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Talk:Farnsworth Paradox

Cleanup debate, 7 May 2004
Farnsworth Paradox - all google hits that I get are about Futurama. Does the paradox only exist on the TV show? Or is it also related to Philo Farnsworth who is a real person?

 * I'd expect this site to provide a "cultural reference" (foot of page) if there was a real world Farnsworth paradox, and on this basis, think it safe to say it is a Futurama invention. --Tagishsimon

 * As far as I can tell it relates only to an episode of "Futurama". I modified the article to reflect that. If it's important in the tv show it should be linked from there. Currently its and orphan other than the link here. Vfd candidate?-- Infrogmation 18:49, 8 May 2004 (UTC)

 * As a Futurama fan, I'd consider this article a candidate for deletion. It isn't even spelled right - the episode title is "The Farnsworth Parabox" not -dox. And there is no specific mention of a "paradox" within the episode dialogue. dmmaus 07:42, 9 May 2004 (UTC)

 * It's not even accurate - the box in the show contains another universe, not the same one. The specifics are covered well enough in the episode guide, so I've redirected it to there. sjorford 09:59, 11 May 2004 (UTC)

title: Talk:Farnsworth Paradox


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Reproducing kernel Hilbert space

In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions $$f$$ and $$g$$ in the RKHS are close in norm, i.e., $$\|f-g\|$$ is small, then $$f$$ and $$g$$ are also pointwise close, i.e., $$|f(x)-g(x)|$$ is small for all $$x$$. The converse does not need to be true. Informally, this can be shown by looking at the supremum norm: the sequence of functions $$\sin^{2n} (x)$$ converges pointwise, but does not converge uniformly i.e. does not converge with respect to the supremum norm. (This is not a counterexample because the supremum norm does not arise from any inner product due to not satisfying the parallelogram law.)

It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Some examples, however, have been found.

L2 spaces are not Hilbert spaces of functions (and hence not RKHSs), but rather Hilbert spaces of equivalence classes of functions (for example, the functions $$f$$ and $$g$$ defined by $$f(x)=0$$ and $$g(x)=1_{\mathbb{Q}}$$ are equivalent in L2). However, there are RKHSs in which the norm is an L2-norm, such as the space of band-limited functions (see the example below).

An RKHS is associated with a kernel that reproduces every function in the space in the sense that for every $$x$$ in the set on which the functions are defined, "evaluation at $$x$$" can be performed by taking an inner product with a function determined by the kernel. Such a reproducing kernel exists if and only if every evaluation functional is continuous.

The reproducing kernel was first introduced in the 1907 work of Stanisław Zaremba concerning boundary value problems for harmonic and biharmonic functions. James Mercer simultaneously examined functions which satisfy the reproducing property in the theory of integral equations. The idea of the reproducing kernel remained untouched for nearly twenty years until it appeared in the dissertations of Gábor Szegő, Stefan Bergman, and Salomon Bochner. The subject was eventually systematically developed in the early 1950s by Nachman Aronszajn and Stefan Bergman.

These spaces have wide applications, including complex analysis, harmonic analysis, and quantum mechanics. Reproducing kernel Hilbert spaces are particularly important in the field of statistical learning theory because of the celebrated representer theorem which states that every function in an RKHS that minimises an empirical risk functional can be written as a linear combination of the kernel function evaluated at the training points. This is a practically useful result as it effectively simplifies the empirical risk minimization problem from an infinite dimensional to a finite dimensional optimization problem.

For ease of understanding, we provide the framework for real-valued Hilbert spaces. The theory can be easily extended to spaces of complex-valued functions and hence include the many important examples of reproducing kernel Hilbert spaces that are spaces of analytic functions.

Definition
Let $$X$$ be an arbitrary set and $$H$$ a Hilbert space of real-valued functions on $$X$$, equipped with pointwise addition and pointwise scalar multiplication. The evaluation functional over the Hilbert space of functions $$H$$ is a linear functional that evaluates each function at a point $$x$$,

 * $$ L_{x} : f \mapsto f(x) \text{   } \forall f \in H. $$

We say that H is a reproducing kernel Hilbert space if, for all $$x$$ in $$X$$, $$ L_x $$ is continuous at every $$f$$ in $$H$$ or, equivalently, if $$ L_x $$ is a bounded operator on $$H$$, i.e. there exists some $$M_x>0$$ such that

Although $$M_x<\infty$$ is assumed for all $$x \in X$$, it might still be the case that $\sup_x M_x = \infty$.

While property ($$) is the weakest condition that ensures both the existence of an inner product and the evaluation of every function in $$H$$ at every point in the domain, it does not lend itself to easy application in practice. A more intuitive definition of the RKHS can be obtained by observing that this property guarantees that the evaluation functional can be represented by taking the inner product of $$ f $$ with a function $$ K_x $$ in $$H$$. This function is the so-called reproducing kernel for the Hilbert space $$H$$ from which the RKHS takes its name. More formally, the Riesz representation theorem implies that for all $$x$$ in $$X$$ there exists a unique element $$ K_x $$ of $$H$$ with the reproducing property,

Since $$ K_x $$ is itself a function defined on $$X$$ with values in the field $$\mathbb{R}$$ (or $$\mathbb{C}$$ in the case of complex Hilbert spaces) and as $$ K_x $$ is in $$H$$ we have that
 * $$ K_x(y) = L_y(K_x)= \langle K_x,\ K_y \rangle_H, $$

where $$K_y\in H$$ is the element in $$H$$ associated to $$L_y$$.

This allows us to define the reproducing kernel of $$H$$ as a function $$ K: X \times X \to \mathbb{R} $$ (or $$\mathbb{C}$$ in the complex case) by

 * $$ K(x,y) = \langle K_x,\ K_y \rangle_H. $$

From this definition it is easy to see that $$ K: X \times X \to \mathbb{R} $$ (or $$\mathbb{C}$$ in the complex case) is both symmetric (resp. conjugate symmetric) and positive definite, i.e.

 * $$ \sum_{i,j =1}^n c_i c_j K(x_i, x_j)=

\sum_{i=1}^n c_i \left\langle K_{x_i}, \sum_{j=1}^n c_j K_{x_j} \right\rangle_{H} = \left\langle \sum_{i=1}^n c_i K_{x_i}, \sum_{j=1}^n c_j K_{x_j} \right\rangle_{H} = \left\|\sum_{i=1}^nc_iK_{x_i}\right\|_H^2 \ge 0 $$

for every $$ n \in \mathbb{N}, x_1, \dots, x_n \in X, \text{ and } c_1, \dots, c_n \in \mathbb{R}. $$ The Moore–Aronszajn theorem (see below) is a sort of converse to this: if a function $$K$$ satisfies these conditions then there is a Hilbert space of functions on $$X$$ for which it is a reproducing kernel.

Example
The space of bandlimited continuous functions $$H$$ is a RKHS, as we now show. Formally, fix some cutoff frequency $$ 0<a < \infty $$ and define the Hilbert space

 * $$ H = \{ f \in C(\mathbb{R}) \mid \operatorname{supp}(F) \subset [-a,a] \} $$

where $$C(\mathbb{R})$$ is the set of continuous square integrable functions, and $ F(\omega) = \int_{-\infty}^\infty f(t) e^{-i\omega t} \, dt $ is the Fourier transform of $$ f$$. As the inner product of this Hilbert space, we use

 * $$\langle f, g\rangle_{L^2} = \int_{-\infty}^\infty f(x) \cdot \overline{g(x)} \, dx.$$

From the Fourier inversion theorem, we have

 * $$ f(x) = \frac{1}{2 \pi} \int_{-a}^a F(\omega) e^{ix \omega} \, d\omega .$$

It then follows by the Cauchy–Schwarz inequality and Plancherel's theorem that, for all $$x$$,

 * $$ |f(x)| \le

\frac{1}{2 \pi} \sqrt{ 2a\int_{-a}^a |F(\omega)|^2 \, d\omega} =\frac{ \sqrt{2a} }{2\pi}\sqrt{\int_{-\infty}^\infty |F(\omega)|^2 \, d\omega} = \sqrt{\frac{a}{\pi}} \|f\|_{L^2}. $$

This inequality shows that the evaluation functional is bounded, proving that $$ H $$ is indeed a RKHS.

The kernel function $$K_x$$ in this case is given by

 * $$K_x(y) = \frac{a}{\pi} \operatorname{sinc}\left ( \frac{a}{\pi} (y-x) \right )=\frac{\sin(a(y-x))}{\pi(y-x)}.$$

The Fourier transform of $$K_x(y)$$ defined above is given by

 * $$\int_{-\infty}^\infty K_x(y)e^{-i \omega y} \, dy =

\begin{cases} e^{-i \omega x} &\text{if } \omega \in [-a, a], \\ 0 &\textrm{otherwise}, \end{cases} $$

which is a consequence of the time-shifting property of the Fourier transform. Consequently, using Plancherel's theorem, we have

 * $$ \langle f, K_x\rangle_{L^2} = \int_{-\infty}^\infty f(y) \cdot \overline{K_x(y)} \, dy

= \frac{1}{2\pi} \int_{-a}^a F(\omega) \cdot e^{i\omega x} \, d\omega = f(x) .$$

Thus we obtain the reproducing property of the kernel.

$$K_x$$ in this case is the "bandlimited version" of the Dirac delta function, and that $$K_x(y)$$ converges to $$\delta(y-x)$$ in the weak sense as the cutoff frequency $$a$$ tends to infinity.

Moore–Aronszajn theorem
We have seen how a reproducing kernel Hilbert space defines a reproducing kernel function that is both symmetric and positive definite. The Moore–Aronszajn theorem goes in the other direction; it states that every symmetric, positive definite kernel defines a unique reproducing kernel Hilbert space. The theorem first appeared in Aronszajn's Theory of Reproducing Kernels, although he attributes it to E. H. Moore.

 * Theorem. Suppose K is a symmetric, positive definite kernel on a set X. Then there is a unique Hilbert space of functions on X for which K is a reproducing kernel.

Proof. For all x in X, define Kx = K(x, ⋅ ). Let H0 be the linear span of {Kx : x ∈ X}. Define an inner product on H0 by

 * $$ \left\langle \sum_{j=1}^n b_j K_{y_j}, \sum_{i=1}^m a_i K_{x_i} \right \rangle_{H_0} = \sum_{i=1}^m \sum_{j=1}^n {a_i} b_j K(y_j, x_i),$$

which implies $$K(x,y)=\left\langle K_{x}, K_{y} \right\rangle_{H_0}$$. The symmetry of this inner product follows from the symmetry of K and the non-degeneracy follows from the fact that K is positive definite.

Let H be the completion of H0 with respect to this inner product. Then H consists of functions of the form

 * $$ f(x) = \sum_{i=1}^\infty a_i K_{x_i} (x) \quad \text{where} \quad \lim_{n \to \infty}\sup_{p\geq0}\left\|\sum_{i=n}^{n+p} a_i K_{x_i}\right\|_{H_0} = 0.$$

Now we can check the reproducing property ($$):

 * $$\langle f, K_x \rangle_H = \sum_{i=1}^\infty a_i\left \langle K_{x_i}, K_x \right \rangle_{H_0}= \sum_{i=1}^\infty a_i K (x_i, x) = f(x).$$

To prove uniqueness, let G be another Hilbert space of functions for which K is a reproducing kernel. For every x and y in X, ($$) implies that

 * $$\langle K_x, K_y \rangle_H = K(x, y) = \langle K_x, K_y \rangle_G.$$

By linearity, $$\langle \cdot, \cdot \rangle_H = \langle \cdot, \cdot \rangle_G$$ on the span of $$\{K_x : x \in X\}$$. Then $$H \subset G$$ because G is complete and contains H0 and hence contains its completion.

Now we need to prove that every element of G is in H. Let $$ f $$ be an element of G. Since H is a closed subspace of G, we can write $$ f=f_H + f_{H^\bot} $$ where $$ f_H \in H $$ and $$ f_{H^\bot} \in H^\bot $$. Now if $$ x \in X $$ then, since K is a reproducing kernel of G and H:

 * $$f(x) = \langle K_x, f \rangle_G = \langle K_x, f_H \rangle_G + \langle K_x, f_{H^\bot} \rangle_G = \langle K_x , f_H \rangle_G = \langle K_x , f_H \rangle_H = f_H(x),   $$

where we have used the fact that $$ K_x $$ belongs to H so that its inner product with $$ f_{H^\bot} $$ in G is zero. This shows that $$ f = f_H $$ in G and concludes the proof.

Integral operators and Mercer's theorem
We may characterize a symmetric positive definite kernel $$K$$ via the integral operator using Mercer's theorem and obtain an additional view of the RKHS. Let $$X$$ be a compact space equipped with a strictly positive finite Borel measure $$\mu$$ and $$K: X \times X \to \R$$ a continuous, symmetric, and positive definite function. Define the integral operator $$T_K: L_2(X) \to L_2(X)$$ as

 * $$ [T_K f](\cdot) =\int_X K(\cdot,t) f(t)\, d\mu(t) $$

where $$L_2(X)$$ is the space of square integrable functions with respect to $$ \mu $$.

Mercer's theorem states that the spectral decomposition of the integral operator $$T_K$$ of $$K$$ yields a series representation of $$K$$ in terms of the eigenvalues and eigenfunctions of $$ T_K $$. This then implies that $$K$$ is a reproducing kernel so that the corresponding RKHS can be defined in terms of these eigenvalues and eigenfunctions. We provide the details below.

Under these assumptions $$T_K$$ is a compact, continuous, self-adjoint, and positive operator. The spectral theorem for self-adjoint operators implies that there is an at most countable decreasing sequence $$(\sigma_i)_i \geq 0 $$ such that $\lim_{i \to \infty}\sigma_i = 0$ and $$T_K\varphi_i(x) = \sigma_i\varphi_i(x)$$, where the $$\{\varphi_i\}$$ form an orthonormal basis of $$L_2(X)$$. By the positivity of $$T_K, \sigma_i > 0$$ for all $$i.$$ One can also show that $$T_K $$ maps continuously into the space of continuous functions $$C(X)$$ and therefore we may choose continuous functions as the eigenvectors, that is, $$\varphi_i \in C(X)$$ for all $$i.$$ Then by Mercer's theorem $$ K $$ may be written in terms of the eigenvalues and continuous eigenfunctions as

 * $$ K(x,y) = \sum_{j=1}^\infty \sigma_j \, \varphi_j(x) \, \varphi_j(y) $$

for all $$x, y \in X$$ such that

 * $$ \lim_{n \to \infty}\sup_{u,v} \left |K(u,v) - \sum_{j=1}^n \sigma_j \, \varphi_j(u) \, \varphi_j(v) \right | = 0. $$

This above series representation is referred to as a Mercer kernel or Mercer representation of $$ K $$.

Furthermore, it can be shown that the RKHS $$ H $$ of $$ K $$ is given by

 * $$ H = \left \{ f \in L_2(X) \,\Bigg\vert\, \sum_{i=1}^\infty \frac{\left\langle f,\varphi_i \right \rangle^2_{L_2}}{\sigma_i} < \infty \right\} $$

where the inner product of $$ H $$ given by

 * $$ \left\langle f,g \right\rangle_H = \sum_{i=1}^\infty \frac{\left\langle f,\varphi_i \right\rangle_{L_2}\left\langle g,\varphi_i \right\rangle_{L_2}}{\sigma_i}. $$

This representation of the RKHS has application in probability and statistics, for example to the Karhunen-Loève representation for stochastic processes and kernel PCA.

Feature maps
A feature map is a map $$ \varphi\colon X \rightarrow F $$, where $$ F $$ is a Hilbert space which we will call the feature space. The first sections presented the connection between bounded/continuous evaluation functions, positive definite functions, and integral operators and in this section we provide another representation of the RKHS in terms of feature maps.

Every feature map defines a kernel via

Clearly $$ K $$ is symmetric and positive definiteness follows from the properties of inner product in $$ F $$. Conversely, every positive definite function and corresponding reproducing kernel Hilbert space has infinitely many associated feature maps such that ($$) holds.

For example, we can trivially take $$ F = H $$ and $$ \varphi(x) = K_x $$ for all $$ x \in X $$. Then ($$) is satisfied by the reproducing property. Another classical example of a feature map relates to the previous section regarding integral operators by taking $$ F = \ell^2 $$ and $$ \varphi(x) = (\sqrt{\sigma_i} \varphi_i(x))_i $$.

This connection between kernels and feature maps provides us with a new way to understand positive definite functions and hence reproducing kernels as inner products in $$ H $$. Moreover, every feature map can naturally define a RKHS by means of the definition of a positive definite function.

Lastly, feature maps allow us to construct function spaces that reveal another perspective on the RKHS. Consider the linear space

 * $$ H_\varphi = \{ f: X \to \mathbb{R} \mid \exists w \in F, f(x) = \langle w, \varphi(x) \rangle_{F}, \forall \text{ } x \in X \} . $$

We can define a norm on $$ H_\varphi $$ by

 * $$ \|f\|_\varphi = \inf \{\|w\|_F : w \in F, f(x) = \langle w, \varphi(x)\rangle_F, \forall \text{ } x \in X \} .$$

It can be shown that $$ H_{\varphi} $$ is a RKHS with kernel defined by $$ K(x,y) = \langle\varphi(x), \varphi(y)\rangle_F $$. This representation implies that the elements of the RKHS are inner products of elements in the feature space and can accordingly be seen as hyperplanes. This view of the RKHS is related to the kernel trick in machine learning.

Properties
Useful properties of RKHSs:

 * Let $$(X_i)_{i=1}^p$$ be a sequence of sets and $$(K_i)_{i=1}^p$$ be a collection of corresponding positive definite functions on $$ (X_i)_{i=1}^p.$$ It then follows that
 * $$K((x_1,\ldots ,x_p),(y_1,\ldots,y_p)) = K_1(x_1,y_1)\cdots K_p(x_p,y_p)$$
 * is a kernel on $$ X = X_1 \times \dots \times X_p.$$
 * Let $$X_0 \subset X,$$ then the restriction of $$ K $$ to $$X_0 \times X_0 $$ is also a reproducing kernel.
 * Consider a normalized kernel $$K$$ such that $$ K(x, x) = 1 $$ for all $$x \in X $$. Define a pseudo-metric on X as
 * $$ d_K(x,y) = \|K_x - K_y\|_H^2 = 2(1-K(x,y)) \qquad \forall x \in X . $$
 * By the Cauchy–Schwarz inequality,
 * $$ K(x,y)^2 \le K(x, x)K(y, y)=1 \qquad \forall x,y \in X.$$
 * This inequality allows us to view $$K$$ as a measure of similarity between inputs. If $$x, y \in X$$ are similar then $$K(x,y)$$ will be closer to 1 while if $$x,y \in X$$ are dissimilar then $$K(x,y)$$ will be closer to 0.

 * The closure of the span of $$ \{ K_x \mid x \in X \} $$ coincides with $$ H $$.

Bilinear kernels

 * $$ K(x,y) = \langle x,y\rangle $$

The RKHS $$H$$ corresponding to this kernel is the dual space, consisting of functions $$f(x) = \langle x,\beta\rangle$$ satisfying $$\|f\|_H^2=\|\beta\|^2$$.

Polynomial kernels

 * $$ K(x,y) = (\alpha\langle x,y \rangle + 1)^d, \qquad \alpha \in \R, d \in \N $$

Radial basis function kernels
These are another common class of kernels which satisfy $$ K(x,y) = K(\|x - y\|)$$. Some examples include:

 * Gaussian or squared exponential kernel:
 * $$ K(x,y) = e^{-\frac{\|x - y\|^2}{2\sigma^2}}, \qquad \sigma > 0 $$
 * Laplacian kernel:
 * $$ K(x,y) = e^{-\frac{\|x - y\|}{\sigma}}, \qquad \sigma > 0 $$
 * The squared norm of a function $$f$$ in the RKHS $$H$$ with this kernel is:
 * $$\|f\|_H^2=\int_{\mathbb R}\Big( \frac1{\sigma} f(x)^2 + \sigma f'(x)^2\Big) \mathrm d x.$$

Bergman kernels
We also provide examples of Bergman kernels. Let X be finite and let H consist of all complex-valued functions on X. Then an element of H can be represented as an array of complex numbers. If the usual inner product is used, then Kx is the function whose value is 1 at x and 0 everywhere else, and $$K(x,y)$$ can be thought of as an identity matrix since

 * $$K(x,y)=\begin{cases} 1 & x=y \\ 0 & x \neq y \end{cases}$$

In this case, H is isomorphic to $$\Complex^n$$.

The case of $$X= \mathbb{D}$$ (where $$\mathbb{D}$$ denotes the unit disc) is more sophisticated. Here the Bergman space $H^2(\mathbb{D})$ is the space of square-integrable holomorphic functions on $$\mathbb{D}$$. It can be shown that the reproducing kernel for $$H^2(\mathbb{D})$$ is

 * $$K(x,y)=\frac{1}{\pi}\frac{1}{(1-x\overline{y})^2}.$$

Lastly, the space of band limited functions in $$ L^2(\R) $$ with bandwidth $$2a$$ is a RKHS with reproducing kernel

 * $$K(x,y)=\frac{\sin a (x - y)}{\pi (x-y)}.$$

Extension to vector-valued functions
In this section we extend the definition of the RKHS to spaces of vector-valued functions as this extension is particularly important in multi-task learning and manifold regularization. The main difference is that the reproducing kernel $$ \Gamma $$ is a symmetric function that is now a positive semi-definite matrix for every $$ x,y $$ in $$ X $$. More formally, we define a vector-valued RKHS (vvRKHS) as a Hilbert space of functions $$ f: X \to \mathbb{R}^T $$ such that for all $$ c \in \mathbb{R}^T $$ and $$ x \in X $$

 * $$ \Gamma_xc(y) = \Gamma(x, y)c \in H \text{ for } y \in X $$

and

 * $$ \langle f, \Gamma_x c \rangle_H = f(x)^\intercal c. $$

This second property parallels the reproducing property for the scalar-valued case. This definition can also be connected to integral operators, bounded evaluation functions, and feature maps as we saw for the scalar-valued RKHS. We can equivalently define the vvRKHS as a vector-valued Hilbert space with a bounded evaluation functional and show that this implies the existence of a unique reproducing kernel by the Riesz Representation theorem. Mercer's theorem can also be extended to address the vector-valued setting and we can therefore obtain a feature map view of the vvRKHS. Lastly, it can also be shown that the closure of the span of $$ \{ \Gamma_xc : x \in X, c \in \mathbb{R}^T \} $$ coincides with $$ H $$, another property similar to the scalar-valued case.

We can gain intuition for the vvRKHS by taking a component-wise perspective on these spaces. In particular, we find that every vvRKHS is isometrically isomorphic to a scalar-valued RKHS on a particular input space. Let $$\Lambda = \{1, \dots, T \} $$. Consider the space $$ X \times \Lambda $$ and the corresponding reproducing kernel

As noted above, the RKHS associated to this reproducing kernel is given by the closure of the span of $$\{ \gamma_{(x,t)} : x \in X, t \in \Lambda \} $$ where $$ \gamma_{(x,t)} (y,s) = \gamma( (x,t), (y,s)) $$ for every set of pairs $ (x,t), (y,s) \in X \times \Lambda $.

The connection to the scalar-valued RKHS can then be made by the fact that every matrix-valued kernel can be identified with a kernel of the form of ($$) via

 * $$ \Gamma(x,y)_{(t,s)} = \gamma((x,t), (y,s)). $$

Moreover, every kernel with the form of ($$) defines a matrix-valued kernel with the above expression. Now letting the map $$ D: H_\Gamma \to H_\gamma $$ be defined as

 * $$ (Df)(x,t) = \langle f(x), e_t \rangle_{\mathbb{R}^T} $$

where $$ e_t $$ is the $$ t^\text{th} $$ component of the canonical basis for $$ \mathbb{R}^T $$, one can show that $$ D $$ is bijective and an isometry between $$ H_\Gamma $$ and $$ H_\gamma $$.

While this view of the vvRKHS can be useful in multi-task learning, this isometry does not reduce the study of the vector-valued case to that of the scalar-valued case. In fact, this isometry procedure can make both the scalar-valued kernel and the input space too difficult to work with in practice as properties of the original kernels are often lost.

An important class of matrix-valued reproducing kernels are separable kernels which can factorized as the product of a scalar valued kernel and a $$T$$-dimensional symmetric positive semi-definite matrix. In light of our previous discussion these kernels are of the form

 * $$ \gamma((x,t),(y,s)) = K(x,y) K_T(t,s) $$

for all $$x,y $$ in $$ X $$ and $$t,s$$ in $$ T $$. As the scalar-valued kernel encodes dependencies between the inputs, we can observe that the matrix-valued kernel encodes dependencies among both the inputs and the outputs.

We lastly remark that the above theory can be further extended to spaces of functions with values in function spaces but obtaining kernels for these spaces is a more difficult task.

Connection between RKHSs and the ReLU function
The ReLU function is commonly defined as $$f(x)=\max \{0, x\}$$ and is a mainstay in the architecture of neural networks where it is used as an activation function. One can construct a ReLU-like nonlinear function using the theory of reproducing kernel Hilbert spaces. Below, we derive this construction and show how it implies the representation power of neural networks with ReLU activations.

We will work with the Hilbert space $$ \mathcal{H}=L^1_2(0)[0, \infty) $$ of absolutely continuous functions with $$f(0) = 0$$ and square integrable (i.e. $$L_2$$) derivative. It has the inner product

 * $$ \langle f,g \rangle_{\mathcal{H}} = \int_0^\infty f'(x)g'(x) \, dx .$$

To construct the reproducing kernel it suffices to consider a dense subspace, so let $$f\in C^1[0, \infty)$$ and $$f(0)=0$$. The Fundamental Theorem of Calculus then gives

 * $$f(y)= \int_0^y f'(x) \, dx = \int_0^\infty G(x,y) f'(x) \, dx = \langle K_y,f \rangle$$

where

 * $$G(x,y)=

\begin{cases} 1, & x < y\\ 0, & \text{otherwise} \end{cases}$$

and $$K_y'(x)= G(x,y),\ K_y(0) = 0$$ i.e.

 * $$K(x, y)=K_y(x)=\int_0^x G(z, y) \, dz=

\begin{cases} x, & 0\leq x<y \\ y, & \text{otherwise.} \end{cases}=\min(x, y)$$

This implies $$K_y=K(\cdot, y)$$ reproduces $$f$$.

Moreover the minimum function on $$ X\times X = [0,\infty)\times [0,\infty) $$ has the following representations with the ReLu function:

 * $$ \min(x,y) = x -\operatorname{ReLU}(x-y) =  y - \operatorname{ReLU}(y-x). $$

Using this formulation, we can apply the representer theorem to the RKHS, letting one prove the optimality of using ReLU activations in neural network settings.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Reproducing kernel Hilbert space


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Cincinnati May Festival

The Cincinnati May Festival is a two-week annual choral festival, held in May in Cincinnati, Ohio, US.

History
The festival's roots go back to the 1840s, when Saengerfests were held in that city, bringing singers from all over the United States and abroad to perform large scale classical works of the day.

In A City That Sings, we find a description of Cincinnati in the late 1800s: "In the final decades of the nineteenth century Cincinnati cemented its position as one of America's cultural leaders with the founding of the still flourishing May Festival (1873), the Cincinnati Art Museum (1886) and its neighbor in Cincinnati, the Cincinnati Art Academy (1887), and the Cincinnati Symphony Orchestra (1895).

Singing societies were very popular in the nineteenth century and the tradition was no different in Cincinnati. Both the English and German societies were evident. As a rule, the differences between the two national choral traditions were in language, origin, composition, and social scope. English societies, the earliest of Cincinnati's choral ensembles, existed primarily to make music. The first German society was founded in 1825, but the majority of these organizations arose after an immigration boom in the late 1840s. They provided social opportunities for eating and drinking, as well as making music. Most of the German societies were men's choruses, but women were sometimes admitted, often as non-singing supporting members, in the last half of the century.

In October 1863, a circular was sent to the 'Musical Public of Cincinnati' to extend a 'cordial invitation to singers, both ladies and gentlemen, who will take an interest in building up a large Choral Society, such as the old societies of Boston and New York, to join.' This mixed chorus approach to presenting oratorio with members drawn from a broad spectrum of Cincinnati's singers regardless of the nationality or society to which they pledged allegiance, must be regarded as an important precedent of the May Festival.

Maria Longworth Nichols Storer the granddaughter of Nicholas Longworth, a banker and winemaker, and wife of Colonel George Ward Nichols took a trip to England in 1871. There she witnessed one of the large English festivals. At home, Mr. Nichols was a member of the Harmonic Society, so the couple was involved in choral music of the English variety. Mrs. Nichols knew Cincinnati's choral tradition was the perfect place to start a musical festival of their own. She wrote to Theodore Thomas, a renowned American, conductor and asked for a meeting with him on his next visit to Cincinnati.

He recounts: 'On my...visit in 1871 [sic—it was 1872] a young married lady, member of one of the leading families, laid before me a plan for a large Musical Festival. She proposed that I should be the conductor of it, saying that I would be responsible for the artistic side, she would find the men who would take charge of the business details. I soon found out that this lady was not only very talented in may ways, but that her taste was not amateurish in anything, and I readily consented to undertake the work she wished me to do.' He agreed to be conductor of the Festival if fifty thousand dollars could be raised for a guarantee fund and a committee could be formed to take care of the business aspects of the event. Mrs. Nichols's biographer, Sr. Rose Angela Boehle, mentions that Maria and Col. Nichols started the guarantee fund of the 1873 Festival with a five-thousand-dollar contribution.

Among the first actions of the executive committee was the hiring of Theordore Thomas as musical director and Carl Barus as director of the chorus. Thomas's touring orchestra would play, as they would continue to do at every festival until 1904. Arthur Mees, a Cincinnati music teacher, was named rehearsal accompanist and later, festival organist. While the artistic forces were almost entirely German, the administrative team that planned the event was mostly of English descent. A circular describing the expectations of chorus members was printed in English and German and distributed to 121 music dealers, 60 post offices, and 144 singing societies. It was also distributed to 1,120 newspapers throughout Ohio, Indiana, Kentucky, and 'the leading papers of the West.' According to the superintendent of the chorus, Samuel D. Carey, by the time the first performance took place, the chorus had 706 members.

The 1873 Festival
"The orchestra is German; the music three-fourths German, and the other fourth exotic; scarcely half the singers were native-born; but, despite all this, it is thoroughly American, for nothing but that attenuated and intensified form of Anglo-Saxon called the Yankee could ever have supplied the nerve-force needed to carry on such a mammoth affair." Performances included repertoire that was overwhelmingly German. Handel, Bach, Beethoven, Haydn, and Mozart comprised the first two concerts. Of the sixty works on the seven concerts, only five were by composers not born in a German-speaking country. The matinee concerts consisted largely of arias, art songs, and light orchestral works, including isolated movements of symphonies, Strauss waltzes, and overtures. Thomas's evening concerts were more formal and consisted of major vocal and choral works, including Handel's Dettingen Te Deum, scenes from Gluck's Orfeo ed Euridice, and Beethoven's Ninth Symphony, along with purely symphonic works and concert arias.

The 1875 Festival
A second May Festival took place at Saengerfest Hall in 1875, and presented the American premieres of Bach's Magnificat and Brahms's Triumphlied. During this festival, Cincinnati's usual springtime thunderstorms pelted the leaky tin roof of the hall with rain and hail, as they had done to a lesser degree in 1873. The cavernous hall acted as a resonating chamber and Thomas had to stop the concerts on more than one occasion to let the storms pass. Two other problems with the hall were mentioned in the press reports of the first festival, which no doubt persisted in 1875. The reporter from the Gazette mentioned that the temperature in the hall was too warm. The situation was 'promptly remedied by a member of the executive committee who crashed out the glass from every window whose frame was wedged too tight in its place to be readily removed.' Another problem arose as the audience, estimated at four thousand with two thousand more in the streets, seems to have grown somewhat frustrated by the crowded conditions in the hall and complained that allowing one thousand people to stand in the aisles 'sorely disappoints ticket-holders who selected their seats on the aisles with especial purpose.'

Music Hall and the 1878 Festival
Kentucky-born railroad entrepreneur Reuben Springer was so annoyed by the hall problems that he made the initial $125,000 donation to create a matching fund to enable the construction of a new music hall.

The area chosen for the new Music Hall was close to the Community Hospital, Insane Asylum, and the Pest House. When inmates or patients expired, they were interred in Potter's Field, the present site of Music Hall. Over two hundred wagon loads of bones were moved to Spring Grove Cemetery and reburied in a mass grave.

Construction for the hall began on May 1, 1877, just a year before the third May Festival was to begin. It was dedicated on May 14 at the first concert of the 1878 May Festival. (125) Tickets for the first festival held in Music Hall were, "$2 for a single reserved seat, $1 for standing room, and $10 dollars for the entire set of seven concert." Cincinnati's May Festival has taken place in Springer Auditorium since 1878.

Edward Elgar in Cincinnati
After the death of Theodore Thomas in 1905, the 1906 May Festival featured British composer Sir Edward Elgar conducting his oratorio The Dream of Gerontius at the final concert. ... Elgar was the most prominent British composer and \his music was popular in the U.S. He conducted the second, fifth, and sixth concerts of the festival, after spending the preceding two weeks rehearsing his music with the performers.

Theodore Thomas had previously conducted Gerontius at the 1904 May Festival, and its success and the fame of the composer may have drawn the board of directors to consider Elgar for a guest conductor and celebrity at the upcoming festival.

Enraging the chorus, Lawrence Maxwell Jr., vice-president of the May Festival Association stated that he was "against the giving of the memorial concert, saying that it had not the sanction of the board, that the program was a log of 'hash,' and that it would make Thomas turn in his grave to listen to it." ...The Festival Board, in an act of either reconciliation or humiliation towards the chorus, planned its own memorial concert to open the 1906 May Festival. The chorus responded in a letter by demanding Maxwell's resignation, adding that "if the Board of Directors do not find this request acceptable, we find no other course open to us than to insist that this communication shall constitute the resignation of the chorus, to take effect at once." The May Festival Chorus was soon disbanded, and the Board of Directors chose Maxwell as the next president of the May Festival Association.

In the midst of the upheaval, the festival reorganized its board and established a marketing plan for the 1906 concerts to choose "[Frank] Van der Stucken as festival conductor, and to depend upon Sir Edward Elgar as a foreign attraction." Frank van der Stucken was the conductor of the recently formed Cincinnati Symphony Orchestra, and the board decided that the Cincinnati orchestra would play in the 1906 May Festival for the first time.

The previous festivals had had open rehearsals and many other public opportunities for Cincinnatians to meet and mingle with the stars, but this would not be the case in 1906. With a completely restructured program, Frank Van der Stucken anticipated rehearsals would be full of personnel problems and musical challenges and considered options that would allow the performers to focus on the music.

The final concert was held on the same evening, and the program consisted of The Dream of Gerontius followed by Beethoven's Ninth Symphony.

The May Festival and Cincinnati School Music
However, perhaps the appearances of massed school choirs most valued by Cincinnatians were those in the then-biennial May Festivals. At the first May Festivals (1873 and 1875) a large chorus of pupils and teachers from the city's intermediate, secondary, and normal schools presented music learned in their public school classes. Beginning in 1882, planners of the festivals began working student choral ensembles into the regular festival repertoire, such as Bach's St. Matthew Passion, Berlioz's Damnation of Faust, and Te Deum, Handel's Judas Maccabeus, or Rubinstein's Tower of Babel. In 1897 a Children's May Festival, utilizing some 2,000 students from the city schools, presented Franz Abt's Cinderella.

Choruses of school children continue to be welcome participants in such major works as Orff's Carmina Burana, Walton's Belshazzar's Feast, Mahler's Symphony No. 3, and Britten's War Requiem. All these accomplishments would have been impossible had not Cincinnati schools, both public and parochial, introduced music into their curricula.

Notable performances and recent history
Perhaps the largest work in the repertoire, Mahler's Symphony No. 8, often called The Symphony of a Thousand, received a then-rare performance at the 1931 festival with a chorus of nine hundred, a massive orchestra, and eight soloists.

African-American composer Robert Nathaniel Dett's oratorio The Ordering of Moses appeared on the 1937 May Festival. In 2014, the May Festival Chorus and Cincinnati Symphony Orchestra gave a full performance of the oratorio in Carnegie Hall.

During times of war and tragedy, the May Festival has traditionally stepped forward to honor heroes and soothe the pain. During World War II, patriotic music held price of place in 1942 and performers donated their services. All proceeds were given to the Cincinnati War Chest. In 1944 a chorus of Women's Army Corps members from Ft. Campbell, Kentucky, performed Frank Loesser's The WAC Hymn and many soldiers attended the festival. Just one month after the September 11, 2001, terrorist attacks in New York, Washington, and Pennsylvania, the May Festival Chorus traveled to New York's Carnegie Hall to perform Britten's powerful War Requiem with the Cincinnati Symphony Orchestra and James Conlon.

When the United States entered World War II, the problems necessitated a three-year span until the next Festival. And then it was reduced to three concerts with a very much abridged program due to the shortage of paper. The Victory Festival in 1946 opened with the Hallelujah Chorus and the whole program was dedicated to the blessings of peace. The Folk Song Symphony by Roy Harris had 500 high school students singing as their costumes divided the chorus stand into three broad stripes of red, white and blue.

Until 1967 the May Festival was held every two years with only a few exceptions, but since then it has been given every year.

Adolphus Hailstork also composed 'Earthrise,' a work for double chorus and orchestra commissioned by the May Festival in 2006 to promote healing following the 2000 riots. The piece featured a white chorus on one side of the risers and an African American chorus on the other side, who by the end form a visual metaphor by interweaving to create a racially harmonic gesture.

May Festival opened its doors to African American singers in 1956.

The all-volunteer May Festival Chorus has been under the direction of Robert Porco since 1989. The chorus rehearses on Tuesday evenings during the academic year from 7-10pm. The chorus performs regularly with the Cincinnati Symphony and Pops Orchestras, singing at Music Hall (Cincinnati) as well as Riverbend Music Center. The May Festival Chorus has garnered critical acclaim for its high level of performance. The May Festival Chamber Chorus performs at the Cathedral Basilica of the Assumption in northern Kentucky during the annual May Festival and at various community concerts throughout the year.

In addition, in 1986 the May Festival Youth Chorus was formed. Made up of youth from area high-schools, the MFYC is conducted by James Bagwell and gives its members (who must audition competitively) the opportunity to perform challenging music well above the average high school level. The MFYC also has the opportunity to perform with the adult May Festival throughout the year, most notably during May Festival.

On 5 December 2017, the Cincinnati May Festival canceled conductor James Levine’s appearance in May 2018, after he was accused by four males of sexual abuse.

Since 12 March 2020, all concerts were cancelled and virtual ones are being held caused by the COVID-19 pandemic.

Music Directors and Chorus Masters
Music Directors

 * Theodore Thomas (1873–1904)
 * Frank van der Stucken (1906–1912)
 * Ernst Kunwald (1914–1916)
 * Eugène Ysaÿe (1918–1920)
 * Frank van der Stucken (1923–1927)
 * Frederick Stock 1929
 * Eugene Aynsley Goossens (1931–1946)
 * Fritz Busch (1948–1950)
 * Josef Krips (1954–1960)
 * Max Rudolf (1963; 1967–1970)
 * Julius Rudel (1971–1972)
 * Leonard Bernstein 1973 (Honorary Music Director)
 * James Levine (1974–1978)
 * James Conlon (1979–2016)

Principal Conductor

 * Juanjo Mena (2017–present)

Chorus Masters

 * Carl Barus (1873, 1882)
 * Otto Singer	 (1875°, 1878°, 1880°)
 * Arthur Mees (1882, 1886°)
 * Edwin Glover	(1884, 1898, 1900, 1902, 1904)
 * Louis Ehrgott	(1888, 1890, 1918)
 * W. L. Blumenschein (1892, 1894, 1896)
 * Alfred Hartzel (1908, 1910, 1912, 1914°, 1916°, 1920, 1922, 1925, 1927, 1929, 1931, 1933, 1935, 1937, 1939, 1941, 1944)
 * Sherwood Kains (1946, 1948)
 * Willis Beckett	(1950, 1952, 1954, 1956, 1958, 1960)
 * Robert Knauf	(1963, 1965, 1967, 1968, 1969)
 * Elmer Thomas (1970–1973; 1975)
 * Earl Rivers (Acting)	1974
 * Thomas Peck	(1976-1978)
 * John Leman	(1979–1988)
 * John H. Williams 1989
 * Robert Porco (1990–present)

Between 1905 and 1946, music directors of the Cincinnati Symphony Orchestra (founded in 1895) also were engaged as music director of the biannual choral festival (Frank Van der Stucken, Ernst Kunwald, Eugène Ysaÿe, Fritz Reiner, and Eugene Goosens held the reins during those years). Fritz Busch and Josef Krips led festivals between 1948 and 1960.

With the appointment of Max Rudolf as director of the festival in 1963, the festival became an annual event in 1967, as it is today. In the 1970s, the centennial of both the festival (1973) and Music Hall (1978) was celebrated with the tenures of Julius Rudel (director from 1971 to 1973) and James Levine (1974 to 1978). James Conlon became director in 1979.

When Leonard Bernstein conducted the festival in its centennial year, he ascribed the true strength of the endeavor with his typical aplomb: "One of the most valuable features of Cincinnati's festival as I see it is that it preserves the involvement of the community chorus…. I don't know how the May Festival does it! … I'm filled with admiration at what they accomplish."

title: Cincinnati May Festival


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Carlina vulgaris

Carlina vulgaris, the carline thistle, is a plant species of the genus Carlina.

It is a biennial that grows on limestone, chalky or other alkaline grasslands or dunes. The flowers are clusters of very small brown florets surrounded by brown-golden bracts. Both feel dry and spiky, so at first glance the plants appear to be dying when in full flower. The green leaves are spiny and may have hairs. Plants are often short but can reach 60 cm.

It originates from Europe, north Africa and Asia but is also found in other parts of the world such as North America. It is considered an invasive species in parts of its introduced range.

title: Carlina vulgaris


text: User talk:68.232.36.247

title: User talk:68.232.36.247


text: User talk:129.130.5.229

title: User talk:129.130.5.229


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Luiz Mott

Luiz Roberto de Barros Mott or Luiz Mott (born 6 May 1946) in São Paulo, is an anthropologist and a gay rights activist in Brazil.

Early life
Luiz Mott graduated in Social Sciences from the University of São Paulo (USP) during the military regime, obtained a master's degree in Ethnography from the Sorbonne and a doctorate in anthropology from the University of Campinas in São Paulo.

Career
Luiz Mott is professor emeritus of the Department of Anthropology of the Federal University of Bahia (UFBA). In his work Mott has traced homosexual desire in a number of indigenous Brazilian tribes such as the Bororo, Guató, Trumai, Tupinambá, Wai Wai and Xavante. He has also explored intimate partner violence between gay, transsexual and transgender people in Salvador, the capital of the Brazilian state of Bahia. In 1995, he declared that national hero Zumbi dos Palmares had been gay.

Alongside his academic work, Mott is also a gay rights activist and founded the Gay Group of Bahia in 1980. It was the first homosexual support group in Bahia and Mott ensured its survival by pressuring the municipal government to fund it. It published records of homophobic attacks, raised awareness of HIV/AIDS and organised a pride parade, workshops, educational events and protests. He also created Centro Baiano Anti-AIDS.

In 1993, he published the first account of the life of Rosa Egipcíaca, a religious mystic and formerly enslaved prostitute, who wrote the book Sagrada Teologia do Amor Divino das Almas Peregrinas, which was the first book to be written by a black woman in Brazil.

In 1995, he was awarded the Felipa de Souza Award by the International Gay and Lesbian Human Rights Commission (now OutRight Action International). Mott commented to the Los Angeles Times in 2015: "Brazil is an extremely contradictory country. On the one hand, we are a pink country, celebrating sexual diversity [..] then, there is another color, the red blood of victims."

title: Luiz Mott


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Social consciousness

Social consciousness or social awareness, is collective consciousness shared by individuals within a society.

Social consciousness is linked to the collective self-awareness and experience of collectively shared social identity. From this viewpoint, social consciousness denotes conscious awareness of being part of an interrelated community of others. The “we feeling” or the “sense of us” may be experienced in members of various cultures and social groups. By the experience of collectively shared social identity, individuals may experience social unity. Social consciousness may also stimulate working towards a common goal.

According to Karl Marx, human beings enter into certain productive, or economic, relations and these relations lead to a form of social consciousness. Marx said:

"'In the social production of their life, men enter into definite relations that are indispensable and independent of their will; these relations of production correspond to a definite stage of development of their material forces of production. The sum total of these relations of production constitutes the economic structure of society — the real foundation, on which rises a legal and political superstructure and to which correspond definite forms of social consciousness. The mode of production of material life determines the social, political and intellectual life process in general. It is not the consciousness of men that determines their being, but, on the contrary, their social being that determines their consciousness.'"

title: Social consciousness


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Catskill High Peaks

The Catskill High Peaks are all of the mountains in New York's Catskill Mountains above 3,500 ft in elevation whose summits are separated either by 1/2 mi or a vertical drop of at least 250 ft between it and the next nearest separate summit. By usual standards, these mountains are rather low and rounded, and mostly covered by vegetation.

Unlike the Adirondack High Peaks, those in the Catskills are more evenly distributed around the eastern half of the range instead of being confined to one small area. All except Bearpen and Vly lie within the Catskill Park Blue Line.

The 3500s
The 3,500 ft elevation was chosen in the early 1950s when Dan Smiley, a descendant of the founders of Mohonk Mountain House and his fellow birders were trying to find the elusive Bicknell's thrush, which prefers to nest and breed in the boreal forests that predominate above that elevation in the Catskills. They drew up a list nearly identical to the one below and published it in a naturalists' journal.

A decade later hikers interested in peak bagging formed the Catskill Mountain 3500 Club, an organization similar to the better-known Adirondack 46ers, to promote the climbing of 34 of the 35 peaks listed below ("Southwest Hunter" was added to the list in 1992). The club implemented the peak-separation rule, modeled on the one already in use by the Appalachian Mountain Club. Prospective members are required to climb all (13 have no established route to the summit, requiring a bushwack) and then do Slide, Blackhead, Balsam and Panther mountains again in wintertime.

Today the New York State Department of Environmental Conservation enforces special rules for all the lands it manages in the Catskills above 3,500 ft, whether tiny plots such as on Rocky or vast ridgetops as on Plateau, Hunter and West Kill, due to the fragility of the ecosystems. In these areas, camping is permitted only in dire emergencies except during winter, when snow shields the earth somewhat, and open fires are banned. Almost all were designated bird conservation areas by Governor George Pataki in 1999, due to Bicknell's and other vulnerable species that return there in springtime to nest and breed.

The high peaks
Surveys of the Catskills date to the 1940s at the newest, and exact elevations have never been officially determined for many of these peaks. In those cases, their elevations are indicated by the highest 20 ft contour shown on the United States Geological Survey (USGS) quadrangles with a plus (+) sign.

 * The name is currently unofficial. It refers to the peak that rises between Hunter and West Kill on the USGS Lexington quad

 * This refers not to "Bearpen Mountain" shown on the USGS Prattsville quad straddling the border between Delaware and Greene counties at 3,520 ft, but rather the unnamed higher area approximately one-half mile (800 m) to the southeast in Greene County.

Potential High Peaks
Without the aforementioned "AND" ruling of the Catskill 3500 Club, the unnamed summit west of Thomas Cole (unofficially referred to as "Camel's Hump") sits within the 3520 ft contour and is more than the required distance from that peak's summit. No one has yet considered it a High Peak, however.

The inexact nature of past USGS surveys of the region have led to speculation that two 3,480 ft summits in the western Catskills might prove to be High Peaks if measurements were redone with more modern technology. GPS readings on some summits have consistently suggested higher elevations than those given on the maps.

Mill Brook Ridge, the named summit of the pair, would displace neighboring Balsam Lake as the westernmost Catskill High Peak were any of the land in its summit contour found to exceed 3,500 feet. However, much of that land is as flat as the map suggests and the peak may come close to that height but not over it.

The other is "East Schoolhouse Mountain," the unofficial name for the higher of the two peaks on the ridge between Balsam Lake and neighboring Graham. Digital Elevation Modeling (DEM) data in some mapping software suggests its summit may just exceed the limit.

There is also a tiny 3500 ft contour indicated more than a half-mile east of West Kill's summit, right next to the drop into Diamond Notch, but exploration of that area has cast doubt on whether it actually exists.

title: Catskill High Peaks


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Jerry Stackhouse

Jerry Darnell Stackhouse (born November 5, 1974) is an American basketball coach and former professional player who is the assistant coach of the Golden State Warriors of the National Basketball Association. Stackhouse played college basketball for the North Carolina Tar Heels and played 18 seasons in the National Basketball Association (NBA) and was a two-time NBA All-Star. He was the head coach of Raptors 905 and an assistant coach for the Toronto Raptors and Memphis Grizzlies. Additionally, he has worked as an NBA TV analyst.

Early career
Stackhouse was a premier player from the time he was a sophomore in high school. He was the state player of the year for North Carolina in 1991–92, leading Kinston (N.C) High School to the state finals. His senior year, he played for Oak Hill Academy with future college teammate Jeff McInnis, leading them to an undefeated season. He was a two-time first team Parade All-America selection, and was the MVP of the McDonald's All-American Game. At the 1992 Nike Camp, he and Rasheed Wallace were considered to be the top players at the camp. There were some who considered Stackhouse the top prep player to come out of North Carolina since Michael Jordan.

Stackhouse attended the University of North Carolina at Chapel Hill, where he was a teammate of future NBA players Wallace, McInnis and Shammond Williams. In his sophomore season at UNC, Stackhouse led the team in scoring with 19.2 points per game and averaged 8.2 rebounds per contest. He led UNC to a Final Four appearance and was named as the National Player of the Year by Sports Illustrated and earned first-team All-America and All-ACC honors. While playing for the Tar Heels, he was coached by Dean Smith. Following the season, Stackhouse declared his eligibility for the 1995 NBA draft.

Although he left UNC after two years, he continued working on his degree and received his bachelor's degree in African American Studies in 1999.

NBA draft
Stackhouse was selected in the first round of the 1995 NBA draft with the third pick by the Philadelphia 76ers. At one time he was hyped as the "Next Jordan" since both players played at North Carolina, went #3 in the draft, were listed at 6'6", looked similar physically, and had similarly acrobatic games. Coincidentally, both had a taller power forward from UNC drafted immediately after them in the #4 spot, Sam Perkins in 1984, and Rasheed Wallace in 1995.

Philadelphia 76ers (1995–1997)
In his first season with the 76ers, Stackhouse led his team with a 19.2 points per game (PPG) average, and was named to the NBA's All-Rookie team. In the 1996–97 season, the 76ers also drafted Allen Iverson. Combined, the two posted 44.2 points per game for the Sixers. Stackhouse represented the 76ers in the 1996 NBA Slam Dunk Contest.

Detroit Pistons (1997–2002)
Midway through the 1997–98 season, Stackhouse was dealt to the Detroit Pistons with Eric Montross for Theo Ratliff, Aaron McKie and future considerations. By the 1999–2000 season, his second full season with the Pistons, Stackhouse was averaging 23.6 points per game. A year later, he had a career-high average of 29.8 points per game. In a late season victory over the Chicago Bulls, he set the Pistons' franchise record and the league's season high for points in a game with 57. In 2001 he led the NBA in total points scored, finishing 2nd in the PPG scoring title to his former 76er teammate, Allen Iverson. Stackhouse saw his final action as a Piston with Detroit's elimination in the second round of the 2001–02 NBA playoffs to the Boston Celtics.

Washington Wizards (2002–2004)
During the 2002 offseason, Stackhouse was traded to the Washington Wizards in a six-player deal, also involving Richard Hamilton.

In his first season with Washington (2002–03), Stackhouse led the Wizards in points and assists per game with 21.5 and 4.5 respectively. Stackhouse became the only teammate to average more points per game than Michael Jordan for an entire season (Jordan averaged 20.0 points per game in 2002-03 which was his last season in the NBA). Stackhouse missed most of the 2003–04 season while recovering from arthroscopic surgery on his right knee, playing in only 26 games.

In 2020 Stackhouse stated that he regretted playing with Jordan, as plays were being designed for an NBA legend who was past his prime, stalling Stackhouse's All-Star momentum.

Dallas Mavericks (2004–2009)

In the 2004 offseason, Stackhouse—along with Christian Laettner and the Wizards' first-round draft pick (Devin Harris)—was traded to the Dallas Mavericks in exchange for former Tar Heel and NBA All-Star Antawn Jamison. He did not play for 41 games during his first two seasons with Dallas due to groin and continued knee problems, and played mostly the role of sixth man. During the 2004–05 playoffs, Stackhouse began wearing pressure stockings during games to keep his legs warm, to aid his groin injury, and hold his thigh sleeves in place; the stockings also allowed for better blood flow to the legs. The practice quickly became a trend among NBA players, with Kobe Bryant, Tracy McGrady, Vince Carter, Dwyane Wade, LeBron James and others adopting pressure stockings the following season.

Stackhouse was still coming off the bench as the 6th man for the Dallas Mavericks during the 2005–06 NBA season. He was a significant factor in the NBA Finals series against the Miami Heat. The Mavericks suffered when Stackhouse was suspended for Game 5 for a flagrant foul on Shaquille O'Neal, and the Heat eventually won the series 4–2. Stackhouse was the third player from the Mavericks suspended during the 2006 playoffs.

Milwaukee Bucks (2010)
Stackhouse was traded to the Memphis Grizzlies on July 8, 2009, in a four-way trade. On the day after the trade, Stackhouse was waived by the Grizzlies. On January 17, 2010, the Milwaukee Bucks signed Stackhouse for the remainder of the 2009–10 season.

Miami Heat (2010)
On October 23, 2010, Stackhouse and the Miami Heat agreed to a contract.

On November 23, 2010, the Heat waived Stackhouse to make room for Erick Dampier who was signed to replace injured forward Udonis Haslem.

Atlanta Hawks (2011–2012)
On December 9, 2011, Stackhouse joined the Atlanta Hawks. Stackhouse was chosen to replace injured teammate Joe Johnson as Atlanta's representative in the Haier Shooting Stars Competition during NBA All-Star weekend.

Brooklyn Nets (2012–2013)
On July 11, 2012, Stackhouse made a verbal agreement to sign a one-year, $1.3 million deal with the Nets. Stackhouse has worn the number 42 in honor of Jackie Robinson, his favorite athlete, and became the first professional athlete to wear the number 42 in Brooklyn since Robinson. On November 26, 2012, the Nets played the New York Knicks for the first time since the Nets had moved to Brooklyn. Stackhouse played 22 minutes and scored 14 points, including a tiebreaking 3-pointer in overtime, and the Nets went on to win. On March 18, 2013, he scored 10 points against the Detroit Pistons, one of his former teams.

After the Nets' first-round playoff loss to the Chicago Bulls, Stackhouse announced his retirement.

Broadcasting career
On November 15, 2013, it was announced that Stackhouse had joined Fox Sports Detroit as a Pistons analyst. He primarily provided studio analysis but also was the road color commentator for Fox Sports Detroit on select road trips. Stackhouse was also a college basketball analyst for the ACC Network and Fox Sports Detroit. By Joining Fox Sports Detroit, Stackhouse reunited with his former Pistons teammate Mateen Cleaves in the studio.

Toronto Raptors
On June 29, 2015, he was hired to serve as an assistant coach by the Toronto Raptors. During his first season with the Raptors, he helped the team finish second in the Eastern Conference along with reaching the Eastern Conference Finals.

Raptors 905
On September 9, 2016, the Raptors named him head coach for Raptors 905, the franchise's NBA Development League team. Stackhouse has aspirations of being an NBA head coach and told The Ringer that he hopes the D-League position will propel him to a head coaching job. Stackhouse led the 905 to a successful year during the 2016–17 season as the team was crowned champion of the NBA D-League. Stackhouse was named NBA D-League Coach of the Year in 2017.

Memphis Grizzlies
Stackhouse served as an assistant coach for the Memphis Grizzlies for the 2018–19 NBA season.

Vanderbilt
On April 5, 2019, Stackhouse was named the head coach of the Vanderbilt Commodores, signing a six-year contract. He was fired on March 14, 2024 after a disappointing 9–23 record in the 2024 season. Throughout his five seasons coaching the Commodores, he led the team to a 70–92 record and failed to make the NCAA Tournament.

Personal life
Stackhouse is the younger brother of former CBA player and Sacramento Kings and Boston Celtics forward Tony Dawson, and he is the uncle of former Wake Forest University guard Craig Dawson.

Stackhouse has performed the U.S. national anthem before Mavericks home games and during the Bucks' 2010 and the Nets' 2013 playoff appearances. Although formerly a Pescatarian, Stackhouse is now back to eating meat.

In 2017, Stackhouse completed the Harvard Business School executive education program on the Business of Entertainment, Media, and Sports.

Achievements

 * Sports Illustrated Player of the Year (1995)
 * Had the highest point total, 2,380, for the 2000–01 NBA season, but was second in scoring average, 29.8.
 * Became the 106th NBA player to score 15,000 career points, only one game after teammate Dirk Nowitzki surpassed 15,000 points.
 * 2017 NBA D-League Coach of the Year

title: Jerry Stackhouse


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User talk:Bepp

Hello and welcome to Wikipedia!

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Riograndenser Hunsrückisch
Bogo-move fixed. Have deleted the redirect. - Ta bu shi da yu 13:03, 19 Oct 2004 (UTC)

Pennsylvaanisch Deitsch
'S gebbt aa en Uffguckbichli (Wikipedia) in eier Schprooch —Preceding unsigned comment added by 91.32.218.103 (talk) 09:55, 6 September 2008 (UTC)

 * Danke schön! Thanks! Bepp (talk) 20:58, 6 September 2008 (UTC)

Carlos Tufvesson
Hi, Bepp!

I'm unable to find any reliable sources that say that Carlos Tufvesson is, in fact, gay - at least not in English. From what I can tell, the article here is a translation you did of the pt article, right? Would you mind taking a look at these: and let me know if any of them say he's gay? Thanks so much! -- SatyrTN (talk / contribs) 04:08, 21 August 2008 (UTC)

 * There is absolutely no question as to Carlos Tufvesson's homosexuality. He routinely appears in Brazilian national TV shows advocating for equal access to civil marriage (gay marriage). And he has published many articles about this subject which also have been read nationwide. Yes he is one of the most important fashion designers in Brazil today, that being the man's profession... however, not to mention his long standing LGBT right's advocacy and permanent relationship with "husbund" and achitect Piva would be, well, let's just say, a tad bit prescriptive... (and NO, I do not work for or with him [smiles]) --Bepp (talk) 05:39, 21 August 2008 (UTC)

 * Okay, but can you provide a source that says so? I mean, I can't just add "everyone knows" to the article :) -- SatyrTN (talk / contribs) 12:53, 21 August 2008 (UTC)

 * (Smiling...) You really are making me do your work for you, no?!

Here is this article by Sérgio Ripardo that was published in 2006 by the FOLHA ONLINE, a major Brazilian newspaper, reading "Carlos Tufvesson dresses celebrities and defends civil unions" ("Carlos Tufvesson veste celebridades e defende união civil")http://www1.folha.uol.com.br/folha/ilustrada/ult90u65848.shtml

 * Umm - *you* wrote the article without a citation, not me :) I'm just glad there's someone around who can speak Portuguese, since English refs are few and far between!  Thanks very much - I'll add that to his article.  Happy editing! -- SatyrTN (talk / contribs) 14:22, 21 August 2008 (UTC)

 * You are right, sorry. Should you ever need any help with Portuguese, do let me know. By the way, I also wanted to show this article here: http://www.farofadigital.com.br/circuito_tuf.htm an article that is titled "Carlos Tufvesson joins [the association] ABRAGAY" (published in March, 2008). The first paragraph reads thusly: "Carlos tufvesson, one of the most impactfull voices in defense of the rights of homosexuals in Brazil is the newest member of ABRAGAY - an association bringing together homossexual men which proposes to fight against homophobia and defend human rights and citizenship for all, in particular for gays." Bepp (talk) 14:34, 21 August 2008 (UTC)

 * Should that be added to the article, too? What's farofadigital?  Looks like a radio station?  Is it a reliable source?  Thanks again! -- SatyrTN (talk / contribs) 16:10, 21 August 2008 (UTC)

 * I think we are fine with the one ref you added... Farofa Digital is an alternative culture portal out of the Northeast of Brazil; and I don't know how stable their pages are. Take care. Bepp (talk) 01:07, 22 August 2008 (UTC)

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 * 1) Leci Brandão -

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 * Thank you for reminding me, I just fixed it! Paul Beppler (talk) 01:55, 13 August 2013 (UTC)

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 * de Ciências Econômicas, Universidade Federal do Rio Grande do Sul. Porto Alegre, Aug. 1, 2011 .] ; and increasingly on national (internal) tourism which in turn revolves around its

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German Brazilian
Hi Bepp. Your edit here refers. It is too much information of the infobox. This whole section:
 * "there are smaller populations with their own variants of Hunsrückisch in Santa Catarina and Espirito Santo; along with other smaller groups of Brazilian German dialects; Brazil has the world's largest population of speakers of Germanic Pomeranian (Pommersch) or East Low German (not to be confused with the slavic Kashubian language of the Lechitic languages spoken in northern Europe)."

is best integrated elsewhere on the page. Regards, Rui ''Gabriel'' Correia (talk) 22:14, 3 May 2014 (UTC)

 * Thanks, Rui. Done! Paul Beppler (talk) 01:39, 5 May 2014 (UTC)

Riograndenser Hunsrückisch
Hi Bepp. Would you be able to find a piece of voice recording of Riograndenser Hunsrückisch. See example here of the standard Hunsrückisch. As a separate issue, I see the German article includes the alternative name "Katharinensisch". Can you comment on that? Then finally, to satisfy my personal curiosity, how do you pronounce the "h" in the word "Hunsrückisch" in Brazil? Like the "r" in "Luciano Ruck (Runsrückisch) or like the normal "h" in German, such as Hanover, Hamburg (Hunsrückisch) or silent as in Portuguese "(h)omem", "(h)orta, (Unsrückisch)? Thanks, regards and bom trabalho. Rui ''Gabriel'' Correia (talk) 19:41, 5 May 2014 (UTC)

Here is a language sample (voice) as per your request: https://www.youtube.com/watch?v=h9ZVddfwdCE (I have others, for example this is me teaching the number in the language https://www.youtube.com/watch?v=azFdlH30Tk0 ; furthermore, here is a tv-report in Portuguese, in which however the Riograndenser Hunsrückisch language is featured and can be heard clearly). I'm unsure what Katharinensisch is suppose to mean exactly but it doesn't mean Riograndenser Hunsrückisch - these two terms really should not be used as synonyms. I will try and edit that sometime soon. Here is a very solid scholarly reference which may come in handy, as far as identifying Riograndenser Hunsrückisch as a German variety on its own: "Hunsrückisch in Rio Grande do Sul - Ein Beitrag zur Beschreibung einer deutschbrasilianischen Dialektvarietät im Kontakt mit dem Portugiesischen" by Cléo Vilson Altenhofen, 1995. Veröffentlichung des Instituts für geschichtliche und Landeskunde an der Universität Mainz E. V. / Mainzer Studien zur Sprach- und Volksforschung 21. Franz Steiner Verlag, Stuttgart - Germany. Detail: Riograndenser Hunsrückisch in the last almost two centuries of its history has spread first west within Rio Grande do Sul state ("Altkolonie", the Old Colony region, then "Neikolonie", in High German "Neukolonie", the New Colony region); but then it also spread organically into Argentina, the western portions of the state of Santa Catarina (as in, please see here: https://en.wikipedia.org/wiki/S%C3%A3o_Jo%C3%A3o_do_Oeste, and Paraná. Meanwhile, during this same period of time there were other much, much smaller settlements of Hunsrückisch speakers, speakers of the same European original Hunsrückisch dialect (usually called Hunsrücker Platt there, where it historically is divided into two main varieties, divided by the St. Goar "dat/das" line: https://en.wikipedia.org/wiki/Sankt_Goar_line ) in other parts of Brazil, like in the East coast of Santa Catarina (please see: https://pt.wikipedia.org/wiki/Ant%C3%B4nio_Carlos_%28Santa_Catarina%29 ) and upwards from the state of Rio de Janeiro, in the state of Espirito Santo. Finally as far as your question regarding the pronunciation of the "h" in Brazil. Generally in southern Brazil, even non German speakers are familiar with German language sounds, if nothing because the vast number and diversity of German last names around. In that context, "Hunsrückisch" is pronounced more or less linke in German: húns-ri-quíx, the "h" being pronounced - unlike in Portuguese when people pronounce "Humberto" in practice as "um-ber-to. However, that said, when people pronounce a Brazilianized version of the name "Hunsrückisch", namely "Hunsriqueano", than the "h" will be silent, like in "Henrique", which sounds like "en-ri-que"; along those lines, "Hunsrück" (the region in southwest Germany which gave name to the dialect) when Brazilianized will be "Hunsrique", pronounced "uns-ri-que". I hope this helps. Feel free to ask me further if you need to. Paul Beppler (talk) 10:59, 16 June 2014 (UTC)
 * Hello Rui, sorry about the wait.

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text: Talk:Mount Bailey (Oregon)

title: Talk:Mount Bailey (Oregon)


text: User talk:68.221.123.80

title: User talk:68.221.123.80


text:
Wasp (disambiguation)

A wasp is a type of flying insect.

Wasp may also refer to:

Fictional entities

 * Several Marvel characters, including:
 * Wasp (character), introduced in 1963
 * Hope van Dyne, the Marvel Cinematic Universe character
 * Nadia van Dyne, who was introduced in 2016
 * Lisbeth Salander, a character in the Millennium series nicknamed Wasp
 * Waspinator or Wasp, several Transformers characters
 * World Aquanaut Security Patrol, an organization in the television series Stingray

Films

 * Wasp (2003 film), a film by Andrea Arnold
 * Wasp (2015 film), a British-French film
 * The Wasp (1915 film), a short by B. Reeves Eason
 * The Wasp (1918 film), a lost silent film comedy drama

Literature

 * Wasp (novel), a novel by Eric Frank Russell
 * The Wasps, a comic play by Aristophanes

Music

 * W.A.S.P. (band), an American rock/metal band
 * W.A.S.P. (album), their 1984 self-titled album
 * "Wasp", a song by Black Sabbath from the album Black Sabbath
 * Wasp (album), a 1980 album by Shaun Cassidy
 * "W.A.S.P.", a song from the Dayglo Abortions album, Two Dogs Fucking
 * The Wasps (Vaughan Williams), a 1909 suite for orchestra composed by Ralph Vaughan Williams
 * "The WASP (Texas Radio and the Big Beat)", a song by The Doors from the album L.A. Woman
 * EDP Wasp, a monophonic synthesizer by Electronic Dream Plant
 * WASP, a library music ensemble founded by Steve Gray

Periodicals

 * The Wasp (magazine), a 19th-century San Francisco magazine
 * The Wasp (newspaper), a 19th-century newspaper published in Nauvoo, Illinois

Radio

 * WASP-LP, a low-power radio station (104.5 FM) licensed to serve Huntington, West Virginia, United States
 * WASP (AM), a defunct radio station (1130 AM) formerly licensed to serve Brownsville, Pennsylvania, United States

Information technology

 * Web Standards Project
 * WAsP, Wind Atlas Analysis and Application Program
 * Wireless application service provider

Weapons

 * AGM-124 Wasp, an American experimental air-to-ground anti-tank missile
 * Wasp 58, a recoilless rocket launcher

Other uses in science and technology

 * Wiskott–Aldrich syndrome protein
 * Wide Angle Search for Planets (WASP), a British group searching for extra-solar planets

Sport

 * Wasps FC, an amateur English rugby union club
 * Wasps Ladies, an English women's rugby union club
 * Wasps RFC, a professional English rugby union club based in the Coventry area (known from 1999 to 2014 as London Wasps)
 * York Wasps, former English rugby league club
 * Chalfont Wasps F.C., English football club
 * Durham Wasps, former British ice–hockey team
 * Newport Wasps, British speedway team
 * Emory and Henry Wasps, NCAA Division III intercollegiate sport team
 * WASP (cricket calculation tool), an algorithm used in limited overs cricket matches to predict the outcome of games

Aviation

 * ABC Wasp, a British World War I aero engine by ABC Motors
 * AeroVironment Wasp III, unmanned aerial vehicle (UAV) developed for United States Air Force special forces
 * Airspeed Queen Wasp, British unmanned target-aircraft
 * Curtiss 18 Wasp, a little-used American triplane fighter of World War I
 * Pratt & Whitney Wasp series, a series of piston engines common in the 1930s and 1940s
 * Westland Wasp, a British-built light shipborne helicopter
 * Women Airforce Service Pilots, World War II American female aviators
 * Williams Aerial Systems Platform, from Williams International
 * Wasp Flight Systems, a British powered hang glider manufacturer
 * Wasp SP Mk2, a British powered hang glider design

Land

 * Hudson Wasp, 1950s, American automobile
 * Martin Wasp, 1910s and 20s, American automobile
 * Wasp Motorcycles, British motorcycle and sidecar manufacturer
 * Wasp, a variant of the Universal Carrier armored vehicle that was armed with a flamethrower

Ships

 * HMS Wasp, several ships of the Royal Navy
 * USS Wasp, several ships of the United States Navy
 * Wasp-class amphibious assault ship, ships of the United States Navy
 * Wasp (1809 ship), a ship that made a whaling voyage in the 1820s
 * Wasp (1776 ship), British slaving vessel

Other uses

 * White Anglo-Saxon Protestants (WASPs), an upper class elite in the United States
 * Wasp, Pleasants County, West Virginia, a community in the United States
 * Workers and Socialist Party, a South African political party

title: Wasp (disambiguation)


text: User talk:65.94.36.94

title: User talk:65.94.36.94


text:
Straight ally

A straight ally, heterosexual ally, or cis ally (often simply called an ally) is a heterosexual and cisgender person who supports equal civil rights, gender equality, and LGBTQ+ social movements. Individuals may meet this designation through their actions without actively identifying as an ally.

Gay-straight alliance
Most LGBTQ+ organizations have straight or cisgender members involved, while others actively encourage straight and cisgender participation. A good example of the change straight allies can help achieve is the gay-straight alliance (GSA) which has been becoming more popular in schools all around the world. A gay–straight alliance (also known as a gender-sexuality alliance) is a student-run club that brings together LGBTQ+ and straight students to create a platform for activism to fight homophobia and transphobia. The goal of most gay-straight alliances is to make their school community safe, facilitate activism on campus, and create a welcoming environment for LGBTQ+ students.

History
The first gay-straight alliance was formed in November 1988 at Concord Academy in Concord, Massachusetts, when Kevin Jennings, a history teacher at the school who had just come out as gay, was approached by Meredith Sterling, a straight student at the school who was upset by the treatment of gay students and others. Jennings recruited some other teachers at the school, thus forming the first gay-straight alliance. Jennings credits students for both the establishment of the club, as well as for setting the agenda of struggling against homophobia, and for changes to CA's nondiscrimination policy. Jennings would go on to co-found the Gay, Lesbian & Straight Education Network (GLSEN) in Boston in 1990.

The GSA Network is an LGBT rights organization founded in 1998 by Carolyn Laub to empower youth activists to start GSA clubs in their respective schools to motivate and inspire fellow students to fight against homophobia and transphobia. Laub initially started working with this movement in 40 GSA clubs in the San Francisco Bay area during 1988–99 and then gradually expanded to other cities and states; by 2005, it began operating programs nationally.

Impact on students
Most of what has been written, academically, about LGBTQ+ youth has focused on non-normative development or risk outcomes. This focus has overshadowed and often-times disregarded the ways in which young LGBTQ+ people and their allies are actively engaged in creating positive change for themselves and their peers; for many young people, this active engagement is achieved through involvement and leadership in high school Gay-Straight Alliances. Adolescence is an important developmental period for individual engagement in community and social concerns; empowerment suggests that young people discover their capacity to become agents of change in issues and causes they care about. Sexuality activism has emerged as an important arena for youth activism, and offers a unique context in which to study youth empowerment.

Those most impacted by gay-straight alliances are LGBTQ+ students; however, gay and straight members alike are benefitted from the sense of community and empowerment that gay-straight alliances provide. In 1994, Anderson categorized gay youth as an "at risk" population. He believes that school-based support groups can help to counter the negative statistics in the lives of gay youth (homelessness, high school dropouts, drug and alcohol abuse, victims of physical violence, and suicide). Participants in this study experienced some of the hopelessness and despair common to gay youth, but they also became empowered young people through their association with the GSA. Being a part of the GSA helped them to move beyond the depressing statistics and gain stronger identities. Their new identities were expressed in their educational lives as well as their personal and social lives. They became empowered by working toward a collective goal: challenging the system in which they previously believed they could not have an impact".

Impact on society
According to many studies, involvement in high school gay-straight alliances leads to more civically active young adults. "The current study demonstrated significant associations between GSA involvement level and forms of civic engagement, including efforts to counter discrimination and raise others' awareness of LGBTQ issues. Our findings highlight the promising role that GSAs could play in building civic engagement capacity among their members. Ultimately, as active and engaged citizens, LGBTQ youth and their allies could play a major role in challenging oppressive systems and promoting social justice for LGBTQ individuals in society".

PFLAG
Parents, Families and Friends of Lesbians and Gays (PFLAG), an international nonprofit organization, works to support LGBTQ+ people and their loved ones. Stemming from parents' desire to be involved in their gay and lesbian children's struggle for equality, PFLAG has been a resource for countless families since 1973. Founded by Jeanne Manford, who is considered the mother of the ally movement, PFLAG unites LGBTQ+ people with parents, families, and allies to gain full civil and legal equality for people in the LGBTQ+ community. In 2007, the organization launched a new project, Straight for Equality to help more allies become engaged with the movement in the workplace, healthcare, and now in faith communities.

"Despite the exclusion of "bisexual" and "transgender" from the organization's name, PFLAG works for the rights of these sexual minorities as well, providing education on gender identification along with sexual orientation. PFLAG's policy statements on such issues as legislation, equality in the workplace, hate crimes, same-gender marriage, religious affiliation, and comprehensive sex education all reflect its deep commitment to ensuring the rights of all lgbtq people".

GLAAD
GLAAD was established by a group of reporters after an article talked down on HIV/AIDS, being officially founded in 1985. GLAAD put pressure on media organizations to end what it saw as homophobic reporting. Over the years, GLAAD has expanded its resources to provide change for LGBTQ+ members and allies. "As a dynamic media force, GLAAD ensures fair, accurate, and inclusive representation that rewrites the script for LGBTQ acceptance. GLAAD tackles tough issues to shape the narrative and provoke dialogue that leads to cultural change. GLAAD protects all that has been accomplished and envisions a world with 100% LGBTQ acceptance. GLAAD works through entertainment, news, and digital media to share stories from the LGBTQ community that accelerate acceptance".

LGBTQQIAAP
A second A was added to the acronym to stand for Ally though this use is sometimes controversial. Sometimes the acronym is spelled LGBTQQIAAP.

The Stonewall Uprising
The Stonewall Uprising, or the Stonewall riots of 1969 (in New York City), is known to be the starting point of the Gay Liberation Front. Protests, advocacy organizations, HIV/AIDS relief groups, etc. collectively have characterized the movement from the start. The Stonewall Uprising, a series of events between police and LGBTQ+ protesters that stretched over six days, became well known due to the media coverage and the subsequent annual Pride traditions. "Stonewall veterans have explicitly stated that they prefer the term Stonewall uprising or rebellion. The reference to these events as riots was initially used by police to justify their use of force. Early publications show that the LGBTQIA+ community largely did not use the term riot until years after the fact".

"The criminalization of homosexuality led many gay establishments to operate sans liquor license, providing an open door for raids and police brutality. Like many gay establishments at the time, the Stonewall Inn was owned by the mafia, and as long as they continued to make a profit, they cared very little what happened to their clientele. Because the owners were still making a profit, they simply adjusted to the raids, and were often tipped off about them ahead of time". The Stonewall Inn remains both as a gay bar and a statement against the violence that it has survived, and has even become a National Historic Landmark. The Stonewall Inn continues to make strides within the LGBTQ+ community with its nonprofit charity. "The Stonewall Inn Gives Back Initiative is ... dedicated to providing educational, strategic and financial assistance to grassroots organizations committed to advocacy for and crucial support to LGBTQ communities and individuals who suffer the indignities and fear arising from social intolerance here in the United States and around the world".

Liberationists
In the 1970s, a divide emerged within the community over opposing beliefs on what liberation truly meant: the "liberationists" and the "homophiles".

Liberationists presented themselves as being considerably radical; their intent lay in going beyond acceptance and transforming traditional constructs of society (homophobia, sexism, militarism, etc.).

In contrast, homophiles aimed only for what was deemed realistic; instead of dismantling an oppressive system, they simply asked for tolerance. Their group maintained an apologist standpoint, where their end goal was living in peaceful coexistence with the oppressor. This assimilationist perspective held particular appeal with members of the community who were able to "blend in" with heterosexual society.

The existence of such a group left room for a movement that was more sensitive to the struggles, needs, and ultimate goals of more marginalized subcultures. The liberationists, who seemed to be drawing in members of such groups, filled this need.

The emergence of the Liberationists allowed for a wider spectrum of sexual-social behavior and identity to be represented without compromise, and with less risk of infighting. Members of the latter group would not feel forced to conform to the more socially conservatives mores of the former group, while they were able to separate themselves from elements of the gay subculture that they had found crass, excessive, decadent, or extreme.

Partnership
Partnership with straight allies has raised challenges as well as benefits for the LGBTQ+ community: there is a perception that such allies evince different levels of 'respect' for the community on whose behalf they advocate, sometimes being patronizing, unaware of their own privilege and power, and crowding out the members. Given that distinguishing between speaking on behalf of a group and speaking for a group is not simple, that line is often crossed without even noticing. This grey area can be referred to as 'positive respect'; a sort of force found in an ally's motives that inhibits the 'servile' (as a result of their internalized oppression) group's freedom to act.

Scrutiny
Another challenge is that straight allies can be easily discouraged, in the face of scrutiny of their motives and approaches. Newer straight allies can become overwhelmed or confused by the complication of their position in the movement. Since newer allies derive their identity from their personal relations with queer-identifying people, this limits their allyship. Some allies can sometimes respond very defensively to criticism from members of the queer community about their understanding of queer issues, which in turn feeds a concern that they are motivated by the praise they anticipate as their moral reward. Additionally, there is a coming out process for being a straight ally that is not explicitly present in other social movements (concerns about being seen as LGBTQ+); this can hinder the level of advocacy an ally does. In other words, allyship requires support that is accompanied by a distinct protocol many find challenging to achieve. Straight allies protesting at Seattle March for Marriage Equality Allies may receive criticism for a variety of reasons. For example, some believe that allies are unable to step outside their own heteronormative world to advocate. Allies are also criticized for using LGBT advocacy as a means to gain popularity and status.

Role in policy change
Studies show that elite allies have a positive effect on the policy goals of a social movement, whatever those goals may be. While allies' main role is to provide wider support for the goals of a social movement, their secondary role of influencing policy is also valuable. The allies' role is to inform policymakers of the struggles endured by a community. Allyship of this kind is often effective, though self-interested; for example, high-ranking, conservative government officials Barry Goldwater and William Weld (former Republican governor of Massachusetts), were motivated by their relations with queer family and friends to provide uncharacteristic support for pro-gay policies.

title: Straight ally