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I have personally done original research in the field of TQFTs, so I can tell you the reasons I find TQFTs interesting. Some that come to mind are: Some "real life" theories are accurately described/approximated by TQFTs; for example, gauge theories such as QCD. If you have a regular QFT that is gapped, then its strongly-coupled regime is almost ...

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In condensed matter physics, topological quantum field theories provide an effective description of (many, but not all) gapped phases of matter at low energies and long distances. A phase of matter is gapped if it costs a finite amount of energy to create any excitation above the ground state. Examples of gapped phases of matter that admit a low-energy TQFT ...

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When we compactify string theory, we are interested in the effective theory in the remaining dimensions, and perform a generalization of Kaluza-Klein reduction. Now, where in flat 10 dimensions the sum over worldsheets in the string amplitudes needed only to sum over the different kinds of worldsheets (their moduli and genus), we now have to distinguish ...

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TQFTs were not discovered by mathematicians - they were actually discovered by physicists, so one should expect there to be physical motivation for the theory. One reason why that this is difficult to discover is that mathematicians have taken over the theory so it is hard to recognise the physical motivation. One reason to study them is as toy models for ...

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Now by the last statement we should have a map from $\mathrm{SO}^+(1,3)$ to $\mathrm{SL}(2,\mathbb C)$. How can we define this map? That's not quite right. What that statement says is that we should have a map from paths$^\ddagger$ on $\mathrm{SO}^+(1,3)$ to paths on $\mathrm{SL}(2,\mathbb C)$. In other words, if we start at some point $p\in \mathrm{SO}^+(... 3 First, a misconception (or rather, an oversimplification). QFTs are local, but that does not mean they are insensitive to global properties. For example a free fermion with anti-periodic boundary conditions has a propagator$\psi(z)\psi(w)=1/(z-w)$but if you choose periodic boundary conditions, the propagator becomes$\psi(z)\psi(w)=(1/2)(\sqrt{z/w}+\sqrt{w/...

2

Producing the frequency spectrum of a time series is nothing but doing a Fourier transform, that is obtaining the amplitudes (and phases) of sinusoidal waves which result in the time series when all summed together. Given a time series $f(t)$ the Fourier amplitude in frequency space is $$F(\omega)=\int_{-\infty}^\infty dt f(t)\:e^{-i \omega t}$$ $F(\omega)$ ...

2

If you compacify to a torus then, in Cartesian coordinates, the spin connection vanishes and so is irrelevent. If you compactify to a sphere, as Fujikawa suggests, it is far less obvious that $\hat A(TM)$ is not needed. That it plays no role is because $\hat A(TM)$ is a genus and so cobordism invariant. This means that the curvature contribution to the ...

2

The heat kernel has four discrete indices in addition to the two continous space indices $x$,$y$ because the gamma matrices have two: $\gamma^{ab}$; and the matrix-valued gauge fields $A_\mu$'s that hide in the Dirac operator have two: $[A_\mu]_{kl}$. The heat kernel is therefore an operator acting on $L^2[{\mathbb R}^4]\otimes S\otimes V$ where $L^2[{... 2 Classically speaking, gauge fields are connections on principle$G$bundles. Matter fields, on the other hand, are sections of associated bundles. To construct an associated bundle, you must choose a representation of$G$. For instance, if$G = U(1)$, then this amounts to choosing an integer. When you construct the associated bundle, this integer is the ... 1 Both deal with theory rather than with experiments. The most striking difference is the level of rigour being put in. For example: In theoretical physics you solve differential equations, in mathematical physics you solve them as well, but you also prove that the solution exists and is unique or, if it is not unique, you consider all possible solutions and ... 1 In the axiomatic algebraic formalism, say of the Ostwalder-Schrader axioms of nets of local algebras, it turns that the local algebra must all be isomorphic to the unique hyperfinite factor of type$III_1$. Hence specific theories depend upon how these algebras embed within each other. For more details, see the paper The Role of Type III Facyors in QFT by ... 1 The answer by Jeanbaptiste Roux contains the exact form of the mapping (Lie group homomorphism) from$\mathrm{SL}(2,\mathbb C)$to$\mathrm{SO}^{+}(1,3;\mathbb{R})$(the latter is also denoted in the literature as$\mathfrak{L}^{+}_{\uparrow}$or$\mathfrak{Lor}(1,3)\$). As a mapping, it is not bijective, so it does not an inverse in the proper mathematical ...

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Calculus Volume 3

1.3Polar Coordinates

Calculus Volume 31.3 Polar Coordinates

Learning Objectives

• 1.3.1. Locate points in a plane by using polar coordinates.
• 1.3.2. Convert points between rectangular and polar coordinates.
• 1.3.3. Sketch polar curves from given equations.
• 1.3.4. Convert equations between rectangular and polar coordinates.
• 1.3.5. Identify symmetry in polar curves and equations.

The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points. This is called a one-to-one mapping from points in the plane to ordered pairs. The polar coordinate system provides an alternative method of mapping points to ordered pairs. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates.

Defining Polar Coordinates

To find the coordinates of a point in the polar coordinate system, consider Figure 1.27. The point $PP$ has Cartesian coordinates $(x,y).(x,y).$ The line segment connecting the origin to the point $PP$ measures the distance from the origin to $PP$ and has length $r.r.$ The angle between the positive $xx$-axis and the line segment has measure $θ.θ.$ This observation suggests a natural correspondence between the coordinate pair $(x,y)(x,y)$ and the values $rr$ and $θ.θ.$ This correspondence is the basis of the polar coordinate system. Note that every point in the Cartesian plane has two values (hence the term ordered pair) associated with it. In the polar coordinate system, each point also has two values associated with it: $rr$ and $θ.θ.$

Figure 1.27 An arbitrary point in the Cartesian plane.

Using right-triangle trigonometry, the following equations are true for the point $P:P:$

$cosθ=xrsox=rcosθcosθ=xrsox=rcosθ$
$sinθ=yrsoy=rsinθ.sinθ=yrsoy=rsinθ.$

Furthermore,

$r2=x2+y2andtanθ=yx.r2=x2+y2andtanθ=yx.$

Each point $(x,y)(x,y)$ in the Cartesian coordinate system can therefore be represented as an ordered pair $(r,θ)(r,θ)$ in the polar coordinate system. The first coordinate is called the radial coordinate and the second coordinate is called the angular coordinate. Every point in the plane can be represented in this form.

Note that the equation $tanθ=y/xtanθ=y/x$ has an infinite number of solutions for any ordered pair $(x,y).(x,y).$ However, if we restrict the solutions to values between $00$ and $2π2π$ then we can assign a unique solution to the quadrant in which the original point $(x,y)(x,y)$ is located. Then the corresponding value of r is positive, so $r2=x2+y2.r2=x2+y2.$

Theorem 1.4

Converting Points between Coordinate Systems

Given a point $PP$ in the plane with Cartesian coordinates $(x,y)(x,y)$ and polar coordinates $(r,θ),(r,θ),$ the following conversion formulas hold true:

$x=rcosθandy=rsinθ,x=rcosθandy=rsinθ,$
1.7
$r2=x2+y2andtanθ=yx.r2=x2+y2andtanθ=yx.$
1.8

These formulas can be used to convert from rectangular to polar or from polar to rectangular coordinates.

Example 1.10

Converting between Rectangular and Polar Coordinates

Convert each of the following points into polar coordinates.

1. $(1,1)(1,1)$
2. $(−3,4)(−3,4)$
3. $(0,3)(0,3)$
4. $(53,−5)(53,−5)$

Convert each of the following points into rectangular coordinates.

1. $(3,π/3)(3,π/3)$
2. $(2,3π/2)(2,3π/2)$
3. $(6,−5π/6)(6,−5π/6)$

Solution

1. Use $x=1x=1$ and $y=1y=1$ in Equation 1.8:
$r2=x2+y2=12+12r=2andtanθ=yx=11=1θ=π4.r2=x2+y2=12+12r=2andtanθ=yx=11=1θ=π4.$

Therefore this point can be represented as $(2,π4)(2,π4)$ in polar coordinates.
2. Use $x=−3x=−3$ and $y=4y=4$ in Equation 1.8:
$r2=x2+y2=(−3)2+(4)2r=5andtanθ=yx=−43θ=−arctan(43)≈2.21.r2=x2+y2=(−3)2+(4)2r=5andtanθ=yx=−43θ=−arctan(43)≈2.21.$

Therefore this point can be represented as $(5,2.21)(5,2.21)$ in polar coordinates.
3. Use $x=0x=0$ and $y=3y=3$ in Equation 1.8:
$r2=x2+y2=(3)2+(0)2=9+0r=3andtanθ=yx=30.r2=x2+y2=(3)2+(0)2=9+0r=3andtanθ=yx=30.$

Direct application of the second equation leads to division by zero. Graphing the point $(0,3)(0,3)$ on the rectangular coordinate system reveals that the point is located on the positive y-axis. The angle between the positive x-axis and the positive y-axis is $π2.π2.$ Therefore this point can be represented as $(3,π2)(3,π2)$ in polar coordinates.
4. Use $x=53x=53$ and $y=−5y=−5$ in Equation 1.8:
$r2=x2+y2=(53)2+(−5)2=75+25r=10andtanθ=yx=−553=−33θ=−π6.r2=x2+y2=(53)2+(−5)2=75+25r=10andtanθ=yx=−553=−33θ=−π6.$

Therefore this point can be represented as $(10,−π6)(10,−π6)$ in polar coordinates.
5. Use $r=3r=3$ and $θ=π3θ=π3$ in Equation 1.7:
$x=rcosθ=3cos(π3)=3(12)=32andy=rsinθ=3sin(π3)=3(32)=332.x=rcosθ=3cos(π3)=3(12)=32andy=rsinθ=3sin(π3)=3(32)=332.$

Therefore this point can be represented as $(32,332)(32,332)$ in rectangular coordinates.
6. Use $r=2r=2$ and $θ=3π2θ=3π2$ in Equation 1.7:
$x=rcosθ=2cos(3π2)=2(0)=0andy=rsinθ=2sin(3π2)=2(−1)=−2.x=rcosθ=2cos(3π2)=2(0)=0andy=rsinθ=2sin(3π2)=2(−1)=−2.$

Therefore this point can be represented as $(0,−2)(0,−2)$ in rectangular coordinates.
7. Use $r=6r=6$ and $θ=−5π6θ=−5π6$ in Equation 1.7:
$x=rcosθ=6cos(−5π6)=6(−32)=−33andy=rsinθ=6sin(−5π6)=6(−12)=−3.x=rcosθ=6cos(−5π6)=6(−32)=−33andy=rsinθ=6sin(−5π6)=6(−12)=−3.$

Therefore this point can be represented as $(−33,−3)(−33,−3)$ in rectangular coordinates.
Checkpoint 1.10

Convert $(−8,−8)(−8,−8)$ into polar coordinates and $(4,2π3)(4,2π3)$ into rectangular coordinates.

The polar representation of a point is not unique. For example, the polar coordinates $(2,π3)(2,π3)$ and $(2,7π3)(2,7π3)$ both represent the point $(1,3)(1,3)$ in the rectangular system. Also, the value of $rr$ can be negative. Therefore, the point with polar coordinates $(−2,4π3)(−2,4π3)$ also represents the point $(1,3)(1,3)$ in the rectangular system, as we can see by using Equation 1.8:

$x=rcosθ=−2cos(4π3)=−2(−12)=1andy=rsinθ=−2sin(4π3)=−2(−32)=3.x=rcosθ=−2cos(4π3)=−2(−12)=1andy=rsinθ=−2sin(4π3)=−2(−32)=3.$

Every point in the plane has an infinite number of representations in polar coordinates. However, each point in the plane has only one representation in the rectangular coordinate system.

Note that the polar representation of a point in the plane also has a visual interpretation. In particular, $rr$ is the directed distance that the point lies from the origin, and $θθ$ measures the angle that the line segment from the origin to the point makes with the positive $xx$-axis. Positive angles are measured in a counterclockwise direction and negative angles are measured in a clockwise direction. The polar coordinate system appears in the following figure.

Figure 1.28 The polar coordinate system.

The line segment starting from the center of the graph going to the right (called the positive x-axis in the Cartesian system) is the polar axis. The center point is the pole, or origin, of the coordinate system, and corresponds to $r=0.r=0.$ The innermost circle shown in Figure 1.28 contains all points a distance of 1 unit from the pole, and is represented by the equation $r=1.r=1.$ Then $r=2r=2$ is the set of points 2 units from the pole, and so on. The line segments emanating from the pole correspond to fixed angles. To plot a point in the polar coordinate system, start with the angle. If the angle is positive, then measure the angle from the polar axis in a counterclockwise direction. If it is negative, then measure it clockwise. If the value of $rr$ is positive, move that distance along the terminal ray of the angle. If it is negative, move along the ray that is opposite the terminal ray of the given angle.

Example 1.11

Plotting Points in the Polar Plane

Plot each of the following points on the polar plane.

1. $(2,π4)(2,π4)$
2. $(−3,2π3)(−3,2π3)$
3. $(4,5π4)(4,5π4)$

Solution

The three points are plotted in the following figure.

Figure 1.29 Three points plotted in the polar coordinate system.
Checkpoint 1.11

Plot $(4,5π3)(4,5π3)$ and $(−3,−7π2)(−3,−7π2)$ on the polar plane.

Polar Curves

Now that we know how to plot points in the polar coordinate system, we can discuss how to plot curves. In the rectangular coordinate system, we can graph a function $y=f(x)y=f(x)$ and create a curve in the Cartesian plane. In a similar fashion, we can graph a curve that is generated by a function $r=f(θ).r=f(θ).$

The general idea behind graphing a function in polar coordinates is the same as graphing a function in rectangular coordinates. Start with a list of values for the independent variable $(θ(θ$ in this case) and calculate the corresponding values of the dependent variable $r.r.$ This process generates a list of ordered pairs, which can be plotted in the polar coordinate system. Finally, connect the points, and take advantage of any patterns that may appear. The function may be periodic, for example, which indicates that only a limited number of values for the independent variable are needed.

Problem-Solving Strategy: Plotting a Curve in Polar Coordinates
1. Create a table with two columns. The first column is for $θ,θ,$ and the second column is for $r.r.$
2. Create a list of values for $θ.θ.$
3. Calculate the corresponding $rr$ values for each $θ.θ.$
4. Plot each ordered pair $(r,θ)(r,θ)$ on the coordinate axes.
5. Connect the points and look for a pattern.

Example 1.12

Graphing a Function in Polar Coordinates

Graph the curve defined by the function $r=4sinθ.r=4sinθ.$ Identify the curve and rewrite the equation in rectangular coordinates.

Solution

Because the function is a multiple of a sine function, it is periodic with period $2π,2π,$ so use values for $θθ$ between 0 and $2π.2π.$ The result of steps 1–3 appear in the following table. Figure 1.30 shows the graph based on this table.

$θθ$ $r=4sinθr=4sinθ$ $θθ$ $r=4sinθr=4sinθ$
0 0 $ππ$ 0
$π6π6$ $22$ $7π67π6$ $−2−2$
$π4π4$ $22≈2.822≈2.8$ $5π45π4$ $−22≈−2.8−22≈−2.8$
$π3π3$ $23≈3.423≈3.4$ $4π34π3$ $−23≈−3.4−23≈−3.4$
$π2π2$ $44$ $3π23π2$ $44$
$2π32π3$ $23≈3.423≈3.4$ $5π35π3$ $−23≈−3.4−23≈−3.4$
$3π43π4$ $22≈2.822≈2.8$ $7π47π4$ $−22≈−2.8−22≈−2.8$
$5π65π6$ $22$ $11π611π6$ $−2−2$
$2π2π$ 0
Figure 1.30 The graph of the function $r=4sinθr=4sinθ$ is a circle.

This is the graph of a circle. The equation $r=4sinθr=4sinθ$ can be converted into rectangular coordinates by first multiplying both sides by $r.r.$ This gives the equation $r2=4rsinθ.r2=4rsinθ.$ Next use the facts that $r2=x2+y2r2=x2+y2$ and $y=rsinθ.y=rsinθ.$ This gives $x2+y2=4y.x2+y2=4y.$ To put this equation into standard form, subtract $4y4y$ from both sides of the equation and complete the square:

$x2+y2−4y=0x2+(y2−4y)=0x2+(y2−4y+4)=0+4x2+(y−2)2=4.x2+y2−4y=0x2+(y2−4y)=0x2+(y2−4y+4)=0+4x2+(y−2)2=4.$

This is the equation of a circle with radius 2 and center $(0,2)(0,2)$ in the rectangular coordinate system.

Checkpoint 1.12

Create a graph of the curve defined by the function $r=4+4cosθ.r=4+4cosθ.$

The graph in Example 1.12 was that of a circle. The equation of the circle can be transformed into rectangular coordinates using the coordinate transformation formulas in Equation 1.8. Example 1.14 gives some more examples of functions for transforming from polar to rectangular coordinates.

Example 1.13

Transforming Polar Equations to Rectangular Coordinates

Rewrite each of the following equations in rectangular coordinates and identify the graph.

1. $θ=π3θ=π3$
2. $r=3r=3$
3. $r=6cosθ−8sinθr=6cosθ−8sinθ$

Solution

1. Take the tangent of both sides. This gives $tanθ=tan(π/3)=3.tanθ=tan(π/3)=3.$ Since $tanθ=y/xtanθ=y/x$ we can replace the left-hand side of this equation by $y/x.y/x.$ This gives $y/x=3,y/x=3,$ which can be rewritten as $y=x3.y=x3.$ This is the equation of a straight line passing through the origin with slope $3.3.$ In general, any polar equation of the form $θ=Kθ=K$ represents a straight line through the pole with slope equal to $tanK.tanK.$
2. First, square both sides of the equation. This gives $r2=9.r2=9.$ Next replace $r2r2$ with $x2+y2.x2+y2.$ This gives the equation $x2+y2=9,x2+y2=9,$ which is the equation of a circle centered at the origin with radius 3. In general, any polar equation of the form $r=kr=k$ where k is a positive constant represents a circle of radius k centered at the origin. (Note: when squaring both sides of an equation it is possible to introduce new points unintentionally. This should always be taken into consideration. However, in this case we do not introduce new points. For example, $(−3,π3)(−3,π3)$ is the same point as $(3,4π3).)(3,4π3).)$
3. Multiply both sides of the equation by $r.r.$ This leads to $r2=6rcosθ−8rsinθ.r2=6rcosθ−8rsinθ.$ Next use the formulas
$r2=x2+y2,x=rcosθ,y=rsinθ.r2=x2+y2,x=rcosθ,y=rsinθ.$

This gives
$r2=6(rcosθ)−8(rsinθ)x2+y2=6x−8y.r2=6(rcosθ)−8(rsinθ)x2+y2=6x−8y.$

To put this equation into standard form, first move the variables from the right-hand side of the equation to the left-hand side, then complete the square.
$x2+y2=6x−8yx2−6x+y2+8y=0(x2−6x)+(y2+8y)=0(x2−6x+9)+(y2+8y+16)=9+16(x−3)2+(y+4)2=25.x2+y2=6x−8yx2−6x+y2+8y=0(x2−6x)+(y2+8y)=0(x2−6x+9)+(y2+8y+16)=9+16(x−3)2+(y+4)2=25.$

This is the equation of a circle with center at $(3,−4)(3,−4)$ and radius 5. Notice that the circle passes through the origin since the center is 5 units away.
Checkpoint 1.13

Rewrite the equation $r=secθtanθr=secθtanθ$ in rectangular coordinates and identify its graph.

We have now seen several examples of drawing graphs of curves defined by polar equations. A summary of some common curves is given in the tables below. In each equation, a and b are arbitrary constants.

Figure 1.31
Figure 1.32

A cardioid is a special case of a limaçon (pronounced “lee-mah-son”), in which $a=ba=b$ or $a=−b.a=−b.$ The rose is a very interesting curve. Notice that the graph of $r=3sin2θr=3sin2θ$ has four petals. However, the graph of $r=3sin3θr=3sin3θ$ has three petals as shown.

Figure 1.33 Graph of $r=3sin3θ.r=3sin3θ.$

If the coefficient of $θθ$ is even, the graph has twice as many petals as the coefficient. If the coefficient of $θθ$ is odd, then the number of petals equals the coefficient. You are encouraged to explore why this happens. Even more interesting graphs emerge when the coefficient of $θθ$ is not an integer. For example, if it is rational, then the curve is closed; that is, it eventually ends where it started (Figure 1.34(a)). However, if the coefficient is irrational, then the curve never closes (Figure 1.34(b)). Although it may appear that the curve is closed, a closer examination reveals that the petals just above the positive x axis are slightly thicker. This is because the petal does not quite match up with the starting point.

Figure 1.34 Polar rose graphs of functions with (a) rational coefficient and (b) irrational coefficient. Note that the rose in part (b) would actually fill the entire circle if plotted in full.

Since the curve defined by the graph of $r=3sin(πθ)r=3sin(πθ)$ never closes, the curve depicted in Figure 1.34(b) is only a partial depiction. In fact, this is an example of a space-filling curve. A space-filling curve is one that in fact occupies a two-dimensional subset of the real plane. In this case the curve occupies the circle of radius 3 centered at the origin.

Example 1.14

Chapter Opener: Describing a Spiral

Recall the chambered nautilus introduced in the chapter opener. This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. Figure 1.35 shows a spiral in rectangular coordinates. How can we describe this curve mathematically?

Figure 1.35 How can we describe a spiral graph mathematically?

Solution

As the point P travels around the spiral in a counterclockwise direction, its distance d from the origin increases. Assume that the distance d is a constant multiple k of the angle $θθ$ that the line segment OP makes with the positive x-axis. Therefore $d(P,O)=kθ,d(P,O)=kθ,$ where $OO$ is the origin. Now use the distance formula and some trigonometry:

$d(P,O)=kθ(x−0)2+(y−0)2=karctan(yx)x2+y2=karctan(yx)arctan(yx)=x2+y2ky=xtan(x2+y2k).d(P,O)=kθ(x−0)2+(y−0)2=karctan(yx)x2+y2=karctan(yx)arctan(yx)=x2+y2ky=xtan(x2+y2k).$

Although this equation describes the spiral, it is not possible to solve it directly for either x or y. However, if we use polar coordinates, the equation becomes much simpler. In particular, $d(P,O)=r,d(P,O)=r,$ and $θθ$ is the second coordinate. Therefore the equation for the spiral becomes $r=kθ.r=kθ.$ Note that when $θ=0θ=0$ we also have $r=0,r=0,$ so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes $r=a+kθr=a+kθ$ for arbitrary constants $aa$ and $k.k.$ This is referred to as an Archimedean spiral, after the Greek mathematician Archimedes.

Another type of spiral is the logarithmic spiral, described by the function $r=a·bθ.r=a·bθ.$ A graph of the function $r=1.2(1.25θ)r=1.2(1.25θ)$ is given in Figure 1.36. This spiral describes the shell shape of the chambered nautilus.

Figure 1.36 A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)

Suppose a curve is described in the polar coordinate system via the function $r=f(θ).r=f(θ).$ Since we have conversion formulas from polar to rectangular coordinates given by

$x=rcosθy=rsinθ,x=rcosθy=rsinθ,$

it is possible to rewrite these formulas using the function

$x=f(θ)cosθy=f(θ)sinθ.x=f(θ)cosθy=f(θ)sinθ.$

This step gives a parameterization of the curve in rectangular coordinates using $θθ$ as the parameter. For example, the spiral formula $r=a+bθr=a+bθ$ from Figure 1.31 becomes

$x=(a+bθ)cosθy=(a+bθ)sinθ.x=(a+bθ)cosθy=(a+bθ)sinθ.$

Letting $θθ$ range from $−∞−∞$ to $∞∞$ generates the entire spiral.

Symmetry in Polar Coordinates

When studying symmetry of functions in rectangular coordinates (i.e., in the form $y=f(x)),y=f(x)),$ we talk about symmetry with respect to the y-axis and symmetry with respect to the origin. In particular, if $f(−x)=f(x)f(−x)=f(x)$ for all $xx$ in the domain of $f,f,$ then $ff$ is an even function and its graph is symmetric with respect to the y-axis. If $f(−x)=−f(x)f(−x)=−f(x)$ for all $xx$ in the domain of $f,f,$ then $ff$ is an odd function and its graph is symmetric with respect to the origin. By determining which types of symmetry a graph exhibits, we can learn more about the shape and appearance of the graph. Symmetry can also reveal other properties of the function that generates the graph. Symmetry in polar curves works in a similar fashion.

Theorem 1.5

Symmetry in Polar Curves and Equations

Consider a curve generated by the function $r=f(θ)r=f(θ)$ in polar coordinates.

1. The curve is symmetric about the polar axis if for every point $(r,θ)(r,θ)$ on the graph, the point $(r,−θ)(r,−θ)$ is also on the graph. Similarly, the equation $r=f(θ)r=f(θ)$ is unchanged by replacing $θθ$ with $−θ.−θ.$
2. The curve is symmetric about the pole if for every point $(r,θ)(r,θ)$ on the graph, the point $(r,π+θ)(r,π+θ)$ is also on the graph. Similarly, the equation $r=f(θ)r=f(θ)$ is unchanged when replacing $rr$ with $−r,−r,$ or $θθ$ with $π+θ.π+θ.$
3. The curve is symmetric about the vertical line $θ=π2θ=π2$ if for every point $(r,θ)(r,θ)$ on the graph, the point $(r,π−θ)(r,π−θ)$ is also on the graph. Similarly, the equation $r=f(θ)r=f(θ)$ is unchanged when $θθ$ is replaced by $π−θ.π−θ.$

The following table shows examples of each type of symmetry.

Example 1.15

Using Symmetry to Graph a Polar Equation

Find the symmetry of the rose defined by the equation $r=3sin(2θ)r=3sin(2θ)$ and create a graph.

Solution

Suppose the point $(r,θ)(r,θ)$ is on the graph of $r=3sin(2θ).r=3sin(2θ).$

1. To test for symmetry about the polar axis, first try replacing $θθ$ with $−θ.−θ.$ This gives $r=3sin(2(−θ))=−3sin(2θ).r=3sin(2(−θ))=−3sin(2θ).$ Since this changes the original equation, this test is not satisfied. However, returning to the original equation and replacing $rr$ with $−r−r$ and $θθ$ with $π−θπ−θ$ yields
$−r=3sin(2(π−θ))−r=3sin(2π−2θ)−r=3sin(−2θ)−r=−3sin2θ.−r=3sin(2(π−θ))−r=3sin(2π−2θ)−r=3sin(−2θ)−r=−3sin2θ.$

Multiplying both sides of this equation by $−1−1$ gives $r=3sin2θ,r=3sin2θ,$ which is the original equation. This demonstrates that the graph is symmetric with respect to the polar axis.
2. To test for symmetry with respect to the pole, first replace $rr$ with $−r,−r,$ which yields $−r=3sin(2θ).−r=3sin(2θ).$ Multiplying both sides by −1 gives $r=−3sin(2θ),r=−3sin(2θ),$ which does not agree with the original equation. Therefore the equation does not pass the test for this symmetry. However, returning to the original equation and replacing $θθ$ with $θ+πθ+π$ gives
$r=3sin(2(θ+π))=3sin(2θ+2π)=3(sin2θcos2π+cos2θsin2π)=3sin2θ.r=3sin(2(θ+π))=3sin(2θ+2π)=3(sin2θcos2π+cos2θsin2π)=3sin2θ.$

Since this agrees with the original equation, the graph is symmetric about the pole.
3. To test for symmetry with respect to the vertical line $θ=π2,θ=π2,$ first replace both $rr$ with $−r−r$ and $θθ$ with $−θ.−θ.$
$−r=3sin(2(−θ))−r=3sin(−2θ)−r=−3sin2θ.−r=3sin(2(−θ))−r=3sin(−2θ)−r=−3sin2θ.$

Multiplying both sides of this equation by $−1−1$ gives $r=3sin2θ,r=3sin2θ,$ which is the original equation. Therefore the graph is symmetric about the vertical line $θ=π2.θ=π2.$

This graph has symmetry with respect to the polar axis, the origin, and the vertical line going through the pole. To graph the function, tabulate values of $θθ$ between 0 and $π/2π/2$ and then reflect the resulting graph.

$θθ$ $rr$
$00$ $00$
$π6π6$ $332≈2.6332≈2.6$
$π4π4$ $33$
$π3π3$ $332≈2.6332≈2.6$
$π2π2$ $00$

This gives one petal of the rose, as shown in the following graph.

Figure 1.37 The graph of the equation between $θ=0θ=0$ and $θ=π/2.θ=π/2.$

Reflecting this image into the other three quadrants gives the entire graph as shown.

Figure 1.38 The entire graph of the equation is called a four-petaled rose.
Checkpoint 1.14

Determine the symmetry of the graph determined by the equation $r=2cos(3θ)r=2cos(3θ)$ and create a graph.

Section 1.3 Exercises

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle $θθ$ and then marking off the distance r along the ray.

125.

$(3,π6)(3,π6)$

126.

$(−2,5π3)(−2,5π3)$

127.

$(0,7π6)(0,7π6)$

128.

$(−4,3π4)(−4,3π4)$

129.

$(1,π4)(1,π4)$

130.

$(2,5π6)(2,5π6)$

131.

$(1,π2)(1,π2)$

For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.

132.

Coordinates of point A.

133.

Coordinates of point B.

134.

Coordinates of point C.

135.

Coordinates of point D.

For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in $(0,2π].(0,2π].$ Round to three decimal places.

136.

$(2,2)(2,2)$

137.

$(3,−4)(3,−4)$ (3, −4)

138.

$(8,15)(8,15)$

139.

$(−6,8)(−6,8)$

140.

$(4,3)(4,3)$

141.

$(3,−3)(3,−3)$

For the following exercises, find rectangular coordinates for the given point in polar coordinates.

142.

$(2,5π4)(2,5π4)$

143.

$(−2,π6)(−2,π6)$

144.

$(5,π3)(5,π3)$

145.

$(1,7π6)(1,7π6)$

146.

$(−3,3π4)(−3,3π4)$

147.

$(0,π2)(0,π2)$

148.

$(−4.5,6.5)(−4.5,6.5)$

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the $xx$-axis, the $yy$-axis, or the origin.

149.

$r=3sin(2θ)r=3sin(2θ)$

150.

$r2=9cosθr2=9cosθ$

151.

$r=cos(θ5)r=cos(θ5)$

152.

$r=2secθr=2secθ$

153.

$r=1+cosθr=1+cosθ$

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

154.

$r=3r=3$

155.

$θ=π4θ=π4$

156.

$r=secθr=secθ$

157.

$r=cscθr=cscθ$

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

158.

$x2+y2=16x2+y2=16$

159.

$x2−y2=16x2−y2=16$

160.

$x=8x=8$

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

161.

$3x−y=23x−y=2$

162.

$y2=4xy2=4x$

For the following exercises, convert the polar equation to rectangular form and sketch its graph.

163.

$r=4sinθr=4sinθ$

164.

$r=6cosθr=6cosθ$

165.

$r=θr=θ$

166.

$r=cotθcscθr=cotθcscθ$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

167.

$r=1+sinθr=1+sinθ$

168.

$r=3−2cosθr=3−2cosθ$

169.

$r=2−2sinθr=2−2sinθ$

170.

$r=5−4sinθr=5−4sinθ$

171.

$r=3cos(2θ)r=3cos(2θ)$

172.

$r=3sin(2θ)r=3sin(2θ)$

173.

$r=2cos(3θ)r=2cos(3θ)$

174.

$r=3cos(θ2)r=3cos(θ2)$

175.

$r2=4cos(2θ)r2=4cos(2θ)$

176.

$r2=4sinθr2=4sinθ$

177.

$r=2θr=2θ$

178.

[T] The graph of $r=2cos(2θ)sec(θ).r=2cos(2θ)sec(θ).$ is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.

179.

[T] Use a graphing utility and sketch the graph of $r=62sinθ−3cosθ.r=62sinθ−3cosθ.$

180.

[T] Use a graphing utility to graph $r=11−cosθ.r=11−cosθ.$

181.

[T] Use technology to graph $r=esin(θ)−2cos(4θ).r=esin(θ)−2cos(4θ).$

182.

[T] Use technology to plot $r=sin(3θ7)r=sin(3θ7)$ (use the interval $0≤θ≤14π).0≤θ≤14π).$

183.

Without using technology, sketch the polar curve $θ=2π3.θ=2π3.$

184.

[T] Use a graphing utility to plot $r=θsinθr=θsinθ$ for $−π≤θ≤π.−π≤θ≤π.$

185.

[T] Use technology to plot $r=e−0.1θr=e−0.1θ$ for $−10≤θ≤10.−10≤θ≤10.$

186.

[T] There is a curve known as the “Black Hole.” Use technology to plot $r=e−0.01θr=e−0.01θ$ for $−100≤θ≤100.−100≤θ≤100.$

187.

[T] Use the results of the preceding two problems to explore the graphs of $r=e−0.001θr=e−0.001θ$ and $r=e−0.0001θr=e−0.0001θ$ for $|θ|>100.|θ|>100.$

format: plain


text:
Article Contents
Article Contents

# Minimum 'energy' approximations of invariant measures for nonsingular transformations

• We study variational methods for rigorous approximation of invariant densities for a nonsingular map $T$ on a Borel measure space. The general method takes the form of a convergent sequence of optimization problems on $L^p$, $1 \leq p < \infty$ with a convex objective and finite moment constraints. Provided $T$ admits an invariant density in the appropriate $L^p$ space, weak convergence of the sequence of optimal solutions is observed; norm convergence can be obtained when the objective is a Kadec functional. No regularity or expansiveness assumptions on $T$ need to be made, and the method applies to maps on multidimensional domains. Objectives leading to norm convergence include Entropy, 'Energy' and 'Positively Constrained Energy'.
Explicit solutions for the finite moment problems in the case of the 'Energy' functional are derived using duality - the optimality condition is then a linear algebra problem. Strong duality is obtained even though the dual functional may not be coercive and the set of moment test functions is not assumed to be pseudo-Haar. Finally, some numerical studies are presented for the case of moment test functions derived from a finite partition of the dynamical phase space and the results are compared with Ulam's method.
Mathematics Subject Classification: Primary: 28D05; Secondary: 37M25, 41A46, 49K27.

 Citation:


text:
# MT Generate Symbol Map (MT Generate Symbol Map (Circular)) (G Dataflow)

Version:

Generates a symbol map, using the matrix of symbol values that you specify, a matrix representing the symbol phases in degrees, and an array representing the radii of each discrete circle. Use the symbol map parameter of the system parameters cluster to configure the operation of the demodulation and modulation nodes.

The symbol values matrix defines how the M-ary binary data are mapped to the discrete I/Q symbols on the constellation diagram. The node maps the specified symbol values to I/Q constellation points using the polar representation $j\pi \frac{{\theta }_{i}}{180}$. The node reads the symbol values/symbol phases matrices row-wise. Each row of symbol values/symbol phases is then mapped onto I/Q symbol points, which are located on a circle with a radius that is specified in the symbol radii array.

Note

The dimensions of the symbol amplitudes matrix must exactly match the dimensions of the symbol phases matrix. Additionally, the number of rows of each matrix must be equal the length of the symbol radii array. If these conditions are violated, the node returns an error.

## symbol values

A matrix of integers that define how the M-ary (bits per symbol) binary data are mapped to discrete I/Q symbols on the constellation diagram. Use negative values to specify holes (missing points) in the constellation diagram.

Note

The binary representation of the symbol values is performed using the LSB first convention. For example, in 8-QAM, the symbol values matrix can be specified as

$\left[\begin{array}{ccc}0& 1& 2\\ 3& -1& 4\\ 5& 6& 7\end{array}\right]$

This matrix corresponds to the following bit values:

$\left[\begin{array}{ccc}000& 100& 010\\ 110& XXX& 001\\ 101& 011& 111\end{array}\right]$

The term XXX here refers to a hole (missing point) on the constellation plot.

Default: empty

## symbol phases

Phases, in degrees, corresponding to the location of the elements of the symbol values matrix on the I/Q constellation diagram.

Default: empty

Radii of discrete concentric circles that contain the I/Q constellation points. There is a one-to-one mapping between every entry on the symbol radii array and every row in the symbol values/symbol phases matrix.

Default: empty

## error in

Error conditions that occur before this node runs.

The node responds to this input according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

Default: No error

## symbol map

An ordered array to be bundled with the system parameters cluster that is passed to the digital demodulation and modulation nodes.

The generated symbol maps must be bundled onto the system parameters cluster prior to being passed to the corresponding modulation node.

## error out

Error information.

The node produces this output according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

## Generic or Combined Amplitude Modulation Scheme

Use MT Generate Symbol Map (Rectangular) to create a custom symbol map for a generic amplitude modulation scheme such as quadrature amplitude modulation (QAM). Use MT Generate Symbol Map (Circular) to create a custom symbol map for a combined amplitude-phase modulation scheme.

## Example: 8-QAM Symbol Map Creation

The following example demonstrates 8-QAM symbol map creation. Define the symbol values as $\left[\begin{array}{ccc}0& 1& 2\\ 3& -1& 4\\ 5& 6& 7\end{array}\right]$.

Define the I symbol amplitude array as $\left[\begin{array}{ccc}-1& 0& 1\end{array}\right]$.

Define the Q symbol amplitude array as $\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]$.

Define gain as 1.0.

These settings result in the following symbol map.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported

Web Server: Not supported in VIs that run in a web application


text:
# Beta(2)-adrenergic receptor desensitization, internalization and phosphorylation in response to full and partial agonists

#### Abstract

The objective of this study is to test the hypothesis that partial agonists produce less desensitization because they generate less of the active conformation of the $\beta\sb2$-adrenergic receptor ($\beta$AR) (R*) and in turn cause less $\beta$AR phosphorylation by beta adrenergic receptor kinase ($\beta$ARK) and less $\beta$AR internalization. In the present work, rates of desensitization, internalization, and phosphorylation caused by a series of $\beta$AR agonists were correlated with a quantitative measure, defined as coupling efficiency, of agonist-dependent $\beta$AR activation of adenylyl cyclase. These studies were preformed in HEK-293 cells overexpressing the $\beta$AR with hemagglutinin (HA) and 6-histidine (6HIS) epitopes introduced into the N- and C-termini respectively. Agonists chosen provided a 95-fold range of coupling efficiencies, and, relative to epinephrine, the best agonist, (100%) were fenoterol (42%), albuterol (4.9%), dobutamine (2.5%) and ephedrine (1.1%). At concentrations of these agonists yielding $>$90% receptor occupancy, the rate and extent of the rapid phase (0-30 min) of agonist induced desensitization of adenylyl cyclase followed the same order as coupling efficiency, that is, epinephrine $\ge$ fitnoterol $>$ albuterol $>$ dobutamine $>$ ephedrine. The rate of internalization, measured by a loss of surface receptors during desensitization, with respect to these agonists also followed the same order as the desensitization and exhibited a slight lag. Like desensitization and internalization, $\beta$AR phosphorylation exhibited a dependency on agonist strength. The two strongest agonists epinephrine and fenoterol provoked 11 to 13 fold increases in the level of $\beta$AR phosphorylation after just 1 min, whereas the weakest agonists dobutamine and ephedrine caused only 3 to 4 fold increases in phosphorylation. With longer treatment times, the level of $\beta$AR phosphorylation declined with the strong agonists, but progressively increased with the weaker partial agonists. The major conclusion drawn from this study is that the occupancy-dependent rate of receptor phosphorylation increases with agonist coupling efficiencies and that this is sufficient to explain the desensitization, internalization, and phosphorylation data obtained. The mechanism of activation and desensitization by the partial $\beta$AR agonist salmeterol was also examined in this study. This drug is extremely hydrophobic and its study presents possibly unique problems. To determine whether salmeterol induces desensitization of the $\beta$AR its action has been studied using our system. Employing the use of reversible antagonists it was found that salmeterol, which has an estimated coupling efficiency near that of albuterol caused $\beta$AR desensitization. This desensitization was much reduced relative to epinephrine. Consistent with its coupling efficiency, it was found to be similar to albuterol in its ability to induce internalization and phosphorylation of the $\beta$AR. (Abstract shortened by UMI.)

Pharmacology

#### Recommended Citation

January, Bridgette Goodwin, "Beta(2)-adrenergic receptor desensitization, internalization and phosphorylation in response to full and partial agonists" (1997). Texas Medical Center Dissertations (via ProQuest). AAI9732770.
https://digitalcommons.library.tmc.edu/dissertations/AAI9732770

COinS


text:
# The figure shows an epicyclic gear train with gears A and B having 40 and 50 teeth, respectively. If the arm C rotates at 200 rpm in anticlockwise direction about the centre of gear A which is fixed, then speed of revolution of gear B in rpm would be

This question was previously asked in
ESE Mechanical 2013 Official Paper - 2
View all UPSC IES Papers >
1. 160
2. 250
3. 360
4. 450

Option 3 : 360

## Detailed Solution

Concept:

$$\frac{N_1}{N_2}~=~\frac{T_2}{T_1}$$

where N1 = rpm of gear 1, N2 = rpm of gear 2, T1 = Number of teeth in gear 1, T2 = Number of teeth in gear 2

Calculation:

Given:

Gear ‘A’ is stationary and Arm ‘C’ rotates at 200 rpm in an anticlockwise direction about the centre of gear A

Therefore, NC = 200 rpm, N= 0

Now, to solve such a problem first we follow the following steps

1. Consider Arm fixed
3. Apply all the given conditions
 Motion Arm A (40) B (50) Arm is Fixed (Let A rotate with x rpm in CW direction) 0 +x (CW) $$-\frac{T_A}{T_B}x~=-~\frac{40}{50}x~(CCW)$$ The arm rotates with +y rpm in CW y y + x (CW) $$y~-~\frac{4}{5}x$$ (CCW)

Now, by applying all the conditions, we have

A is fixed hence NA = 0

x + y = 0

x = - y        ----(1)

Also, the rpm of Arm C is 200

NC = 100

y = 200

x = - 200 rpm

The speed of B is

NB = 200 - 0.8(-200) = 360 rpm


text:
# zbMATH — the first resource for mathematics

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#### Citations contained in zbMATH Open

1,617 Publications have been cited 8,730 times in 7,718 Documents Cited by Year
A generalized Drazin inverse. Zbl 0897.47002
Koliha, J. J.
1996
On the distance of the composition of two derivations to the generalized derivations. Zbl 0731.47037
Brešar, Matej
1991
Some properties of non-commutative regular graded rings. Zbl 0824.16032
Levasseur, Thierry
1992
On products of idempotent matrices. Zbl 0157.07101
Erdos, J. A.
1967
Slant submanifolds in Sasakian manifolds. Zbl 0957.53022
Cabrerizo, J. L.; Carriazo, A.; Fernández, L. M.; Fernández, M.
2000
On an integral operator from the Zygmund space to the Bloch-type space on the unit ball. Zbl 1176.47029
Stević, Stevo
2009
Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux. (Generalization of the Kato decomposition to paranormal and spectral operators). Zbl 0657.47038
Mbekhta, Mostafa
1987
Rings in which elements are uniquely the sum of an idempotent and a unit. Zbl 1057.16007
Nicholson, W. K.; Zhou, Y.
2004
On quasi-duo rings. Zbl 0819.16001
Yu, Hua-Ping
1995
On $$\ell$$-adic representations attached to modular forms. II. Zbl 0596.10027
Ribet, Kenneth A.
1985
A non-abelian tensor product of Lie algebras. Zbl 0724.17016
Ellis, Graham J.
1991
A potential well theory for the wave equation with nonlinear source and boundary damping terms. Zbl 1016.35048
Vitillaro, Enzo
2002
Remarks on weak compactness in $$L_1(\mu,X)$$. Zbl 0342.46020
Diestel, J.
1977
$$c$$-supplemented subgroups of finite groups. Zbl 0968.20009
Ballester-Bolinches, A.; Wang, Yanming; Guo, Xiuyun
2000
On a periodic neutral logistic equation. Zbl 0737.34050
Gopalsamy, K.; He, Xue-Zhong; Wen, Lizhi
1991
The discrete and continuous Painlevé VI hierarchy and the Garnier systems. Zbl 0990.39015
Nijhoff, F. W.; Walker, A. J.
2001
On semi B-Fredholm operators. Zbl 0995.47008
Berkani, M.; Sarih, M.
2001
A common generalization of local cohomology theories. Zbl 0438.13009
1980
Sur la lacunarité des puissances de $$\eta$$ . (Lacunarity of powers of $$\eta$$ ). Zbl 0583.10015
Serre, Jean-Pierre
1985
Proof of a conjecture of Ramanujan. Zbl 0163.04302
Atkin, A. O. L.
1967
Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions. Zbl 0962.53015
Chen, Bang-Yen
1999
Convexity conditions for non-locally convex lattices. Zbl 0564.46004
Kalton, N. J.
1984
Minimal theta functions. Zbl 0639.10017
Montgomery, Hugh L.
1988
Asymptotic behavior of eigenvalues of greatest common divisor matrices. Zbl 1083.11021
Hong, Shaofang; Loewy, Raphael
2004
Mean curvature and shape operator of isometric immersions in real-space-forms. Zbl 0866.53038
Chen, Bang-Yen
1996
On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations. Zbl 1037.34041
Lipovan, Olivia
2003
Large automorphism groups of compact Klein surfaces with boundary. I. Zbl 0363.14008
May, Coy L.
1977
Hardy-Bloch type spaces and lacunary series on the polydisk. Zbl 1123.32004
Avetisyan, K. L.
2007
Character modules, dimension and purity. Zbl 0258.16028
Fieldhouse, David J.
1972
On the equivalence of Browder’s and generalized Browder’s theorem. Zbl 1097.47012
Amouch, M.; Zguitti, H.
2006
Hook representations of the symmetric groups. Zbl 0235.20012
Peel, M. H.
1971
Regular $$\omega$$-semigroups. Zbl 0165.33504
Munn, W. D.
1968
On successive approximations for nonexpansive mappings in Banach spaces. Zbl 0223.47024
Kirk, W. A.
1971
Automorphisms of compact non-orientable Riemann surfaces. Zbl 0232.30012
Singerman, D.
1971
Matched pairs of Lie algebroids. Zbl 0886.22012
Mokri, Tahar
1997
Approximate point spectrum and commuting compact perturbations. Zbl 0602.47003
1986
Some convexity theorems for matrices. Zbl 0256.15006
Fillmore, P. A.; Williams, J. P.
1971
Browder’s theorems and spectral continuity. Zbl 0979.47004
Djordjević, Slaviša; Han, Young Min
2000
On dual Baer modules. Zbl 1215.16007
Keskin Tütüncü, Derya; Tribak, Rachid
2010
Congrucences and Green’s relations on regular semigroups. Zbl 0257.20057
Hall, T. E.
1972
Asymptotic behavior of eigenvalues of reciprocal power LCM matrices. Zbl 1186.11011
Hong, Shaofang; Enoch Lee, K. S.
2008
A proof of some identities of Ramanujan using modular forms. Zbl 0679.10018
Biagioli, Anthony J. F.
1989
Solving the truncated moment problem solves the full moment problem. Zbl 0995.44004
Stochel, Jan
2001
Duo modules. Zbl 1116.16003
Özcan, A. Ç.; Harmanci, A.; Smith, P. F.
2006
A bilinear approach to a Pfaffian self-dual Yang-Mills equation. Zbl 0986.35109
Ohta, Y.; Nimmo, J. J. C.; Gilson, C. R.
2001
Hopf images and inner faithful representations. Zbl 1243.16033
Banica, Teodor; Bichon, Julien
2010
Poor modules: the opposite of injectivity. Zbl 1228.16004
2010
Valuations of near polygons. Zbl 1085.51009
De Bruyn, Bart; Vandecasteele, Pieter
2005
Axiomatisations of the average and a further generalisation of monotonic sequences. Zbl 0291.40003
Bibby, John
1974
On Littlewood-Paley functions associated with Bessel operators. Zbl 1151.42008
Betancor, J. J.; Fariña, J. C.; Sanabria, A.
2009
Star-complexes, and the dependence problems for hyperbolic complexes. Zbl 0684.20022
Pride, Stephen J.
1988
On non-Hurwitz groups and non-congruence subgroups of the modular group. Zbl 0452.20049
Cohen, Jeffrey
1981
Extended eigenvalues and the Volterra operator. Zbl 1037.47013
Biswas, Animikh; Lambert, Alan; Petrovic, Srdjan
2002
Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip. Zbl 1142.35067
Zelik, Sergey
2007
Stability of mappings on multi-normed spaces. Zbl 1125.39023
Dales, H. G.; Moslehian, Mohammad Sal
2007
A spectral mapping theorem for the Weyl spectrum. Zbl 0869.47017
Lee, Woo Young; Lee, Sang Hoon
1996
Consistency conditions for dimer models. Zbl 1244.14042
Bocklandt, Raf
2012
The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space. Zbl 0970.53015
Izumiya, Shyuichi; Pei, Donghe; Sano, Takasi
2000
Non-abelian exterior products of groups and exact sequences in the homology of groups. Zbl 0631.20040
Ellis, G. J.
1987
Weak compactness in locally convex spaces. Zbl 0159.41802
Tweddle, I.
1968
Lower bounds for blow-up time in some non-linear parabolic problems under Neumann boundary conditions. Zbl 1225.35037
Enache, Cristian
2011
Free right type A semigroups. Zbl 0737.20032
Fountain, John
1991
On the Gauss map of ruled surfaces. Zbl 0762.53004
Baikoussis, Christos; Blair, David E.
1992
The non-Abelian tensor product of groups and related constructions. Zbl 0681.20034
Gilbert, N. D.; Higgins, P. J.
1989
Class numbers and quadratic residues. Zbl 0323.12006
1976
$$\pi$$-domains, overrings, and divisorial ideals. Zbl 0379.13011
Anderson, D. D.
1978
Idempotents in completely O-simple semigroups. Zbl 0393.20044
Howie, John M.
1978
Prime numbers of the form $$[n^2]$$. Zbl 0987.11052
Rivat, Joël; Wu, Jie
2001
A generalization of PP-rings and p.q.-Baer rings. Zbl 1110.16003
Liu, Zhongkui; Zhao, Renyu
2006
The translational hull of a completely O-simple semigroup. Zbl 0179.04101
Petrich, M.
1968
Isoperimetric inequalities for $$L_{p}$$ geominimal surface area. Zbl 1235.52019
Zhu, Baocheng; Li, Ni; Zhou, Jiazu
2011
Perfect Pell powers. Zbl 0852.11014
Cohn, J. H. E.
1996
Maps preserving the local spectrum of product of operators. Zbl 1326.47041
2015
A note on inhomogeneous Diophantine approximation. Zbl 1039.11048
Bugeaud, Yann
2003
Biembeddings of Latin squares and Hamiltonian decompositions. Zbl 1062.05030
Grannell, M. J.; Griggs, T. S.; Knor, M.
2004
On the family of Diophantine triples $$\{k-1, k+1, 16k^3 - 4k\}$$. Zbl 1168.11007
Bugeaud, Yann; Dujella, Andrej; Mignotte, Maurice
2007
Automorphism groups of Riemann surfaces on genus $$p+1$$, where $$p$$ is prime. Zbl 1084.30050
Belolipetsky, Mikhail; Jones, Gareth A.
2005
The solution of some integral equations and their connection with dual integral equations and series. Zbl 0193.39901
Cooke, J. C.
1970
Duals of Banach spaces of entire functions. Zbl 0769.46011
Anderson, J. M.; Duncan, J.
1990
Quotient rings of semiprime rings with bounded index. Zbl 0472.16002
Hannah, John
1982
Almost factorisable inverse semigroups. Zbl 0820.20070
Lawson, M. V.
1994
Two-generator two-groups of class two and their nonabelian tensor squares. Zbl 0970.20012
Kappe, Luise-Charlotte; Visscher, Matthew P.; Sarmin, Nor Haniza
1999
Enumerative proofs of certain q-identities. Zbl 0146.01205
Andrews, G. E.
1967
Ptolemaic spaces and CAT(0). Zbl 1171.53024
Buckley, S. M.; Falk, K.; Wraith, D. J.
2009
Comparison theorems on the oscillation and asymptotic behaviour of higher-order neutral differential equations. Zbl 1216.34073
Karpuz, Başak; Öcalan, Özkan; Öztürk, Sermin
2010
The Boussinesq integrable system: compatible lattice and continuum structures. Zbl 1085.35125
Tongas, Anastasios; Nijhoff, Frank
2005
The irrationality of certain infinite series. Zbl 0629.10027
1987
Boundaries of nonpositively curved groups of the form $$G\times \mathbb{Z}^ n$$. Zbl 0861.53044
Bowers, Philip L.; Ruane, Kim
1996
A bipartite Ramsey problem and the Zarankiewicz numbers. Zbl 0367.05051
Irving, Robert W.
1978
On Weyl and Browder spectra of tensor products. Zbl 1136.47013
Kubrusly, C. S.; Duggal, B. P.
2008
A simple proof of the Goldberg-Straus theorem on numerical radii. Zbl 0605.15019
Tam, Bitshun
1986
Kernels of representations and coideal subalgebras of Hopf algebras. Zbl 1245.16024
Burciu, Sebastian
2012
Ore extensions of weak zip rings. Zbl 1186.16017
Ouyang, Lunqun
2009
Hermite function expansions versus Hermite polynomial expansions. Zbl 1129.42409
Abu-Falahah, I.; Torrea, J. L.
2006
An algebraically closed field. Zbl 0159.05302
Rayner, F. J.
1968
Free products of inverse semigroups. II. Zbl 0742.20057
Jones, Peter R.; Margolis, Stuart W.; Meakin, John; Stephen, Joseph B.
1991
Interpolating sequences for the derivatives of Bloch functions. Zbl 0751.30032
Attele, K. R. M.
1992
Low dimensional homotopy for monoids. II: Groups. Zbl 0932.20046
Pride, Stephen J.
1999
Groups with few non-nilpotent subgroups. Zbl 0883.20018
Smith, Howard
1997
On commutative Noetherian rings which satisfy the radical formula. Zbl 0921.13001
Leung, Ka Hin; Man, Shing Hing
1997
Detecting Steiner and linear isometries operads. Zbl 1464.55018
Rubin, Jonathan
2021
Closed ideals and Lie ideals of minimal tensor product of certain $$C$$*-algebras. Zbl 07333635
Talwar, Bharat; Jain, Ranjana
2021
On nonlocal nonlinear elliptic problems with the fractional Laplacian. Zbl 1429.35092
Ma, Li
2020
Flat ring epimorphisms of countable type. Zbl 1445.16002
Positselski, Leonid
2020
Local negativity of surfaces with non-negative Kodaira dimension and transversal configurations of curves. Zbl 1445.14016
Laface, Roberto; Pokora, Piotr
2020
Oriented flip graphs, noncrossing tree partitions, and representation theory of tiling algebras. Zbl 07141357
Garver, Alexander; McConville, Thomas
2020
The unital Ext-groups and classification of $$C^*$$-algebras. Zbl 1440.46051
Gabe, James; Ruiz, Efren
2020
Epsilon-strongly groupoid-graded rings, the Picard inverse category and cohomology. Zbl 07141361
Nystedt, Patrik; Öinert, Johan; Pinedo, Héctor
2020
Class number formula for dihedral extensions. Zbl 1450.11117
Caputo, Luca; Nuccio Mortarino Majno Di Capriglio, Filippo A. E.
2020
Decomposing the tube category. Zbl 1448.18033
Hardiman, Leonard; King, Alastair
2020
Nakayama automorphisms of Ore extensions over polynomial algebras. Zbl 1455.16010
Liu, Liyu; Ma, Wen
2020
Configuration categories and homotopy automorphisms. Zbl 1436.57036
Weiss, Michael S.
2020
Characterizations of Berger spheres from the viewpoint of submanifold theory. Zbl 1430.53064
Kim, Byung Hak; Kim, In-Bae; Maeda, Sadahiro
2020
How small can polynomials be in an interval of given length? Zbl 1439.11175
Bernik, Vasili; Mc Guire, Stephen
2020
A note on $$A$$-annihilated generators of $$H_* QX$$. Zbl 1442.55013
2020
On the arithmetic of Mori monoids and domains. Zbl 1440.13005
Zhong, Qinghai
2020
JSJ decompositions of doubles of free groups. Zbl 07194255
Heil, Simon
2020
Localizations of the hearts of cotorsion pairs. Zbl 07236510
Liu, Yu
2020
Fourier-type transforms on rearrangement-invariant quasi-Banach function spaces. Zbl 1440.42018
Ho, Kwok-Pun
2019
Exhaustion of the curve graph via rigid expansions. Zbl 06988392
Hernández Hernández, Jesús
2019
Distortion in the finite determination result for embeddings of locally finite metric spaces into Banach spaces. Zbl 1459.46028
Ostrovska, S.; Ostrovskii, M. I.
2019
Some results of the $$\mathcal{K}_{\mathcal{A}}$$-approximation property for Banach spaces. Zbl 1430.46013
Kim, Ju Myung
2019
Frequently hypercyclic bilateral shifts. Zbl 07114290
Grosse-Erdmann, Karl-G.
2019
Homotopy types of gauge groups over non-simplyconnected closed 4-manifolds. Zbl 1429.55008
So, Tseleung
2019
Focal surfaces of wave fronts in the Euclidean 3-space. Zbl 1429.57028
Teramoto, Keisuke
2019
Complex of relatively hyperbolic groups. Zbl 07093785
Pal, Abhijit; Paul, Suman
2019
Frame-less Hilbert C*-modules. Zbl 1458.46046
Asadi, M. B.; Frank, M.; Hassanpour-Yakhdani, Z.
2019
Prime spectra of ambiskew polynomial rings. Zbl 1445.16024
Fish, Christopher D.; Jordan, David A.
2019
Complex product structures on Hom-Lie algebras. Zbl 06988387
2019
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Bernatska, Julia; Leykin, Dmitry
2019
Higgs bundles over elliptic curves for complex reductive groups. Zbl 1433.14028
2019
Strong approximation theorem for absolutely irreducible varieties over the compositum of all symmetric extensions of a global field. Zbl 1439.12008
Jarden, Moshe; Razon, Aharon
2019
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Müller, Jürgen; Sarkar, Siddhartha
2019
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Tanaka, Yuuji
2019
Extreme points for combinatorial Banach spaces. Zbl 1435.46009
Beanland, Kevin; Duncan, Noah; Holt, Michael; Quigley, James
2019
Stratifying systems for exact categories. Zbl 1441.18011
Santiago, Valente
2019
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Gongopadhyay, Krishnendu; Mukherjee, Abhishek; Sardar, Sujit Kumar
2019
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Herscovich, Estanislao
2019
$$r$$-Fredholm theory in Banach algebras. Zbl 1431.46030
Benjamin, Ronalda; Laustsen, Niels Jakob; Mouton, Sonja
2019
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Goldbring, Isaac; Sinclair, Thomas
2019
Nonincreasing depth functions of monomial ideals. Zbl 1421.13001
Matsuda, Kazunori; Suzuki, Tao; Tsuchiya, Akiyoshi
2018
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Zhang, Xiaohui; Dong, Lihong
2018
Pure-injectivity from a different perspective. Zbl 1441.16004
López-Permouth, S. R.; Mastromatteo, J.; Tolooei, Y.; Ungor, B.
2018
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Berger, Roland; Lambre, Thierry; Solotar, Andrea
2018
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Rebolledo, Marusia; Wuthrich, Christian
2018
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Beil, Charlie
2018
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Nuño-Ballesteros, J. J.; Oréfice-Okamoto, B.; Tomazella, J. N.
2018
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Kalaj, David
2018
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Finite groups with Engel sinks of bounded rank. Zbl 06909856
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2018
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Tarcsay, Zsigmond; Titkos, Tamás
2018
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Masbaum, Gregor
2018
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Pooya, Sanaz; Valette, Alain
2018
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Lin, Hezi
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2018
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Polycyclic, metabelian or soluble of type $$(\mathrm{FP})_\infty$$ groups with Boolean algebra of rational sets and biautomatic soluble groups are virtually abelian. Zbl 1427.20041
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2017
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 61 Stević, Stevo 42 Duggal, Bhagwati Prashad 36 Mosić, Dijana 32 Zhang, James J. 31 Gould, Victoria A. R. 30 Zhou, Yiqiang 29 Petrich, Mario 25 Chen, Huanyin 25 De Bruyn, Bart 24 de Giovanni, Francesco 23 Bujalance, Emilio 23 Dolinka, Igor 22 Costa Gonzáles, Antonio Félix 21 Chen, Jianlong 21 Moussavi, Ahmad 21 Niroomand, Peyman 21 O’Regan, Donal 20 Jones, Peter R. 19 Aiena, Pietro 19 Araújo, João 19 Ding, Nanqing 19 Đorđević, Dragan S. 18 East, James 18 Konieczny, Janusz 18 Li, Songxiao 17 Agarwal, Ravi P. 17 Hu, Xingbiao 17 Ladra González, Manuel 17 Siraj-Uddin 16 Betancor Perez, Jorge Juan 16 Etayo Gordejuela, José Javier 16 Hong, Shaofang 16 Lee, Yang 16 Martinez, Ernesto C. 16 May, Coy L. 15 Conder, Marston D. E. 15 Gomes, Gracinda M. S. 15 Li, Chi-Kwong 15 López-Permouth, Sergio Roberto 15 Mao, Lixin 15 Paykan, Kamal 15 Rashid, Mohammad Hussein Mohammad 15 Nadeem-Ur-Rehman 15 Salemkar, Ali Reza 14 Grannell, Michael John 14 Ricker, Werner Joseph 14 Tuganbaev, Askar Akanovich 13 Bétermin, Laurent 13 Facchini, Alberto 13 Guo, Xiaojiang 13 Johari, Farangis 13 Koşan, Muhammet Tamer 13 Lee, Tsiu-Kwen 12 Dieulefait, Luis Victor 12 Ellis, Graham J. 12 Fountain, John Barringham 12 González Ortiz, Manuel 12 Huynh, Dinh Van 12 Jeribi, Aref 12 Kirkman, Ellen E. 12 Liu, Cheng-Kai 12 Ono, Ken 12 Papageorgiou, Nikolaos S. 12 Puczyłowski, Edmund R. 12 Shum, Kar-Ping 12 Shumyatsky, Pavel 12 Yousif, Mohamed F. 12 Zhang, Dajun 12 Zhong, Huaijie 11 Albiac, Fernando 11 Birkenmeier, Gary Francis 11 Brown, Kenneth Alexander 11 Deng, Chunyuan 11 Goodearl, Kenneth R. 11 Griggs, Terry S. 11 Guo, Wenbin 11 Harte, Robin E. 11 Izquierdo, Milagros 11 Jespers, Eric 11 Kurdachenko, Leonid Andriĭovych 11 Liu, Zhong-kui 11 Mihai, Ion 11 Park, Jae Keol 11 Parvizi, Mohsen 11 Širáň, Jozef 11 Smith, Patrick Fleming 11 Szendrei, Mária B. 11 Togbé, Alain 11 Tsing, Nam-Kiu 11 Wei, Yimin 10 Álvarez Seco, Teresa 10 Beidar, Konstantin Igorevich 10 Bourhim, Abdellatif 10 Casas Miras, José Manuel 10 Castro, Alejandro J. 10 Chen, Bang-Yen 10 de Lima, Henrique F. 10 Dixon, Martyn Russell 10 Djordjević, Slaviša V. 10 Guo, Xiuyun ...and 7,427 more Authors
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#### Cited in 590 Journals

 489 Communications in Algebra 375 Journal of Algebra 318 Journal of Mathematical Analysis and Applications 208 Proceedings of the American Mathematical Society 197 Semigroup Forum 157 Journal of Pure and Applied Algebra 141 Linear Algebra and its Applications 135 Journal of Algebra and its Applications 132 Glasgow Mathematical Journal 123 Transactions of the American Mathematical Society 107 Journal of Number Theory 103 Applied Mathematics and Computation 94 Linear and Multilinear Algebra 89 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 83 Advances in Mathematics 81 Mediterranean Journal of Mathematics 76 Archiv der Mathematik 71 Mathematische Zeitschrift 66 Rocky Mountain Journal of Mathematics 63 Israel Journal of Mathematics 63 Journal of Functional Analysis 61 Rendiconti del Circolo Matemàtico di Palermo. Serie II 60 International Journal of Algebra and Computation 56 Monatshefte für Mathematik 53 Topology and its Applications 52 Bulletin of the Australian Mathematical Society 52 Journal of Mathematical Sciences (New York) 51 Integral Equations and Operator Theory 50 Discrete Mathematics 50 Journal of Geometry and Physics 50 Czechoslovak Mathematical Journal 50 Quaestiones Mathematicae 50 Abstract and Applied Analysis 48 Acta Mathematica Sinica. English Series 48 International Journal of Number Theory 45 Algebra Colloquium 44 The Ramanujan Journal 43 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 43 Journal of Inequalities and Applications 41 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 40 Journal of Mathematical Physics 40 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 39 Manuscripta Mathematica 37 Mathematische Annalen 37 Results in Mathematics 35 Indagationes Mathematicae. New Series 35 Algebras and Representation Theory 32 Acta Mathematica Hungarica 32 Journal of the Australian Mathematical Society 31 Journal of Soviet Mathematics 31 Turkish Journal of Mathematics 31 Positivity 30 Geometriae Dedicata 30 Bulletin of the Iranian Mathematical Society 30 Applied Mathematics Letters 29 Complex Analysis and Operator Theory 28 Mathematika 28 Comptes Rendus. Mathématique. Académie des Sciences, Paris 28 Asian-European Journal of Mathematics 27 Ukrainian Mathematical Journal 26 Annali di Matematica Pura ed Applicata. Serie Quarta 26 Mathematische Nachrichten 26 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 25 Communications in Mathematical Physics 25 Beiträge zur Algebra und Geometrie 25 Algebra Universalis 25 Finite Fields and their Applications 24 Journal of Differential Equations 24 Proceedings of the Edinburgh Mathematical Society. Series II 24 Differential Geometry and its Applications 23 Periodica Mathematica Hungarica 23 European Journal of Combinatorics 23 Journal of Group Theory 22 International Journal of Mathematics and Mathematical Sciences 22 Rendiconti del Seminario Matematico della Università di Padova 22 Siberian Mathematical Journal 22 Frontiers of Mathematics in China 21 Compositio Mathematica 20 Indian Journal of Pure & Applied Mathematics 20 Mathematical Notes 20 Journal of Combinatorial Theory. Series A 20 Journal of Geometry 20 Numerical Functional Analysis and Optimization 20 International Journal of Mathematics 20 Science China. Mathematics 19 Computers & Mathematics with Applications 19 Illinois Journal of Mathematics 19 Tohoku Mathematical Journal. Second Series 19 Journal de Théorie des Nombres de Bordeaux 19 Complex Variables and Elliptic Equations 18 Proceedings of the Japan Academy. Series A 18 The Journal of Geometric Analysis 18 Open Mathematics 17 Mathematical Methods in the Applied Sciences 17 Mathematical Proceedings of the Cambridge Philosophical Society 17 Algebraic & Geometric Topology 17 Fixed Point Theory and Applications 17 Advances in Difference Equations 16 Inventiones Mathematicae 16 Revista Matemática Iberoamericana ...and 490 more Journals
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#### Cited in 63 Fields

 1,497 Group theory and generalizations (20-XX) 1,324 Associative rings and algebras (16-XX) 1,100 Operator theory (47-XX) 903 Number theory (11-XX) 809 Functional analysis (46-XX) 550 Differential geometry (53-XX) 464 Partial differential equations (35-XX) 387 Combinatorics (05-XX) 382 Algebraic geometry (14-XX) 345 Functions of a complex variable (30-XX) 336 Commutative algebra (13-XX) 325 Nonassociative rings and algebras (17-XX) 306 Linear and multilinear algebra; matrix theory (15-XX) 246 Ordinary differential equations (34-XX) 235 Dynamical systems and ergodic theory (37-XX) 224 Manifolds and cell complexes (57-XX) 210 Category theory; homological algebra (18-XX) 166 Several complex variables and analytic spaces (32-XX) 160 Order, lattices, ordered algebraic structures (06-XX) 141 Global analysis, analysis on manifolds (58-XX) 120 Harmonic analysis on Euclidean spaces (42-XX) 116 Topological groups, Lie groups (22-XX) 113 General algebraic systems (08-XX) 113 General topology (54-XX) 111 Difference and functional equations (39-XX) 97 Measure and integration (28-XX) 96 Algebraic topology (55-XX) 92 Real functions (26-XX) 81 Special functions (33-XX) 79 Quantum theory (81-XX) 73 Geometry (51-XX) 68 Computer science (68-XX) 67 Probability theory and stochastic processes (60-XX) 66 Numerical analysis (65-XX) 64 Field theory and polynomials (12-XX) 62 Mathematical logic and foundations (03-XX) 62 Convex and discrete geometry (52-XX) 62 Fluid mechanics (76-XX) 61 Abstract harmonic analysis (43-XX) 56 Integral transforms, operational calculus (44-XX) 55 Integral equations (45-XX) 53 Biology and other natural sciences (92-XX) 50 Mechanics of deformable solids (74-XX) 47 Calculus of variations and optimal control; optimization (49-XX) 44 Statistical mechanics, structure of matter (82-XX) 43 Potential theory (31-XX) 36 Systems theory; control (93-XX) 27 $$K$$-theory (19-XX) 26 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 22 Information and communication theory, circuits (94-XX) 21 Sequences, series, summability (40-XX) 20 Approximations and expansions (41-XX) 20 Operations research, mathematical programming (90-XX) 18 History and biography (01-XX) 16 Relativity and gravitational theory (83-XX) 9 Mechanics of particles and systems (70-XX) 8 General and overarching topics; collections (00-XX) 7 Statistics (62-XX) 4 Optics, electromagnetic theory (78-XX) 3 Classical thermodynamics, heat transfer (80-XX) 2 Geophysics (86-XX) 1 Astronomy and astrophysics (85-XX) 1 Mathematics education (97-XX)


text:
# Always Related?

Logic Level 3

True or False?

In classical logic, for any truth values of two propositions $$p$$ and $$q$$, at least one of the following statements is true:

$\begin{eqnarray} p &\implies& q \\ q &\implies& p. \end{eqnarray}$

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