystyle \Vert P\mathbf{v}\Vert \le \Vert \mathbf{v}\Vert }$.

For finite-dimensional complex or real vector spaces, the standard inner product can be substituted for ${\displaystyle ⟨\cdot ,\cdot ⟩}$.

Formulas

A simple case occurs when the orthogonal projection is onto a line. If ${\displaystyle \mathbf{u}}$ is a unit vector on the line, then the projection is given by the outer product

$${\displaystyle P_{\mathbf{u}}=\mathbf{u}{\mathbf{u}}^{\mathsf{T}}.}$$

(If ${\displaystyle \mathbf{u}}$ 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 ${\displaystyle \mathbf{u}}$, 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 ${\displaystyle \mathbf{x}}$ as the sum of a component on the line (i.e. the projected vector we seek) and another perpendicular to it, ${\displaystyle \mathbf{x}={\mathbf{x}}_{\parallel }+{\mathbf{x}}_{\perp }}$. Applying projection, we get

$${\displaystyle P_{\mathbf{u}}\mathbf{x}=\mathbf{u}{\mathbf{u}}^{\mathsf{T}}{\mathbf{x}}_{\parallel }+\mathbf{u}{\mathbf{u}}^{\mathsf{T}}{\mathbf{x}}_{\perp }=\mathbf{u}\left(\mathrm{sgn}\left({\mathbf{u}}^{\mathsf{T}}{\mathbf{x}}_{\parallel }\right)\Vert {\mathbf{x}}_{\parallel }\Vert \right)+\mathbf{u}\cdot \mathbf{0}={\mathbf{x}}_{\parallel }}$$

by 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 ${\displaystyle {\mathbf{u}}_1,\dots ,{\mathbf{u}}_k}$ be an orthonormal basis of the subspace ${\displaystyle U}$, with the assumption that the integer ${\displaystyle k\ge 1}$, and let ${\displaystyle A}$ denote the ${\displaystyle n\times k}$ matrix whose columns are ${\displaystyle {\mathbf{u}}_1,\dots ,{\mathbf{u}}_k}$, i.e., ${\displaystyle A=\left[\begin{array}{ccc}{\mathbf{u}}_1& \cdots & {\mathbf{u}}_k\end{array}\right]}$. Then the projection is given by:

$${\displaystyle P_A=A{A}^{\mathsf{T}}}$$

which can be rewritten as

$${\displaystyle P_A=\sum _i⟨{\mathbf{u}}_i,\cdot ⟩{\mathbf{u}}_i.}$$

The matrix ${\displaystyle A^{\mathsf{T}}}$ is the partial isometry that vanishes on the orthogonal complement of ${\displaystyle U}$, and ${\displaystyle A}$ is the isometry that embeds ${\displaystyle U}$ into the underlying vector space. The range of ${\displaystyle P_A}$ is therefore the final space of ${\displaystyle A}$. It is also clear that ${\displaystyle A{A}^{\mathsf{T}}}$ is the identity operator on ${\displaystyle U}$.

The orthonormality condition can also be dropped. If ${\displaystyle {\mathbf{u}}_1,\dots ,{\mathbf{u}}_k}$ is a (not necessarily orthonormal) basis with ${\displaystyle k\ge 1}$, and ${\displaystyle A}$ is the matrix with these vectors as columns, then the projection is:

$${\displaystyle P_A=A{\left(A^{\mathsf{T}}A\right)}^{-1}{A}^{\mathsf{T}}.}$$

The matrix ${\displaystyle A}$ still embeds ${\displaystyle U}$ into the underlying vector space but is no longer an isometry in general. The matrix ${\displaystyle {\left(A^{\mathsf{T}}A\right)}^{-1}}$ is a "normalizing factor" that recovers the norm. For example, the rank-1 operator ${\displaystyle \mathbf{u}{\mathbf{u}}^{\mathsf{T}}}$ is not a projection if ${\displaystyle \Vert \mathbf{u}\Vert \ne 1.}$ After dividing by ${\displaystyle {\mathbf{u}}^{\mathsf{T}}\mathbf{u}={\Vert \mathbf{u}\Vert }^2,}$ we obtain the projection ${\displaystyle \mathbf{u}{\left({\mathbf{u}}^{\mathsf{T}}\mathbf{u}\right)}^{-1}{\mathbf{u}}^{\mathsf{T}}}$ onto the subspace spanned by ${\displaystyle u}$.

In the general case, we can have an arbitrary positive definite matrix ${\displaystyle D}$ defining an inner product ${\displaystyle ⟨x,y{⟩}_D=y^{†}Dx}$, and the projection ${\displaystyle P_A}$ is given by $P_{A}x={\mathrm{argmin}}_{y\in \mathrm{range}(A)}{\Vert x-y\Vert }_D^2$. Then

$${\displaystyle P_A=A{\left(A^{\mathsf{T}}DA\right)}^{-1}{A}^{\mathsf{T}}D.}$$

When 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: ${\displaystyle P_A=A{A}^{+}}$. Here ${\displaystyle A^{+}}$ stands for the Moore–Penrose pseudoinverse. This is just one of many ways to construct the projection operator.

If ${\displaystyle \left[\begin{array}{cc}A& B\end{array}\right]}$ is a non-singular matrix and ${\displaystyle A^{\mathsf{T}}B=0}$ (i.e., ${\displaystyle B}$ is the null space matrix of ${\displaystyle A}$), the following holds:

$${\displaystyle \begin{array}{rl}I& =\left[\begin{array}{cc}A& B\end{array}\right]{\left[\begin{array}{cc}A& B\end{array}\right]}^{-1}{\left[\begin{array}{c}{A}^{\mathsf{T}}\\ B^{\mathsf{T}}\end{array}\right]}^{-1}\left[\begin{array}{c}{A}^{\mathsf{T}}\\ B^{\mathsf{T}}\end{array}\right]\\ & =\left[\begin{array}{cc}A& B\end{array}\right]{\left(\left[\begin{array}{c}{A}^{\mathsf{T}}\\ B^{\mathsf{T}}\end{array}\right]\left[\begin{array}{cc}A& B\end{array}\right]\right)}^{-1}\left[\begin{array}{c}{A}^{\mathsf{T}}\\ B^{\mathsf{T}}\end{array}\right]\\ & =\left[\begin{array}{cc}A& B\end{array}\right]{\left[\begin{array}{cc}{A}^{\mathsf{T}}A& O\\ O& B^{\mathsf{T}}B\end{array}\right]}^{-1}\left[\begin{array}{c}{A}^{\mathsf{T}}\\ B^{\mathsf{T}}\end{array}\right]\\ & =A{\left(A^{\mathsf{T}}A\right)}^{-1}{A}^{\mathsf{T}}+B{\left(B^{\mathsf{T}}B\right)}^{-1}{B}^{\mathsf{T}}\end{array}}$$

If the orthogonal condition is enhanced to ${\displaystyle A^{\mathsf{T}}WB=A^{\mathsf{T}}{W}^{\mathsf{T}}B=0}$ with ${\displaystyle W}$ non-singular, the following holds:

$${\displaystyle I=\left[\begin{array}{cc}A& B\end{array}\right]\left[\begin{array}{c}{\left(A^{\mathsf{T}}WA\right)}^{-1}{A}^{\mathsf{T}}\\ {\left(B^{\mathsf{T}}WB\right)}^{-1}{B}^{\mathsf{T}}\end{array}\right]W.}$$

All these formulas also hold for complex inner product spaces, provided that 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.

Oblique projections

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 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 is an orthogonal projection. When these basis vectors are not orthogonal to the kernel, the projection is an oblique projection, or just a projection.

A matrix representation formula for a nonzero projection operator

Let ${\displaystyle P}$ be a linear operator, ${\displaystyle P:V\to V,}$ such that ${\displaystyle P^2=P}$ and assume that ${\displaystyle P:V\to V}$ is not the zero operator. Let the vectors ${\displaystyle {\mathbf{u}}_1,\dots ,{\mathbf{u}}_k}$ form a basis for the range of ${\displaystyle P}$, and assemble these vectors in the ${\displaystyle n\times k}$ matrix ${\displaystyle A}$. Therefore the integer ${\displaystyle k\ge 1}$, otherwise ${\displaystyle k=0}$ and ${\displaystyle P}$ is the zero operator. The range and the kernel are complementary spaces, so the kernel has dimension ${\displaystyle n-k}$. It follows that the orthogonal complement of the kernel has dimension ${\displaystyle k}$. Let ${\displaystyle {\mathbf{v}}_1,\dots ,{\mathbf{v}}_k}$ form a basis for the orthogonal complement of the kernel of the projection, and assemble these vectors in the matrix ${\displaystyle B}$. Then the projection ${\displaystyle P}$ (with the condition ${\displaystyle k\ge 1}$) is given by

$${\displaystyle P=A{\left(B^{\mathsf{T}}A\right)}^{-1}{B}^{\mathsf{T}}.}$$

This expression generalizes the formula for orthogonal projections given above. A standard proof of this expression is the following. For any vector ${\displaystyle \mathbf{x}}$ in the vector space ${\displaystyle V}$, we can decompose ${\displaystyle \mathbf{x}={\mathbf{x}}_1+{\mathbf{x}}_2}$, where vector ${\displaystyle {\mathbf{x}}_1=P(\mathbf{x})}$ is in the image of ${\displaystyle P}$, and vector ${\displaystyle {\mathbf{x}}_2=\mathbf{x}-P(\mathbf{x}).}$ So ${\displaystyle P({\mathbf{x}}_2)=P(\mathbf{x})-P^2(\mathbf{x})=\mathbf{0}}$, and then ${\displaystyle {\mathbf{x}}_2}$ is in the kernel of ${\displaystyle P}$, which is the null space of ${\displaystyle A.}$ In other words, the vector ${\displaystyle {\mathbf{x}}_1}$ is in the column space of ${\displaystyle A,}$ so ${\displaystyle {\mathbf{x}}_1=A\mathbf{w}}$ for some ${\displaystyle k}$ dimension vector ${\displaystyle \mathbf{w}}$ and the vector ${\displaystyle {\mathbf{x}}_2}$ satisfies ${\displaystyle B^{\mathsf{T}}{\mathbf{x}}_2=\mathbf{0}}$ by the construction of ${\displaystyle B}$. Put these conditions together, and we find a vector ${\displaystyle \mathbf{w}}$ so that ${\displaystyle B^{\mathsf{T}}(\mathbf{x}-A\mathbf{w})=\mathbf{0}}$. Since matrices ${\displaystyle A}$ and ${\displaystyle B}$ are of full rank ${\displaystyle k}$ by their construction, the ${\displaystyle k\times k}$-matrix ${\displaystyle B^{\mathsf{T}}A}$ is invertible. So the equation ${\displaystyle B^{\mathsf{T}}(\mathbf{x}-A\mathbf{w})=\mathbf{0}}$ gives the vector ${\displaystyle \mathbf{w}=(B^{\mathsf{T}}A{)}^{-1}{B}^{\mathsf{T}}\mathbf{x}.}$ In this way, ${\displaystyle P\mathbf{x}={\mathbf{x}}_1=A\mathbf{w}=A(B^{\mathsf{T}}A{)}^{-1}{B}^{\mathsf{T}}\mathbf{x}}$ for any vector ${\displaystyle \mathbf{x}\in V}$ and hence ${\displaystyle P=A(B^{\mathsf{T}}A{)}^{-1}{B}^{\mathsf{T}}}$.

In the case that ${\displaystyle P}$ is an orthogonal projection, we can take ${\displaystyle A=B}$, and it follows that ${\displaystyle P=A{\left(A^{\mathsf{T}}A\right)}^{-1}{A}^{\mathsf{T}}}$. By using this formula, one can easily check that ${\displaystyle P=P^{\mathsf{T}}}$. In general, if the vector space is over complex number field, one then uses the Hermitian transpose ${\displaystyle A^{\ast }}$ and has the formula ${\displaystyle P=A{\left(A^{\ast }A\right)}^{-1}{A}^{\ast }}$. Recall that one can define the Moore–Penrose inverse of the matrix ${\displaystyle A}$ by ${\displaystyle A^{+}=(A^{\ast }A{)}^{-1}{A}^{\ast }}$ since ${\displaystyle A}$ has full column rank, so ${\displaystyle P=A{A}^{+}}$.

Singular values

Note that ${\displaystyle I-P}$ is also an oblique projection. The singular values of ${\displaystyle P}$ and ${\displaystyle I-P}$ can be computed by an orthonormal basis of ${\displaystyle A}$. Let ${\displaystyle Q_A}$ be an orthonormal basis of ${\displaystyle A}$ and let ${\displaystyle Q_A^{\perp }}$ be the orthogonal complement of ${\displaystyle Q_A}$. Denote the singular values of the matrix ${\displaystyle Q_A^{T}A(B^{T}A{)}^{-1}{B}^T{Q}_A^{\perp }}$ by the positive values ${\displaystyle {\gamma }_1\ge {\gamma }_2\ge \dots \ge {\gamma }_k}$. With this, the singular values for ${\displaystyle P}$ are:

$${\displaystyle {\sigma }_i=\{\begin{array}{ll}\sqrt{1+{\gamma }_i^2}& 1\le i\le k\\ 0& \text{otherwise}\end{array}}$$

and the singular values for ${\displaystyle I-P}$ are

$${\displaystyle {\sigma }_i=\{\begin{array}{ll}\sqrt{1+{\gamma }_i^2}& 1\le i\le k\\ 1& k+1\le i\le n-k\\ 0& \text{otherwise}\end{array}}$$

This implies that the largest singular values of ${\displaystyle P}$ and ${\displaystyle I-P}$ are equal, and thus that the matrix norm of the oblique projections are the same. However, the condition number satisfies the relation ${\displaystyle \kappa (I-P)=\frac{{\sigma }_1}{1}\ge \frac{{\sigma }_1}{{\sigma }_k}=\kappa (P)}$, and is therefore not necessarily equal.

Finding projection with an inner product

Let ${\displaystyle V}$ be a vector space (in this case a plane) spanned by orthogonal vectors ${\displaystyle {\mathbf{u}}_1,{\mathbf{u}}_2,\dots ,{\mathbf{u}}_p}$. Let ${\displaystyle y}$ be a vector. One can define a projection of ${\displaystyle \mathbf{y}}$ onto ${\displaystyle V}$ as

$${\displaystyle {\mathrm{proj}}_V\mathbf{y}=\frac{\mathbf{y}\cdot {\mathbf{u}}^i}{{\mathbf{u}}^i\cdot {\mathbf{u}}^i}{\mathbf{u}}^i}$$

where repeated indices are summed over (Einstein sum notation). The vector ${\displaystyle \mathbf{y}}$ can be written as an orthogonal sum such that ${\displaystyle \mathbf{y}={\mathrm{proj}}_V\mathbf{y}+\mathbf{z}}$. ${\displaystyle {\mathrm{proj}}_V\mathbf{y}}$ is sometimes denoted as ${\displaystyle \hat{\mathbf{y}}}$. There is a theorem in linear algebra that states that this ${\displaystyle \mathbf{z}}$ is the smallest distance (the orthogonal distance) from ${\displaystyle \mathbf{y}}$ to ${\displaystyle V}$ and is commonly used in areas such as machine learning.

y is being projected onto the vector space V.

Canonical forms

Any projection ${\displaystyle P=P^2}$ on a vector space of dimension ${\displaystyle d}$ over a field is a diagonalizable matrix, since its minimal polynomial divides ${\displaystyle x^2-x}$, which splits into distinct linear factors. Thus there exists a basis in which ${\displaystyle P}$ has the form

${\displaystyle P=I_r\oplus 0_{d-r}}$

where ${\displaystyle r}$ is the rank of ${\displaystyle P}$. Here ${\displaystyle I_r}$ is the identity matrix of size ${\displaystyle r}$, ${\displaystyle 0_{d-r}}$ is the zero matrix of size ${\displaystyle d-r}$, 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

${\displaystyle P=\left[\begin{array}{cc}1& {\sigma }_1\\ 0& 0\end{array}\right]\oplus \cdots \oplus \left[\begin{array}{cc}1& {\sigma }_k\\ 0& 0\end{array}\right]\oplus I_m\oplus 0_s.}$

where ${\displaystyle {\sigma }_1\ge {\sigma }_2\ge \cdots \ge {\sigma }_k>0}$. The integers ${\displaystyle k,s,m}$ and the real numbers ${\displaystyle {\sigma }_i}$ are uniquely determined. Note that ${\displaystyle 2k+s+m=d}$. The factor ${\displaystyle I_m\oplus 0_s}$ corresponds to the maximal invariant subspace on which ${\displaystyle P}$ acts as an orthogonal projection (so that P itself is orthogonal if and only if ${\displaystyle k=0}$) and the ${\displaystyle {\sigma }_i}$-blocks correspond to the oblique components.

Projections on normed vector spaces

When the underlying vector space ${\displaystyle X}$ is a (not necessarily finite-dimensional) normed vector space, analytic questions, irrelevant in the finite-dimensional case, need to be considered. Assume now ${\displaystyle X}$ is a Banach space.

Many of the algebraic results discussed above survive the passage to this context. A given direct sum decomposition of ${\displaystyle X}$ into complementary subspaces still specifies a projection, and vice versa. If ${\displaystyle X}$ is the direct sum ${\displaystyle X=U\oplus V}$, then the operator defined by ${\displaystyle P(u+v)=u}$ is still a projection with range ${\displaystyle U}$ and kernel ${\displaystyle V}$. It is also clear that ${\displaystyle P^2=P}$. Conversely, if ${\displaystyle P}$ is projection on ${\displaystyle X}$, i.e. ${\displaystyle P^2=P}$, then it is easily verified that ${\displaystyle (1-P{)}^2=(1-P)}$. In other words, ${\displaystyle 1-P}$ is also a projection. The relation ${\displaystyle P^2=P}$ implies ${\displaystyle 1=P+(1-P)}$ and ${\displaystyle X}$ is the direct sum ${\displaystyle \mathrm{rg}(P)\oplus \mathrm{rg}(1-P)}$.

However, in contrast to the finite-dimensional case, projections need not be continuous in general. If a subspace ${\displaystyle U}$ of ${\displaystyle X}$ is not closed in the norm topology, then the projection onto ${\displaystyle U}$ is not continuous. In other words, the range of a continuous projection ${\displaystyle P}$ must be a closed subspace. Furthermore, the kernel of a continuous projection (in fact, a continuous linear operator in general) is closed. Thus a continuous projection ${\displaystyle P}$ gives a decomposition of ${\displaystyle X}$ into two complementary closed subspaces: ${\displaystyle X=\mathrm{rg}(P)\oplus \ker (P)=\ker (1-P)\oplus \ker (P)}$.

The converse holds also, with an additional assumption. Suppose ${\displaystyle U}$ is a closed subspace of ${\displaystyle X}$. If there exists a closed subspace ${\displaystyle V}$ such that X = U ⊕ V, then the projection ${\displaystyle P}$ with range ${\displaystyle U}$ and kernel ${\displaystyle V}$ is continuous. This follows from the closed graph theorem. Suppose x_n → x and Px_n → y. One needs to show that ${\displaystyle Px=y}$. Since ${\displaystyle U}$ is closed and {Px_n} ⊂ U, y lies in ${\displaystyle U}$, i.e. Py = y. Also, x_n − Px_n = (I − P)x_n → x − y. Because ${\displaystyle V}$ is closed and {(I − P)x_n} ⊂ V, we have ${\displaystyle x-y\in V}$, i.e. ${\displaystyle P(x-y)=Px-Py=Px-y=0}$, which proves the claim.

The above argument makes use of the assumption that both ${\displaystyle U}$ and ${\displaystyle V}$ are closed. In general, given a closed subspace ${\displaystyle U}$, there need not exist a complementary closed subspace ${\displaystyle V}$, although for Hilbert spaces this can always be done by taking 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 ${\displaystyle U}$ be the linear span of ${\displaystyle u}$. By Hahn–Banach, there exists a bounded linear functional ${\displaystyle \phi }$ such that φ(u) = 1. The operator ${\displaystyle P(x)=\phi (x)u}$ satisfies ${\displaystyle P^2=P}$, i.e. it is a projection. Boundedness of ${\displaystyle \phi }$ implies continuity of ${\displaystyle P}$ and therefore ${\displaystyle \ker (P)=\mathrm{rg}(I-P)}$ is a closed complementary subspace of ${\displaystyle U}$.

Applications and further considerations

Projections (orthogonal and otherwise) play a major role in algorithms for certain linear algebra problems:

• QR decomposition (see Householder transformation and Gram–Schmidt decomposition);
• Singular value decomposition
• Reduction to Hessenberg form (the first step in many eigenvalue algorithms)
• Linear regression
• Projective elements of matrix algebras are used in the construction of certain K-groups in Operator K-theory

As stated above, projections are a special case of idempotents. Analytically, orthogonal projections are non-commutative generalizations of characteristic functions. Idempotents are used in classifying, for instance, semisimple algebras, while measure theory begins with considering characteristic functions of measurable sets. Therefore, as one can imagine, projections are very often encountered in the context of operator algebras. In particular, a von Neumann algebra is generated by its complete lattice of projections.

Generalizations

More generally, given a map between normed vector spaces ${\displaystyle T:V\to W,}$ one can analogously ask for this map to be an isometry on the orthogonal complement of the kernel: that ${\displaystyle (\ker T{)}^{\perp }\to W}$ be an isometry (compare Partial isometry); in particular it must be onto. The case of an orthogonal projection is when W is a subspace of V. In Riemannian geometry, this is used in the definition of a Riemannian submersion.

Industrial agriculture

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Industrial_agriculture 

 

See also: Intensive agriculture

Industrial agriculture is a form of modern farming that refers to the industrialized production of crops and animals and animal products like eggs or milk. The methods of industrial agriculture include innovation in agricultural machinery and farming methods, genetic technology, techniques for achieving economies of scale in production, the creation of new markets for consumption, the application of patent protection to genetic information, and global trade. These methods are widespread in developed nations and increasingly prevalent worldwide. Most of the meat, dairy, eggs, fruits and vegetables available in supermarkets are produced in this way.

Historical development and future prospects

Main article: History of agriculture

Industrial agriculture arose hand in hand with the Industrial Revolution in general. The identification of nitrogen, potassium and phosphorus (referred to by the acronym NPK) as critical factors in plant growth led to the manufacture of synthetic fertilizers, making possible more intensive types of agriculture. The discovery of vitamins and their role in animal nutrition, in the first two decades of the 20th century, led to vitamin supplements, which in the 1920s allowed certain livestock to be raised indoors, reducing their exposure to adverse natural elements. The discovery of antibiotics and vaccines facilitated raising livestock in concentrated, controlled animal feed operations by reducing diseases caused by crowding. Chemicals developed for use in World War II gave rise to synthetic pesticides. Developments in shipping networks and technology have made long-distance distribution of agricultural produce feasible.

Agricultural production across the world doubled four times between 1820 and 1975 (it doubled between 1820 and 1920; between 1920 and 1950; between 1950 and 1965; and again between 1965 and 1975) to feed a global population of one billion human beings in 1800 and 6.5 billion in 2002. During the same period, the number of people involved in farming dropped as the process became more automated. In the 1930s, 24 percent of the American population worked in agriculture compared to 1.5 percent in 2002; in 1940, each farm worker supplied 11 consumers, whereas in 2002, each worker supplied 90 consumers. The number of farms has also decreased, and their ownership is more concentrated.For example, in the 2000s; the price of farmland in the United States increased due to the Midwest farming crisis. The number of small- and medium-scale farming operations decreased due to the increased production and farmland costs. This forced farmers to find alternatives by taking advantage of new products of industrial agriculture such as financialization.

Financialization takes place through the process of ongoing monetization. An example of monetization involves financial institutions expanding and gain authority in the market. Financialization affects all aspects of farm operations, including the structure of the work, the value of it and the social organizations. Farmers turned to land availability in the Brazilian Cerrado through the help of investors and other capital gaining methods needed for financialization. investors wanted to get involved because the investment appears low-risk with high rewards. For example, investors would gain inside information on the market in Brazil. In the article Financialization of work, value, and social organization among transnational soy farmers in the Brazilian Cerrado Ofstehage gives examples of how industrialized farming has evolved into a management model.

A management model entails the structure and rules that ensure work of management is completed. Work is reliant on outsourcing in order to complete labor farming tasks, but is also an essential part in the way management and financial work is completed. Social value system of farming changed when using a management model. Farmers have to take into consideration the division between good and bad farming tactics under the new management model. Many farmers were reluctant to mobilize because of the effect this would have on their family business. The separation between the management styles of farmers comes down to two approaches; farming as a lifestyle versus farming solely for profit. In the Brazilian Cerrado the farming model is strictly based on increased profit margins which dictates decisions involving management and labor related work.

In the U.S., four companies produce 81 percent of cows, 73 percent of sheep, 57 percent of pigs, and produce 50 percent of chickens, cited as an example of "vertical integration" by the president of the U.S. National Farmers' Union. In 1967, there were one million pig farms in America; as of 2002, there were 114,000 with 80 million pigs (out of 95 million) produced each year on factory farms, according to the U.S. National Pork Producers Council. According to the Worldwatch Institute, 74 percent of the world's poultry, 43 percent of beef and 68 percent of eggs are produced this way.

British agricultural revolution

Main article: British Agricultural Revolution

The British agricultural revolution describes a period of agricultural development in Britain between the 16th century and the mid-19th century, which saw a massive increase in agricultural productivity and net output. This in turn supported unprecedented population growth, freeing up a significant percentage of the workforce, and thereby helped drive the Industrial Revolution. How this came about is not entirely clear. In recent decades, historians cited four key changes in agricultural practices, enclosure, mechanization, four-field crop rotation and selective breeding, and gave credit to a relatively few individuals.

Challenges and issues

See also: Agricultural policy, Agribusiness, and Factory farming

The challenges and issues of industrial agriculture for global and local society, for the industrial agriculture sector, for the individual industrial agriculture farm, and for animal rights include the costs and benefits of both current practices and proposed changes to those practices. This is a continuation of thousands of years of the invention and use of technologies in feeding ever growing populations.

[W]hen hunter-gatherers with growing populations depleted the stocks of game and wild foods across the Near East, they were forced to introduce agriculture. But agriculture brought much longer hours of work and a less rich diet than hunter-gatherers enjoyed. Further population growth among shifting slash-and-burn farmers led to shorter fallow periods, falling yields and soil erosion. Plowing and fertilizers were introduced to deal with these problems—but once again involved longer hours of work and degradation of soil resources(Boserup, The Conditions of Agricultural Growth, Allen and Unwin, 1965, expanded and updated in Population and Technology, Blackwell, 1980.).

While the point of industrial agriculture is lower cost products to create greater productivity thus a higher standard of living as measured by available goods and services, industrial methods have side effects both good and bad. Further, industrial agriculture is not some single indivisible thing, but instead is composed of numerous separate elements, each of which can be modified, and in fact is modified in response to market conditions, government regulation and scientific advances. So the question then becomes for each specific element that goes into an industrial agriculture method or technique or process: What bad side effects are bad enough that the financial gain and good side effects are outweighed? Different interest groups not only reach different conclusions on this, but also recommend differing solutions, which then become factors in changing both market conditions and government regulations.

Society

The major challenges and issues faced by society concerning industrial agriculture include:

Maximizing the benefits:

• Cheap and abundant food
• Convenience for the consumer
• The contribution to our economy on many levels, from growers to harvesters to processors to sellers

while minimizing the downsides:

• Environmental and social costs
• Antibiotic resistance 
• Damage to fisheries
• Cleanup of surface and groundwater polluted with animal waste
• Increased health risks from pesticides
• Increased ozone pollution via methane byproducts of animals
• Global warming from heavy use of fossil fuels

Benefits

An example of industrial agriculture providing cheap and plentiful food is the U.S.'s "most successful program of agricultural development of any country in the world". Between 1930 and 2000 U.S. agricultural productivity (output divided by all inputs) rose by an average of about 2 percent annually causing food prices paid by consumers to decrease. "The percentage of U.S. disposable income spent on food prepared at home decreased, from 22 percent as late as 1950 to 7 percent by the end of the century."

Liabilities

Economic

Economic liabilities for industrial agriculture include the dependence on finite non-renewable fossil fuel energy resources, as an input in farm mechanization (equipment, machinery), for food processing and transportation, and as an input in agricultural chemicals. A future increase in energy prices as projected by the International Energy Agency is therefore expected to result in increase in food prices; and there is therefore a need to 'de-couple' non-renewable energy usage from agricultural production. Other liabilities include peak phosphate as finite phosphate reserves are currently a key input into chemical fertilizer for industrial agriculture.

Environment

Main article: Environmental science

Industrial agriculture uses huge amounts of water, energy, and industrial chemicals; increasing pollution in the arable land, usable water and atmosphere. Herbicides, insecticides, fertilizers and animal waste products are accumulating in ground and surface waters. "Many of the negative effects of industrial agriculture are remote from fields and farms. Nitrogen compounds from the Midwest, for example, travel down the Mississippi to degrade coastal fisheries in the Gulf of Mexico. But other adverse effects are showing up within agricultural production systems—for example, the rapidly developing resistance among pests is rendering our arsenal of herbicides and insecticides increasingly ineffective.". Chemicals used in industrial agriculture, as well as the practice of monoculture, have also been implicated in Colony Collapse Disorder which has led to a collapse in bee populations. Agricultural production is highly dependent on bee pollination to pollinate many varieties of plants, fruits and vegetables.

Social

Main article: Rural sociology

A study done for the U.S. Office of Technology Assessment conducted by the UC Davis Macrosocial Accounting Project concluded that industrial agriculture is associated with substantial deterioration of human living conditions in nearby rural communities.

Future increase in food commodity prices, driven by the energy price rises under peak oil and dependency of industrial agriculture on fossil fuels is expected to lead to increase in food prices which has particular impacts on poor people. An example of this can be seen in the 2007–2008 world food price crisis. Food price increases have a disproportionate impact on the poor as they spend a large proportion of their income on food.

Animals

Main article: Industrial agriculture (animals)

Governor Tom Wolf hosts the 102nd Pennsylvania Farm Show for Industrial Agriculture

"Concentrated animal feeding operations" or "intensive livestock operations", can hold large numbers (some up to hundreds of thousands) of animals, often indoors. These animals are typically cows, hogs, turkeys, or chickens. The distinctive characteristics of such farms is the concentration of livestock in a given space. The aim of the operation is to produce as much meat, eggs, or milk at the lowest possible cost and with the greatest level of food safety.

Food and water are supplied in place, and artificial methods are often employed to maintain animal health and improve production, such as therapeutic use of antimicrobial agents, vitamin supplements and growth hormones. Growth hormones are not used in chicken meat production nor are they used in the European Union for any animal. In meat production, methods are also sometimes employed to control undesirable behaviours often related to stresses of being confined in restricted areas with other animals. More docile breeds are sought (with natural dominant behaviours bred out for example), physical restraints to stop interaction, such as individual cages for chickens, or animals physically modified, such as the de-beaking of chickens to reduce the harm of fighting. Weight gain is encouraged by the provision of plentiful supplies of food to animals breed for weight gain.

The designation "confined animal feeding operation" in the U.S. resulted from that country's 1972 Federal Clean Water Act, which was enacted to protect and restore lakes and rivers to a "fishable, swimmable" quality. The United States Environmental Protection Agency (EPA) identified certain animal feeding operations, along with many other types of industry, as point source polluters of groundwater. These operations were designated as CAFOs and subject to special anti-pollution regulation.

In 17 states in the U.S., isolated cases of groundwater contamination has been linked to CAFOs. For example, the ten million hogs in North Carolina generate 19 million tons of waste per year. The U.S. federal government acknowledges the waste disposal issue and requires that animal waste be stored in lagoons. These lagoons can be as large as 7.5 acres (30,000 m²). Lagoons not protected with an impermeable liner can leak waste into groundwater under some conditions, as can runoff from manure spread back onto fields as fertilizer in the case of an unforeseen heavy rainfall. A lagoon that burst in 1995 released 25 million gallons of nitrous sludge in North Carolina's New River. The spill allegedly killed eight to ten million fish.

The large concentration of animals, animal waste and dead animals in a small space poses ethical issues to some consumers. Animal rights and animal welfare activists have charged that intensive animal rearing is cruel to animals. As they become more common, so do concerns about air pollution and ground water contamination, and the effects on human health of the pollution and the use of antibiotics and growth hormones.

According to the U.S. Centers for Disease Control and Prevention (CDC), farms on which animals are intensively reared can cause adverse health reactions in farm workers. Workers may develop acute and chronic lung disease, musculoskeletal injuries, and may catch infections that transmit from animals to human beings. These type of transmissions, however, are extremely rare, as zoonotic diseases are uncommon.

Crops

Main article: Industrial agriculture (crops)

The projects within the Green Revolution spread technologies that had already existed, but had not been widely used outside of industrialized nations. These technologies included pesticides, irrigation projects and synthetic nitrogen fertilizer.

The novel technological development of the Green Revolution was the production of what some referred to as "miracle seeds." Scientists created strains of maize, wheat and rice that are generally referred to as HYVs or "high-yielding varieties." HYVs have an increased nitrogen-absorbing potential compared to other varieties. Since cereals that absorbed extra nitrogen would typically lodge, or fall over before harvest, semi-dwarfing genes were bred into their genomes. Norin 10 wheat, a variety developed by Orville Vogel from Japanese dwarf wheat varieties, was instrumental in developing Green Revolution wheat cultivars. IR8, the first widely implemented HYV rice to be developed by the International Rice Research Institute, was created through a cross between an Indonesian variety named "Peta" and a Chinese variety named "Dee Geo Woo Gen."

With the availability of molecular genetics in Arabidopsis and rice the mutant genes responsible (reduced height(rh), gibberellin insensitive (gai1) and slender rice (slr1)) have been cloned and identified as cellular signaling components of gibberellic acid, a phytohormone involved in regulating stem growth via its effect on cell division. Stem growth in the mutant background is significantly reduced leading to the dwarf phenotype. Photosynthetic investment in the stem is reduced dramatically as the shorter plants are inherently more stable mechanically. Assimilates become redirected to grain production, amplifying in particular the effect of chemical fertilizers on commercial yield.

HYVs significantly outperform traditional varieties in the presence of adequate irrigation, pesticides and fertilizers. In the absence of these inputs, traditional varieties may outperform HYVs. One criticism of HYVs is that they were developed as F1 hybrids, meaning they need to be purchased by a farmer every season rather than saved from previous seasons, thus increasing a farmer's cost of production.

Sustainable agriculture

Main article: Sustainable agriculture

The idea and practice of sustainable agriculture has arisen in response to the problems of industrial agriculture. Sustainable agriculture integrates three main goals: environmental stewardship, farm profitability and prosperous farming communities. These goals have been defined by a variety of disciplines and may be looked at from the vantage point of the farmer or the consumer.

Organic farming methods

Main article: Organic farming

See also: Integrated multi-trophic aquaculture

Organic farming methods combine some aspects of scientific knowledge and highly limited modern technology with traditional farming practices; accepting some of the methods of industrial agriculture while rejecting others. Organic methods rely on naturally occurring biological processes, which often take place over extended periods of time, and a holistic approach; while chemical-based farming focuses on immediate, isolated effects and reductionist strategies.

Integrated Multi-Trophic Aquaculture is an example of this holistic approach. Integrated Multi-Trophic Aquaculture (IMTA) is a practice in which the by-products (wastes) from one species are recycled to become inputs (fertilizers, food) for another. Fed aquaculture (e.g. fish, shrimp) is combined with inorganic extractive (e.g. seaweed) and organic extractive (e.g. shellfish) aquaculture to create balanced systems for environmental sustainability (bio-mitigation), economic stability (product diversification and risk reduction) and social acceptability (better management practices).

Lead paint

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Lead_paint

Dutch Boy Paint logo (front)

Dutch Boy Paint logo (rear)

Lead paint or lead-based paint is paint containing lead. As pigment, lead(II) chromate (PbCrO
₄, "chrome yellow"), lead(II,IV) oxide, (Pb
₃O
₄, "red lead"), and lead(II) carbonate (PbCO
₃, "white lead") are the most common forms. Lead is added to paint to accelerate drying, increase durability, maintain a fresh appearance, and resist moisture that causes corrosion. It is one of the main health and environmental hazards associated with paint. Lead paint has been generally phased out of use due to the toxic nature of lead. Alternatives such as water-based, lead-free traffic paint are readily available.

In some countries, lead continues to be added to paint intended for domestic use, whereas countries such as the United States and the United Kingdom have regulations prohibiting its use. However, lead paint may still be found in older properties painted prior to the introduction of such regulations. Although lead has been banned from household paints in the United States since 1978, it may still be found in road marking paint.

History

White lead was being produced during the 4th century BC; the process is described by Pliny the Elder, Vitruvius, and the ancient Greek author Theophrastus.

The traditional method of making the pigment was called the stack process. Hundreds or thousands of earthenware pots containing vinegar and lead were embedded in a layer of either tan bark or cow dung. The pots were designed so that the vinegar and lead were in separate compartments, but the lead was in contact with the vapor of the vinegar. The lead was usually coiled into a spiral and placed on a ledge inside the pot. The pot was loosely covered with a grid of lead, which allowed the carbon dioxide formed by the fermentation of the tan bark or the dung to circulate in the pot. Each layer of pots was covered by a new layer of tan, then another layer of pots. The heat created by the fermentation, acetic acid vapor, and carbon dioxide within the stack did their work, and within a month the lead coils were covered with a crust of white lead. This crust was separated from the lead, washed, and ground for pigment. This was an extremely dangerous process for the workmen. Medieval texts warned of the danger of "apoplexy, epilepsy, and paralysis" from working with lead white.

In 1786, Benjamin Franklin wrote a letter warning a friend about the hazards of lead and lead paint, which he considered well-established. Despite the risks, the pigment was very popular with artists because of its density and opacity; a small amount could cover a large surface. It was widely used by artists until the 19th century, when it was replaced by zinc white and titanium white.

The dangers of lead paint were considered well-established by the beginning of the 20th century. In the July 1904 edition of its monthly publication, Sherwin-Williams reported the dangers of paint containing lead, noting that a French expert had deemed lead paint "poisonous in a large degree, both for the workmen and for the inhabitants of a house painted with lead colors". As early as 1886, German health laws prohibited women and children from working in factories processing lead paint and lead sugar.

The League of Nations began efforts to ban lead paint in 1921.

Toxicity

Further information: Lead poisoning

Lead paint can crack and form flakes, which then contaminate the surrounding environment.

Lead paint is hazardous. It can cause nervous system damage, stunted growth, kidney damage, and delayed development. It is associated with high violent crime rates. It is dangerous to children because it tastes sweet, therefore encouraging children to put lead chips and toys with lead dust in their mouths. Lead paint can cause reproductive problems, including a decrease in sperm concentration in men. Lead is also considered a likely carcinogen. High levels of exposure can be lethal.

Regulation

As of 30 December 2021, these are the places with confirmed lead paint laws according to the WHO Global Health Observatory Database:

Africa	Algeria Cameroon Ethiopia Kenya Morocco (new since 1/1/2021) South Africa United Republic of Tanzania
Latin America and the Caribbean	Argentina Brazil Chile Colombia Costa Rica Cuba Dominic Ecuador Guyana Mexico Panama Peru (new since 1/1/2021) Trinidad and Tobago Uruguay
West Asia	Iraq Israel Jordan (existing laws revised) Lebanon Qatar Oman
Asia and the Pacific	Australia Bangladesh China India Lao People's Democratic Republic (new since 1/1/2021) Nepal New Zealand Pakistan Philippines Sri Lanka Thailand Viet Nam
Europe	Armenia Austria Belarus Belgium Bulgaria Croatia Cyprus Czech Republic Denmark Estonia Finland France Georgia (new since 1/1/2021) Germany Greece Hungary Iceland Ireland Italy Kyrgyzstan Latvia Liechtenstein Lithuania Luxembourg Malta Monaco Montenegro Netherlands North Macedonia Norway Poland Portugal Romania Russian Federation Serbia Slovakia Slovenia Spain Sweden Switzerland Ukraine (new since 1/1/2021) United Kingdom
North America	Canada United States of America

Canada

In Canada, regulations were first enacted under the Hazardous Products Act in 1976 that limited lead content of paints and other liquid coatings on furniture, household products, children's products, and exterior and interior surfaces of any building frequented by children to 0.5% by weight. New regulations on surface coating materials, which came into force in 2005, further limit lead to its background level for both interior and exterior paints sold to consumers. Canadian paint manufacturers have been conforming to this background level in their interior and exterior consumer paints since 1991. Nevertheless, a Canadian company, Dominion Colour Corporation, is "the largest manufacturer of lead-based paint pigments in the world" and has faced public criticism for obtaining permission from the European Chemicals Agency to continue to export lead chromate paints from its Dutch subsidiary to countries where its uses are not tightly regulated.

China

New regulation effective from December 1, 2020 updates an older lead paint standard introduced in the 1980s, which measured soluble lead in products instead of total lead. Measuring soluble lead is considered to be a less accurate method for measuring the amount of lead paint exposure in children. The new standards set a 90 ppm total lead limit for woodware coatings and architectural wall coatings. For vehicle and industrial coatings the new total lead limit is 1,000 ppm.

European Union

Lead paint is banned in the European Union by the 2003 Restriction of Hazardous Substances Directive (RoHS), which forbids hazardous substances in consumer goods, including paint. This act superseded and harmonized existing laws of the member states, many of which had banned lead paint years before.

To protect the health of painters, France had passed in 1909 a law banning the use of paints containing lead for the painting of the interior and exterior of all buildings.

Hong Kong

As of 2023, there are no regulation and legislation on lead content in paints. Furthermore, unlike the U.S., which implemented stricter rules in 2010, renovators in Hong Kong do not need to be certified when performing lead paint related works. Methods used to remove lead-based-paint (e.g., use of power tools) are not regulated as well. The use of HEPA-filtered vacuum or a HEPA filtered dust collection system is also not mandatory. No dust test on lead level is required upon the end of any renovation or remodeling job.

India

Lead paint was not prohibited in India until 2016. A 2015 study found that over 31% of household paints in India (small brands manufactured by small and medium enterprises in India, with limited local reach and distribution) had lead concentration above 10,000 parts per million (ppm), which far exceeds the BIS standard of 90 ppm for lead in paint. The Regulation on Lead Contents in Household and Decorative Paint Rules came into effect on 1 November 2017, according to which the paints should have lead less than 90 ppm and their label should say so. However, two years later, an analysis of 32 locally-manufactured paint samples from nine states found lead content ranging from 10 ppm to 186,062 ppm, with 90% of samples having lead levels above 90 ppm.

Philippines

The Philippines banned lead paint in 2013, but in 2017, 15% of the paint still was not certified. The EcoWaste Coalition and the Philippine Association of Paint Manufacturers declared on 1 January 2020 that the Philippines has phased-out lead paint following the implementation of Department of Environment and Natural Resources (DENR) Administrative Order 2013–24, or the Chemical Control Order for Lead and Lead Compounds, which directed manufacturers of lead-containing paints for industrial uses to phase out such paints by 31 December 2019.

Singapore

Since 1 Feb 1995, labelling is required for paints with total lead concentrations exceeding 600 ppm. From 3 Jan 2022, the manufacture, import and sale of paints exceeding 90ppm total lead concentration for local use were banned, except for zinc-based anti-corrosion paints and copper-based anti-fouling paints. For export and re-export a Hazardous Substance Licence is required ( except for zinc-based anti-corrosion paints and copper-based anti-fouling paints). For local sale of zinc-based anti-corrosion paints and copper-based anti-fouling paints exceeding 90ppm total lead concentration labelling is required and only industrial uses are allowed.

South Africa

In South Africa, the Hazardous Substances Act of 2009 classifies lead as a hazardous substance and limits its use in paint to 600 parts per million (ppm). A proposed amendment will modify this to 90 ppm, thereby almost completely eradicating lead from paint. The amendment would also include all industrial paints, which were previously excluded.

United Kingdom

Main article: Lead-based paint in the United Kingdom

Lead paint was banned in the United Kingdom in 1992.

United States

EPA poster on protecting children from lead poisoning

The U.S. Consumer Product Safety Commission (CPSC) banned lead paint in 1977 in residential properties and public buildings (16 CFR 1303), along with toys and furniture containing lead paint. The cited reason was "to reduce the risk of lead poisoning in children who may ingest paint chips or peelings". For manufacturers, the CPSC instituted the Consumer Product Safety Improvement Act of 2008, which changed the cap on lead content in paint from 0.06% to 0.009% starting 14 August 2009. In 2018 the State of Delaware banned the use of lead paint on outdoor structures. Also, the Residential Lead-Based Paint Hazard Reduction Act (a.k.a. the "Lead Paint Act") was created in order to ensure that the disclosure of any lead-based hazards in a building be discussed with potential buyers or renters of units. While EPA and HUD have defined LBP as being 1.0 mg/cm² (as measure by XRF) or 0.5% lead by dry weight (aka 5,000 ppm), some states and municipalities gone beyond this. For example, New York City's Local Law 66 of 2019 defines LBP as 0.500 mg/cm² (XRF) or 0.25% lead dry weight (2,500 ppm). 

In April 2010 the U.S. Environmental Protection Agency (EPA) required that all renovators working in homes built before 1978 and disturbing more than 6 square feet (0.56 m²) of lead paint inside the home or 20 square feet (1.9 m²) outside the home be certified. EPA's Lead Renovation, Repair and Painting Rule (RRP Rule) lowers the risk of lead contamination from home renovation activities. It requires that firms performing renovation, repair, and painting projects that disturb lead-based paint in homes, child care facilities and pre-schools (any child occupied facility) built before 1978 be certified by EPA and use certified renovators who are trained by EPA-approved training providers to follow lead-safe work practices.

Careful stabilization of any deteriorated (peeling, chipping, cracking, etc.) paint in a lead-safe manner is also encouraged. The government (HUD, United States Department of Housing and Urban Development) prohibits lead-based-paint removal by dry scraping, dry sanding, torching and burning, the use of heat guns over 1100°F, and machine-sanding / grinding without a HEPA-filtered vacuum or a HEPA filtered dust collection system, as these methods have been proven to produce significant amount of lead dust during renovation, remodeling and painting.

At the end of any remodeling or repainting job, a dust test performed by an independent third-party professional is also required by HUD for "clearance". Lead evaluations are done using a method called X-Ray fluorescence (XRF), which gives a result in 4–8 seconds with a 95% accuracy at the 2-sigma level.

As of 2018, there are an estimated 37 million homes and apartments with lead paint in the United States.

Lead paint in art

Oil paints

In art, white lead paint is known as "flake white" or "Cremnitz white". It is valued for the ease of handling and resilience the lead confers to oil paints. Lead white paint dries relatively quickly to form a strong, flexible paint film. Lead-based white is one of the oldest manufactured pigments. It was the only white pigment available to artists in appreciable quantities until the twentieth century, when zinc white and titanium white became available. Industrially produced lead white, the typical pigment from the 19th century until its ban, was thought to be inferior to traditionally fabricated forms, which had larger "flake" particles that conferred ease of handling.

Titanium and zinc whites are far less toxic than lead white and have largely supplanted it in most fine arts applications. Safety regulations have also made lead white more expensive and difficult to obtain in some regions, such as the EU. Lead white oil paints are still produced and in use by artists who prefer their unique handling, mixing, and structural qualities. Lead white has also shown to have extended longevity compared to zinc and titanium, which will crack much earlier. 

Flake white has various drawbacks, including a tendency to become transparent over time. It also blackens in the presence of certain atmospheric pollutants, although this can be reversed.

Water-based paints

Lead is not a traditional pigment in water media, as zinc is superior for works on paper, as is calcium hydroxide (slaked lime) for frescos. Lead-based paints, when used on paper, often cause the work to become discolored after long periods; the paint's lead carbonate reacts with hydrogen sulfide in the air and with acids, which often come from fingerprints.

Substitutes

Titanium

Paint manufacturers have replaced white lead with a less toxic substitute, titanium dioxide, which was first used in paints in the 19th century. Titanium dioxide is considered safe enough to use as a food coloring and in toothpaste, and is a common ingredient in sunscreen. Titanium white has far greater opacity and tinting strength than lead white, and it can easily overpower most other pigments if not mixed carefully. Titanium white has been criticized for leading to "chalkiness" in mixtures.

Zinc

Zinc white is less opaque and weaker in tinting strength than either titanium white or lead white. It is commonly used to lighten mixtures subtly while maintaining transparency. Although zinc white is the standard white in watercolors, its structural soundness in oils has been debated. Zinc white dries slowly and creates a relatively inflexible paint film. Critics of the pigment argue that its use leads to excessive cracking and delamination, even when used sparingly.

Insect ecology

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Insect_ecology

Insect ecology is the scientific study of how insects, individually or as a community, interact with the surrounding environment or ecosystem.

Insects play significant roles in the ecology of the world due to their vast diversity of form, function and lifestyle; their considerable biomass; and their interaction with plant life, other organisms and the environment. Since they are the major contributor to biodiversity in the majority of habitats, except in the sea, they accordingly play a variety of extremely important ecological roles in the many functions of an ecosystem. Taking the case of nutrient recycling, insects contribute to this vital function by degrading or consuming leaf litter, wood, carrion and dung and by dispersal of fungi.

Insects form an important part of the food chain, especially for entomophagous vertebrates such as many mammals, birds, amphibians and reptiles. Insects play an important role in maintaining community structure and composition; in the case of animals by transmission of diseases, predation and parasitism, and in the case of plants, through phytophagy and by plant propagation through pollination and seed dispersal. From an anthropocentric point of view, insects compete with humans; they consume as much as 10% of the food produced by man and infect one in six humans with a pathogen.

Community ecology

Community ecology is the process by which a group of organisms which live in the same location interact. There is direct interaction, which takes the form of symbiosis, competition and predation, which are the most easily notable. There is also indirect interaction, such as reproduction, foraging patterns and decaying. Every organism at its most basic state could be a consumer in some situations, and a producer in others. The culmination of all these interactions is what defines a community and what differentiates one from another. Insects often play several roles in these communities, though these roles vary widely based on what species is present.

Decomposers

Dung beetles (Scarabaeus laticollis) and dung ball

Decomposer insects are ones that feed on dead or rotten bodies of plant or animal life. These insects are called saprophages and fall into three main categories: those that feed on dead or dying plant matter, those that feed on dead animals (carrion), and those that feed on excrement (feces) of other animals. As dead plants are eaten away, more surface area is exposed, allowing the plants to decay faster due to an increase in microorganisms eating the plant. These insects are largely responsible for helping to create a layer of humus on the soil that provides an ideal environment for various fungi, microorganisms and bacteria. These organisms produce much of the nitrogen, carbon, and minerals that plants need for growth. Carrion feeders include several beetles, ants, mites, wasps, fly larvae (maggots), and others. These insects occupy the dead body for a short period of time but rapidly consume and/or bury the carcass. Typically, some species of fly are the first to eat the body, but the order of insects that follows is predictable and known as the faunal succession. Many dung beetles and manure flies are attracted to the smell of animal feces. The adults often lay eggs on fresh excrement and the larvae will feed on the organic matter. Many species of dung-feeders have evolved so they will only feed on feces from a specific species. There is even a type of dung-beetle that will roll feces into a ball, push it into a pre-dug hole, lay an egg in that dung and then cover it with fresh dirt to provide a perfect nursery for their larvae.

Carnivores

Carnivorous insects survive by eating other living animals, be it through hunting, sucking blood, or as an internal parasite. These insects fall into three basic categories: predators, parasites, and parasitoids.

Predatory insects are typically larger as their survival is dependent upon their ability to hunt, kill/immobilize, and eat their prey. However, there are several exceptions, with ants being the most notable. Ants, and other colony insects, can use their sheer numbers to overwhelm their prey even if the ants are significantly smaller. They often have specialized mandibles (mouthparts) for this task, some causing excruciating pain, paralysis, or simply having a high bite force. Conversely, insects that live on their own must be able to reliably bring down their prey and as such have developed a myriad of unique hunting methods. Some actively travel, seeking out their prey, while others wait in an ambush. Others may release chemicals to attract specific creatures and others still will eat anything they can.

Parasites infest the victim's body and eat it from the inside out. The presence of the parasite is often not noticed by the host as the size discrepancy is typically so vast. Parasites vary widely in how they survive in their host; some complete their full life cycle within the body while others may only stay in for the duration of their larval stage. There is as great of variation in methodology and species in parasites as in any other type of insect. The most threatening parasites to humans are ones that live outside the host and consume the host's blood. These species transmit viruses, disease, and even other, smaller parasites to the host, spreading these throughout the populations of many third world countries with poor health care.

A subcategory of parasites, called parasitoids, is one that feeds on the host body so much so that the host is eventually eaten. One species of wasp, the spider wasp, will paralyze spiders before bringing them back to their nest and injecting them with a wasp larvae. The larvae will eat its way out, secreting a numbing and paralyzing agent until there is nothing left of the spider other than the exoskeleton, then go through a metamorphism and become an adult wasp.

Herbivores

Out of all described eukaryotes almost one third are herbivorous insects, about 500,000. They feed on living plant matter or the products of a plant. These insects may eat essential parts of the plant, such as the leaves or sap, or they may survive on the pollen and nectar produced by the plant. Herbivorous insects often use olfactory or visual cues to determine a potential host plant. A visual cue could simply be the outline of a certain type of leaf, or the high contrast between the petals of a flower and the leaves surrounding it. These are typically associated with the olfactory signal an insect may receive from their intended meal. The olfactory cue could be the scent of the nectar produced by a flower, a certain chemical excreted to repel unwanted predators, or the exposed sap of a cherry tree. Either of these two senses could be the driving force behind an insect choosing to consume a certain plant, but it is only after it takes the first bite, and the confirmation of this food is made by its sense of taste, that it truly feeds. After a herbivorous insect is finished feeding on a plant, it will either wait there until hungry again, or move on to another task, be it finding more food, a mate, or shelter. Herbivorous insects bring significantly more danger to a plant than that of consumption; they are among the most prominent disease-carrying creatures in the insect world. There are numerous diseases, fungi, and parasites that can be carried by nearly any herbivorous insect, many of which fatal to the plant infected. Some diseases even produce a sweet smelling, sticky secretion from the infected plant to attract more insects and spread farther. In return plants have their own defenses. Some of these defenses are toxic secondary metabolites to deter insects. These toxins limit the diet breadth of herbivores, and evolving mechanisms to nonetheless continue herbivory is an important part of maintaining diet breadth in insects, and so in their evolutionary history as a whole. Both pleiotropy and epistasis have complex effects in this regard, with the simulations of Griswold 2006 showing that more genes provide the benefit of more targets for adaptive mutations, while Fisher 1930 showed that a mutation can improve one trait while epistasis causes it to also trigger negative effects - slowing down adaptation.

Schoonhoven and associates, from Blaney et al 1985 to Schoonhoven et al 1992, illuminate the interplay between chemoreceptor stimuli in Lepidoptera and Orthoptera. They used Helicoverpa armigera, Spodoptera littoralis, S. frugiperda, Chloridea virescens, and grasshoppers. They find that most insects respond immediately and roughly equally to phagostimulant – indicating good food – and phagodeterrent – indicating a food to be avoided, or a material which is not food – substances. They also present some divergent examples, both delayed response – suggesting that food decisions were mediated by cognition and not just simple chemoreception – and unequal chemoreceptor stimulation – with gustatory cells firing equally when presented with any material, but deterrent cells firing to a greater degree for undesirable materials. (They also investigate similar questions of seeking/avoidance in common questions of dietary balance of protein and carbohydrate – i.e. less risky dietary choices where toxins are not the deciding factor – and find similar results, with some insects eating solely by chemoreception and some showing delayed decisions, suggesting cognition.) Both salicin and caffeine are antifeedants, and some of the Schoonhoven group's investigations test both the deterrence they produce and habituation to them. The Glendinning group has done some similar work. They find Manduca sexta's habituation to salicin to be cognitively mediated because deterrent sensory cell stimulation barely decreases even when avoidance ceases. On the other hand Glendinning et al 1999 finds M. sexta habituation to caffeine to be due to change in chemoreceptor activation because it decreases significantly, and at the same time as cessation of feeding avoidance. The same work tests the cross-effects of habituation between the two chemicals, finding that they probably share a second messenger. For both phagostimulus and deterrence stimuli they find that the effects of multiple stimulations by multiple substances – upon the same cells, simultaneously – produce additive effects, up to the cell's firing rate ceiling.

Climate change is expected to change herbivory relationships. Liu et al 2011 finds no change in distribution in one example, but instead the same herbivore switched primary hosts due to altered flowering time. Gillespie et al 2012 found host mismatch due to temperature shift. (These methodologies in herbivory could be applied to study the same question in climate change + pollination. As of 2014 however this remains to be tried.)

Coevolution

Coevolution is the ecological process by which two species exclusively affect each other’s evolution. This concept is essential to the study of insect ecology. Coevolution is particularly important in how it can lead to both micro- and macro-evolutionary changes. Micro-evolutionary changes include shifts in genome and alleles while macro-evolution is the emergence of a new species, also called speciation. Two species that coevolve experience reciprocal evolution and go through biological changes as a result of the other species. One example of this in insect ecology is the coevolution of Dasyscolia ciliata, a species of wasp, and Ophrys speculum, a species of orchid. These two species have both evolved in such a way that the wasp is the only known pollinator of the plant. This relationship can be seen in other species of flowering plants and pollinating insects, but a more distinct example is the coevolution of ants and acacias. The acacia ant (Pseudomyrmex ferruginea) is an insect that has been discovered to protect five different species of acacia trees. The ant provides protection to the plant while the acacias reciprocate by supplying food and shelter. Over generations, these two species have adapted to accommodate each other, an example of coevolution.

Interspecific relationships

Due to their diverse functions, diets, and lifestyles, insects are integral components of terrestrial ecological communities. Beyond functioning as decomposers, carnivores, and herbivores, insects often participate in other species interactions. These interactions can both positively and adversely affect plants, mammals, and other insects. More specifically, insects participate in mutualism, amensalism, commensalism, predation and parasitism.

Pollination of a flowering plant by a bee.

Mutualism

Mutualism is a symbiotic relationship between two or more species in which each benefits. Common mutualistic relationships include cleaning symbiosis, animal induced pollination, or protection from predators. One example of insect mutualism is the pollination of flowering plants by insects, a field of study known as anthecology. Primarily, various bee species work as pollinators of flowering plants, feeding on their nectar and in turn picking up their pollen and spreading it to other flowers. Another example of insect mutualism is the process by which ants shelter and feed aphids in their anthills and feed off of their honeydew in return.

Amensalism

Amensalism is a non-symbiotic species interaction in which one organism negatively affects the other organism but is unaffected by that organism. This type of species interaction is common in nature, and an example in insect ecology is between goats and insects. The two individuals compete for the same food source, but goats will deprive the latter from feeding. The goat is completely unaffected by the interaction, but the insect is left hungry.

Mites benefiting from the movement of Nicrophorus humator.

Commensalism

Commensalism is a different type of ecological interaction between species in which one species gains benefits while the other is neither harmed nor benefited. Two examples of commensalism that can be seen in insect ecology are phoresy, an interaction in which one attaches itself to another for transportation, and inquilinism, the use of another organism for shelter. Ticks and mites have adapted to latch onto beetles, flies, and bees (as well as other organisms) for transportation, an example of phoresy. In terms of inquilinism, insects commonly establish themselves in human garages or shelters of other animals for protection against predators and weather.

Parasitoid insects

Parasitoids are insects that live intimately with a host, feed off of the host like a parasite, but eventually kill the host. This specific type of species interaction is exclusive to insects and is employed most commonly by wasps. An example of this is when parasitoid wasps inject their eggs into aphids. The eggs will eventually hatch and produce wasp larvae that feed on and consume the organism. Additionally, some parasitoids chemically affect the host to propagate the development of parasitic offspring. Parasitoid wasps typically prey on a specific insect or spider species, and the host life-stage at which the wasp deposits its seed differs. In regard to humans, parasitoid insects are favored because they can be used as biological pest controls for farmers, preying on other insects that damage crops.