text:
###### Waste less time on Facebook — follow Brilliant.
×
Back to all chapters

# Sum and Difference Trigonometric Formulas

These formulas explain how to add and subtract trigonometric functions (and their arguments). If you've got sum time, see what a difference these formulas will make for your trig toolkit.

# Tangent - Sum and Difference Formulas

Which of the following is equal to

$\frac{ \tan 43 ^ \circ - \tan 14 ^ \circ } { 1 + \tan 43 ^ \circ \times \tan 14 ^ \circ } ?$

Which of the following is equal to

$\frac{ \tan 19 ^ \circ + \tan 26 ^ \circ } { 1 - \tan 19 ^ \circ \times \tan 26 ^ \circ } ?$

If $$\tan x=\frac{1}{14}$$, the value of $$\tan \left(x+\frac{\pi}{4}\right)$$ can be expressed as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b$$?

If $$\tan N ^ \circ = \tan 12 ^ \circ + \sec 12 ^ \circ$$, where $$0 \leq N \leq 90$$, what is the value of $$N$$?

Which of the following expressions is equal to $$\tan 2 \theta$$?

×

Problem Loading...

Note Loading...

Set Loading...


text:
# Ring structure of non-modular group cohomology

I know if $k$ is a field such that $char(k)$ divides $|G|$ (a finite group), then finding the ring structure on $H^\ast(G,k)$ can be very, very hard.

But what about when $k=\mathbb{Z}$? Is the computation easier? I know of several "modern" techniques to compute the cohomology groups, but have never encountered a detailed computation of the ring structure.

Thanks!

-

If $G$ is a finite group and $k$ is a field such that $\operatorname{char}k\nmid|G|$, then $H^p(G,k)=0$ for all $p>0$, so that the only non-zero cohomology group us $H^0(G,k)\cong k$, and in fact this isomorphism is an isomorphism of $k$-algebras.
ah, sorry, I messed that up. I meant to be talking about over $\mathbb{Z}$. –  user641 Apr 24 '12 at 5:17

format: plain


text:
# Impulse response from difference equation without partial fractions

I have a system with input $x[n]$ and output $y[n]$, described by the difference equation

$$y[n] - 0.3y[n-1] + 0.1 y[n-2] = 3x[n] - x[n -1]$$

and I am trying to find the frequency and impulse responses. The frequency response I found was

$$H(e^{jw}) = \frac{3-e^{-jw}}{1 - 0.3e^{-jw} + 0.1e^{-j2w}}$$

but I am having trouble converting this to $h[n]$ because the denominator can't be factored using partial fractions.

Also, I tried using the recursion method to find $h[n]$ but I am unsure whether the system is causal or not, so I don't know where to start for the recursion. Could anyone give me some tips on what to do next?

First of all you can apply partial fraction expansion method to get the inverse DTFT to find the impulse resposne h[n] for this system. You have to assume casual ot anti-causal system.

$$H(e^{jw}) = \frac{3-e^{-jw}}{1 - 0.3e^{-jw} + 0.1e^{-j2w}} = \frac{3-e^{-jw}}{ (1 - \alpha e^{-jw})(1 - \alpha^* e^{-jw})}$$

$$H(e^{jw})= \frac{A}{1 - \alpha e^{-jw}} + \frac{B}{1 - \alpha^* e^{-jw}}$$

Where $\alpha = 0.1500 + j0.2784$ for this particular case. Hence assuming causality and using inverse DTFT look-up tables, deduce as:

$$\boxed{ h[n] = A \alpha^n u[n] + B (\alpha^*)^n u[n] }$$

Time domain recursion is also possible, requiring only two auxiliary values. For example assuming causality one can use recursion from the LCCDE to obtain $y[n]$ for all $n \geq 0$ from the given LCCDE $$y[n] - 0.3y[n-1] + 0.1 y[n-2] = 3x[n] - x[n -1]$$

by first rearranging it for $y[n]$ as $$y[n] = 0.3y[n-1] - 0.1 y[n-2] + 3x[n] - x[n -1]$$

Then the recursion follows for $n \geq 0$ assuming that $x[n]=\delta[n]$ \begin{align} y[0] &= 0.3y[-1] - 0.1 y[-2] + 3x[0] - x[-1] \\ y[1] &= 0.3y[0] - 0.1 y[-1] + 3x[1] - x[0] \\ y[2] &= 0.3y[1] - 0.1 y[0] + 3x[2] - x[1] \\ y[n] &= 0.3y[n-1] - 0.1 y[n-2] \\ \end{align}

Note that in computing $y[0]$ you need $y[-1]$ and $y[-2]$ as auxiliary (initial) conditions. Also note that $x[-1]=0$ and $x[0]=1$ and $x[n]=0$ for all remaining $n>0$.

A similar analysis fallows for the anti-causal case.


text:
Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# Fraunhofer diffraction

Joseph von Fraunhofer (1787-1826)

Fraunhofer diffraction occurs when the distances from the object to the source and the object to the image are so great that the incoming and outgoing waves are effectively planar. It is named after Joseph von Fraunhofer, who studied diffraction, and even built the first diffraction grating. The condition is given by

$${1 \over 2}\left( {{1 \over d} + {1 \over L}} \right){w^2} \ll \lambda$$

where d is the distance from the source to mask, L is the distance from mask to image, w is the width of the slit and λ is the wavelength of the radiation. If the condition is not met, then curvature of the waves is involved and the resulting effect is known as Fresnel diffraction.


text:
Surface (mathematics)

Last updated

In mathematics, a surface is a generalization of a plane which doesn't need to be flat – that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study.

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that it may be curved.

Contents

The mathematical concept of surface is an idealization of what is meant by surface in common language, science, and computer graphics.

Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.

Computer graphics are pictures and films created using computers. Usually, the term refers to computer-generated image data created with the help of specialized graphical hardware and software. It is a vast and recently developed area of computer science. The phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes erroneously referred to as computer-generated imagery (CGI).

Definitions

Often, a surface is defined by equations that are satisfied by the coordinates of its points. This is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, which is called an implicit surface. [1] If the defining three-variate function is a polynomial, the surface is an algebraic surface. For example, the unit sphere is an algebraic surface, as it may be defined by the implicit equation

In mathematics, an equation is a statement that asserts the equality of two expressions. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English any equality is an equation.

In mathematics, the graph of a function f is, formally, the set of all ordered pairs (x, f ), such that x is in the domain of the function f. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in the Euclidean plane and form thus a subset of this plane, which is a curve in the case of a continuous function. This graphical representation of the function is also called the graph of the function.

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.

${\displaystyle x^{2}+y^{2}+z^{2}-1=0.}$

A surface may also be defined as the image, in some space of dimension at least 3, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters. For example, the unit sphere may be parametrized by the Euler angles, also called longitude u and latitude v by

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.

A parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

{\displaystyle {\begin{aligned}x&=\cos(u)\cos(v)\\y&=\sin(u)\cos(v)\\z&=\sin(v)\,.\end{aligned}}}

Parametric equations of surfaces are often irregular at some points. For example, all but two points of the unit sphere, are the image, by the above parametrization, of exactly one pair of Euler angles (modulo 2π). For the remaining two points (the north and south poles), one has cos v = 0, and the longitude u may take any values. Also, there are surfaces for which there cannot exist a single parametrization that covers the whole surface. Therefore, one often considers surfaces which are parametrized by several parametric equations, whose images cover the surface. This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane (see Surface (topology) and Surface (differential geometry)). This allows defining surfaces in spaces of dimension higher than three, and even abstract surfaces, which are not contained in any other space. On the other hand, this excludes surfaces that have singularities, such as the vertex of a conical surface or points where a surface crosses itself.

In computing, the modulo operation finds the remainder after division of one number by another.

The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is defined as the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface.

The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on the surface of Earth and lies on the opposite side of Earth from the North Pole.

In classical geometry, a surface is generally defined as a locus of a point or a line. For example, a sphere is the locus of a point which is at a given distance of a fixed point, called the center; a conical surface is the locus of a line passing through a fixed point and crossing a curve; a surface of revolution is the locus of a curve rotating around a line. A ruled surface is the locus of a moving line satisfying some constraints; in modern terminology, a ruled surface is a surface, which is a union of lines.

In geometry, a locus is a set of all points, whose location satisfies or is determined by one or more specified conditions.

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.

In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. Each of those lines is called a generatrix of the surface.

Terminology

In this article, several kinds of surfaces are considered and compared. An unambiguous terminology is thus necessary to distinguish them. Therefore, we call topological surfaces the surfaces that are manifolds of dimension two (the surfaces considered in Surface (topology)). We call differential surfaces the surfaces that are differentiable manifolds (the surfaces considered in Surface (differential geometry)). Every differential surface is a topological surface, but the converse is false.

For simplicity, unless otherwise stated, "surface" will mean a surface in the Euclidean space of dimension 3 or in R3. A surface that is not supposed to be included in another space is called an abstract surface.

Examples

• The graph of a continuous function of two variables, defined over a connected open subset of R2 is a topological surface. If the function is differentiable, the graph is a differential surface.
• A plane is both an algebraic surface and a differentiable surface. It is also a ruled surface and a surface of revolution.
• A circular cylinder (that is, the locus of a line crossing a circle and parallel to a given direction) is an algebraic surface and a differential surface.
• A circular cone (locus of a line crossing a circle, and passing through a fixed point, the apex, which is outside the plane of the circle) is an algebraic surface which is not a differential surface. If one removes the apex, the remainder of the cone is the union of two differential surfaces.
• The surface of a polyhedron is a topological surface, which is neither a differential surface nor an algebraic surface.
• A hyperbolic paraboloid (the graph of the function z = xy) is a differential surface and an algebraic surface. It is also a ruled surface, and, for this reason, is often used in architecture.
• A two-sheet hyperboloid is an algebraic surface and the union of two non-intersecting differential surfaces.

Parametric surface

A parametric surface is the image of an open subset of the Euclidean plane (typically ${\displaystyle \mathbb {R} ^{2}}$) by a continuous function, in a topological space, generally a Euclidean space of dimension at least three. Usually the function is supposed to be continuously differentiable, and this will be always the case in this article.

Specifically, a parametric surface in ${\displaystyle \mathbb {R} ^{3}}$ is given by three functions of two variables u and v, called parameters

{\displaystyle {\begin{aligned}x&=f_{1}(u,v)\\y&=f_{2}(u,v)\\z&=f_{3}(u,v)\,.\end{aligned}}}

As the image of such a function may be a curve (for example if the three functions are constant with respect to v), a further condition is required, generally that, for almost all values of the parameters, the Jacobian matrix

${\displaystyle {\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial u}}&{\dfrac {\partial f_{1}}{\partial v}}\\{\dfrac {\partial f_{2}}{\partial u}}&{\dfrac {\partial f_{2}}{\partial v}}\\{\dfrac {\partial f_{3}}{\partial u}}&{\dfrac {\partial f_{3}}{\partial v}}\\\end{bmatrix}}}$

has rank two. Here "almost all" means that the values of the parameters where the rank is two contain a dense open subset of the range of the parametrization. For surfaces in a space of higher dimension, the condition is the same, except for the number of columns of the Jacobian matrix.

Tangent plane and normal vector

A point p where the above Jacobian matrix has rank two is called regular, or, more properly, the parametrization is called regular at p.

The tangent plane at a regular point p is the unique plane passing through p and having a direction parallel to the two row vectors of the Jacobian matrix. The tangent plane is an affine concept, because its definition is independent of the choice of a metric. In other words, any affine transformation maps the tangent plane to the surface at a point to the tangent plane to the image of the surface at the image of the point.

The normal line , or simply normal at a point of a surface is the unique line passing through the point and perpendicular to the tangent plane. A normal vector is a vector which is parallel to the normal.

For other differential invariants of surfaces, in the neighborhood of a point, see Differential geometry of surfaces.

Irregular point and singular point

A point of a parametric surface which is not regular is irregular. There are several kinds of irregular points.

It may occur that an irregular point becomes regular, if one changes the parametrization. This is the case of the poles in the parametrization of the unit sphere by Euler angles: it suffices to permute the role of the different coordinate axes for changing the poles.

On the other hand, consider the circular cone of parametric equation

{\displaystyle {\begin{aligned}x&=t\cos(u)\\y&=t\sin(u)\\z&=t\,.\end{aligned}}}

The apex of the cone is the origin (0, 0, 0), and is obtained for t = 0. It is an irregular point that remains irregular, whichever parametrization is chosen (otherwise, there would exist a unique tangent plane). Such an irregular point, where the tangent plane is undefined, is said singular.

There is another kind of singular points. There are the self-crossing points, that is the points where the surface crosses itself. In other words, these are the points which are obtained for (at least) two different values of the parameters.

Graph of a bivariate function

Let z = f(x, y) be a function of two real variables. This is a parametric surface, parametrized as

{\displaystyle {\begin{aligned}x&=t\\y&=u\\z&=f(t,u)\,.\end{aligned}}}

Every point of this surface is regular, as the two first columns of the Jacobian matrix form the identity matrix of rank two.

Rational surface

A rational surface is a surface that may be parametrized by rational functions of two variables. That is, if fi(t, u) are, for i = 0, 1, 2, 3, polynomials in two indeterminates, then the parametric surface, defined by

{\displaystyle {\begin{aligned}x&={\frac {f_{1}(t,u)}{f_{0}(t,u)}}\\y&={\frac {f_{2}(t,u)}{f_{0}(t,u)}}\\z&={\frac {f_{3}(t,u)}{f_{0}(t,u)}}\,,\end{aligned}}}

is a rational surface.

A rational surface is an algebraic surface, but most algebraic surfaces are not rational.

Implicit surface

An implicit surface in a Euclidean space (or, more generally, in an affine space) of dimension 3 is the set of the common zeros of a differentiable function of three variables

${\displaystyle f(x,y,z)=0.}$

Implicit means that the equation defines implicitly one of the variables as a function of the other variables. This is made more exact by the implicit function theorem: if f(x0, y0, z0) = 0, and the partial derivative in z of f is not zero at (x0, y0, z0), then there exists a differentiable function φ(x, y) such that

${\displaystyle f(x,y,\varphi (x,y))=0}$

in a neighbourhood of (x0, y0, z0). In other words, the implicit surface is the graph of a function near a point of the surface where the partial derivative in z is nonzero. An implicit surface has thus, locally, a parametric representation, except at the points of the surface where the three partial derivatives are zero.

Regular points and tangent plane

A point of the surface where at least one partial derivative of f is nonzero is called regular. At such a point ${\displaystyle (x_{0},y_{0},z_{0})}$, the tangent plane and the direction of the normal are well defined, and may be deduced, with the implicit function theorem from the definition given above, in § Tangent plane and normal vector. The direction of the normal is the gradient, that is the vector

${\displaystyle \left[{\frac {\partial f}{\partial x}}(x_{0},y_{0},z_{0}),{\frac {\partial f}{\partial y}}(x_{0},y_{0},z_{0}),{\frac {\partial f}{\partial z}}(x_{0},y_{0},z_{0})\right].}$

The tangent plane is defined by its implicit equation

${\displaystyle {\frac {\partial f}{\partial x}}(x_{0},y_{0},z_{0})(x-x_{0})+{\frac {\partial f}{\partial y}}(x_{0},y_{0},z_{0})(y-y_{0})+{\frac {\partial f}{\partial z}}(x_{0},y_{0},z_{0})(z-z_{0})=0.}$

Singular point

A singular point of an implicit surface (in ${\displaystyle \mathbb {R} ^{3}}$) is a point of the surface where the implicit equation holds and the three partial derivatives of its defining function are all zero. Therefore, the singular points are the solutions of a system of four equations in three indeterminates. As most such systems have no solution, many surfaces do not have any singular point. A surface with no singular point is called regular or non-singular.

The study of surfaces near their singular points and the classification of the singular points is singularity theory. A singular point is isolated if there is no other singular point in a neighborhood of it. Otherwise, the singular points may form a curve. This is in particular the case for self-crossing surfaces.

Algebraic surface

Originally, an algebraic surface was a surface which may be defined by an implicit equation

${\displaystyle f(x,y,z)=0,}$

where f is a polynomial in three indeterminates, with real coefficients.

The concept has been extended in several directions, by defining surfaces over arbitrary fields, and by considering surfaces in spaces of arbitrary dimension or in projective spaces. Abstract algebraic surfaces, which are not explicitly embedded in another space, are also considered.

Surfaces over arbitrary fields

Polynomials with coefficients in any field are accepted for defining an algebraic surface. However, the field of coefficients of a polynomial is not well defined, as, for example, a polynomial with rational coefficients may also be considered as a polynomial with real or complex coefficients. Therefore, the concept of point of the surface has been generalized in the following way: [2]

Given a polynomial f(x, y, z), let k be the smallest field containing the coefficients, and K be an algebraically closed extension of k, of infinite transcendence degree. [3] Then a point of the surface is an element of K3 which is a solution of the equation

${\displaystyle f(x,y,z)=0.}$

If the polynomial has real coefficients, the field K is the complex field, and a point of the surface that belongs to ${\displaystyle \mathbb {R} ^{3}}$ (a usual point) is called a real point. A point that belongs to k3 is called rational over k, or simply a rational point, if k is the field of rational numbers.

Projective surface

A projective surface in a projective space of dimension three is the set of points whose homogeneous coordinates are zeros of a single homogeneous polynomial in four variables. More generally, a projective surface is a subset of a projective space, which is a projective variety of dimension two.

Projective surfaces are strongly related to affine surfaces (that is, ordinary algebraic surfaces). One passes from a projective surface to the corresponding affine surface by setting to one some coordinate or indeterminate of the defining polynomials (usually the last one). Conversely, one passes from an affine surface to its associated projective surface (called projective completion) by homogenizing the defining polynomial (in case of surfaces in a space of dimension three), or by homogenizing all polynomials of the defining ideal (for surfaces in a space of higher dimension).

In higher dimensional spaces

One cannot define the concept of algebraic surface in a space of dimension higher than three without a general definition of an algebraic variety and of the dimension of an algebraic variety. In fact, an algebraic surface is an algebraic variety of dimension two.

More precisely, an algebraic surface in a space of dimension n is the set of the common zeros of at least n – 2 polynomials, but these polynomials must satisfy further conditions that may be not immediate to verify. Firstly, the polynomials must not define a variety or an algebraic set of higher dimension, which is typically the case if one of the polynomials is in the ideal generated by the others. Generally, n – 2 polynomials define an algebraic set of dimension two or higher. If the dimension is two, the algebraic set may have several irreducible components. If there is only one component the n – 2 polynomials define a surface, which is a complete intersection. If there are several components, then one needs further polynomials for selecting a specific component.

Most authors consider as an algebraic surface only algebraic varieties of dimension two, but some also consider as surfaces algebraic sets all of whose irreducible components have the dimension two.

In the case of surfaces in a space of dimension three, every surface is a complete intersection, and a surface is defined by a single polynomial, which is irreducible or not, depending on whether non-irreducible algebraic sets of dimension two are considered as surfaces or not.

Topological surface

In topology, a surface is generally defined as a manifold of dimension two. This means that a topological surface is a topological space such that every point has a neighborhood that is homeomorphic to an open subset of a Euclidean plane.

Every topological surface is homeomorphic to a polyhedral surface such that all facets are triangles. The combinatorial study of such arrangements of triangles (or, more generally, of higher-dimensional simplexes) is the starting object of algebraic topology. This allows the characterization of the properties of surfaces in terms of purely algebraic invariants, such as the genus and homology groups.

The homeomorphism classes of surfaces have been completely described (see Surface (topology)).

Notes

1. Here implicit does not refer to a property of the surface, which may be defined by other means, but it refers to how it is defined. Thus this term is an abbreviation of "surface defined by an implicit equation".
2. Weil, André (1946), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, 29, Providence, R.I.: American Mathematical Society, MR   0023093
3. The infinite degree of transcendence is a technical condition, which allows an accurate definition of the concept of generic point.

Related Research Articles

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passes through the point (c, f ) on the curve and has slope f'(c) where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object such as a surface deviates from being a flat plane, or a curve from being straight as in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space – in a way that relates to the radius of curvature of circles that touch the object – and intrinsic curvature, which is defined in terms of the lengths of curves within a Riemannian manifold.

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In mathematics, a affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve can be restricted to an affine algebraic plane curve by replacing by one some indeterminate of the defining homogeneous polynomial. As these two operations are each inverse to the other, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally, and sometimes globally.

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object.

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan and Hermann Weyl. The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space Rn by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.

In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by . Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth hypersurface in a Riemannian manifold and a smooth choice of the unit normal vector at each point.

In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves, and algebraic plane curves. Plane curves also include the Jordan curves and the graphs of continuous functions.

In the mathematical field of algebraic geometry, a singular point of an algebraic varietyV is a point P that is 'special', in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.

In mathematics an implicit surface is a surface in Euclidean space defined by an equation

Critical point is a wide term used in a lot of branches of mathematics, but is always connected to the derivative of a function or mapping.

In projective geometry, a dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is algebraic then so is its dual and the degree of the dual is known as the class of the original curve. The equation of the dual of C, given in line coordinates, is known as the tangential equation of C.

In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.


text:
## Links and Pfaffian equation. (Entrelacements et équations de Pfaff.)(French)Zbl 0573.58022

Astérisque 107/108, 87-161 (1983).
The text gives a proof for the existence of nonisomorphic contact structures on the 3-dimensional Euclidean space. Principal themes: contact topology in dimension 3, linking properties of curves and plane fields, geometry of intersections of a surface with the Reeb foliation. Digressions: theory of braids, and analytic curves in the complex plane.
For the entire collection see [Zbl 0516.00018].

### MSC:

 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 57R30 Foliations in differential topology; geometric theory 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)


text:
## Annals of Functional Analysis

### Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications

#### Abstract

In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderón–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.

#### Article information

Source
Ann. Funct. Anal., Volume 9, Number 3 (2018), 413-425.

Dates
Accepted: 5 September 2017
First available in Project Euclid: 6 February 2018

https://projecteuclid.org/euclid.afa/1517886227

Digital Object Identifier
doi:10.1215/20088752-2017-0056

Mathematical Reviews number (MathSciNet)
MR3835228

Zentralblatt MATH identifier
06946365

#### Citation

Fiorenza, Alberto; Kokilashvili, Vakhtang. Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications. Ann. Funct. Anal. 9 (2018), no. 3, 413--425. doi:10.1215/20088752-2017-0056. https://projecteuclid.org/euclid.afa/1517886227

#### References

• [1] A. Böttcher and Y. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, Progr. Math. 154, Birkhäuser, Basel, 1997.
• [2] A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA 74 (1977), no. 4, 1324–1327.
• [3] C. Capone, M. R. Formica, and R. Giova, Grand Lebesgue spaces with respect to measurable functions, Nonlinear Anal. 85 (2013), 125–131.
• [4] D’Onofrio, C. Sbordone, and R. Schiattarella, Grand Sobolev spaces and their applications in geometric function theory and PDEs, J. Fixed Point Theory Appl. 13 (2013), no. 2, 309–340.
• [5] E. M. Dyn’kin, Methods of the theory of singular integrals (the Hilbert transform and Calderón-Zygmund theory), Itogi Nauki. Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz. 15 (1987), 197–202.
• [6] A. Fiorenza, M. R. Formica, and J. M. Rakotoson, Pointwise estimates for $G\Gamma$-functions and applications, Differential Integral Equations 30 (2017), no. 11-12, 809–824.
• [7] A. Fiorenza and J. M. Rakotoson, Some estimates in $G\Gamma(p,m,w)$ spaces, J. Math. Anal. Appl. 340 (2008), no. 2, 793–805.
• [8] N. Fusco, P.-L. Lions, and C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc. 124 (1996), no. 2, 561–565.
• [9] E. Greco, T. Iwaniec, and C. Sbordone, Inverting the $p$-harmonic operator, Manuscripta Math. 92 (1997), no. 2, 249–258.
• [10] V. P. Havin, Boundary properties of integrals of Cauchy type and of conjugate harmonic functions in regions with rectifiable boundary, Math. Sb. (N.S.) 68 (1965), no. 4, 499–517.
• [11] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Ration. Mech. Anal. 119 (1992), no. 2, 129–143.
• [12] G. Khuskivadze, V. Kokilashvili, and V. Paatashvili, Boundary value problems for analytic and harmonic functions in domain with nonsmooth boundaries: Application to conformal mappings, Mem. Differential Equations Math. Phys. 14 (1998), 1–195.
• [13] V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko, Integral Operators in Non-Standard Function Spaces, II: Variable Exponent Hölder, Morrey-Campanato and grand spaces, Oper. Theory Adv. Appl. 249, Birkhäuser, Cham, 2016.
• [14] V. Kokilashvili and V. Paatashvili, Boundary Value Problems for Analytic and Harmonic Functions in Nonstandard Banach Function Spaces, Nova Science, New York, 2012.
• [15] I. I. Privalov, Randeigenschaften analytischer Funktionen, II, Hochschulbücher für Mathematik 25, VEB Dt. Verl. d. Wiss., Berlin, 1956.
• [16] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989.


text:
# Unbiased Estimator for $EX_1EY_1$

Suppose you have a set of i.i.d. variables $X_1,...X_n$ and $Y_1, ..Y_n$. The correlation between $X_i$ and $Y_i$ is $0.5$ for $i = 1,...n$ and $0$ for $X_i$ and $Y_j$ for any $i \neq j$. Provide an unbiased estimator for $\phi = \textbf{E}X_1\textbf{E}Y_1$.

I started testing whether using sample mean give us an unbiased estimator but it does not. So far, since the covariance formula does have an instance of $EX_iEY_i$ I found that encouraging but not sure how to procede from there.

Since $X_i$ and $Y_i$ are dependent, $\mathbb{E}[X_iY_i]\ne\mathbb{E}[X_i]\mathbb{E}[Y_i]$. However, since $X_i$ and $Y_j$ are independent when $j\ne i$, $$\mathbb{E}[X_iY_j]=\mathbb{E}[X_i]\mathbb{E}[Y_j]$$ Therefore $$\frac{1}{n}\sum_{i=1}^n X_i\frac{1}{n-1}\sum_{\stackrel{j=1}{j\ne i}}^n Y_j=\frac{1}{n(n-1)}\sum_{i=1}^n X_i\left\{\sum_{j=1}^n Y_j-Y_i\right\}=\frac{n}{n-1}\bar{X}\bar{Y}-\frac{\sum_{i=1}^n X_iY_i}{n(n-1)}$$ is an unbiased estimator of $\mathbb{E}[X_i]\mathbb{E}[Y_i]$. (Which in addition provides an estimate of the bias of $\bar{X}\bar{Y}$.)

There is a more general solution to this problem.

In a bivariate world, with random variables $(X,Y)$:

$$\text{an unbiased estimator of } \quad \mathbb{E}[X^a Y^b] \; \; \mathbb{E}[X^r Y^t] \quad \text{ is} \quad \frac{s_{a,b} s_{r,t}-s_{a+r,b+t}}{n(n-1)}$$

where $s_{c,d}$ denotes the bivariate power rum $s_{c,d}=\sum _{i=1}^n X_i^c Y_i^d$.

In the OP's case, $a =1$, $b = 0$, $r = 0$ and $t=1$, the solution is thus:

$$\frac{s_{1,0} s_{0,1} - s_{1,1}}{n(n-1)}$$

... which is equal to the solution given by Xi'an.

In the current version of mathStatica, there is a function called PolyRaw that automatically calculates unbiased estimators of products of raw moments, for any dimension.


text:
# Probability of winning a game between players A and B?

The following problem is from A First Course in Probability by Sheldon Ross, and it was assigned as homework by my professor. I was wondering if you guys could help me find a answer to the problem.

$A$ and $B$ flip coins. $A$ starts and continues flipping until a tails occurs, at which point $B$ starts flipping and continues until there is a tail. Then $A$ takes over, and so on. Let $P_1$ be the probability of the coin's landing on heads when $A$ flips and $P_2$ when $B$ flips. The winner of the game is the first one to get a total of two heads. What is the probability that $A$ wins? What is the probability that $A$ wins if the winner of the game is the first one to get a total of three heads?

• What work have you done so far on this? What issues are you running into? – DylanSp Feb 8 '16 at 17:55
• I tried using what Ross calls "conditioning" to solve it. Pretty much what you do is you use the identity $P(E) = P(E|F)P(F) + P(E|G)P(G) + \ldots + P(E|H)P(H)$ where $S = F \cup G \cup \ldots \cup H$. I used this identity with $F$ being on round 1, $G$ being on round 2, etc and $E$ being winning. It's just that I realized there is probably a much easier solution to the problem. – Birdman2246 Feb 8 '16 at 18:02

For $i,j=0,1,\;$ define:
$$a_{ij} = P(\text{A wins where A starts with i heads and B starts with j heads}).$$
So we seek $a_{00}$ and by conditioning we have the following relations:
\begin{align} a_{11} &= P_1 + (1-P_1)(1-P_2)a_{11} \\ & \qquad\text{since if $A$ gets $H$ he wins and if he gets $T$ and then so does $B$} \\ & \qquad\text{we are back where we started} \\ & \\ \therefore\quad a_{11} &= \dfrac{P_1}{P_1+P_2-P_1P_2}. \\ & \\ a_{10} &= P_1 + (1-P_1)\left[P_2(1-P_2)a_{11} + (1-P_2)a_{10}\right] \\ & \qquad\text{since if $A$ gets $H$ he wins and if he gets $T$ and then so does $B$} \\ & \qquad\text{we are back where we started and if he gets T and $B$ gets $TH$ then we have $a_{11}$} \\ & \\ & \text{Substituting for $a_{11}$ and re-arranging, we get:} \\ a_{10} &= \dfrac{P_1^2P_2^2-2P1^2P_2+P_1^2-P_1P_2^2+2P_1P_2}{(P_1+P_2-P_1P_2)^2}. \\ & \\ a_{01} &= P_1 a_{11} + (1-P_1)(1-P_2)a_{01} \\ & \\ \therefore\quad a_{01} &= \dfrac{P_1^2}{(P_1+P_2-P_1P_2)^2}. \\ & \\ a_{00} &= P_1 a_{10} + (1-P_1)\left[P_2(1-P_2)a_{01} + (1-P_2)a_{00}\right] \\ & \\ & \text{Substituting for $a_{10}$ and $a_{01}$ and re-arranging, we get:} \\ a_{00} &= \dfrac{P_1^2(2P_1P_2^2-3P_1P_2+P_1-2P_2^2+3P_2)}{(P_1+P_2-P_1P_2)^3}. \\ \end{align}


text:
# Geometry

posted by .

How do I find the distance between (-9,3) and (-1,-5)

By using the Pythagorean theorem. The line between the two points is the hypotenuse of a triangle with perpendicular side lengths of 8 in the x direction and 8 in the y direction. The 8's come from the differences in x and y coordinates of the two points.

Plotting the points on a graph should be helpful to you for visualizing what is going on.

So Distance = sqrt[(8)^2 + (8)^2]
That can be reduced to 8 sqrt 2

• Geometry -

(-3,-10)and(7-8)

## Similar Questions

1. ### isosceles triangle

An isosceles triangle is a triangle that has two equal sides. Find AB, BC and AC. Determine if the triangle is isoseles or not based on the lengths of the three sides. can someone show me how to solve?
2. ### Algebra

Need help with these-- 21. Find the possible values of a, of the distance between the points is 5, and the coordinates are (1, 1) and (4,a) 22. The Pythagorean Theorem works only for _______ triangles. Right?
3. ### geometry

A. Derive the distance formula (d) shown below for points A = (x1, y1) and B = (x2, y2). 2 2 1 2 2 1 d = (x − x ) + ( y − y ) Note: Look for an application of the Pythagorean theorem where the red line, segment AB, is the …
4. ### Geometry

A. Derive the distance formula (d) shown below for points A = (x1, y1) and B = (x2, y2). 2 2 1 2 2 1 d = (x − x ) + ( y − y ) Note: Look for an application of the Pythagorean theorem where the red line, segment AB, is the …
5. ### Geometry (Checking)

2.) Which is true about both Pappus's Theorem and Desargues' Theorem?

The radius of this circle is one unit. Find the exact lengths of the legs in the right triangle if: Sides a is twice as long as side b. And Sides b is twice as long as side a. Basically the hypotenuse of this triangle is 1 and we are …
7. ### Math

The radius of this circle is one unit. Find the exact lengths of the legs in the right triangle if: Sides a is twice as long as side b. And Sides b is twice as long as side a. Basically the hypotenuse of this triangle is 1 and we are …
8. ### Math Help!

The radius of this circle is one unit. Find the exact lengths of the legs in the right triangle if: Sides a is twice as long as side b. And Sides b is twice as long as side a. Basically the hypotenuse of this triangle is 1 and we are …
9. ### geometry

In triangle $ABC$, $BC = 4$, $AC = 3 \sqrt{2}$, and $\angle C = 45^\circ$. Altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. Find $AH:HD$. I tried using the pythagorean theorem to find side lengths, algebraically, but …

More Similar Questions

Post a New Question


text:
# Dynamic boundary condition

I need to compute the dynamic boundary condition for a small drop slowly spreading on a completely wetting, solid substrate . We are using cylindrical coordinates (s,z), and there is no flow in the $\phi$ direction. The shape of the drop is given by h(s,t). The velocity field in s is u(s,z), and the velocity field in the z direction is w(s,z).

The stress tensor $\sigma_1$ is in cylindrical coordinates given by:

$\sigma_{ss}=2\eta \nabla_{s}u-p$ $\\$

$\sigma_{zz}=2\eta \nabla_{z}w-p$ $\\$

$\sigma_{sz}=\eta( \nabla_{z}u+\nabla_{s}w)$ $\\$

The dynamic boundary condition is the continuity of the stress vector across an interface i.e. $\ \sigma_{1}n_{1}-\sigma_2n_2=0$=> $\sigma_{1}n_{1}=\sigma_2n_2$

I guess a normal vector could be $n_1=n_2=(1,0,0)$.

I am a bit unsure if that works.

The other thing I am not sure about is the stress tensor for air $\sigma_2$.

So I am not sure how to compute this dynamic BC.

The second thing I need to show is that in lowest order of $h'=\nabla_sh$ the conditions can be written as:

$\nabla_{z}u(s,z=h) \approx 0$ and $p(s,z=h) \approx P.$ where P is the laplace pressure

Any ideas ?

• The normal is related to dh/ds. – Chet Miller Apr 28 '18 at 21:43
• How is $\nabla _z$ operator defined? – Deep Apr 29 '18 at 3:57
• $\nabla_z$ is defined as : $\nabla_z= \frac{\partial }{\partial z}$ – Elias S. Apr 29 '18 at 11:35

If you apply the Cauchy stress relationship at the free surface and take the traction in the air to be zero (i.e., zero gauge pressure (p=0) and zero tangential stress), then the stress boundary conditions at the free surface are: $$\sigma_{zz}=\sigma_{sz}\frac{\partial h}{\partial s}$$and$$\sigma_{sz}=\sigma_{ss}\frac{\partial h}{\partial s}$$ This neglects surface tension.

• I forgot to mention that P is the Laplace pressure from surface tension. – Elias S. Apr 29 '18 at 12:31

The shape of the drop is given by the equation $q(s,z,t)=z-h(s,t)=0$, in which $s$ is the radial coordinate, $z$ is the axial coordinate referred to a cylindrical coordinate system whose axis coincides with the symmetry axis of the spreading drop, $h$ and $q$ are functions, and $t$ is time. The outward normal to this surface (i.e. normal pointing into the air) is given by: $$\mathbf{n}=\frac{\nabla q}{|\nabla q|}$$

Now $\nabla q=(-\frac{\partial h}{\partial s},0,1)$ in cylindrical coordinates. Therefore: $$\frac{1}{|\nabla q|}=\left(1+\left(\frac{\partial h}{\partial s}\right)^2\right)^{-1/2}=1-\frac{1}{2}\left(\frac{\partial h}{\partial s}\right)^2+O\left(\left(\frac{\partial h}{\partial s}\right)^4\right)$$

Therefore if $\frac{\partial h}{\partial s}\ll 1$ then correct to linear order in $\frac{\partial h}{\partial s}$ we have: $\mathbf{n}=\nabla q$.

If $\sigma_{0}, \sigma,$ are stress tensors inside air and the drop respectively, then at the drop surface the corresponding stress vectors are $\mathbf{f}_0=\sigma_{0}\cdot\mathbf{n}$ and $\mathbf{f}=\sigma\cdot(-\mathbf{n})$. There is a jump in the normal stress due to surface tension of the drop-air interface. If $P(t)$ be the associated Laplace pressure then $-P\mathbf{n}$ is the force exerted by surface tension. Then force balance gives: $$\mathbf{f}_0+\mathbf{f}-P\mathbf{n}=\mathbf{0}\\\Rightarrow\sigma_{0}\cdot\mathbf{n}-\sigma\cdot\mathbf{n}=P\mathbf{n}$$ which is the dynamic boundary condition you need.

The stress tensor for air is a required input for further calculations. Usually the shear stress exerted by air on the drop is assumed to be zero. If you make this assumption (which must be correct to at least linear order in $\frac{\partial h}{\partial s}$), then the stress tensor for air becomes isotropic: $\sigma_0=-p_0\mathbf{I}$, in which $\mathbf{I}$ is the identity tensor and $p_0$ is the constant hydrostatic pressure due to air. If the shear stress exerted by air is zero, then from a force balance in the tangential direction at the drop surface, it follows that shear stress on the liquid side (evaluated at the drop surface) must also be zero. Therefore $\sigma=-p\mathbf{I}$ (when evaluated at the drop surface). Substituting into the dynamic boundary condition, and dotting the resulting expression with $\mathbf{n}$, we get: $$p(t)-p_0=P(t)$$ If you are considering only gauge pressures then you may set $p_0=0$ in the equation above.

P.S. If $\mathbf{t}$ is a tangent to the drop surface then we must have $\mathbf{t}\cdot\mathbf{n}=0$. Since $\mathbf{n}=(-\frac{\partial h}{\partial s},0,1)$, a tangent would be $\mathbf{t}=(1,0,\frac{\partial h}{\partial s})$. Another orthogonal tangent would then be $\tilde{\mathbf{t}}=\mathbf{n}\times\mathbf{t}=(0,1,0)$. Now dotting the dynamic boundary condition with $\mathbf{t}$ and $\tilde{\mathbf{t}}$ in turn, and using the last result, you can perhaps derive the other identities you need correct to linear order in $\frac{\partial h}{\partial s}$. I didn't try because it involves tedious algebra.


text:
Give the molecular formula, the condensed formula, and indicate the number of hydrogens each carbon has in the octane molecule.

Solution:

An 8-carbon alkane has 18 hydrogens and its molecular formula is: C 8 H 18 . The condensed formula is given by: CH 2 (CH 2 ) n CH 3 .

The formula developed in which the carbons and hydrogens that the molecule has are indicated is:


text:
# If $K_{14}$ is colored with two colors, there will be a monochromatic quadrangle.

This question is from Problem Solving Strategies by Engel, Chapter 4 question 50.

If $$K_{14}$$ is colored with two colors, there will be a monochromatic quadrangle.

Here, $$K_{14}$$ is the complete graph with 14 vertices, a coloring means to assign each edge a color (lets say either red or blue) and I am assuming quadrangle means a cycle of length 4.

I know the technique to prove that $$R(3,3) = 6$$ (the Ramsey number), and I tried to apply it but with no success. For example, if I have a red path of length 3, then I can force the last edge to be blue. I also started by considering that each vertex has 13 neighbors, so each vertex has either more red or blue edges. So there exists at least 7 vertices, each with at least 7 edges of the same color. But I don't see a way to proceed.

Let's try to avoid a monochromatic quadrangle ($$4$$-cycle):

As you noted each vertex has at least $$7$$ incident edges with the same color. Let $$x$$ be a vertex and let $$a_1$$, $$a_2$$, $$a_3$$, $$a_4$$ be four vertices for which the edges from $$x$$ to each $$a_i$$ are all the same color ($$i=1,2,3,4$$). Without loss of generality, we may assume that the color of those edges is red.

Now let $$y$$ be a different vertex (not $$x$$ and not any of $$a_1,...,a_4$$). At least three of the edges $$(y,a_1), (y,a_2), (y,a_3), (y,a_4)$$ must be blue. (If any two are red, you get a red quadrangle using $$x,a_i,y,a_j$$ where the edges $$(y,a_i)$$ and $$(y,a_j)$$ are red.) Without loss of generality, we may assume $$(y,a_1), (y,a_2), (y,a_3)$$ are all blue edges.

If the edge from $$a_1$$ to $$a_2$$ is red, then the edge from $$a_2$$ to $$a_3$$ must be blue (otherwise $$x,a_1,a_2,a_3$$ is a red quadrangle). But then the edge from $$a_1$$ to $$a_3$$ will complete a monochromatic quadrangle (if red: $$x,a_2,a_1,a_3$$; if blue: $$y,a_1,a_3,a_2$$).

Similarly if the edge from $$a_1$$ to $$a_2$$ is blue, you will again get a monochromatic quadrangle.

Note: It appears to me that this argument could be used to show that a monochromatic quadrangle must exist on an edge 2-colored $$K_9$$.

• That is a really nice proof! Seems like the crux move is considering the vertices $x$ and $y$, each with 3 edges of their respective colors connecting to the three $a_i$. Thanks! – eatfood May 10 at 15:26
• This also works for $K_7$, right? – Santana Afton May 10 at 15:35
• I don't think it works for $K_7$ because you need the $x$ and $y$ vertices separate from $7$ other vertices. – paw88789 May 10 at 15:50
• It sure does work for $K_7$. If you $2$-color the edges of $K_7$ there must (by pigeonhole & parity) be a vertex with $4$ incident edges of the same color; say $xa_1$, $xa_2$, $xa_3$, $xa_4$ are distinct red edges. Choose a vertex $y\notin\{x,a_1,a_2,a_3,a_4\}$, etc. @SantanaAfton This is what you had in mind, right? – bof May 11 at 5:07
• @bof Yup! Exactly. – Santana Afton May 11 at 5:14

A sharper result shows that any $$2$$-coloring of $$K_6$$ contains a monochromatic $$C_4$$.

We know $$R(3,3)=6$$, so any $$2$$-coloring of $$K_6$$ contains a monochromatic triangle. Let $$\{x,y,z\}$$ be the vertices of the triangle, and without loss of generality let red be the color of edges $$xy, xz, yz$$.

If there is another vertex $$w$$ with red edges to at least two vertices among $$\{x,y,z\}$$, then we get a red $$C_4$$. Say $$wx$$ and $$wy$$ are red: then the edges $$wx, xz, zy, yw$$ form a red cycle.

Otherwise, each of the three vertices outside $$\{x,y,z\}$$ has at most one red edge to $$\{x,y,z\}$$ - and at least two blue edges to $$\{x,y,z\}$$. If there are two vertices $$w_1, w_2$$ with blue edges to the same two vertices among $$\{x,y,z\}$$, then we get a blue $$C_4$$. Say $$w_1x, w_1y, w_2x, w_2y$$ are blue: then the edges $$w_1x, xw_2, w_2y, yw_1$$ form a blue cycle.

The remaining possibility is that the other three vertices all have exactly two blue edges to $$\{x,y,z\}$$, and it's a different set of blue edges for each one. We can label the remaining vertices $$\{x', y', z'\}$$ such that edges $$xx', yy', zz'$$ are red while $$xy', xz', yx', yz', zx', zy'$$ are blue, getting the partial coloring below:

Now we look at the colors of the three remaining edges $$x'y', x'z', y'z'$$.

• If $$x'y'$$ is red, then the edges $$x'y', y'y, yx, xx'$$ form a red cycle.
• If $$y'z'$$ is red, then the edges $$y'z', z'z, zy, yy'$$ form a red cycle.
• If $$x'y'$$ and $$y'z'$$ are both blue, then the edges $$x'y', y'z', z'y, yx'$$ form a blue cycle.

The complete graph $$K_{14}$$ is much bigger than needed for this result; here are three proper subgraphs with the same property.

I. Every $$2$$-coloring of the complete bipartite graph $$K_{3,7}$$ contains a monochromatic $$C_4$$.

Let $$K_{3,7}$$ have partite sets $$V_1,V_2$$ with $$|V_1|=3$$ and $$|V_2|=7$$. Each vertex in $$V_2$$ is joined by edges of one color to two vertices in $$V_1$$. By the pigeonhole principle, two vertices in $$V_2$$ are joined by edges of the same color to the same two vertices in $$V_1$$.

II. Every $$2$$-coloring of the complete graph $$K_6$$ contains a monochromatic $$C_4$$. (This was shown in Misha Lavrov's answer; the following argument is perhaps slightly simpler.)

We may assume that there is a vertex $$x$$ which is incident with at most two blue edges. In fact, we may assume that $$x$$ is incident with exactly two blue edges; otherwise $$x$$ would be incident with four red edges, call them $$xa_1$$, $$xa_2$$, $$xa_3$$, $$xa_4$$, and then we could consider a vertex $$y\notin\{x,a_1,a_2,a_3,a_4\}$$ and proceed as in paw88789's answer. (It may be even easier to observe that the subgraph induced by $$\{a_1,a_2,a_3,a_4\}$$ contains either a red $$P_3$$ or a blue $$C_4$$.)

So let $$x$$ be incident with exactly two blue edges, $$xy$$ and $$xz$$. If one of the remaining three vertices is joined by blue to both $$y$$ and $$z$$, then we have a blue $$C_4$$. On the other hand, if each of those three vertices is joined by red to a vertex in $$\{y,z\}$$, then two of them are joined by red to the same vertex in $$\{y,z\}$$, and also to $$x$$, making a red $$C_4$$.

III. Every $$2$$-coloring of the complete bipartite graph $$K_{5,5}$$ contains a monochromatic $$C_4$$.

The proof is left as an exercise for the reader.


text:
# A finite unital and commutative ring with exactly one maximal ideal has $p^{n}$ elements.

Suppose $R$ is a finite unital and commutative ring that has exactly one maximal ideal. Prove that $\left | R \right |=p^{n}$ where $p$ is a prime number. If $R$ will be non-commutative, do we have the desired result?

Suppose $I$ is a maximal ideal of $R$, so $R/I$ is a field, because $R$ is finite, thus $| R/I|=p^{m}$, where $p$ is prime. Now I don't know what I should do next, so please help me.

• $R$ being vector space under $R/I$?? – user87543 Dec 25 '13 at 10:21

First show that the maximal ideal of $R$, let say $M$, is nilpotent, that is, $M^n=0$ for some $n\ge 1$. If $n=1$ you are done, since then $R$ is a finite field. If $n=2$ use the following exact sequence $$0\to M/M^2\to R/M^2\to R/M\to 0.$$ Here $M/M^2$ is an $R/M$-vector space, so it has as number of elements a power of $p$, the characteristic of the finite field $R/M$. Then $R/M^2$ has also a power of $p$ as its cardinality.


text:
This document describes Marginal Maximum Likelihood (MML) estimation for student test data in the Dire package. In these models, failing to account for the measurement variance can bias the regression or variance estimates.

The student test data are assumed to have been generated by an Item Response Theory (IRT) model, where students’ responses are correct or incorrect (have an increasing score) when student $$i$$ has higher ability ($$\theta_i$$) and decreasing when the item is more difficult ($$d_j$$), so the probability of a correct response increases as the quantity $$\theta_i - d_j$$ increases. See the appendix for the likelihood functions for various response models.

This package considers a regression model of the following form (one case in Cohen & Jiang 1999): \begin{align} \bm{\theta} = \bm{X\beta} + \bm{\epsilon} \end{align} where $$\bm{\theta}$$ is a vector of student abilities, $$\bm{X}$$ is a matrix of covariates with unknown parameters $$\bm{\beta}$$ and residual variance $$\bm{\epsilon}$$. For estimation, we assume that the residual variance is normally distributed without covariance across observations (students) sharing variance of unknown level $$\sigma^2$$ so that \begin{align} \bm{\epsilon} \sim N(\bm{0},\sigma^2 \bm{I}) \end{align} where $$N(\bm{0}, \sigma^2 \bm{I})$$ is the normal distribution with mean zero, and covariance $$\sigma^2 \bm{I}$$, and $$\bm{I}$$ is the identity matrix. The variance estimation then allows for covariances between students (e.g., in a two-stage sample or clustered within schools).

The next section describes the estimation of $$\bm{\beta}$$ and $$\sigma^2$$. The final section describes five methods for variance estimation available in MML estimation, including the traditional (consistent) method, two heteroskedasticity robust methods, and two methods appropriate to a two-stage survey sample, such as the National Assessment of Educational Progress (NAEP).

# Parameter Estimation

Student test data consist of a series of items on which a student receives a score. The matrix $$\bm{R}$$ has row $$i$$ regarding a student and column $$j$$ regarding an item so that $$R_{ij}$$ is student $$i$$’s score on item $$j$$ and takes on integer values from 0 to the maximum score on the item. Many possible models exist for the $$\bm{R}$$ matrix data, which are covered, briefly, in the appendix to this document. The rest of this document simply assumes that item parameters have been estimated with a consistent estimator and are treated as being estimated without error.

In an MML model for test data for $$N$$ individuals, conditional on a set of parameters for a set of $$K$$ test items, the likelihood of a regression equation is \begin{align} \mathcal{L} (\bm{\beta}, \sigma|\bm{w}, \bm{R}, \bm{X}, \bm{P}) = \prod_{i=1}^N \left[ \int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2\pi}} \exp \frac{-(\theta_i - \bm{X}_i \beta)^2}{2\sigma^2} \, \prod_{j=1}^K {\rm Pr}(\bm{R}_{ij}|\theta_i,\bm{P}_j) d\theta_i \right]^{\bm{w}_i} \end{align} where $$\mathcal{L}$$ is the likelihood of the regression parameters $$\bm{\beta}$$ with full sample weights $$\bm{w}_i$$ conditional on item score matrix $$\bm{R}$$, student covariate matrix $$\bm{X}$$, and item parameter data $$\bm{P}$$; $$\sigma^2$$ is the variance of the regression residual; $$\theta_i$$ is the $$i$$th student’s latent ability measure that is being integrated out; $${\rm Pr}(\bm{R}_{ij}|\theta_i, \bm{P}_j)$$ is the probability of individual $$i$$’s score on test item $$j$$, conditional on the student’s ability and item parameters $$\bm{P}_j$$—see the appendix for example forms of $${\rm Pr}(\bm{R}_{ij}|\theta_i, \bm{P}_j)$$. Note that if the user is only interested in the population mean, it can be regarded as a special case; $$\bm{X}$$ is a vector of all ones, and the value of $$\bm{\beta}$$ has only one element that is the mean estimate.

The integral is evaluated using the trapezoid rule at quadrature points $$t_q$$ and quadrature weights $$\delta$$ so that \begin{align} \mathcal{L} (\bm{\beta}, \sigma|\bm{w}, \bm{R}, \bm{X}, \bm{P}) &= \prod_{i=1}^N \left[ \sum_{q=1}^Q \delta \frac{1}{\sigma \sqrt{2\pi}} \exp \frac{-(t_q - \bm{X}_i \bm{\beta})^2}{2\sigma^2} \prod_{j=1}^K {\rm Pr}(\bm{R}_{ij}|t_q, \bm{P}_j)\right]^{\bm{w}_i} \end{align} where $$\delta$$ is the distance between any two uniformly spaced quadrature points so that $$\delta = t_{q+1} - t_{q}$$ for any $$q$$ that is at least one and less than $$Q$$. The range and value of $$Q$$ parameterize the quadrature, and its accuracy and should be varied to ensure convergence. The advantage of the trapezoidal rule is that the fixed quadrature points allow the values of the probability to be calculated once per student.

The variance formulas use the log-likelihood, which is given by \begin{align} \ell (\bm{\beta}, \sigma|\bm{w}, \bm{R}, \bm{X}, \bm{P}) &= \sum_{i=1}^N \bm{w}_i \, {\rm log} \left[\delta \sum_{q=1}^Q \frac{1}{\sigma \sqrt{2\pi}} \exp \frac{-(t_q - \bm{X}_i \bm{\beta})^2}{2\sigma^2} \prod_{j=1}^K {\rm Pr}(\bm{R}_{ij}|t_q, \bm{P}_j) \right] \end{align} Note that $$\delta$$ can be removed for optimization, and its presence adds $${\rm log}(\delta) \sum \bm{w}_i$$ to the log-likelihood.

## Composite Scores

When the outcome of interest is composite scores, the parameters are estimated by separately estimating the coefficients for each subscale ($$\bm{\beta}_s$$ for subscale $$s$$) and then calculating the composite scores ($$\bm{\beta}_c$$) using subscale weights ($$\omega_s$$). \begin{align} \bm{\beta}_c &= \sum_{s=1}^S \omega_s \bm{\beta}_s \label{eq:composite} \end{align} where there are $$S$$ subscales.

For variance estimation, the covariance matrix ($$\bm{\Sigma}$$) between subscales is of interest. The covariance terms are estimated one at a time using the submatrix \begin{align} \bm{\Sigma}_{ij} = \left[ \begin{array}{cc} s_i & s_{ij} \\ s_{ij} & s_j \end{array} \right] \end{align}

so that the two are jointly bivariate normally distributed \begin{align} \left( \begin{array}{c} \beta_i \\ \beta_j \end{array} \right) | \bm{\Sigma}_{ij}, \bm{w}, \bm{R}, \bm{X}, \bm{P} & \sim {\rm MVN} \left( \left. \left( \begin{array}{c} \beta_i \\ \beta_j \end{array} \right), \bm{\Sigma}_{ij} \right| \bm{w}, \bm{R}, \bm{X}, \bm{P} \right) \label{eq:compcov} \end{align} where $${\rm MVN}(u, S|\cdot)$$ is the multivariate normal density function with mean $$u$$ and covariance $$S$$, conditional on $$\cdot$$, which are additional parameters.

The likelihood is then \begin{align} \ell \left( s_{ij} \left| \beta_i, \beta_j , s_i, s_j; \bm{w}, \bm{R}, \bm{X}, \bm{P}\right. \right) &= \sum_{n=1}^N \bm{w}_n \, {\rm log} \left\{ \delta^2 \sum_{q_i=1}^Q \sum_{q_j=1}^Q \frac{1}{\sqrt{2\pi}} \frac{1}{\sqrt{|\bm{\Sigma}|}}\exp \left( \bm{r}_{q_1q_2}^T \bm{\Sigma}^{-1} \bm{r}_{q_1 q_2} \right) \right. \\ &\mathrel{\phantom{=}} \left. \times \left[ \prod_{k=1}^K {\rm Pr}(\bm{R}_{nk}|t_{q_1}, \bm{P}_k) \right] \left[ \prod_{k=1}^K {\rm Pr}(\bm{R}_{nk}|t_{q_2}, \bm{P}_k) \right] \right\} \label{eq:compcovlnl} \end{align} where $$|\bm{\Sigma}|$$ is the determinant of $$\bm{\Sigma}$$, and the residual term is defined as \begin{align} \bm{r}_{q_1 q_2} = \left( \begin{array}{c} t_{q_1} - \bm{X}_n \bm{\beta}_i \\ t_{q_2} - \bm{X}_n \bm{\beta}_j \end{array} \right) \end{align} Notice that the parameters $$\bm{\beta}_i$$, $$\bm{\beta}_j$$, $$s_i$$, and $$s_j$$ are used from the by-subscale estimation and optimization of the density function is exclusively over the covariance term $$s_{ij}$$.

The joint distribution of the vector \begin{align} \bm{\beta}_{\cdot} = \left( \begin{array}{c} \bm{\beta}_1 \\ \vdots \\ \bm{\beta}_S \end{array} \right) \end{align} is then \begin{align} \bm{\beta}_{\cdot} ,\bm{\Sigma} | \bm{w}, \bm{R}, \bm{X}, \bm{P} &= {\rm MVN}(\left. \bm{\beta}_{\cdot}, \bm{\Sigma} \right| \bm{w}, \bm{R}, \bm{X}, \bm{P}) \label{eq:lnlcomposite} \end{align} which has an intractably high dimensional log-likelihood because it involves $$S$$ sums inside the log-likelihood.

# Variance Estimation

Estimating variance of the parameters $$\bm{\beta}$$ can be done in one of several ways.

The inverse Hessian matrix is a consistent estimator when the estimator of $$\bm{\beta}$$ is consistent (Green, 2003, p. 520): \begin{align} {\rm Var}(\bm{\beta}) = -\bm{H}(\bm{\beta})^{-1} = - \left[\frac{\partial^2 \ell(\bm{\beta}, \sigma|\bm{w}, \bm{R}, \bm{X})}{\partial \bm{\beta}^2} \right]^{-1} \label{eq:vbeta} \end{align} This variance is returned when the variance method is set to consistent or left as the default.

A class of variance estimators typically called “sandwich” or “robust” variance estimators allow for variation in the residual and are of the form \begin{align} {\rm Var}(\bm{\beta}) = H(\bm{\beta})^{-1} \bm{V} H(\bm{\beta})^{-1} \label{eq:sandwich} \end{align} where $$V$$ is an estimate of the variance of the summed score function (Binder, 1983).

For a convenience sample, we provide two robust estimators. First, the so-called robust (Huber or Huber-White) variance estimator uses \begin{align} \bm{V} &= \sum_{i=1}^N \left[ \frac{\partial \ell(\beta, \sigma|\bm{w}_i, \bm{R}_i, \bm{X}_i)}{\partial \beta} \right] \left[ \frac{\partial \ell(\beta, \sigma|\bm{w}_i, \bm{R}_i, \bm{X}_i)}{\partial \beta} \right]^{'} \end{align}

Second, for the cluster robust case, the partial derivatives are summed within the cluster so that \begin{align} \bm{V} &= \sum_{c=1}^{n^\prime} \left[ \frac{\partial \ell(\beta, \sigma|\bm{w}_c, \bm{R}_c, \bm{X}_c)}{\partial \beta} \right] \left[ \frac{\partial \ell(\beta, \sigma|\bm{w}_c, \bm{R}_c, \bm{X}_c)}{\partial \beta} \right]^{'} \end{align} where there are $$n^\prime$$ clusters, indexed by $$c$$, and the partial derivatives are summed within the group of which there are $$n_c$$ members: \begin{align} \frac{\partial \ell(\beta, \sigma|\bm{w}_c, \bm{R}_c, \bm{X}_c)}{\partial \beta} &= \sum_{i=1}^{n_c} \frac{\partial \ell(\beta, \sigma|\bm{w}_i, \bm{R}_i, \bm{X}_i)}{\partial \beta} \end{align}

We also provide two survey sampling variance estimation techniques. The first one uses replicate weights, either from the jackknife, including Fay’s method for the jackknife, or from balanced repeated replication. In this approach, the typical method of estimating sampling variance still works, and the sampling covariance matrix can be calculated as \begin{align} {\rm Var}(\bm{\beta}) &= \sum_{j=1}^J \left(\bm{\beta}_j - \bm{\beta}_0 \right) \left(\bm{\beta}_j - \bm{\beta}_0 \right)^{'} \end{align} where there are $$J$$ replicate weights and the result of applying direct estimation under the set of weights $$j$$ is $$\bm{\beta}_j$$, whereas $$\bm{\beta}_0$$ is the estimate of $$\bm{\beta}$$ under the full sample weights. We recomend using this method when replicate variance estimation is requested.

The second survey sampling method is called the Taylor series method and uses the same formula as Eq. , but $$\bm{V}$$ is the estimate of the variance of the score vector (Binder, 1983). Our implementation assumes a two-stage design with $$n_a$$ primary sampling units (PSUs) in stratum $$a$$ and summed across the $$A$$ strata according to \begin{align} \bm{V} &= \sum_{a=1}^A \bm{V}_a \end{align} where $$\bm{V}_a$$ is a variance estimate for stratum $$a$$ and is defined by \begin{align} \bm{V}_a &= \frac{n_a}{n_a -1} \sum_{p=1}^{n_a} \left( \bm{s}_p - \bar{\bm{s}}_a \right)\left( \bm{s}_p - \bar{\bm{s}}_a \right)' \label{eq:Va} \end{align} where $$s_p$$ is the sum of the weighted (or pseudo-) score vector that includes all units in PSU $$p$$ in stratum $$a$$ and $$\bar{\bm{s}}_a$$ is the (unweighted) mean of the $$\bm{s}_p$$ terms in stratum $$a$$ so that \begin{align} s_p &=\sum_{i \in {\rm PSU} \ p}\frac{\partial \ell(\beta, \sigma|\bm{w}_i, \bm{R}_i, \bm{X}_i)}{\partial \beta} & \bar{\bm{s}}_a&= \frac{1}{n_a} \sum_{p \in {\rm stratum} \ a} s_p \end{align}

When a stratum has only one PSU, $$\bm{V}_a$$ is undefined. The best approach is for the analyst to adjust the strata and PSU identifiers, in a manner consistent with the sampling approach, to avoid singleton strata. Two simpler but less defensible options are available. First, the strata with single PSUs can be dropped from the variance estimation, yielding an underestimate of the variance.

The second option is for the singleton stratum to use the overall mean of $$s_p$$ in place of $$\bar{s}_a$$. So, \begin{align} \bar{\bm{s}} &= \frac{1}{n^\prime} \sum s_p \end{align} where the sum is across all PSUs, and $$n^\prime$$ is the number of PSUs across all strata. Then, for each singleton stratum, Eq.  becomes \begin{align} \bm{V}_a &= 2 \left( \bm{s}_p - \bar{\bm{s}} \right)\left( \bm{s}_p - \bar{\bm{s}} \right)' \label{eq:Va1} \end{align} where the value 2 is used in place of $$\frac{n_a}{n_a-1}$$, which is undefined when $$n_a=1$$. This option can underestimate the variance but is thought to more likely overestimate it.

## Composite Scores

The likelihood of composite scores (Eq. ) is additively separable, the covariances (including the variances) can be calculated in two steps using Eq. . First, the covariance matrix of $$\bm{\xi}$$ is formed, and then the composite covariance terms are estimated as the variance of a linear combination of the elements of $$\bm{\xi}$$.

In the first step, any of the methods in the section “Variance Estimation” are applied to Eq. , treating $$\bm{\xi}$$ in the same fashion Eq.  treats $$\bm{\beta}$$. This step results in a block diagonal inverse Hessian matrix, with a block for each subscale, and a potentially dense matrix for $$\bm{V}$$. Each matrix is square and has $$S \cdot (\zeta+1)$$ rows and columns, where $$\zeta$$ is the number of elements in the regression formula (each subscale), to which one is added for the $$\sigma$$ terms.

This step results in the following matrix: \begin{align} {\rm Var}(\bm{\xi})=H(\bm{\xi})^{-1} \bm{V} H(\bm{\xi})^{-1} \end{align}

For the second step, the composite coefficient then has an $$i$$th variance term of \begin{align} {\rm Var}(\bm{\xi}_{ci}) &= \bm{e}_i H(\bm{\xi})^{-1} \bm{V} H(\bm{\xi})^{-1} \bm{e}_i \end{align} where $$\bm{\xi}_{ci}$$ is the composite coefficient for the $$i$$th coefficient, and $$\bm{e}_{i}$$ is the vector of weights arranged such that \begin{align} \bm{\xi}_{ci} = \bm{e}_{i}^T \bm{\xi} \end{align} The covariance between two terms, $$i$$ and $$j$$, is a simple extension \begin{align} {\rm Cov}(\bm{\beta}_{ci}, \bm{\beta}_{cj}) &= \bm{e}_i H(\bm{\beta})^{-1} \bm{V} H(\bm{\beta})^{-1} \bm{e}_j \end{align} which uses the definition, \begin{align} \bm{\xi}_{cj} &= \bm{e}_{j}^T \bm{\xi} \end{align}

A simple example may help clarify. Imagine a composite score composed of two subscales, 1 and 2, with weights $$\omega_1 = 0.4$$ and $$\omega_2=0.6$$. Supposed a user is interested in a regression of the form \begin{align} \theta = a + x_1 \cdot b + \epsilon \label{eq:compex} \\ \epsilon \sim N(0,\sigma) \end{align} Then the regression in Eq.  would be fit once for subscale 1 and once for subscale 2; the first fit would yield estimated values $$\left\{ a_1, b_1, \sigma_1 \right\}$$, and the second fit would yield $$\left\{ a_2, b_2, \sigma_2 \right\}$$. The estimated value, for example, $$a_c$$, would be $$a_c = 0.4\cdot\alpha_1 + 0.6\cdot\alpha_2$$. By stacking the estimates together, \begin{align} \bm{\theta} &= \begin{bmatrix} a_1 \\ b_1 \\ \sigma_1\\ a_2 \\ b_2\\ \sigma_2 \end{bmatrix} \end{align} the covariance matrix can then be estimated and will result in a matrix $$\bm{\Omega} \equiv {\rm Var}(\bm{\beta})$$ from Eq.  that has six rows and six columns. Using the vector \begin{align} \bm{e}_1 &= \begin{bmatrix} 0.4 \\ 0 \\ 0\\ 0.6 \\ 0\\ 0 \end{bmatrix} \end{align} it can easily be confirmed that $$a_c = \bm{e}_1^T \bm{\xi}$$, so $${\rm Var}(a_c)= \bm{e}_1^T \bm{\Omega} \bm{e}_1$$.

# References

Binder, D. A. (1983). On the variances of asymptotically normal estimators from complex surveys. International Statistical Review, 51(3), 279–292.

Black, P. E. (2019). Big-O notation. In P. E. Black (Ed.), Dictionary of algorithms and data structures. Washington, DC: National Institute of Standards and Technology. Retrieved from https://www.nist.gov/dads/HTML/bigOnotation.html

Cohen, J. D., & Jiang, T. (1999). Comparison of partially measured latent traits across nominal subgroups. Journal of the American Statistical Association, 94(448), 1035–1044.

Green, W. H. (2003). Econometric analysis Upper Saddle River, NJ: Prentice Hall.

Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium of Mathematical Statistics and Probability, Vol. I: Statistics (pp. 221–233). Berkeley, CA: University of California Press.

McCullagh, P. & Nelder, J. A. (1989). Generalized linear models. (2nd ed.). London, UK: Chapman & Hall/CRC.

NAEP. (2008). The generalized partial credit model [NAEP Technical Documentation Website]. Retrieved from https://nces.ed.gov/nationsreportcard/tdw/analysis/scaling_models_gen.aspx.

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817–838.

# Appendix. Test Probability Density Functions

For all cases scored as either correct or incorrect, we use the three parameter logit (3PL) model: \begin{align} \Pr(\bm{R}_{ij}|\theta_i, \bm{P}_j) &= g_j + \frac{1-g_j}{1+\exp\left[ -D \, a_j \, (\theta_i - d_j)\right]} \label{eq:3PL} \end{align} where $$g_j$$ is the guessing parameter, $$a_j$$ is the discrimination factor, $$d_j$$ is the item difficulty, and $$D$$ is a constant, usually set to 1.7, to map the $$\theta_i$$ and $$d_j$$ terms to a probit-like space; this term is applied by tradition.

When a two parameter logit (2PL) is used, Eq.  is modified to omit $$g_j$$ (effectively setting it to zero): \begin{align} \Pr(\bm{R}_{ij}|\theta_i, \bm{P}_j) &= \frac{1}{1+\exp\left[ -D \, a_j \, (\theta_i - d_j)\right]} \label{eq:2PL} \end{align}

When a Rasch model is used, Eq.  is further modified to set all $$a_j$$ to a single $$a$$, and $$D$$ is set to one. \begin{align} \Pr(\bm{R}_{ij}|\theta_i, \bm{P}_j) &= \frac{1}{1+\exp\left[ - a \, (\theta_i - d_j)\right]} \label{eq:Rasch} \end{align}

The Graded Response Model (GRM) has a probability density that generalizes an ordered logit (McCullagh & Nelder, 1989): \begin{align} \Pr(\bm{R}_{ij}|\theta_i, \bm{P}_j) &= \frac{1}{1+\exp\left[-D\, a_j \, (\theta_i - d_{R_{ij},j})\right]} - \frac{1}{1+\exp\left[-D\, a_j \, (\theta_i - d_{1+R_{ij},j})\right]} \label{eq:grm} \end{align} Here the parameters $$\bm{P}_j$$ are the cut points $$d_{cj}$$, where $$d_{0j}=-\infty$$ and $$d_{C+1,j}=\infty$$. In the first term on the right side of Eq. , the subscript $$R_{ij}$$ on $$d_{R_{ij},j}$$ indicates it is the cut point associated with the response level to item $$j$$ for person $$i$$, whereas the last subscript ($$j$$) indicates that it is the $$d$$ term for item $$j$$. In the second term, the cut point above that cut point is used.

The Generalized Partial Credit Model (GPCM) has a probability density that generalizes a multinomial logit (McCullagh & Nelder, 1989) \begin{align} \Pr(\bm{R}_{ij}|\theta_i, \bm{P}_j) &= \frac{\exp \left[ \sum_{c=0}^{R_{ij}} D a_j (\theta_i - d_{cj}) \right]}{\sum_{r=0}^{C} \exp \left[ \sum_{c=0}^{r} D a_j (\theta_i - d_{cj}) \right]} \end{align} where $$c$$ indexes cut points, of which there are $$C$$, and $$j$$ indexes the item.

The GPCM equation has an indeterminancy because all $$d_j$$ terms could increase and make the values of the probability the same. We can solve the indeterminacy in several ways.

NAEP (2008) uses a mean difficulty ($$b_j$$), and the $$d_j$$ values are then given by \begin{align} d_{0j} &= 0 & d_{cj} &= b_j - \delta_{jc} \, ; \, 1 \leq c \leq C \end{align} where the $$\delta_{jc}$$ values are estimated so that $$0=\sum_{c=1}^{C} \delta_{jc}$$. In this package, when the polyParamTab has an itemLocation, it serves as b. When there is no itemLocation, the package uses the $$\delta$$ values directly \begin{align} d_{0j} &= 0 & d_{cj} &= \delta_{jc} \, ; \, 1 \leq c \leq C \end{align}

When a Partial Credit Model (PCM) is used, and the value of $$D$$ is set to one, whereas $$a_j$$ is again shared across all items. So \begin{align} \Pr(\bm{R}_{ij}|\theta_i, \bm{P}_j) &= \frac{\exp \left[ \sum_{c=0}^{R_{ij}} a (\theta_i - d_{cj}) \right]}{\sum_{r=0}^{C} \exp \left[ \sum_{c=0}^{r} a (\theta_i - d_{cj}) \right]} \end{align}