text:
# Cohomological Automorphic Representations on Unitary Groups

Consider the family of automorphic representations on some unitary group with fixed (possibly non-tempered) cohomological representation π0 at infinity and level dividing some finite upper bound. We compute statistics of this family as the level restriction goes to infinity. For unramified unitary groups and a large class of π0, we are able to compute the exact leading term for both counts of representations and averages of Satake parameters. We get bounds on our error term similar to previous work by Shin-Templier that studied the case of discrete series at infinity. We also discuss in-progress computations towards expected corollaries related to the Sarnak-Xue density conjecture, average Sato-Tate equidistribution in families, and growth of cohomology for towers of locally symmetric spaces.

The main technical tool is an extension of an inductive argument that was originally developed by Taïbi to count unramified representations on Sp and SO and used the endoscopic classification of representations (which our case requires for non-quasisplit unitary groups).

This is joint work with Mathilde Gerbelli-Gauthier

### Affiliation

Johns Hopkins University

Rahul Dalal


text:
## Circuit Lab B/C

azboy1910
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### Re: Circuit Lab B/C

Alright,

1. Explain how an inductor functions in a circuit.
2. A 7-volt battery is in series with a 2.5k-ohm resistor and a 55 mF capacitor, forming an RC circuit. How long will it take for the capacitor to be charged to 3.2 volts? What is the charge on the capacitor at this state?
3. At the peak of a standard AC U.S. graph, what is the voltage?
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### Re: Circuit Lab B/C

azboy1910 wrote:
October 25th, 2020, 11:51 am
Alright,

1. Explain how an inductor functions in a circuit.
2. A 7-volt battery is in series with a 2.5k-ohm resistor and a 55 mF capacitor, forming an RC circuit. How long will it take for the capacitor to be charged to 3.2 volts? What is the charge on the capacitor at this state?
3. At the peak of a standard AC U.S. graph, what is the voltage?
1. It acts to resist changes in current by providing back EMFs through Faraday's law.
2. $3.2 = 7\left(1-e^{-\frac{t}{2500\cdot0.055}}\right)$, $t = 84\,\mathrm{s}.$
$Q = CV = 0.055 \cdot 3.2 = 0.18\,\mathrm{C}.$
3. 60root2 V? Is this for household application or high-voltage wires?

azboy1910
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### Re: Circuit Lab B/C

UTF-8 U+6211 U+662F wrote:
October 25th, 2020, 1:38 pm
azboy1910 wrote:
October 25th, 2020, 11:51 am
Alright,

1. Explain how an inductor functions in a circuit.
2. A 7-volt battery is in series with a 2.5k-ohm resistor and a 55 mF capacitor, forming an RC circuit. How long will it take for the capacitor to be charged to 3.2 volts? What is the charge on the capacitor at this state?
3. At the peak of a standard AC U.S. graph, what is the voltage?
1. It acts to resist changes in current by providing back EMFs through Faraday's law.
2. $3.2 = 7\left(1-e^{-\frac{t}{2500\cdot0.055}}\right)$, $t = 84\,\mathrm{s}.$
$Q = CV = 0.055 \cdot 3.2 = 0.18\,\mathrm{C}.$
3. 60root2 V? Is this for household application or high-voltage wires?

1. Correct
2. Correct
3. I'm sorry for not being clear on this, this is for household applications, and not when the voltage is tens of thousands of volts before the use of step-down transformers.
Jay M. Robinson Middle School '18-now
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UTF-8 U+6211 U+662F
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### Re: Circuit Lab B/C

azboy1910 wrote:
October 25th, 2020, 3:59 pm
UTF-8 U+6211 U+662F wrote:
October 25th, 2020, 1:38 pm
azboy1910 wrote:
October 25th, 2020, 11:51 am
Alright,

1. Explain how an inductor functions in a circuit.
2. A 7-volt battery is in series with a 2.5k-ohm resistor and a 55 mF capacitor, forming an RC circuit. How long will it take for the capacitor to be charged to 3.2 volts? What is the charge on the capacitor at this state?
3. At the peak of a standard AC U.S. graph, what is the voltage?
1. It acts to resist changes in current by providing back EMFs through Faraday's law.
2. $3.2 = 7\left(1-e^{-\frac{t}{2500\cdot0.055}}\right)$, $t = 84\,\mathrm{s}.$
$Q = CV = 0.055 \cdot 3.2 = 0.18\,\mathrm{C}.$
3. 60root2 V? Is this for household application or high-voltage wires?

1. Correct
2. Correct
3. I'm sorry for not being clear on this, this is for household applications, and not when the voltage is tens of thousands of volts before the use of step-down transformers.
3. 20 root 2 V? I have no idea

azboy1910
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### Re: Circuit Lab B/C

UTF-8 U+6211 U+662F wrote:
October 25th, 2020, 4:22 pm
azboy1910 wrote:
October 25th, 2020, 3:59 pm
UTF-8 U+6211 U+662F wrote:
October 25th, 2020, 1:38 pm
1. It acts to resist changes in current by providing back EMFs through Faraday's law.
2. $3.2 = 7\left(1-e^{-\frac{t}{2500\cdot0.055}}\right)$, $t = 84\,\mathrm{s}.$
$Q = CV = 0.055 \cdot 3.2 = 0.18\,\mathrm{C}.$
3. 60root2 V? Is this for household application or high-voltage wires?

1. Correct
2. Correct
3. I'm sorry for not being clear on this, this is for household applications, and not when the voltage is tens of thousands of volts before the use of step-down transformers.
3. 20 root 2 V? I have no idea
For question 3, the answer is 170 volts.
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### Re: Circuit Lab B/C

Around how many amps must flow through the heart in order to cause ventricular fibrillation (and death)? What are three ways to reduce this possible current when working with high voltage applications?

Creationist127
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### Re: Circuit Lab B/C

UTF-8 U+6211 U+662F wrote:
October 25th, 2020, 4:48 pm
Around how many amps must flow through the heart in order to cause ventricular fibrillation (and death)? What are three ways to reduce this possible current when working with high voltage applications?
.075 amps, or 75 mA. To reduce it, wear nonconductive PPE (e.g. rubber boots and gloves), make sure anything you touch is not "hot" (charged or connected to power), and don't work with electricity when you're wet.
2018: Hovercraft, Thermo, Coaster, Solar System
2019: Thermo, Circuit Lab, Sounds, Wright Stuff
2020: Circuit Lab, Wright Stuff, Machines
2021: Circuit Lab, Machines, WIDI, anything but Wright Stuff

Can I request that we delete 2020 from our memories and do it over again?

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### Re: Circuit Lab B/C

Creationist127 wrote:
October 30th, 2020, 7:43 am
UTF-8 U+6211 U+662F wrote:
October 25th, 2020, 4:48 pm
Around how many amps must flow through the heart in order to cause ventricular fibrillation (and death)? What are three ways to reduce this possible current when working with high voltage applications?
.075 amps, or 75 mA. To reduce it, wear nonconductive PPE (e.g. rubber boots and gloves), make sure anything you touch is not "hot" (charged or connected to power), and don't work with electricity when you're wet.

Creationist127
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### Re: Circuit Lab B/C

1. A capacitor has 1 cm^2 plates, .5 mm apart from each other (with nothing in between). What is its capacitance?
2. A resistor is a cylinder with 5 mm diameter and is 1 cm long, and has resistivity 1 Ω*m. What is its resistance?
3. The resistor and capacitor are placed in series with each other and a 9-volt battery. How long will it take for the capacitor to charge to 8.5 volts?

EDIT: Fixed wording
Last edited by Creationist127 on November 5th, 2020, 9:20 am, edited 1 time in total.
2018: Hovercraft, Thermo, Coaster, Solar System
2019: Thermo, Circuit Lab, Sounds, Wright Stuff
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Can I request that we delete 2020 from our memories and do it over again?

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### Re: Circuit Lab B/C

Creationist127 wrote:
November 4th, 2020, 7:24 am
1. A capacitor has 1 cm-square plates, .5 mm apart from each other (with nothing in between). What is its capacitance?
2. A resistor is a cylinder with 5 mm diameter and is 1 cm long, and has resistivity 1 Ω*m. What is its resistance?
3. The resistor and capacitor are placed in series with each other and a 9-volt battery. How long will it take for the capacitor to charge to 8.5 volts?
1. 1.77E(-10) F or 0.177 nF
2. 509.3 ohms
3. about 2.89 time constants
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text:
# with vector $\mathbf{x}=[x_1,x_2,..,x_n]$, is $f_i(\mathbf{x})=x_i \exp(-(x_1+x_2+...x_i))$ concave for vector $\mathbf{x}$ ????

It looks not that complicated but I'm stuck in the middle.

$\mathbf{x}=[x_1,x_2, \cdots ,x_n]$.

1. $g(\mathbf{x})=\exp(-\mathbf{x})$ is a decreasing, and convex function.
2. $h(\mathbf{x})=x_1+x_2+x_3+\dotsm\;$ is a linear, increasing function, convex/concave.

So, $g(h(\mathbf{x}))$ is a convex function. Am I right so far?

and the problem is that $x_i$ is multiplied to $g(h(\mathbf{x}))$. $x_i$'s are positive and between 0 to 1 real value.

How can I prove $f_i(\mathbf{x})=x_i g(h(\mathbf{x}))$ is concave?

Thanks a lot.

You can do the hessian H of $f_i(x)$ and prove that H is negative semidefinite (if x'Hx ≤ 0 for all x).

the Hessian of $f_i$ is positive semi definite in this case which means that $f_i$ is convex if $x_i$ >0.

• Thank you for your answer. I tried by deriving Hessian H of $f_i(\mathbf{x})$, but I failed to prove $\mathbf{v}'H\mathbf{v}\leq 0$ for all $\mathbf{v}$. Is there other way to prove this? I guess this function will be concave in 0<x<1.. at least quasiconcave. Jul 21 '18 at 6:11
• Try: $f_i( \lambda x +(1- \lambda )y) \leq \lambda f_i(x) +(1-\lambda) f_i(y)$ $g(x)=exp(-x)$ convex and you can prove that easily in $R^n$ So, $f_i( \lambda x+(1-\lambda) y) \leq x_i*(\lambda g_i(x)+ (1- \lambda) g_i(y) )$ Jul 21 '18 at 19:27
• I hope that this helps !! Jul 21 '18 at 19:32
• Thank you for your kindness!!, for $x,y\in R^n$ and $f_i(x)=x_i g(x)$, $f_i(\lambda x+ (1-\lambda)y) = (\lambda x_i +(1-\lambda) y_i) g_i(\lambda x + (1-\lambda)y) \leq (\lambda x_i +(1-\lambda) y_i) [\lambda g_i(x) +(1-\lambda)g_i(y)]$ Jul 22 '18 at 4:01
• Then, i get $\lambda^2 x_i g_i(x) +(1-\lambda)^2 y_i g_i(y) +\lambda (1-\lambda)[y_i g_i(x) + x_i g_i(y)]=\lambda^2 f_i(x) + (1-\lambda)^2 f_i(y) + \lambda (1-\lambda)[y_i g_i(x) + x_i g_i(y)]$.. Here I stuck again..I don't know how the cross term $\lambda(1-\lambda)x_i f_i(y)$ can be dealt with. ;-( Jul 22 '18 at 4:08

Note that $f(x) = x e^{-x}$ is strictly convex for $x \ge 2$ (just compute $f''(x)$).

• you are right. $f''(x) = (2-x)e^{-x}$. My consideration is only in $x\in[0,1]$ :) Thank you for your answer Jul 22 '18 at 4:34
• You might want to add that to your question, otherwise this is an answer in the negative. Jul 22 '18 at 4:39


text:
### Tournament of Towns 2001-10 (ToT) 69p (UC)

geometry problems from International Matehmatics Tournament of Towns
with aops links in the names

[O = Easier, A = Harder]

2001-2010
(missing all 2004, 2006 fall , 2008 spring, 2009 spring)

2001

2001 ToT Spring Junior O P2
One of the midlines of a triangle is longer than one of its medians. Prove that the triangle has an obtuse angle.

2001 ToT Spring Junior A P3
Point $A$ lies inside an angle with vertex $M$. A ray issuing from point $A$ is reflected in one side of the angle at point $B$, then in the other side at point $C$ and then returns back to point $A$ (the ordinary rule of reflection holds). Prove that the center of the circle circumscribed about triangle $\triangle BCM$ lies on line $AM$.

2001 ToT Spring Junior A P6 , Senior A P3
Let $AH_A$, $BH_B$ and $CH_C$ be the altitudes of triangle $\triangle ABC$. Prove that the triangle whose vertices are the intersection points of the altitudes of triangles $\triangle AH_BH_C$, $\triangle BH_AH_C$ and $\triangle CH_AH_B$ is equal to triangle $\triangle H_AH_BH_C$.

2001 ToT Spring Senior O P3
Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $\triangle ABC$. The segments $AY$ and $CX$ intersect at the point $Z$. Given that $AY = YC$ and $AB = ZC$, prove that the points $B$, $X$, $Z$, and $Y$ lie on the same circle.

2001 ToT Fall Junior O P1
In the quadrilateral $ABCD$, $AD$ is parallel to $BC$.  $K$ is a point on $AB$. Draw the line through $A$ parallel to $KC$ and the line through $B$ parallel to $KD$. Prove that these two lines intersect at some point on $CD$.

2001 ToT Fall Senior O P1
An altitude of a pentagon is the perpendicular drop from a vertex to the opposite side. A median of a pentagon is the line joining a vertex to the midpoint of the opposite side. If the five altitudes and the five medians all have the same length,prove that the pentagon is regular.

2002

2002 ToT Spring Junior O P4
Quadrilateral $ABCD$ is circumscribed about a circle $\Gamma$ and $K,L,M,N$ are points of tangency of sides $AB,BC,CD,DA$ with $\Gamma$ respectively. Let $S\equiv KM\cap LN$. If quadrilateral $SKBL$ is cyclic then show that $SNDM$ is also cyclic.

Let $E$ and $F$ be the respective midpoints of $BC,CD$ of a convex quadrilateral $ABCD$. Segments $AE,AF,EF$ cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$.

2002 ToT Spring Senior O P2
$\Delta ABC$ and its mirror reflection $\Delta A^{\prime}B^{\prime}C^{\prime}$ is arbitrarily placed on the plane. Prove the midpoints of $AA^{\prime},BB^{\prime},CC^{\prime}$ are collinear.

2002 ToT Spring Senior A P5
Let $AA_1,BB_1,CC_1$ be the altitudes of acute $\Delta ABC$. Let $O_a,O_b,O_c$ be the incentres of $\Delta AB_1C_1,\Delta BC_1A_1,\Delta CA_1B_1$ respectively. Also let $T_a,T_b,T_c$ be the points of tangency of the incircle of $\Delta ABC$ with $BC,CA,AB$ respectively. Prove that $T_aO_cT_bO_aT_cO_b$ is an equilateral hexagon.

2002 ToT Fall Junior O P5
An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?

2002 ToT Fall Junior A P4
Point $P$ is chosen in the plane of triangle $ABC$ such that $\angle{ABP}$ is congruent to $\angle{ACP}$ and $\angle{CBP}$ is congruent to $\angle{CAP}$. Show $P$ is the orthocentre.

2002 ToT Fall Senior A P5
Two circles $\Gamma_1,\Gamma_2$ intersect at $A,B$. Through $B$ a straight line $\ell$ is drawn and $\ell\cap \Gamma_1=K,\ell\cap\Gamma_2=M\;(K,M\neq B)$. We are given $\ell_1\parallel AM$ is tangent to $\Gamma_1$ at $Q$. $QA\cap \Gamma_2=R\;(\neq A)$ and further $\ell_2$ is tangent to $\Gamma_2$ at $R$. Prove that:
• $\ell_2\parallel AK$
• $\ell,\ell_1,\ell_2$ have a common point.

2003

2003 ToT Spring Junior O P3
Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ($AB = BC$) so that $AK +LC = KL$. A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$, intersecting side $AC$ at point $N$. Find the value of $\angle KNL$.

2003 ToT Spring Junior A P2
Triangle $ABC$ is given. Prove that $\frac{R}{r} > \frac{a}{h}$, where $R$ is the radius of the circumscribed circle, $r$ is the radius of the inscribed circle, $a$ is the length of the longest side, $h$ is the length of the shortest altitude.

2003 ToT Spring Junior A P6
A trapezoid with bases $AD$ and $BC$ is circumscribed about a circle, $E$ is the intersection point of the diagonals. Prove that $\angle AED$ is not acute.

2003 ToT Spring Senior O P3
Point $M$ is chosen in triangle $ABC$ so that the radii of the circumcircles of triangles $AMC, BMC$, and $BMA$ are no smaller than the radius of the circumcircle of $ABC$. Prove that all four radii are equal.

2003 ToT Spring Senior A P1
A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).

2003 ToT Spring Senior A P4
A right triangle $ABC$ with hypotenuse $AB$ is inscribed in a circle. Let $K$ be the midpoint of the arc $BC$ not containing $A, N$ the midpoint of side $AC$, and $M$ a point of intersection of ray $KN$ with the circle. Let $E$ be a point of intersection of tangents to the circle at points $A$ and $C$. Prove that $\angle EMK = 90^\circ$

2003 ToT Fall Junior O P2
In $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_7, A_6A_1$ and $A_7A_2$ are congruent to each other and diagonals $A_1A_4, A_2A_5, A_3A_6, A_4A_7, A_5A_1, A_6A_2$ and $A_7A_3$ are also congruent to each other. Is the polygon necessarily regular?

2003 ToT Fall Junior A P5
A point $O$ lies inside of the square $ABCD$. Prove that the difference between the sum of angles $OAB, OBC, OCD , ODA$ and $180^{\circ}$ does not exceed $45^{\circ}$.

2003 ToT Fall Senior O P4
Each side of $1 \times 1$ square is a hypothenuse of an exterior right triangle. Let $A, B, C, D$ be the vertices of the right angles and $O_1, O_2, O_3, O_4$ be the centers of the incircles of these triangles. Prove that
a) The area of quadrilateral $ABCD$ does not exceed $2$;
b) The area of quadrilateral $O_1O_2O_3O_4$ does not exceed $1$.

2003 ToT Fall Senior A P4
In a triangle $ABC$, let $H$ be the point of intersection of altitudes, $I$ the center of incircle, $O$ the center of circumcircle, $K$ the point where the incircle touches $BC$. Given that $IO$ is parallel to $BC$, prove that $AO$ is parallel to $HK$.

2003 ToT Fall Senior A P6
Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.

2004 problems are missing from aops

2004

2004 ToT Spring Junior O P1
In triangle ABC the bisector of angle A, the perpendicular to side AB from its midpoint, and the altitude from vertex B, intersect in the same point. Prove that the bisector of angle A, the perpendicular to side AC from its midpoint, and the altitude from vertex C also intersect in the same point.

2004 ToT Spring Junior Α P4
Two circles intersect in points A and B. Their common tangent nearer B touches the circles at points E and F, and intersects the extension of AB at the point M. The point K is chosen on the extention of AM so that KM = MA. The line KE intersects the circle containing E again at the point C. The line KF intersects the circle containing F again at the point D. Prove that the points A, C and D are collinear.

2004 ToT Spring Senior O P1
Segments AB, BC and CD of the broken line ABCD are equal and are tangent to a circle with centre at the point O. Prove that the point of contact of this circle with BC, the point O and the intersection point of AC and BD are collinear.

2004 ToT Spring Senior O P3
Perimeter of a convex quadrilateral is 2004 and one of its diagonals is 1001. Can another diagonal be 1? 2? 1001?

2004 ToT Spring Senior Α P3
The perpendicular projection of a triangular pyramid on some plane has the largest possible area. Prove that this plane is parallel to either a face or two opposite edges of the pyramid.

2004 ToT Spring Senior Α P5
The parabola y = x^2 intersects a circle at exactly two points A and B. If their tangents at A coincide, must their tangents at B also coincide?

2004 ToT Fall Junior O P4
A circle and a straight line with no common points are given. With compass and straightedge construct a square with two adjacent vertices on the circle and two other vertices on the line (it is known that such a square exists).

2004 ToT Fall Junior Α P2
An incircle of triangle ABC touches the sides BC, CA and AB at points A', B' and C' respectively. Is it necessarily true that triangle ABC is equilateral if AA' = BB' = CC'?

2004 ToT Fall Junior Α P5
Point K belongs to side BC of triangle ABC. Incircles of triangles ABK and ACK touch
BC at points M and N respectively. Prove that BM · CN > KM · KN.

2004 ToT Fall Senior O P1
Three circles pass through point X and A, B, C are their intersection points (other than X). Let A'  be the second point of intersection of straight line AX and the circle circumscribed around triangle BCX. Define simiarly points B' , C' . Prove that triangles ABC' , AB' C, and A' BC are similar.

2004 ToT Fall Senior Α P4
A circle with the center I is entirely inside of a circle with center O. Consider all possible chords AB of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle AIB.

2004 ToT Fall Senior Α P7
Let \angleAOB be obtained from \angle COD by rotation (ray AO transforms into ray CO). Let E and F be the points of intersection of the circles inscribed into these angles. Prove that \angle AOE = \angle DOF.

2005

2005 ToT Spring Junior O P4, Senior O P3
$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$.

The altitudes $AD$ and $BE$ of triangle $ABC$ meet at its orthocentre $H$. The midpoints of $AB$ and $CH$ are $X$ and $Y$, respectively. Prove that $XY$ is perpendicular to $DE$.

2005 ToT Spring Senior A P2
A circle $\omega_1$ with centre $O_1$ passes through the centre $O_2$ of a second circle $\omega_2$. The tangent lines to $\omega_2$ from a point $C$ on $\omega_1$ intersect $\omega_1$ again at points $A$ and $B$ respectively. Prove that $AB$ is perpendicular to $O_1O_2$.

2005 ToT Spring Senior A P5
Prove that if a regular icosahedron and a regular dodecahedron have a common circumsphere, then they have a common insphere.

2005 ToT Fall Junior O P1
In triangle $ABC$, points $M_1, M_2$ and $M_3$ are midpoints of sides $AB$, $BC$ and $AC$, respectively, while points $H_1, H_2$ and $H_3$ are bases of altitudes drawn from $C$, $A$ and $B$, respectively. Prove that one can construct a triangle from segments $H_1M_2, H_2M_3$ and $H_3M_1$.

2005 ToT Fall Junior A P2
The extensions of sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ intersect at $K$. It is known that $AD = BC$. Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$. Prove that the triangle $MNK$ is obtuse.

2005 ToT Fall Senior O P2
A segment of length $\sqrt2 + \sqrt3 + \sqrt5$ is drawn. Is it possible to draw a segment of unit length using a compass and a straightedge?

2005 ToT Fall Senior O P4
On all three sides of a right triangle $ABC$ external squares are constructed; their centers denoted by $D$, $E$, $F$. Show that the ratio of the area of triangle $DEF$ to the area of triangle $ABC$ is:
a) greater than $1$;
b)  at least $2$.

2005 ToT Fall Senior A P5
In triangle $ABC$ bisectors $AA_1, BB_1$ and $CC_1$ are drawn. Given $\angle A : \angle B : \angle C = 4 : 2 : 1$, prove that $A_1B_1 = A_1C_1$.

2006

2006 ToT Spring Junior O P1
Let $\angle A$ in a triangle $ABC$ be $60^\circ$. Let point $N$ be the intersection of $AC$ and perpendicular bisector to the side $AB$ while point $M$ be the intersection of $AB$ and perpendicular bisector to the side $AC$. Prove that $CB = MN$.

2006 ToT Spring Junior A P3
On sides $AB$ and $BC$ of an acute triangle $ABC$ two congruent rectangles $ABMN$ and $LBCK$ are constructed (outside of the triangle), so that $AB = LB$. Prove that straight lines $AL, CM$ and $NK$ intersect at the same point.

2006 ToT Spring Senior O P4
Quadrilateral $ABCD$ is a cyclic, $AB = AD$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ respectfully so that $\angle MAN =1/2 (\angle BAD)$. Prove that  $MN = BM + ND$.

2006 ToT Spring Senior A P4
In triangle $ABC$ let $X$ be some fixed point on bisector $AA'$ while point $B'$  be intersection of  $BX$ and $AC$ and point $C'$ be intersection of $CX$ and $AB$. Let point $P$ be intersection of segments $A'B'$ and $CC'$ while point $Q$  be intersection of segments $A'C'$ and $BB'$. Prove τhat $\angle PAC = \angle QAB$.

2006 fall problems are missing from aops

2006 ToT Fall Junior O P4
Given triangle ABC, BC is extended beyond B to the point D such that BD = BA. The bisectors of the exterior angles at vertices B and C intersect at the point M. Prove that quadrilateral ADMC is cyclic.

2006 ToT Fall Junior A P1
Two regular polygons, a 7-gon and a 17-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal.

2006 ToT Fall Junior A P4
A circle of radius R is inscribed into an acute triangle. Three tangents to the circle split the triangle into three right angle triangles and a hexagon that has perimeter Q. Find the sum of diameters of circles inscribed into the three right triangles.

2006 ToT Fall Senior O P2
The incircle of the quadrilateral ABCD touches AB, BC, CD and DA at E, F, G and H respectively. Prove that the line joining the incentres of triangles HAE and FCG is perpendicular to the line joining the incentres of triangles EBF and GDH.

2006 ToT Fall Senior O P5
Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges?

2006 ToT Fall Senior A P2
Suppose ABC is an acute triangle. Points A1, B1 and C1 are chosen on sides BC, AC and AB
respectively so that the rays A1A, B1B and C1C are bisectors of triangle A1B1C1. Prove that
AA1, BB1 and CC1 are altitudes of triangle ABC.

2007

2007 ToT Spring Junior O P1
The sides of a convex pentagon are extended on both sides to form five triangles. If these triangles are congruent to one another, does it follow that the pentagon is regular?

2007 ToT Spring Junior A P2
$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$, respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to  $AB$. The perimeter of triangle $KLB$ is equal to $1$. What is the area of triangle $MND$?

2007 ToT Spring Junior A P6
In the quadrilateral $ABCD$, $AB = BC = CD$ and $\angle BMC = 90^\circ$, where $M$ is the midpoint of $AD$. Determine the acute angle between the lines $AC$ and $BD$.

2007 ToT Spring Senior O P3
$B$ is a point on the line which is tangent to a circle at the point $A$. The line segment $AB$ is rotated about the centre of the circle through some angle to the line segment $A'B'$. Prove that the line $AA'$ passes through the midpoint of $BB'$.

2007 ToT Spring Senior A P7
$T$ is a point on the plane of triangle $ABC$ such that $\angle ATB = \angle BTC = \angle CTA = 120^\circ$. Prove that the lines symmetric to $AT, BT$ and $CT$ with respect to $BC, CA$ and $AB$, respectively, are concurrent.

2007 ToT Fall Junior O P3
$D$ is the midpoint of the side $BC$ of triangle $ABC$. $E$ and $F$ are points on $CA$ and $AB$ respectively, such that $BE$ is perpendicular to $CA$ and $CF$ is perpendicular to $AB$. If $DEF$ is an equilateral triangle, does it follow that $ABC$ is also equilateral?

2007 ToT Fall Junior A P1
Let $ABCD$ be a rhombus. Let $K$ be a point on the line $CD$, other than $C$ or $D$, such that $AD = BK$. Let $P$ be the point of intersection of $BD$ with the perpendicular bisector of $BC$. Prove that $A, K$ and $P$ are collinear.

2007 ToT Fall Junior P P2
Let us call a triangle “almost right angle triangle” if one of its angles differs from $90^\circ$ by no more than $15^\circ$. Let us call a triangle “almost isosceles triangle” if two of its angles differs from each other by no more than $15^\circ$. Is it true that that any acute triangle is either “almost right angle triangle” or “almost isosceles triangle”?

2007 ToT Fall Junior P P3
A triangle with sides $a, b, c$ is folded along a line $\ell$ so that a vertex $C$ is on side $c$. Find the segments on which point $C$ divides $c$, given that the angles adjacent to $\ell$ are equal.

2007 ToT Fall Senior O P3
Give a construction by straight-edge and compass of a point $C$ on a line $\ell$ parallel to a segment $AB$, such that the product $AC \cdot BC$ is minimum.

2007 ToT Fall Senior A P2
Let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$ of a cyclic quadrilateral $ABCD$. Let $P$ be the point of intersection of $AC$ and $BD$. Prove that the circumradii of triangles $PKL, PLM, PMN$ and $PNK$ are equal to one another.

2007 ToT Fall Senior P P4
Jim and Jane divide a triangular cake between themselves. Jim chooses any point in the cake and Jane makes a straight cut through this point and chooses the piece. Find the size of the piece that each of them can guarantee for himself/herself (both of them want to get as much as possible).
2008

2008 spring problems are missing from aops

2008 ToT Spring Junior O P1
In the convex hexagon ABCDEF, AB, BC and CD are respectively parallel to DE, EF and FA. If AB = DE, prove that BC = EF and CD = FA.

2008 ToT Spring Junior A P2
A line parallel to the side AC of triangle ABC cuts the side AB at K and the side BC at M. O is the point of intersection of AM and CK. If AK = AO and KM = MC, prove that AM = KB.

2008 ToT Spring Junior A P7
A convex quadrilateral ABCD has no parallel sides. The angles between the diagonal AC and the four sides are 55^o, 55^o, 19^o and 16^o in some order. Determine all possible values of the acute angle between AC and BD.

2008 ToT Spring Senior O P3
In triangle ABC, \angle A = 90^o. M is the midpoint of BC and H is the foot of the altitude from
A to BC. The line passing through M and perpendicular to AC meets the circumcircle of
triangle AMC again at P. If BP intersects AH at K, prove that AK = KH.

2008 ToT Spring Senior A P4
Each of Peter and Basil draws a convex quadrilateral with no parallel sides. The angles between a diagonal and the four sides of Peter's quadrilateral are \alpha, \alpha, \beta and \gamma  in some order. The angles between a diagonal and the four sides of Basil's quadrilateral are also \alpha, \alpha, \beta and \gamma   in some order. Prove that the acute angle between the diagonals of Peter's quadrilateral is equal to the acute angle between the diagonals of Basil's quadrilateral.

2008 ToT Spring Senior A P7
Each of three lines cuts chords of equal lengths in two given circles. The points of intersection of these lines form a triangle. Prove that its circumcircle passes through the midpoint of the segment joining the centres of the circles.

2008 ToT Fall Junior O P3
Acute triangle $A_1A_2A_3$ is inscribed in a circle of radius $2$. Prove that one can choose points $B_1, B_2, B_3$ on the arcs $A_1A_2, A_2A_3, A_3A_1$ respectively, such that the numerical value of the area of the hexagon $A_1B_1A_2B_2A_3B_3$ is equal to the numerical value of the perimeter of the triangle $A_1A_2A_3.$

2008 ToT Fall Junior A P3
In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio).

2008 ToT Fall Junior A P6
Let $ABC$ be a non-isosceles triangle. Two isosceles triangles $AB'C$ with base $AC$ and $CA'B$ with base $BC$ are constructed outside of triangle $ABC$. Both triangles have the same base angle $\varphi$. Let $C_1$ be a point of intersection of the perpendicular from $C$ to $A'B'$ and the perpendicular bisector of the segment $AB$. Determine the value of $\angle AC_1B.$

2008 ToT Fall Senior O P3
A $30$-gon $A_1A_2\cdots A_{30}$ is inscribed in a circle of radius $2$. Prove that one can choose a point $B_k$ on the arc $A_kA_{k+1}$ for $1 \leq k \leq 29$ and a point $B_{30}$ on the arc $A_{30}A_1$, such that the numerical value of the area of the $60$-gon $A_1B_1A_2B_2 \dots A_{30}B_{30}$ is equal to the numerical value of the perimeter of the original $30$-gon.

2008 ToT Fall Senior A P4
Let $ABCD$ be a non-isosceles trapezoid. Define a point $A1$ as intersection of circumcircle of triangle $BCD$ and line $AC$. (Choose $A_1$ distinct from $C$). Points $B_1, C_1, D_1$ are defined in similar way. Prove that $A_1B_1C_1D_1$ is a trapezoid as well.

2009

2009 spring problems are missing from aops

2009 ToT Spring Junior O P5
In rhombus ABCD, angle A equals 120^ο. Points M and N are chosen on sides BC and CD so that angle NAM equals 30^ο. Prove that the circumcenter of triangle NAM lies on a diagonal of of the rhombus.

2009 ToT Spring Junior A P7
Angle C of an isosceles triangle ABC equals 120^ο. Each of two rays emitting from vertex C (inwards the triangle) meets AB at some point (P_i) reflects according to the rule the angle of incidence equals the angle of refkection" and meets lateral side of triangle ABC at point Q_i (i = 1,2). Given that angle between the rays equals 60^ο, prove that area of triangle P_1CP_2 equals the sum of areas of triangles AQ_1P_1 and BQ_2P_2 (AP_1 < AP_2).

2009 ToT Spring Senior O P5
Suppose that X is an arbitrary point inside a tetrahedron. Through each vertex of the tetrahedron, draw a straight line that is parallel to the line segment connecting X with the intersection point of the medians of the opposite face. Prove that these four lines meet at the same point.

2009 ToT Spring Senior A P4
Three planes dissect a parallelepiped into eight hexahedrons such that all of their faces are quadrilaterals (each plane intersects two corresponding pairs of opposite faces of the parallelepiped and does not intersect the remaining two faces). One of the hexahedrons has a circumscribed sphere. Prove that each of these hexahedrons has a circumscribed sphere.

2009 ToT Fall Junior A P4
Let $ABCD$ be a rhombus. $P$ is a point on side $BC$ and $Q$ is a point on side $CD$ such that $BP = CQ$. Prove that centroid of triangle $APQ$ lies on the segment $BD.$

2009 ToT Fall Senior O P2
$A, B, C, D, E$ and $F$ are points in space such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, $CD$ is parallel to $FA$, but $AB \neq DE$. Prove that all six points lie in the same plane.

2009 ToT Fall Senior A P3
Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent.

2009 ToT Fall Senior A P5
Let $XY Z$ be a triangle. The convex hexagon $ABCDEF$ is such that $AB; CD$ and $EF$ are parallel and equal to $XY; Y Z$ and $ZX$, respectively. Prove that area of triangle with vertices at the midpoints of $BC; DE$ and $FA$ is no less than area of triangle $XY Z.$

2010

2010 ToT Spring Junior O P3
An angle is given in a plane. Using only a compass, one must find out
(a) if this angle is acute. Find the minimal number of circles one must draw to be sure.
(b) if this angle equals $31^{\circ}$.(One may draw as many circles as one needs).

2010 ToT Spring Junior A P2
Let $M$ be the midpoint of side $AC$ of the triangle $ABC$. Let $P$ be a point on the side $BC$. If $O$ is the point of intersection of $AP$ and $BM$ and $BO = BP$, determine the ratio $\frac{OM}{PC}$ .

2010 ToT Spring Senior O P5
A needle (a segment) lies on a plane. One can rotate it $45^{\circ}$ round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?

2010 ToT Spring Senior A P6
Quadrilateral $ABCD$ is circumscribed around the circle with centre $I$. Let points $M$ and $N$ be the midpoints of sides $AB$ and $CD$ respectively and let $\frac{IM}{AB} = \frac{IN}{CD}$. Prove that $ABCD$ is either a trapezoid or a parallelogram.

2010 ToT Fall Junior O P2
In a quadrilateral $ABCD$ with an incircle, $AB = CD; BC < AD$ and $BC$ is parallel to $AD$. Prove that the bisector of $\angle C$ bisects the area of $ABCD$

2010 ToT Fall Junior A P1
A round coin may be used to construct a circle passing through one or two given points on the plane. Given a line on the plane, show how to use this coin to construct two points such that they dene a line perpendicular to the given line. Note that the coin may not be used to construct a circle tangent to the given line.

2010 ToT Fall Junior A P6
In acute triangle $ABC$, an arbitrary point $P$ is chosen on altitude $AH$. Points $E$ and $F$ are the midpoints of sides $CA$ and $AB$ respectively. The perpendiculars from $E$ to $CP$ and from $F$ to $BP$ meet at point $K$. Prove that $KB = KC$.

The diagonals of a convex quadrilateral $ABCD$ are perpendicular to each other and intersect at the point $O$. The sum of the inradii of triangles $AOB$ and $COD$ is equal to the sum of the inradii of triangles $BOC$ and $DOA$.
(a) Prove that $ABCD$ has an incircle.
(b) Prove that $ABCD$ is symmetric about one of its diagonals.

2010 ToT Fall Senior A P5
The quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$ and $BD$ do not pass through $O$. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.

2011

2011 ToT Spring Junior O P
2011 ToT Spring Junior Α P
2011 ToT Spring Senior O P
2011 ToT Spring Senior O P

2011 ToT Fall Junior O P
2011 ToT Fall Junior Α P
2011 ToT Fall Senior O P
2011 ToT Fall Senior O P

2012

2012 ToT Spring Junior O P
2012 ToT Spring Junior Α P
2012 ToT Spring Senior O P
2012 ToT Spring Senior O P

2012 ToT Fall Junior O P
2012 ToT Fall Junior Α P
2012 ToT Fall Senior O P
2012 ToT Fall Senior O P

2013

2013 ToT Spring Junior O P
2013 ToT Spring Junior Α P
2013 ToT Spring Senior O P
2013 ToT Spring Senior O P

2013 ToT Fall Junior O P
2013 ToT Fall Junior Α P
2013 ToT Fall Senior O P
2013 ToT Fall Senior O P

2014

2014 ToT Spring Junior O P
2014 ToT Spring Junior Α P
2014 ToT Spring Senior O P
2014 ToT Spring Senior O P

2014 ToT Fall Junior O P
2014 ToT Fall Junior Α P
2014 ToT Fall Senior O P
2014 ToT Fall Senior O P

2015

2015 ToT Spring Junior O P
2015 ToT Spring Junior Α P
2015 ToT Spring Senior O P
2015 ToT Spring Senior O P

2015 ToT Fall Junior O P
2015 ToT Fall Junior Α P
2015 ToT Fall Senior O P
2015 ToT Fall Senior O P

2016

2016 ToT Spring Junior O P
2016 ToT Spring Junior Α P
2016 ToT Spring Senior O P
2016 ToT Spring Senior O P

2016 ToT Fall Junior O P
2016 ToT Fall Junior Α P
2016 ToT Fall Senior O P
2016 ToT Fall Senior O P

2017

2017 ToT Spring Junior O P
2017 ToT Spring Junior Α P
2017 ToT Spring Senior O P
2017 ToT Spring Senior O P

2017 ToT Fall Junior O P
2017 ToT Fall Junior Α P
2017 ToT Fall Senior O P
2017 ToT Fall Senior O P

sources:
www.math.toronto.edu/oz/turgor/
www.turgor.ru/en


text:
# Statement of Purpose Template

I am trying, loosely based on this answer to make a template for my Statement of Purpose. How could I remove the empty extra line between the horizontal rule and the text? Moreover, how could be the first letter be of larger font in the title?

Any other suggestion is more than welcomed.

\documentclass{article}

\usepackage[left=1in,right=1in,top=1in,bottom=1in]{geometry}

\newcommand{\HRule}{\rule{\linewidth}{0.5mm}}
\newcommand{\Hrule}{\rule{\linewidth}{0.3mm}}

\makeatletter% since there's an at-sign (@) in the command name
\renewcommand{\@maketitle}{
\parindent=0pt% don't indent paragraphs in the title block
\begin{center}
\MakeUppercase{\Large \bf \@title}
\HRule%
\end{center}%
\textit{\@author \hfill \@date}
\par
}
\makeatother% resets the meaning of the at-sign (@)

\title{Statement of Purpose}
\author{name}
\date{Ph.D. Applicant}

\begin{document}

\maketitle% prints the title block

Emergency medicine has always been a passion of mine\ldots

\end{document}

-

Replace the center environment with the \centering switch to avoid additional vertical space. For the second issue simply use small capitals for the title. See code below for the modifications.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[margin=1in]{geometry}

\newcommand{\HRule}{\rule{\linewidth}{0.5mm}}
\newcommand{\Hrule}{\rule{\linewidth}{0.3mm}}

\makeatletter% since there's an at-sign (@) in the command name
\renewcommand{\@maketitle}{%
\parindent=0pt% don't indent paragraphs in the title block
\centering
{\Large \bfseries\textsc{\@title}}
\HRule\par%
\textit{\@author \hfill \@date}
\par
}
\makeatother% resets the meaning of the at-sign (@)

\title{Statement of Purpose}
\author{name}
\date{Ph.D. Applicant}

\begin{document}
\maketitle% prints the title block

Emergency medicine has always been a passion of mine\ldots
\end{document}

-

For the extra line, you can tweak it manually by using negative vertical spaces :

\vspace{-0.2in}

However, be careful as this can mess up your document if you give too big numbers.

-


text:
# Is every 3-Sasakian a Sasakian-Einstein manifold?

Hi, a short question: Is every 3-Sasakian manifold a Sasaki-Einstein manifold? If not, do you have an example? If yes, how can I prove this?

Thanks and best regards

-

Yes, because a manifold is Sasaki-Einstein if and only if its metric cone is Ricci-flat Kähler, whereas the cone of a 3-sasakian manifold is hyperkähler.

See, for instance, Bär's "Real Killing spinors and holonomy" published in CMP.

-
Okay, I thought this... If someone is hyperkähler, is this Ricci-flat Kähler, too? What I mean: What is the connection between hyperkähler and Ricci-flat Kähler? Thanks! –  user7028 Nov 22 '10 at 18:27
A hyperkähler manifold is Ricci-flat Kähler in a variety of ways -- that variety being $\mathbb{CP}^1$. –  José Figueroa-O'Farrill Nov 22 '10 at 21:41


text:
## DingusBrain Group Title A neutral atom can be negatively charged by? A-transferring an electron to another atom. B-transferring a neutron to another atom. C- receiving a neutron from another atom. D-receiving an electron from another atom. one year ago one year ago

1. poopsiedoodle Group Title

D-receiving an electron from another atom.

2. poopsiedoodle Group Title

$$\Huge\text{Duh.}$$

3. DingusBrain Group Title

Thanks for the help not......


text:
## Gibbs sampling with incompatible conditionals

Posted in Books, Kids, R, Statistics with tags , , , , , , on July 23, 2019 by xi'an

An interesting question (with no clear motivation) on X validated wondering why a Gibbs sampler produces NAs… Interesting because multi-layered:

1. The attached R code indeed produces NAs because it calls the Negative Binomial Neg(x¹,p) random generator with a zero success parameter, x¹=0, which automatically returns NAs. This can be escaped by returning a one (1) instead.
2. The Gibbs sampler is based on a Bin(x²,p) conditional for X¹ and a Neg(x¹,p) conditional for X². When using the most standard version of the Negative Binomial random variate as the number of failures, hence supported on 0,1,2…. these two conditionals are incompatible, i.e., there cannot be a joint distribution behind that returns these as conditionals, which makes the limiting behaviour of the Markov chain harder to study. It however seems to converge to a distribution close to zero, which is not contradictory with the incompatibility property: the stationary joint distribution simply does not enjoy the conditionals used by the Gibbs sampler as its conditionals.
3. When using the less standard version of the Negative Binomial random variate understood as a number of attempts for the conditional on X², the two conditionals are compatible and correspond to a joint measure proportional to $x_1^{-1} {x_1 \choose x_2} p^{x_2} (1-p)^{x_1-x_2}$, however this pmf does not sum up to a finite quantity (as in the original Gibbs for Kids example!), hence the resulting Markov chain is at best null recurrent, which seems to be the case for p different from ½. This is unclear to me for p=½.

## automatic adaptation of MCMC algorithms

Posted in pictures, Statistics with tags , , , , , , , on March 4, 2019 by xi'an

“A typical adaptive MCMC sampler will approximately optimize performance given the kind of sampler chosen in the first place, but it will not optimize among the variety of samplers that could have been chosen.”

Last February (2018), Dao Nguyen and five co-authors arXived a paper that I missed. On a new version of adaptive MCMC that aims at selecting a wider range of proposal kernels. Still requiring a by-hand selection of this collection of kernels… Among the points addressed, beyond the theoretical guarantees that the adaptive scheme does not jeopardize convergence to the proper target, are a meta-exploration of the set of combinations of samplers and integration of the computational speed in the assessment of each sampler. Including the very difficulty of assessing mixing. One could deem the index of the proposal as an extra (cyber-)parameter to its generic parameter (like the scale in the random walk), but the discreteness of this index makes the extension more delicate than expected. And justifies the distinction between internal and external parameters. The notion of a worst-mixing dimension is quite appealing and connects with the long-term hope that one could spend the maximum fraction of the sampler runtime over the directions that are poorly mixing, while still keeping the target as should be. The adaptive scheme is illustrated on several realistic models with rather convincing gains in efficiency and time.

The convergence tools are inspired from Roberts and Rosenthal (2007), with an assumption of uniform ergodicity over all kernels considered therein which is both strong and delicate to assess in practical settings. Efficiency is rather unfortunately defined in terms of effective sample size, which is a measure of correlation or lack thereof, but which does not relate to the speed of escape from the basin of attraction of the starting point. I also wonder at the pertinence of the estimation of the effective sample size when the chain is based on different successive kernels, since the lack of correlation may be due to another kernel. Another calibration issue is the internal clock that relates to the average number of iterations required to tune properly a specific kernel, which again may be difficult to assess in a realistic situation. A last query is whether or not this scheme could be compared with an asynchronous (and valid) MCMC approach that exploits parallel capacities of the computer.

## Gibbs for incompatible kids

Posted in Books, Statistics, University life with tags , , , , , , , , , , on September 27, 2018 by xi'an

In continuation of my earlier post on Bayesian GANs, which resort to strongly incompatible conditionals, I read a 2015 paper of Chen and Ip that I had missed. (Published in the Journal of Statistical Computation and Simulation which I first confused with JCGS and which I do not know at all. Actually, when looking at its editorial board,  I recognised only one name.) But the study therein is quite disappointing and not helping as it considers Markov chains on finite state spaces, meaning that the transition distributions are matrices, meaning also that convergence is ensured if these matrices have no null probability term. And while the paper is motivated by realistic situations where incompatible conditionals can reasonably appear, the paper only produces illustrations on two and three states Markov chains. Not that helpful, in the end… The game is still afoot!

## Bayesian gan [gan style]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , on June 26, 2018 by xi'an

In their paper Bayesian GANS, arXived a year ago, Saatchi and Wilson consider a Bayesian version of generative adversarial networks, putting priors on both the model and the discriminator parameters. While the prospect seems somewhat remote from genuine statistical inference, if the following statement is representative

“GANs transform white noise through a deep neural network to generate candidate samples from a data distribution. A discriminator learns, in a supervised manner, how to tune its parameters so as to correctly classify whether a given sample has come from the generator or the true data distribution. Meanwhile, the generator updates its parameters so as to fool the discriminator. As long as the generator has sufficient capacity, it can approximate the cdf inverse-cdf composition required to sample from a data distribution of interest.”

I figure the concept can also apply to a standard statistical model, where x=G(z,θ) rephrases the distributional assumption x~F(x;θ) via a white noise z. This makes resorting to a prior distribution on θ more relevant in the sense of using potential prior information on θ (although the successes of probabilistic numerics show formal priors can be used on purely numerical ground).

The “posterior distribution” that is central to the notion of Bayesian GANs is however unorthodox in that the distribution is associated with the following conditional posteriors

where D(x,θ) is the “discriminator”, that is, in GAN lingo, the probability to be allocated to the “true” data generating mechanism rather than to the one associated with G(·,θ). The generative conditional posterior (1) then aims at fooling the discriminator, i.e. favours generative parameter values that raise the probability of wrong allocation of the pseudo-data. The discriminative conditional posterior (2) is a standard Bayesian posterior based on the original sample and the generated sample. The authors then iteratively sample from these posteriors, effectively implementing a two-stage Gibbs sampler.

“By iteratively sampling from (1) and (2) at every step of an epoch one can, in the limit, obtain samples from the approximate posteriors over [both sets of parameters].”

What worries me about this approach is that  just cannot work, in the sense that (1) and (2) cannot be compatible conditional (posterior) distributions. There is no joint distribution for which (1) and (2) would be the conditionals, since the pseudo-data appears in D for (1) and (1-D) in (2). This means that the convergence of a Gibbs sampler is at best to a stationary σ-finite measure. And hence that the meaning of the chain is delicate to ascertain… Am I missing any fundamental point?! [I checked the reviews on NIPS webpage and could not spot this issue being raised.]

## Gibbs for kidds

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , , , on February 12, 2018 by xi'an

A chance (?) question on X validated brought me to re-read Gibbs for Kids, 25 years after it was written (by my close friends George and Ed). The originator of the question had difficulties with the implementation, apparently missing the cyclic pattern of the sampler, as in equations (2.3) and (2.4), and with the convergence, which is only processed for a finite support in the American Statistician paper. The paper [which did not appear in American Statistician under this title!, but inspired an animal bredeer, Dan Gianola, to write a “Gibbs for pigs” presentation in 1993 at the 44th Annual Meeting of the European Association for Animal Production, Aarhus, Denmark!!!] most appropriately only contains toy examples since those can be processed and compared to know stationary measures. This is for instance the case for the auto-exponential model

$f(x,y) \propto exp(-xy)$

which is only defined as a probability density for a compact support. (The paper does not identify the model as a special case of auto-exponential model, which apparently made the originator of the model, Julian Besag in 1974, unhappy, as George and I found out when visiting Bath, where Julian was spending the final year of his life, many years later.) I use the limiting case all the time in class to point out that a Gibbs sampler can be devised and operate without a stationary probability distribution. However, being picky!, I would like to point out that, contrary, to a comment made in the paper, the Gibbs sampler does not “fail” but on the contrary still “converges” in this case, in the sense that a conditional ergodic theorem applies, i.e., the ratio of the frequencies of visits to two sets A and B with finite measure do converge to the ratio of these measures. For instance, running the Gibbs sampler 10⁶ steps and ckecking for the relative frequencies of x’s in (1,2) and (1,3) gives 0.685, versus log(2)/log(3)=0.63, since 1/x is the stationary measure. One important and influential feature of the paper is to stress that proper conditionals do not imply proper joints. George would work much further on that topic, in particular with his PhD student at the time, my friend Jim Hobert.

With regard to the convergence issue, Gibbs for Kids points out to Schervish and Carlin (1990), which came quite early when considering Gelfand and Smith published their initial paper the very same year, but which also adopts a functional approach to convergence, along the paper’s fixed point perspective, somehow complicating the matter. Later papers by Tierney (1994), Besag (1995), and Mengersen and Tweedie (1996) considerably simplified the answer, which is that irreducibility is a necessary and sufficient condition for convergence. (Incidentally, the reference list includes a technical report of mine’s on latent variable model MCMC implementation that never got published.)


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# Tag Info

As you point out, there is the DHGV 2010 scheme over the integers based on the approximate GCD problem but asymptotics are not great with this scheme, for eg. one of the parameters for DHGV is around $2^{\mathcal{O}(\theta^{5})}$ where $\theta$ is the security parameter. Of the so called second generation schemes, I would say that BGV has been pretty well ...


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# Looking for an operator on polynomials

I have a small, self-contained, math question, whose motivation is from theoretical computer science (specifically, list decoding of algebraic codes, derivative/multiplicity codes, etc). I wonder whether someone might have an idea.

I'm looking for an operator T that can be applied to m-variate polynomials over a finite field. When applied to a polynomial, the operator should yield a polynomial of not much higher degree. The operator should satisfy the following property: for every m-variate polynomial $F$, and m univariate non-constant polynomials $g_1,...,g_m$, if you know $g_1,...,g_m$ and $F(g_1(t),...,g_m(t))$ for a parameter t, then you can also compute $TF(g_1(t),...,g_m(t))$. The operator should be non-trivial, in the sense that for a fixing $x_1,..,x_m$, the value $F(x_1,...,x_m)$ does not determine the value of $TF(x_1,...,x_m)$.

For $m=1$, the derivative operator gives exactly that: By the chain rule $(F(g(t)))' = F'(g(t))\cdot g'(t)$, which implies that $F'(g(t)) = (F(g(t)))'/g'(t)$. My question is whether there is an operator that works for general m.

• Why total derivative wouldn't fit your requirements? – Tegiri Nenashi Mar 28 '12 at 17:43
• Suppose you take $TF(g_1(t),..,g_m(t)) = \sum_{i=1}^{m} g_i'(t) dF(x_1,...,x_m)/dx_i$ (known as the total derivative). How you do compute $dF(x_1,...,x_m)/dx_i$ from $F(g_1(t),...,g_m(t))$? – Dana Moshkovitz Mar 28 '12 at 19:24
• Your example in one dimensional case is not flawless. What if $g'(t)=0$ ? – Tegiri Nenashi Mar 29 '12 at 15:27
• F(g(t))' and g'(t) are formal polynomials in t. You can divide the formal polynomials. – Dana Moshkovitz Mar 29 '12 at 19:58
• Honestly, I asked this question so long ago, I only barely remember what I needed it for :-)... I believe that the issue is that multiplying by a fixed polynomial is "trivial" in the sense that to compute the multiplication at a point, you only need to know the value of F at the point (and nothing about F's inner-workings). – Dana Moshkovitz Oct 30 '12 at 21:56


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# Methods of Head Modeling - Question on Efficency

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Okay... I''ve been modeling heads till I puke in my current 3d animation class. Used a couple of different methods... Starting with a Sphere, starting with a cube, and creating the geometry as you go along. Then there is also the difference between starting with a cube that already is segmented, or segmenting a 6 sided cube yourself with Knife cuts. Sphere makes shaping the head better, but makes the face harder to model. Cube makes the face easier to model, but the head much harder. Creating the geometry yourself gives very pleasing results (particularly around the eye) but is a pain in the @ and often you get some "sticky" areas. And, as far as the last method I mentioned, I never have any luck with starting with a cube that is already segmented, yet most tutorials I find suggest starting that way? So, how does everyone else model heads? Pros/Cons of each method etc. Right now, I''m leaning towards starting with a 6 polygon cube. - T

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## Runborg, Olof

Compute Distance To:
 Author ID: runborg.olof Published as: Runborg, Olof; Runborg, O. External Links: MGP
 Documents Indexed: 52 Publications since 1993 3 Contributions as Editor Co-Authors: 44 Co-Authors with 48 Joint Publications 1,421 Co-Co-Authors
all top 5

### Co-Authors

 7 single-authored 14 Engquist, Bjorn E. 7 Motamed, Mohammad 3 Gosse, Laurent 3 Kevrekidis, Ioannis George 3 Liu, Hailiang 3 Tanushev, Nicolay M. 2 Arjmand, Doghonay 2 Daubechies, Ingrid Chantal 2 Häggblad, Jon 2 Holst, Henrik 2 Kevrekidis, Panayotis G. 2 Leitenmaier, Lena 2 Lötstedt, Per 2 Malenová, Gabriela 2 Popović, Jelena 2 Theodoropoulos, Constantinos 2 Tornberg, Anna-Karin 2 Tsai, Yen-Hsi Richard 1 Andersson, Ulf 1 Appelö, Daniel 1 Arlinger, Bert G. 1 Benamou, Jean-David 1 Castella, François 1 Collino, Francis 1 Garcia, Fortino 1 Gear, Charles William 1 Hyman, James M. 1 Izzo, Federico 1 Jarlebring, Elias 1 Kieri, Emil 1 Kreiss, Gunilla 1 Ledfelt, Gunnar 1 Lehtimäki, Reijo 1 Marin, Oana 1 Mele, Giampaolo 1 Mineau, Pascal 1 Oppelstrup, Jesper 1 Persson, Per-Olof 1 Perthame, Benoît 1 Ralston, James V. 1 Sweldens, Wim 1 Tempone, Raúl F. 1 Weinerfelt, Per Å. 1 Zou, Jing
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### Serials

 7 Journal of Computational Physics 7 Communications in Mathematical Sciences 3 Multiscale Modeling & Simulation 3 Lecture Notes in Computational Science and Engineering 2 BIT 2 Numerische Mathematik 2 SIAM Journal on Applied Mathematics 2 SIAM Journal on Scientific Computing 1 Computers & Mathematics with Applications 1 IMA Journal of Numerical Analysis 1 Nonlinearity 1 Wave Motion 1 Mathematics of Computation 1 Journal of Computational and Applied Mathematics 1 Constructive Approximation 1 Communications in Partial Differential Equations 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Applied and Computational Harmonic Analysis 1 M2AN. Mathematical Modelling and Numerical Analysis. ESAIM, European Series in Applied and Industrial Mathematics 1 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 Acta Numerica 1 Communications in Computational Physics 1 Advances in Applied Mathematics and Mechanics 1 SIAM/ASA Journal on Uncertainty Quantification 1 Research in the Mathematical Sciences
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### Fields

 45 Numerical analysis (65-XX) 34 Partial differential equations (35-XX) 19 Optics, electromagnetic theory (78-XX) 5 Ordinary differential equations (34-XX) 5 Fluid mechanics (76-XX) 4 Approximations and expansions (41-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 3 General and overarching topics; collections (00-XX) 3 Integral transforms, operational calculus (44-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Probability theory and stochastic processes (60-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Differential geometry (53-XX) 1 Computer science (68-XX) 1 Mechanics of deformable solids (74-XX) 1 Quantum theory (81-XX) 1 Geophysics (86-XX) 1 Systems theory; control (93-XX)

### Citations contained in zbMATH Open

45 Publications have been cited 651 times in 463 Documents Cited by Year
Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Zbl 1086.65066
Kevrekidis, Ioannis G.; Gear, C. William; Hyman, James M.; Kevrekidis, Panagiotis G.; Runborg, Olof; Theodoropoulos, Constantinos
2003
Computational high frequency wave propagation. Zbl 1049.65098
Engquist, Björn; Runborg, Olof
2003
Multi-phase computations in geometrical optics. Zbl 0947.78001
Engquist, Björn; Runborg, Olof
1996
High-frequency wave propagation by the segment projection method. Zbl 0996.78001
Engquist, Björn; Runborg, Olof; Tornberg, Anna-Karin
2002
Normal multiresolution approximation of curves. Zbl 1051.42025
Daubechies, Ingrid; Runborg, Olof; Sweldens, Wim
2004
Mathematical models and numerical methods for high frequency waves. Zbl 1164.78300
Runborg, Olof
2007
Some new results in multiphase geometrical optics. Zbl 0972.78001
Runborg, Olof
2000
Taylor expansion and discretization errors in Gaussian beam superposition. Zbl 1231.78006
2010
Multi-scale methods for wave propagation in heterogeneous media. Zbl 1281.65110
Engquist, Björn; Holst, Henrik; Runborg, Olof
2011
Error estimates for Gaussian beam superpositions. Zbl 1277.35236
Liu, Hailiang; Runborg, Olof; Tanushev, Nicolay M.
2013
Effective bifurcation analysis: a time-stepper-based approach. Zbl 1005.37055
Runborg, Olof; Theodoropoulos, Constantinos; Kevrekidis, Ioannis G.
2002
Multiscale methods for wave propagation in heterogeneous media over long time. Zbl 1246.65149
Engquist, Björn; Holst, Henrik; Runborg, Olof
2012
Wavelet-based numerical homogenization with applications. Zbl 0989.65117
Engquist, Björn; Runborg, Olof
2002
Analysis of heterogeneous multiscale methods for long time wave propagation problems. Zbl 1315.65083
Arjmand, Doghonay; Runborg, Olof
2014
A contribution to wavelet-bases subgrid modeling. Zbl 0937.65113
Andersson, Ulf; Engquist, Björn; Ledfelt, Gunnar; Runborg, Olof
1999
Equation-free, effective computation for discrete systems: a time stepper based approach. Zbl 1140.93422
Möller, J.; Runborg, O.; Kevrekidis, P. G.; Lust, K.; Kevrekidis, I. G.
2005
Corrected trapezoidal rules for a class of singular functions. Zbl 1304.65115
Marin, Oana; Runborg, Olof; Tornberg, Anna-Karin
2014
A time dependent approach for removing the cell boundary error in elliptic homogenization problems. Zbl 1349.65689
Arjmand, Doghonay; Runborg, Olof
2016
The waveguide eigenvalue problem and the tensor infinite Arnoldi method. Zbl 1366.65104
Jarlebring, Elias; Mele, Giampaolo; Runborg, Olof
2017
High frequency limit of the Helmholtz equation. II: Source on a general smooth manifold. Zbl 1290.35262
Castella, F.; Perthame, B.; Runborg, O.
2002
A sparse spectral method for homogenization multiscale problems. Zbl 1152.65099
Daubechies, Ingrid; Runborg, Olof; Zou, Jing
2007
Numerical microlocal analysis of harmonic wavefields. Zbl 1067.78017
Benamou, Jean-David; Collino, Francis; Runborg, Olof
2004
Fast interface tracking via a multiresolution representation of curves and surfaces. Zbl 1181.65025
Runborg, Olof
2009
Finite moment problems and applications to multiphase computations in geometric optics. Zbl 1100.78006
Gosse, Laurent; Runborg, Olof
2005
Multiphase computations in geometrical optics. Zbl 0963.78004
Engquist, Björn; Runborg, Olof
1999
Approximation of high frequency wave propagation problems. Zbl 1179.78113
2009
Gaussian beam methods for the Helmholtz equation. Zbl 1302.35135
Liu, Hailiang; Ralston, James; Runborg, Olof; Tanushev, Nicolay M.
2014
Analysis of a fast method for solving the high frequency Helmholtz equation in one dimension. Zbl 1260.65067
Popovic, Jelena; Runborg, Olof
2011
Existence, uniqueness, and a constructive solution algorithm for a class of finite Markov moment problems. Zbl 1158.65010
Gosse, Laurent; Runborg, Olof
2008
Analysis of high order fast interface tracking methods. Zbl 1305.65176
Runborg, Olof
2014
A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems. Zbl 1133.65089
2007
Resolution of the finite Markov moment problem. Zbl 1081.65121
Gosse, Laurent; Runborg, Olof
2005
WaveHoltz: iterative solution of the Helmholtz equation via the wave equation. Zbl 1453.65396
Appelö, Daniel; Garcia, Fortino; Runborg, Olof
2020
Sobolev and max norm error estimates for Gaussian beam superpositions. Zbl 1372.35263
Liu, Hailiang; Runborg, Olof; Tanushev, Nicolay M.
2016
A sparse stochastic collocation technique for high-frequency wave propagation with uncertainty. Zbl 1362.35344
Malenova, G.; Motamed, M.; Runborg, O.; Tempone, R.
2016
A fast phase space method for computing creeping rays. Zbl 1111.78013
2006
Introduction to normal multiresolution approximation. Zbl 1105.65358
Runborg, Olof
2005
On energy preserving consistent boundary conditions for the Yee scheme in 2D. Zbl 1257.78021
Engquist, B.; Häggblad, J.; Runborg, O.
2012
Wavelets and wavelet based numerical homogenization. Zbl 1159.65373
Runborg, Olof
2009
Projection generated homogenization. Zbl 1015.35012
Engquist, Björn; Runborg, Olof
2002
Multiscale methods in science and engineering. Papers presented at the conference, Uppsala, Sweden, January 26–28, 2004. Zbl 1070.65500
2005
Stochastic regularity of a quadratic observable of high-frequency waves. Zbl 1387.60108
Malenová, G.; Motamed, M.; Runborg, O.
2017
Numerical analysis of multiscale computations. Proceedings of a winter workshop at the Banff International Research Station 2009, Canada, December 6–11, 2011. Zbl 1229.65003
2012
Simulation of a waveguide filter using wavelet-based numerical homogenization. Zbl 1040.78019
2001
Accuracy of staircase approximations in finite-difference methods for wave propagation. Zbl 1318.65050
2014
WaveHoltz: iterative solution of the Helmholtz equation via the wave equation. Zbl 1453.65396
Appelö, Daniel; Garcia, Fortino; Runborg, Olof
2020
The waveguide eigenvalue problem and the tensor infinite Arnoldi method. Zbl 1366.65104
Jarlebring, Elias; Mele, Giampaolo; Runborg, Olof
2017
Stochastic regularity of a quadratic observable of high-frequency waves. Zbl 1387.60108
Malenová, G.; Motamed, M.; Runborg, O.
2017
A time dependent approach for removing the cell boundary error in elliptic homogenization problems. Zbl 1349.65689
Arjmand, Doghonay; Runborg, Olof
2016
Sobolev and max norm error estimates for Gaussian beam superpositions. Zbl 1372.35263
Liu, Hailiang; Runborg, Olof; Tanushev, Nicolay M.
2016
A sparse stochastic collocation technique for high-frequency wave propagation with uncertainty. Zbl 1362.35344
Malenova, G.; Motamed, M.; Runborg, O.; Tempone, R.
2016
Analysis of heterogeneous multiscale methods for long time wave propagation problems. Zbl 1315.65083
Arjmand, Doghonay; Runborg, Olof
2014
Corrected trapezoidal rules for a class of singular functions. Zbl 1304.65115
Marin, Oana; Runborg, Olof; Tornberg, Anna-Karin
2014
Gaussian beam methods for the Helmholtz equation. Zbl 1302.35135
Liu, Hailiang; Ralston, James; Runborg, Olof; Tanushev, Nicolay M.
2014
Analysis of high order fast interface tracking methods. Zbl 1305.65176
Runborg, Olof
2014
Accuracy of staircase approximations in finite-difference methods for wave propagation. Zbl 1318.65050
2014
Error estimates for Gaussian beam superpositions. Zbl 1277.35236
Liu, Hailiang; Runborg, Olof; Tanushev, Nicolay M.
2013
Multiscale methods for wave propagation in heterogeneous media over long time. Zbl 1246.65149
Engquist, Björn; Holst, Henrik; Runborg, Olof
2012
On energy preserving consistent boundary conditions for the Yee scheme in 2D. Zbl 1257.78021
Engquist, B.; Häggblad, J.; Runborg, O.
2012
Numerical analysis of multiscale computations. Proceedings of a winter workshop at the Banff International Research Station 2009, Canada, December 6–11, 2011. Zbl 1229.65003
2012
Multi-scale methods for wave propagation in heterogeneous media. Zbl 1281.65110
Engquist, Björn; Holst, Henrik; Runborg, Olof
2011
Analysis of a fast method for solving the high frequency Helmholtz equation in one dimension. Zbl 1260.65067
Popovic, Jelena; Runborg, Olof
2011
Taylor expansion and discretization errors in Gaussian beam superposition. Zbl 1231.78006
2010
Fast interface tracking via a multiresolution representation of curves and surfaces. Zbl 1181.65025
Runborg, Olof
2009
Approximation of high frequency wave propagation problems. Zbl 1179.78113
2009
Wavelets and wavelet based numerical homogenization. Zbl 1159.65373
Runborg, Olof
2009
Existence, uniqueness, and a constructive solution algorithm for a class of finite Markov moment problems. Zbl 1158.65010
Gosse, Laurent; Runborg, Olof
2008
Mathematical models and numerical methods for high frequency waves. Zbl 1164.78300
Runborg, Olof
2007
A sparse spectral method for homogenization multiscale problems. Zbl 1152.65099
Daubechies, Ingrid; Runborg, Olof; Zou, Jing
2007
A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems. Zbl 1133.65089
2007
A fast phase space method for computing creeping rays. Zbl 1111.78013
2006
Equation-free, effective computation for discrete systems: a time stepper based approach. Zbl 1140.93422
Möller, J.; Runborg, O.; Kevrekidis, P. G.; Lust, K.; Kevrekidis, I. G.
2005
Finite moment problems and applications to multiphase computations in geometric optics. Zbl 1100.78006
Gosse, Laurent; Runborg, Olof
2005
Resolution of the finite Markov moment problem. Zbl 1081.65121
Gosse, Laurent; Runborg, Olof
2005
Introduction to normal multiresolution approximation. Zbl 1105.65358
Runborg, Olof
2005
Multiscale methods in science and engineering. Papers presented at the conference, Uppsala, Sweden, January 26–28, 2004. Zbl 1070.65500
2005
Normal multiresolution approximation of curves. Zbl 1051.42025
Daubechies, Ingrid; Runborg, Olof; Sweldens, Wim
2004
Numerical microlocal analysis of harmonic wavefields. Zbl 1067.78017
Benamou, Jean-David; Collino, Francis; Runborg, Olof
2004
Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Zbl 1086.65066
Kevrekidis, Ioannis G.; Gear, C. William; Hyman, James M.; Kevrekidis, Panagiotis G.; Runborg, Olof; Theodoropoulos, Constantinos
2003
Computational high frequency wave propagation. Zbl 1049.65098
Engquist, Björn; Runborg, Olof
2003
High-frequency wave propagation by the segment projection method. Zbl 0996.78001
Engquist, Björn; Runborg, Olof; Tornberg, Anna-Karin
2002
Effective bifurcation analysis: a time-stepper-based approach. Zbl 1005.37055
Runborg, Olof; Theodoropoulos, Constantinos; Kevrekidis, Ioannis G.
2002
Wavelet-based numerical homogenization with applications. Zbl 0989.65117
Engquist, Björn; Runborg, Olof
2002
High frequency limit of the Helmholtz equation. II: Source on a general smooth manifold. Zbl 1290.35262
Castella, F.; Perthame, B.; Runborg, O.
2002
Projection generated homogenization. Zbl 1015.35012
Engquist, Björn; Runborg, Olof
2002
Simulation of a waveguide filter using wavelet-based numerical homogenization. Zbl 1040.78019
2001
Some new results in multiphase geometrical optics. Zbl 0972.78001
Runborg, Olof
2000
A contribution to wavelet-bases subgrid modeling. Zbl 0937.65113
Andersson, Ulf; Engquist, Björn; Ledfelt, Gunnar; Runborg, Olof
1999
Multiphase computations in geometrical optics. Zbl 0963.78004
Engquist, Björn; Runborg, Olof
1999
Multi-phase computations in geometrical optics. Zbl 0947.78001
Engquist, Björn; Runborg, Olof
1996
all top 5

### Cited by 659 Authors

 34 Kevrekidis, Ioannis George 19 Runborg, Olof 16 Jin, Shi 15 Qian, Jianliang 14 Abdulle, Assyr 12 Amat, Sergio P. 12 Samaey, Giovanni 11 Liandrat, Jacques 11 Liu, Hailiang 10 Chung, Tsz Shun Eric 10 Siettos, Constantinos I. 10 Yang, Xu 9 Luo, Songting 8 Engquist, Bjorn E. 8 Gosse, Laurent 8 Osher, Stanley Joel 8 Tsai, Yen-Hsi Richard 7 Arjmand, Doghonay 7 Benamou, Jean-David 7 Gear, Charles William 7 Kutz, J. Nathan 7 Roberts, Anthony John 6 Chappell, David J. 6 Efendiev, Yalchin R. 6 Jarlebring, Elias 6 Roose, Dirk 6 Ying, Lexing 5 Brunton, Steven L. 5 Bunder, J. E. 5 Dadourian, Karine 5 Dehghan Takht Fooladi, Mehdi 5 Erban, Radek 5 Fox, Rodney O. 5 Kavousanakis, Michail E. 5 Laing, Carlo R. 5 Motamed, Mohammad 5 Salehi, Rezvan 5 Tanner, Gregor K. 5 Trillo, Juan Carlos 5 Xiu, Dongbin 4 Boudouvis, Andreas G. 4 Cheng, Li-Tien 4 Karniadakis, George Em 4 Kreiss, Gunilla 4 Leung, Shingyu 4 Pouchon, Timothée 4 Vandekerckhove, Christophe 4 Wang, Zhongming 4 Wen, Xin 4 Wu, Haijun 4 Zhang, Yanyan 4 Zhang, Zhiwen 4 Zielinski, Przemyslaw 3 Acharya, Amit 3 Anand, Akash 3 Bajars, Janis 3 Burridge, Robert 3 Castella, François 3 Cockburn, Bernardo 3 Deng, Weibing 3 E, Weinan 3 Fu, Shubin 3 Gottwald, Georg A. 3 Gravemeier, Volker 3 Grohs, Philipp 3 Grote, Marcus J. 3 Henning, Patrick 3 Hoang, Viet Ha 3 Hou, Thomas Yizhao 3 Koutsourelakis, Phaedon-Stelios 3 Lelièvre, Tony 3 Leung, Wingtat 3 Lu, Jianfeng 3 Ma, Dingjiong 3 Mele, Giampaolo 3 Melis, Ward 3 Owhadi, Houman 3 Pavliotis, Grigorios A. 3 Proctor, Joshua L. 3 Rajendran, Karthikeyan 3 Reitich, Fernando 3 Russo, Lucia 3 Tadmor, Eitan 3 Tanushev, Nicolay M. 3 Tornberg, Anna-Karin 3 Wall, Wolfgang A. 3 Yang, Dinghui 3 Yin, Dongsheng 3 Zhang, Yu 3 Zhao, Hongkai 3 Zhou, Zhennan 3 Zou, Yu 2 Ariel, Gil 2 Babuška, Ivo 2 Barton, Nathan R. 2 Berra, Michele 2 Bold, Katherine A. 2 Byrne, Helen Mary 2 Candès, Emmanuel J. 2 Chen, Yifan ...and 559 more Authors
all top 5

### Cited in 110 Serials

 113 Journal of Computational Physics 23 Multiscale Modeling & Simulation 19 Journal of Scientific Computing 18 Journal of Computational and Applied Mathematics 15 Computer Methods in Applied Mechanics and Engineering 14 SIAM Journal on Scientific Computing 10 Numerical Algorithms 8 Mathematics of Computation 8 Physica D 8 Applied Numerical Mathematics 8 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 8 Advances in Computational Mathematics 8 SIAM Journal on Applied Dynamical Systems 7 Computers & Mathematics with Applications 7 Wave Motion 7 Numerische Mathematik 7 Applied and Computational Harmonic Analysis 6 Applied Mathematics and Computation 6 SIAM Journal on Numerical Analysis 5 Computers and Fluids 5 BIT 5 International Journal for Numerical Methods in Engineering 5 Foundations of Computational Mathematics 5 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 4 Journal of Mathematical Biology 4 Journal of Differential Equations 4 Comptes Rendus. Mathématique. Académie des Sciences, Paris 4 Journal of Computational Dynamics 3 Communications on Pure and Applied Mathematics 3 Constructive Approximation 3 Applied Mathematical Modelling 3 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 3 Acta Numerica 3 Communications in Computational Physics 3 Journal of Theoretical Biology 3 SIAM/ASA Journal on Uncertainty Quantification 2 Biological Cybernetics 2 International Journal for Numerical Methods in Fluids 2 Inverse Problems 2 Journal of Fluid Mechanics 2 Journal of Mathematical Physics 2 Journal of Functional Analysis 2 Computational Mechanics 2 Numerical Methods for Partial Differential Equations 2 SIAM Journal on Matrix Analysis and Applications 2 Proceedings of the National Academy of Sciences of the United States of America 2 SIAM Review 2 Physics of Fluids 2 Chaos 2 M2AN. Mathematical Modelling and Numerical Analysis. ESAIM, European Series in Applied and Industrial Mathematics 2 Combustion Theory and Modelling 2 International Journal of Numerical Analysis and Modeling 2 S$$\vec{\text{e}}$$MA Journal 2 Research in the Mathematical Sciences 1 Archive for Rational Mechanics and Analysis 1 Journal of Engineering Mathematics 1 Journal of the Franklin Institute 1 Journal of Statistical Physics 1 Rocky Mountain Journal of Mathematics 1 Molecular Simulation 1 Bulletin of Mathematical Biology 1 Automatica 1 Journal of Approximation Theory 1 Mathematics and Computers in Simulation 1 Mathematische Nachrichten 1 Rendiconti di Matematica e delle sue Applicazioni. Serie VII 1 Acta Applicandae Mathematicae 1 Computer Aided Geometric Design 1 Acta Mathematicae Applicatae Sinica. English Series 1 ACM Transactions on Graphics 1 Revista Matemática Iberoamericana 1 Journal of the American Mathematical Society 1 Mathematical and Computer Modelling 1 International Journal of Computer Mathematics 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Journal of Non-Newtonian Fluid Mechanics 1 Linear Algebra and its Applications 1 Stochastic Processes and their Applications 1 Bulletin of the American Mathematical Society. New Series 1 International Journal of Robust and Nonlinear Control 1 Journal of Mathematical Imaging and Vision 1 Experimental Mathematics 1 Journal of Computational Neuroscience 1 Engineering Analysis with Boundary Elements 1 Journal of Mathematical Chemistry 1 Mathematical Problems in Engineering 1 International Journal of Computational Fluid Dynamics 1 Abstract and Applied Analysis 1 Mathematical and Computer Modelling of Dynamical Systems 1 Acta Mathematica Sinica. English Series 1 Communications in Nonlinear Science and Numerical Simulation 1 Far East Journal of Mathematical Sciences 1 The ANZIAM Journal 1 Nonlinear Analysis. Real World Applications 1 Mathematical Modelling and Analysis 1 Entropy 1 Journal of Machine Learning Research (JMLR) 1 Communications on Pure and Applied Analysis 1 Optimization Letters 1 Mathematical Modelling of Natural Phenomena ...and 10 more Serials
all top 5

### Cited in 37 Fields

 312 Numerical analysis (65-XX) 187 Partial differential equations (35-XX) 63 Fluid mechanics (76-XX) 54 Optics, electromagnetic theory (78-XX) 45 Dynamical systems and ergodic theory (37-XX) 38 Statistical mechanics, structure of matter (82-XX) 37 Mechanics of deformable solids (74-XX) 33 Biology and other natural sciences (92-XX) 31 Ordinary differential equations (34-XX) 30 Probability theory and stochastic processes (60-XX) 30 Quantum theory (81-XX) 21 Approximations and expansions (41-XX) 21 Computer science (68-XX) 16 Statistics (62-XX) 15 Information and communication theory, circuits (94-XX) 10 Harmonic analysis on Euclidean spaces (42-XX) 10 Systems theory; control (93-XX) 8 Linear and multilinear algebra; matrix theory (15-XX) 8 Calculus of variations and optimal control; optimization (49-XX) 8 Mechanics of particles and systems (70-XX) 8 Classical thermodynamics, heat transfer (80-XX) 8 Geophysics (86-XX) 6 Operations research, mathematical programming (90-XX) 5 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 4 Combinatorics (05-XX) 4 Integral transforms, operational calculus (44-XX) 4 Operator theory (47-XX) 3 Real functions (26-XX) 3 Integral equations (45-XX) 2 Functional analysis (46-XX) 2 Differential geometry (53-XX) 2 Global analysis, analysis on manifolds (58-XX) 1 Mathematical logic and foundations (03-XX) 1 Functions of a complex variable (30-XX) 1 Potential theory (31-XX) 1 Difference and functional equations (39-XX) 1 Relativity and gravitational theory (83-XX)


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How do you solve 18x^2+48x=-32?

Double-root: at $x = - \frac{4}{3}$
$f \left(x\right) = 9 {x}^{2} + 24 x + 16 = {\left(3 x + 4\right)}^{2} = 0$
The 2 roots are: $\pm \left(3 x + 4\right) = 0$
There is double-root at $x = - \frac{4}{3}$


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# Christmas Streak 10/88: Two Strange To Be Strange

Probability Level 3

I have two distinct, fair, six-sided dice that have positive integers on their sides. Neither of the dice is a "normal" die (i.e., with 1, 2, 3, 4, 5, and 6 on the sides).

Interestingly, the probability distribution of the sum of rolling these two dice is exactly the same as that of the sum of rolling 2 "normal" dice.

Let the numbers on the first die be $(a_1, a_2, a_3, a_4, a_5, a_6),$ and those on the second $(b_1, b_2, b_3, b_4, b_5, b_6),$ where $a_1\leq a_2 \leq\cdots\leq a_6$ and $b_1\leq b_2 \leq \cdots\leq b_6.$

Submit your answer as the product of these two 6-digit integers: $\overline{a_1 a_2 a_3 a_4 a_5 a_6} \times \overline{b_1 b_2 b_3 b_4 b_5 b_6 }.$

The dice above have 1,1,2,2,4,8 and 1,2,3,3,4,4 on their sides. Their sum can produce the same numbers (2-12) as the sum of two ordinary dice, but not with the same probability distribution.

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# Time reversal and the solutions to Maxwell equations

The standard Lienard-Wiechert potentials describe the electromagnetic field at a point $P$ at time $t$ due to an arbitrarily moving charge $q$ at the retarded time $t-r/c$. An electromagnetic influence is assumed to travel, with the velocity of light $c$, from the moving charge $q$ at time $t-r/c$ to the field point $P$ at time $t$ while the charge moves along its path. This situation is illustrated on the left-hand side of the diagram. Physically the charge $q$ is in arbitrary motion whereas the field point $P$ is at rest in an inertial frame.

But the laws of electromagnetism are assumed to be time reversible. Therefore the time reverse of this process should occur in Nature. This is shown on the right-hand side of the diagram. Have I got this right? Basically all velocities are reversed including that of the electromagnetic influence.

Thus we have a picture in which an electromagnetic influence travels from a field point $P$ at time $t$ and intercepts a moving charged particle $q$ at a later time time $t+r/c$.

Note that I am not assuming that a signal goes backwards in time. I am "only" assuming that a physical process exists in which cause and effect are switched around in the temporal order. Instead of first moving charge then later field effect we have first field effect then later moving charge.

So to what physical situation does the advanced description correspond?

I believe that it might describe the situation in which the charge $q$ is at rest in an inertial frame and the point $P$ is in arbitrary motion. Thus, relative to an observer moving with point $P$, charge $q$ has an apparent motion illustrated by the right-hand diagram. The observer, at point $P$ and time $t$, measures a kind of inertial electromagnetic field that is determined by the apparent position of the charge at the advanced time $t+r/c$.

Could this be the answer to the meaning of the advanced solutions to Maxwell's equations?

The laws of electromagnetism are indeed time-reversal invariant, but this can sometimes be in a fairly restricted sense, particularly when dealing with causality as regards the Liénard-Wiechert potentials.

Consider the following situation. A charged particle is travelling under the influence of some force $\mathbf F_\text{ext}(t)$ which is due to a time-reversible interaction with an external system. At a certain time $t_0$ this interaction causes a bend in the particle's trajectory. This acceleration causes it to emit radiation, which we know will be described by the retarded Liénard-Wiechert potentials. This radiation will exert a back-reaction force on the particle, which alters the trajectory from what the single force $\mathbf F_\text{ext}(t)$ would have shaped.

$\qquad\qquad$

Let's look, on the other hand, at the time-reversed situation. The particle now traverses its trajectory in the opposite sense, but the really important thing is that your initial conditions now include an incoming pulse of radiation. This pulse will coincide with the particle at the bend in the trajectory. After that, it will destructively interfere with the pulse radiated by the particle, and that will leave no radiation in the end. The combination of both pulses will exert the extra force to make the particle what it should, instead of what the time-reversed $\mathbf F_\text{ext}$ would have shaped.

Your dynamics are therefore completely time-reversal invariant. By a weird "coincidence", you can describe the incoming pulse of radiation in the time-reversed version using the advanced Liénard-Wiechert potentials. Here it's important to note that this is because the pulse would have been described by the potential $$\text{advanced LWP}-\text{retarded LWP},$$ which is a regular solution to the Maxwell equations, and has no singularity at the particle.

Thus far for the classical understanding of this topic. The above shows that classical EM is indeed time-reversal invariant, but that this invariance requires that you consider the fields to be physical entities in themselves, and time-reverse them as well, boundary conditions and all. While this is not unsatisfactory, it has been a persistent itch on the side of a number of theorists because you can't measure the fields directly. Should it not be possible to make a time-reversal invariant description of electromagnetism purely in terms of the position of the charges?

This is the essence of the Wheeler-Feynman absorber theory, which has its own problems and is not really taken very seriously AFAIK, but is quite beautiful. This postulates that all charges radiate both forward and backward in time, and that the radiation reaction force on a given particle is not an interaction with its own field, but with the advanced radiation 'sent from the future' by the particles that will be influenced by the 'physical' retarded radiation. Again, not something that's currently a part of standard physics, but it's definitely something to read up on.

• But in my example there is not necessarily any radiation emitted. I am seeking a physical understanding of a reversed electromagnetic influence from observation point P back to the source without assuming the Wheeler-Feynman theory with its advanced em waves actually travelling backwards in time. – John Eastmond Sep 14 '13 at 7:07
• There is radiation emitted: that's what your "electromagnetic influence at point P" really is. – Emilio Pisanty Sep 14 '13 at 11:29
• But what happens if the charge is moving with constant velocity? In that case any electromagnetic influence emanating from it is not really radiation. – John Eastmond Sep 14 '13 at 17:02
• Good question. I'll have to think about that one. – Emilio Pisanty Sep 14 '13 at 17:29
• @Emilio I do not think there is a problem. Classically, looking at the advanced Lienard solution, the charge must accelerate on receiving the wavepacket, not before. So it can move with constant velocity until it receives the packet and just during (and after) that time lapse it accelerates. – Valter Moretti Apr 15 '20 at 7:29