text:
## Binary Linear Diophantine Equations

Euclid’s Algorithm (previous post) provides a means of finding solutions, if solutions exist, of the binary linear diophantine equation

$a_{1}x_{1} + a_{2}x_{2} = a_{0}$

where the ak are unequal integers (if they are equal we are essentially dealing with an equation in one variable) and solutions in integers are sought.

We begin by showing that if n1 > n2 > 0 are integers we can construct integers a1 and a2 such that

$a_{1}n_{1}+a_{2}n_{2}=(n_{1}, n_{2})$

where (n1, n2) is the greatest common divisor of n1 and n2.  We perform the Euclidean algorithm (as described in the previous post) on n1 and n2.  If the algorithm has only k = 1 step we have

$n_{1} = q_{1}n_{2}$ (q1 an integer) so $n_{1} + (1-q_{1})n_{2} = n_{2} = (n_{1}, n_{2})$

If the algorithm has k = 2 steps then

$n_{1}-q_{1}n_{2}=(n_{1},n_{2})$

More generally it is easy to show by induction that if the algorithm completes in k ≥ 2 steps then

$a_{1}=c_{k-1}$  and  $a_{2} = d_{k-1}$ where

$c_{1}=1$  ;  $d_{1}=-q_{1}$  ;  $c_{2} = -q_{2}$  ;  $d_{2}=1+q_{1}q_{2}$ and

$c_{i}=c_{i-2}-q_{i-2}c_{i-1}$  ;  $d_{i}=d_{i-2}-q_{i-2}d_{i-1}$ for i ≥ 3

Proof:  The proposition is true for k = 2 and is easily shown to be true for k = 3.  Let Pi be the proposition that

$n_{i+2} = c_{i+2}n_{1}+d_{i+2}n_{2}$

with ci and di defined as in the proposition, and for 1 ≤ i ≤ k-1.

It is easily seen that P1 is true. Suppose the proposition is true for all values of i up to some fixed value i ≤ k – 2.  Then

$n_{i+3}=n_{i+1}-q_{i+1}n_{i+2}=(c_{i+1}-q_{i+1}c_{i+2})n_{1}+(d_{i+1}-q_{i+1}d_{i+2})n_{2}$

So Pi+1 is true. Consequently Pk-1 is true and nk+1 = (n1, n2) can be expressed in the form proposed.

The construction can be easily extended to all (unequal) non-zero integers n1 and n2. We simply find integers a’1 and a’2 such that a’1|n1| + a’2|n2| = (n1, n2) then set a’1=(n1/|n1|)a1 and a’2=(n2/|n2|)a2.

The following is an obvious corollary:

Let p, m, n be non-zero integers. If p divides mn and (p, m) = 1 then p divides n.  (There are integers x, y such that xp + my = (p, m) = 1.  Hence xpn + mny = n.  Since p divides the left hand side it must divide n.)

We can now state a criterion for determining whether a solution to the diophantine equation exists and, if it does, provide a method of constructing a solution:

The linear diophantine equation a1x1 + a2x2 = a0 (with a1 and a2 non-zero and unequal) has a solution only if (a1, a2) divides a0.  If x’1 and x’2 are integers such that a1x’1 + a2x’2 = (a1, a2) then

$\displaystyle x_{1} = \frac {a_{0}x'_{1}}{(a_{1},a_{2})}$  ;   $\displaystyle x_{2} = \frac {a_{0}x'_{2}}{(a_{1},a_{2})}$  is a solution.

Proof:  if there is a solution then since (a1, a2) divides a1x1 + a2x2 it must divide a0. The rest is clear on substituting.

Knowing one solution, we can now identify all solutions:

If x10, x20 is a solution of a1x1 + a2x2 = a0 (with a1 and a2 non-zero and unequal) then x1, x2 is a solution if and only if

$\displaystyle x_{1}=x_{1}^{0}+ \frac {a_{2}t} {(a_{1},a_{2})}$  ;   $\displaystyle x_{2}=x_{2}^{0}- \frac {a_{1}t} {(a_{1},a_{2})}$   where t is any integer

Proof:  the if part is easily seen by substitution.  For the converse, consider first the case a0 = 0, noting that x1 = 0, x2 = 0 is a solution. Then

$\displaystyle x_{1}= \frac {-a_{2}x_{2}} {a_{1}} = \frac {-a'_{2}x_{2}} {a'_{1}}$  where a1 = a’1(a1, a2) ; a2 = a’2(a1, a2)

Since (a’1, a’2) = 1 (if this were not so and there were a common divisor k > 1 then k(a1, a2) would divide a1 and a2, a contradiction) then a’1 divides x2 so

$\displaystyle x_{2} = \frac {-a_{1}t} {(a_{1}, a_{2})}$ where t is some integer and $\displaystyle x_{1} = \frac {a_{2}t} {(a_{1}, a_{2})}$

The general case follows easily from the special case on noting that if x10, x20 and x1, x2 are solutions then a1(x10 -x1) + a2( x20 – x2) = 0


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# Intervals and boxes¶

We will first only focus on static items that are not evolving with time. The vectors $$\mathbf{x}$$ and matrices $$\mathbf{X}$$, presented in the previous pages, have their interval counterpart. This page provides the main uses of these sets. They will be involved afterwards for solving problems.

Note

These interval variables come from the IBEX library. They are briefly presented here for the sake of consistency. For more information, please refer to the IBEX documentation for C++ use or to the pyIbex manual for Python usage.

## Intervals, boxes and interval matrices¶

• Interval x(double lb, double ub) will define the interval $$[x]$$.
It is made of its lower and upper bounds $$[x^{-},x^{+}]$$.
x = Interval(0, 10)                         # [0,10]
x = Interval(1, oo)                         # [1,∞]
x = Interval(-oo, -10)                      # [-∞,-10]

Interval x(0, 10);                          // [0,10]
Interval x(1, oo);                          // [1,∞]
Interval x(-oo, -10);                       // [-∞,-10]

Some pre-defined values are also at hand:

x = Interval()                              # [-∞,∞] (default value)
x = Interval.EMPTY_SET                      # ∅
x = Interval.PI                             # [π]
x = Interval.TWO_PI                         # [2π]
x = Interval.HALF_PI                        # [π/2]

Interval x;                                 // [-∞,∞] (default value)
Interval x = Interval::EMPTY_SET;           // ∅
Interval x = Interval::PI;                  // [π]
Interval x = Interval::TWO_PI;              // [2π]
Interval x = Interval::HALF_PI;             // [π/2]

Note that the constant $$[\pi]$$ is a reliable enclosure of the $$\pi$$ value, that cannot be exactly represented in a computer with a single floating-point value.

x = Interval.PI                             # [π]
# x = [3.141592653589793, 3.141592653589794]

Interval x = Interval::PI;                  // [π]
// x = [3.141592653589793, 3.141592653589794]

• IntervalVector x(int n) is used for $$n$$-d vectors of intervals, also called boxes.
For instance:
x = IntervalVector(2, [-1,3])               # creates [x]=[-1,3]×[-1,3]=[-1,3]^2
y = IntervalVector([[3,4],[4,6]])           # creates [y]= [3,4]×[4,6]
z = IntervalVector(3, Interval(0,oo))       # creates [z]=[0,∞]^3
w = IntervalVector(y)                       # creates a copy: [w]=[y]

v = (0.42,0.42,0.42)                        # one vector (0.42;0.42;0.42)
iv = IntervalVector(v)                      # creates one box that wraps v:
#   [0.42,0.42]×[0.42,0.42]×[0.42,0.42]

IntervalVector x(2, Interval(-1,3));        // creates [x]=[-1,3]×[-1,3]=[-1,3]^2
IntervalVector y{{3,4},{4,6}};              // creates [y]= [3,4]×[4,6]
IntervalVector z(3, Interval(0,oo));        // creates [z]=[0,∞]^3
IntervalVector w(y);                        // creates a copy: [w]=[y]

Vector v(3, 0.42);                          // one vector (0.42;0.42;0.42)
IntervalVector iv(v);                       // creates one box that wraps v:
//   [0.42,0.42]×[0.42,0.42]×[0.42,0.42]

One can access vector components as we do for Vector objects:

x[1] = Interval(0,10)                       # updates to [x]=[-1,3]×[0,10]

x[1] = Interval(0,10);                      // updates to [x]=[-1,3]×[0,10]

The vector operations to handle Vector objects can also be used for boxes:

n = x.size()                  # box dimension (number of components): 2
x.resize(5)                   # updates [x] to [-1,3]×[0,10]×[-∞,∞]×[-∞,∞]×[-∞,∞]
m = x.subvector(1,2)          # creates [m]=[0,10]×[-∞,∞]
x.put(2,y)                    # updates [x] to [-1,3]×[0,10]×[3,4]×[4,6]×[-∞,∞]

int n = x.size();             // box dimension (number of components): 2
x.resize(5);                  // updates [x] to [-1,3]×[0,10]×[-∞,∞]×[-∞,∞]×[-∞,∞]
IntervalVector m = x.subvector(1,2); // creates [m]=[0,10]×[-∞,∞]
x.put(2,y);                   // updates [x] to [-1,3]×[0,10]×[3,4]×[4,6]×[-∞,∞]

Lastly, the concatenation of two IntervalVector can be done with the cart_prod function:

a = IntervalVector([[0,1],[2,3]])
b = IntervalVector([[4,5],[6,7]])
c = cart_prod(a,b)
# c: ([0, 1] ; [2, 3] ; [4, 5] ; [6, 7])

IntervalVector a({{0,1},{2,3}});
IntervalVector b({{4,5},{6,7}});
IntervalVector c = cart_prod(a,b);
// c: ([0, 1] ; [2, 3] ; [4, 5] ; [6, 7])

• IntervalMatrix is also available.

## The empty set¶

In mathematics, the empty set is the unique set having no elements; it corresponds to one entity while in Tubex (as in IBEX/pyIbex) there exists one empty set representation for each class of domain.

Note

In our framework, empty sets correspond to domains that do not contain feasible solutions. This may be the result of a too restrictive definition of the problem, for instance due to some errors in the model or because of outliers in the dataset.

The empty set of an Interval object is given by:

x = Interval.EMPTY_SET                      # ∅

Interval x = Interval::EMPTY_SET;           // ∅

For boxes (interval vectors), we have to specify their dimension even in case of empty set. This differs from mathematical definitions, but allows simple operations when programming with boxes.

x = IntervalVector.empty(3) # ∅×∅×∅

IntervalVector x = IntervalVector::empty(3); // ∅×∅×∅

## Set operations¶

Set operations are available for Interval and IntervalVector objects (see the official reference). In the following table, if $$[x]$$ is an interval object, $$d$$ is a real value.

Code

Meaning

x==y

$$[x]=[y]$$

x!=y

$$[x]\neq [y]$$

x.is_empty()

$$[x]=\varnothing$$

x.is_unbounded()

true iff $$[x]$$ has one of its bounds infinite

x.is_subset(y)

$$[x]\subseteq [y]$$

x.is_strict_subset(y)

$$[x]\subseteq [y]\wedge [x]\neq [y]$$

x.is_superset(y)

$$[x]\supseteq [y]$$

x.is_strict_superset(y)

$$[x]\supseteq [y]\wedge [x]\neq [y]$$

x.contains(p)

$$d\in [x]$$

x.intersects(y)

$$[x]\cap [y]\neq\varnothing$$

x.is_disjoint(y)

$$[x]\cap [y]=\varnothing$$

x.overlaps(y)

$$\mathring{[x]}\cap \mathring{[y]}\neq\varnothing$$

Where $$\mathring{[x]}$$ denotes the interior of $$[x]$$.
In addition of these test functions, operations on sets are available:

Code

Meaning

x&y

$$[x]\cap [y]$$

x|y

$$[x]\sqcup[y]$$

x.set_empty()

$$[x]\leftarrow \varnothing$$

x=y

$$[x]\leftarrow [y]$$

x&=y

$$[x]\leftarrow ([x]\cap [y])$$

x|=y

$$[x]\leftarrow ([x]\sqcup[y])$$

Finally, one can also access properties of the sets. First for Interval:

Return type

Code

Meaning

double

x.lb()

$$\underline{x}$$, the lower (left) bound of $$[x]$$

double

x.ub()

$$\overline{x}$$, the upper (right) bound of $$[x]$$

double

x.diam()

diameter, $$|\overline{x}-\underline{x}|$$

double

x.rad()

double

x.mid()

the midpoint, ($$(\underline{x}+\overline{x})/2$$)

Interval

x.inflate(eps)

an interval with the same midpoint and radius increased by eps

Then for IntervalVector:

Return type

Code

Meaning

Vector

x.lb()

lower-left corner (vector of lower bounds of $$[x]$$)

Vector

x.ub()

upper-right corner (vector of upper bounds of $$[x]$$)

Vector

x.diam()

vector of diameters, $$|\overline{x_i}-\underline{x_i}|$$

double

x.min_diam()

minimal diameter, among all components of [x]

double

x.max_diam()

maximal diameter, among all components of [x]

Vector

x.rad()

vector of radii (halves of diameters)

Vector

x.mid()

the midpoint, ($$(\underline{x}+\overline{x})/2$$)

double

x.volume()

the volume of the box

bool

x.is_flat()

true if the volume is null (one dimension is degenerated)

IntervalVector

x.inflate(eps)

new box: same midpoint and each radius increased by eps

bool

x.is_unbounded()

true iff $$[x]$$ has one of its bounds infinite

## Interval arithmetic¶

Interval analysis is based on the extension of all classical real arithmetic operators. Consider two intervals $$[x]$$ and $$[y]$$ and an operator $$\diamond\in\left\{+,-,\cdot,/\right\}$$. We define $$[x]\diamond[y]$$ as the smallest interval containing all feasible values for $$x\diamond y$$, assuming that $$x\in[x]$$ and $$y\in[y]$$.

$\begin{split}[x]\diamond[y]&=&\left[\left\{x\diamond y\in\mathbb{R} \mid x\in[x],y\in[y]\right\}\right],\\ \left[x\right]\diamond\varnothing&=&\varnothing.\end{split}$

Dealing with closed intervals, most of the operations can rely on their bounds. It is for instance the case of addition, difference, union, etc.:

$\begin{split}\begin{eqnarray} [x]+[y]&=&\left[\underline{x}+\underline{y},\overline{x}+\overline{y}\right],\\ \left[x\right]-\left[y\right]& = &\left[\underline{x}-\overline{y},\overline{x}-\underline{y}\right],\\ \left[x\right]\sqcup\left[y\right]& = &\left[\min\left(\underline{x},\underline{y}\right),\max\left(\overline{x},\overline{y}\right)\right],\\ \left[x\right]\cap\left[y\right]& = &\left[\max\left(\underline{x},\underline{y}\right),\min\left(\overline{x},\overline{y}\right)\right] \textrm{if} \max\left\{\underline{x},\underline{y}\right\}\leqslant\min\left\{\overline{x},\overline{y}\right\},\nonumber\\ ~ & = & \varnothing \textrm{ otherwise}. \end{eqnarray}\end{split}$

Low-level libraries upon which Tubex has been built provide functionalities for computing arithmetic on intervals, involving basic operations as well as non-linear functions. The following functions can be used:

Code

Meaning

sqr(x)

$$[x]^2$$

sqrt(x)

$$\sqrt{[x]}$$

pow(x,n)

$$[x]^n$$

pow(x,y)

$$[x]^{[y]} = e^{[y]\log([x])}$$

root(x,n)

$$\sqrt[n]{[x]}$$

exp(x)

$$\exp([x])$$

log(x)

$$\log([x])$$

cos(x)

$$\cos([x])$$

sin(x)

$$\sin([x])$$

tan(x)

$$\tan([x])$$

acos(x)

$$\textrm{acos}([x])$$

asin(x)

$$\textrm{asin}([x])$$

atan(x)

$$\textrm{atan}([x])$$

atan2(y,x)

$$\textrm{atan2}([y],[x])$$

The use on intervals is transparent:

a = Interval(-2,4) * Interval(1,3)                              # a = [-6,12]
b = Interval(-2,4) & Interval(6,7)                              # b = [empty] (intersection)
c = max(Interval(2,7), Interval(1,9))                           # c = [2,9]
d = max(Interval.EMPTY_SET, Interval(1,2))                      # d = [empty]
e = Interval(-1,3) / Interval(0,oo)                             # e = [-oo,oo]
f = (Interval(1,2) * Interval(-1,3)) \
+ max(Interval(1,3) & Interval(6,7), Interval(1,2))           # f = [empty]

Interval a = Interval(-2,4) * Interval(1,3);                    // a = [-6,12]
Interval b = Interval(-2,4) & Interval(6,7);                    // b = [empty] (intersection)
Interval c = max(Interval(2,7), Interval(1,9));                 // c = [2,9]
Interval d = max(Interval::EMPTY_SET, Interval(1,2));           // d = [empty]
Interval e = Interval(-1,3) / Interval(0,oo);                   // e = [-oo,oo]
Interval f = (Interval(1,2) * Interval(-1,3))
+ max(Interval(1,3) & Interval(6,7), Interval(1,2)); // f = [empty]

If intervals and boxes are used to handle static variables, tubes provide a way to deal with trajectories.

format: plain


text:
Precalculus (6th Edition) Blitzer

$\frac{1}{\sqrt {4+x}+2}$
Multiplying the conjugate of the numerator, we have $\frac{\sqrt {4+x}-2}{x}=\frac{\sqrt {4+x}-2}{x}\times\frac{\sqrt {4+x}+2}{\sqrt {4+x}+2}=\frac{4+x-4}{x(\sqrt {4+x}+2)}=\frac{1}{\sqrt {4+x}+2}$


text:
Question

# The postulate or theorem that can be used to prove each pair of triangles congruent.Given:The given figure is:

Polynomial factorization

The postulate or theorem that can be used to prove each pair of triangles congruent.
Given:
The given figure is:

2021-04-04

Calculation:
If the 2 angles and the non-included side of one triangle is congruent to the 2 angles and non-included side of another triangle, then the two triangles are congruent.
Label the diagram given:

Angle A is congruent to angle C and angle BDA is congruent to angle CDA.
According to the reflexive property of congruence, line BD is congruent to BD.
Therefore,
$$\displaystyle\triangle{B}{D}{A}\stackrel{\sim}{=}\triangle{B}{D}{C}$$ (AAA Congruence Theorem).


text:
Resources tagged with: Iteration

Filter by: Content type:
Age range:
Challenge level:

There are 13 results

Broad Topics > Patterns, Sequences and Structure > Iteration

First Forward Into Logo 10: Count up - Count Down

Age 11 to 18 Challenge Level:

What happens when a procedure calls itself?

First Forward Into Logo 1: Square Five

Age 7 to 16 Challenge Level:

A Short introduction to using Logo. This is the first in a twelve part series.

Differs

Age 11 to 14 Challenge Level:

Choose any 4 whole numbers and take the difference between consecutive numbers, ending with the difference between the first and the last numbers. What happens when you repeat this process over and. . . .

Stringing it Out

Age 14 to 16 Challenge Level:

Explore the transformations and comment on what you find.

Difference Dynamics

Age 14 to 18 Challenge Level:

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

Loopy

Age 14 to 16 Challenge Level:

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Dalmatians

Age 14 to 18 Challenge Level:

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

Slippage

Age 14 to 16 Challenge Level:

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

Converging Means

Age 14 to 16 Challenge Level:

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

Triangle Incircle Iteration

Age 14 to 16 Challenge Level:

Keep constructing triangles in the incircle of the previous triangle. What happens?

The Golden Ratio, Fibonacci Numbers and Continued Fractions.

Age 14 to 16

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

Peaches in General

Age 14 to 16 Challenge Level:

It's like 'Peaches Today, Peaches Tomorrow' but interestingly generalized.

Stretching Fractions

Age 14 to 16 Challenge Level:

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?


text:
# Enriched Factorization Systems

• Published 2014

#### Abstract

In a paper of 1974, Brian Day employed a notion of factorization system in the context of enriched category theory, replacing the usual diagonal lifting property with a corresponding criterion phrased in terms of hom-objects. We set forth the basic theory of such enriched factorization systems. In particular, we establish stability properties for enriched prefactorization systems, we examine the relation of enriched to ordinary factorization systems, and we provide general results for obtaining enriched factorizations by means of wide (co)intersections. As a special case, we prove results on the existence of enriched factorization systems involving enriched strong monomorphisms or strong epimorphisms.

### Cite this paper

@inproceedings{LucyshynWright2014EnrichedFS, title={Enriched Factorization Systems}, author={Rory B. B. Lucyshyn-Wright}, year={2014} }


text:
# Discrepancy in radial displacement governing eq for a sphere from Dr. Allen Bower's text in section 4.1.3

Attached is a pdf file indicating a discrepancy in the governing eq for radial displacements in a sphere.  I am + Dr. Bower's text is correct but I cannot reconcile my derivation with his so I am assuming there must be some underlying assumption or approximation I am not aware of.

Can anyone provide me with greater insight?

my email: jberg3@unl.edu

AttachmentSize
161.07 KB

### Re:Discrepancy in Bower

Check your equilibrium equations in spherical coordinates.  You should get an equation of the form

\sigma_{rr,r} + 1/r(2\sigma_{rr} - \sigma_{tt} - \sigma_{pp}) = 0

-- Biswajit

### I am not exactly familiar with your notation-->

What does the forward slash \ denote?

I did apply the divergence of the stress field to = 0.  I presume you saw my attachment.  Did I not execute the divergence in spherical coord's correctly?

### Re: Bower and unfamiliarity with notation

What does the forward slash \ denote?

The \ is to make it easy for you to process it in Latex (with Lyx for example) since iMechanica does not allow Latex embedding.

I did apply the divergence of the stress field to = 0.  Did I not execute the divergence in spherical coord's correctly?

That is correct.  See e.g. Sadd, p. 70; Slaughter p. 189; Stulazec, p. 28; Richards, p. 58;   These turn up for a Google scholar search with keywords "equilibrium spherical coordinates".

-- Biswajit

### Re: Divergence operator in spherical coordinates

Just thought that I should remind students of mechanics that the divergence operator that appears in the equilibrium equations cannot be written in curvilinear coordinates as

sigma_{ij,i} = 0

Spherical coordinates are a special form of orthogonal curvilinear coordinate.  The appropriate form of the divergence operator for general curvilinear coordinates can be found at

It's a good exercise to work out the details for the special case of spherical coordinates.

-- Biswajit


text:
1. 315430.897721
Consider this Thomistic-style doctrine: God’s believing that a contingent entity x exists is the cause of x’s existing. Let B be God’s believing that I exist. Then, either B exists in all possible worlds or B exists in all and only the worlds where I exist. …
Found 3 days, 15 hours ago on Alexander Pruss's Blog
2. 546043.897789
It wasn't just the positivists who thought there was a tight connection between meaning and truth in the case of a priori propositions: However, it seems to me that nevertheless one ingredient of this wrong theory of mathematical truth [i.e. …
Found 6 days, 7 hours ago on Tristan Haze's blog
3. 595882.897827
It is widely believed that the semantic contents of some linguistic and mental representations are determined by factors independent of a person’s bodily makeup. Arguments derived from Hilary Putnam’s seminal Twin Earth thought experiment have been especially influential in establishing that belief. I claim that there is a neglected version of the mind-body relation which undermines those arguments and also excludes the possibility of zombies. It has been neglected because it is counterintuitive but I show that it can nonetheless be intelligibly worked out in detail and all obvious objections met. This suggests that we may be faced with a choice between embracing a counterintuitive interpretation of the mind-body relation or accepting that a currently very promising theory in cognitive science, Prediction Error Minimization, faces a fundamental problem. Furthermore, blocking that threat entails that any physicalist/materialst theory of mind is freed from the spectre of zombie worlds. The proposal also makes the ideas of personal teleportation of mind uploading more plausible.
Found 6 days, 21 hours ago on PhilSci Archive
4. 768946.897867
There are four modal paradigms in ancient philosophy: the frequency interpretation of modality, the model of possibility as a potency, the model of antecedent necessities and possibilities with respect to a certain moment of time (diachronic modalities), and the model of possibility as non-contradictoriness. None of these conceptions, which were well known to early medieval thinkers through the works of Boethius, was based on the idea of modality as involving reference to simultaneous alternatives. This new paradigm was introduced into Western thought in early twelfth-century discussions influenced by Augustine’s theological conception of God as acting by choice between alternative histories.
Found 1 week, 1 day ago on Stanford Encyclopedia of Philosophy
5. 818620.897897
I argue that there is an important similarity between causation and grounding. In particular I argue that, just as there is a type of scientific explanation that appeals to causal mechanisms—causal-mechanical explanation—there is a type of metaphysical explanation that appeals to grounding mechanisms—grounding-mechanical explanation. The upshot is that the role that grounding mechanisms play in certain metaphysical explanations mirrors the role that causal mechanisms play in certain scientific explanations. In this light, it becomes clear that grounding-mechanical explanations make crucial contributions to the evaluation of a variety of important philosophical theses, including priority monism and physicalism.
Found 1 week, 2 days ago on PhilPapers
6. 933888.897922
An odd dissensus between confident metaphysicians and neo-pragmatist antimetaphysicians pervades early twenty-first century analytic philosophy. Each faction is convinced their side has won the day, but both are mistaken about the philosophical legacy of the twentieth century. More historical awareness is needed to overcome the current dissensus. Lewis and his possible-world system are lionised by metaphysicians; Quine’s pragmatist scruples about heavy-duty metaphysics inspire anti-metaphysicians. But Lewis developed his system under the influence of his teacher Quine, inheriting from him his empiricism, his physicalism, his meta-ontology, and, I will show in this paper, also his Humeanism. Using published as well as never-before-seen unpublished sources, I will make apparent that both heavy-duty metaphysicians and neo-pragmatist anti-metaphysicians are wrong about the roles Quine and Lewis played in the development of twentieth-century philosophy. The two are much more alike than is commonly supposed, and Quine much more instrumental to the pedigree of current metaphysics.
Found 1 week, 3 days ago on PhilPapers
7. 980884.897946
On a sunny Saturday afternoon, Socrates is walking through the streets of Western Heights, a small town in the country of New Zealand. Feeling rather hungry, he decides to visit a café for food and coffee. As he is about to enter, he bumps into his old friend Greg, the owner of the café.
Found 1 week, 4 days ago on Brent Silby's site
8. 1007112.897973
J. D. Hamkins and M. Kikuchi, “The inclusion relations of the countable models of set theory are all isomorphic.” (manuscript under review)   Citation arχiv @ARTICLE{HamkinsKikuchi:The-inclusion-relations-of-the-countable-models-of-set-theory-are-all-isomorphic, author = {Joel David Hamkins and Makoto Kikuchi}, title = {The inclusion relations of the countable models of set theory are all isomorphic}, journal = {}, editor = {}, year = {}, volume = {}, number = {}, pages = {}, month = {}, doi = {}, note = {manuscript under review}, eprint = {1704.04480}, archivePrefix = {arXiv}, primaryClass = {math.LO}, url = {http://jdh.hamkins.org/inclusion-relations-are-all-isomorphic}, abstract = {}, keywords = {}, source = {}, } Abstract. …
Found 1 week, 4 days ago on Joel David Hamkins's blog
9. 1046285.898001
In our ASSC20 symposium, “Does unconscious perception really exist?”, the four of us asked some difficult questions about the purported phenomenon of unconscious perception, disagreeing on a number of points. This disagreement reflected the objective of the symposium: not only to come together to discuss a single topic of keen interest to the ASSC community, but to do so in a way that would fairly and comprehensively represent the heterogeneity of ideas, opinions, and evidence that exists concerning this contentious topic. The crux of this controversy rests in no small part on disagreement about what is meant by the terms of the debate and how to determine empirically whether a state is unconscious or not.
Found 1 week, 5 days ago on Ned Block's site
10. 1220798.898043
This paper presents a systematic approach for analyzing and explaining the nature of social groups. I argue against prominent views that attempt to unify all social groups or to divide them into simple typologies. Instead I argue that social groups are enormously diverse, but show how we can investigate their natures nonetheless. I analyze social groups from a bottom-up perspective, constructing profiles of the metaphysical features of groups of specific kinds. We can characterize any given kind of social group with four complementary profiles: its “construction profile,” its “extra essentials” profile, its “anchor” profile, and its “accident” profile. Together these provide a framework for understanding the nature of groups, help classify and categorize groups, and shed light on group agency.
Found 2 weeks ago on Brian Epstein's site
11. 1270083.898079
In his discussion of the four causes, Aristotle claims that ‘the hypotheses are material causes of the conclusion’ (Physics 2. 3, Metaphysics Δ 2). This claim has puzzled commentators since antiquity. It is usually taken to mean that the premisses of any deduction are material causes of the conclusion. By contrast, I argue that the claim does not apply to deductions in general but only to scientific demonstrations. For Aristotle, the theorems of a given science are composites consisting of the indemonstrable premisses from which they are demonstrated. Accordingly, these premisses are elements, and hence material causes, of the theorems. In this way, Aristotle’s claim can be shown to be well-motivated and illuminating.
Found 2 weeks ago on Marko Malink's site
12. 1287821.898115
Fatalism is the thesis that human acts occur by necessity and hence are unfree. Theological fatalism is the thesis that infallible foreknowledge of a human act makes the act necessary and hence unfree. If there is a being who knows the entire future infallibly, then no human act is free. Fatalism seems to be entailed by infallible foreknowledge by the following informal line of reasoning: For any future act you will perform, if some being infallibly believed in the past that the act would occur, there is nothing you can do now about the fact that he believed what he believed since nobody has any control over past events; nor can you make him mistaken in his belief, given that he is infallible.
Found 2 weeks ago on Stanford Encyclopedia of Philosophy
13. 1352980.898143
Found 2 weeks, 1 day ago on The Prosblogion
14. 1461077.898205
We provide a sufficient frame-theoretic condition for a super-bi-intuitionistic logic to have Maksimova’s variable separation property. We conclude that bi-intuitionistic logic enjoys the property. Furthermore, we offer an algebraic characterization of the super-bi-intuitionistic logics with Maksimova’s property.
Found 2 weeks, 2 days ago on The Australasian Journal of Logic
15. 1461133.89824
A theorem from Archimedes on the area of a circle is proved in a setting where some inconsistency is permissible, by using paraconsistent reasoning. The new proof emphasizes that the famous method of exhaustion gives approximations of areas closer than any consistent quantity. This is equivalent to the classical theorem in a classical context, but not in a context where it is possible that there are inconsistent infinitesimals. The area of the circle is taken ‘up to inconsistency’. The fact that the core of Archimedes’s proof still works in a weaker logic is evidence that the integral calculus and analysis more generally are still practicable even in the event of inconsistency.
Found 2 weeks, 2 days ago on The Australasian Journal of Logic
16. 1461259.898267
Circular definitions have primarily been studied in revision theory in the classical scheme. I present systems of circular definitions in the Strong Kleene and supervaluation schemes and provide complete proof systems for them. One class of definitions, the intrinsic definitions, naturally arises in both schemes. I survey some of the features of this class of definitions.
Found 2 weeks, 2 days ago on The Australasian Journal of Logic
17. 1525887.898295
If $M$ is a model of ZFC set theory, let $I$ be the definable cut of its ordinals, the collection of ordinals that are below an ordinal $\delta$ of $M$ that is definable in $M$ without parameters. This would include all the ordinals of $M$, if the definable ordinals happen to be unbounded in $M$, but one can also construct examples where the definable cut is bounded in $M$. …
Found 2 weeks, 3 days ago on Joel David Hamkins's blog
18. 1567097.89833
I argue that perceptual consciousness is constituted by a mental activity. The mental activity in question is the activity of employing perceptual capacities, such as discriminatory, selective capacities. This is a radical view, but I hope to make it plausible. In arguing for this mental activist view, I reject orthodox views on which perceptual consciousness is analyzed in terms of (sensory awareness relations to) peculiar entities, such as, phenomenal properties, external mind-independent properties, propositions, sense-data, qualia, or intentional objects.
Found 2 weeks, 4 days ago on Susanna Schellenberg's site
19. 1918656.898368
Chapter 8 of Idealism and Christian Theology is “Jonathan Edwards, Idealism, and Christology” by Oliver Crisp. This is the second of the two previously published essays, having appeared in another edited volume in 2011. …
Found 3 weeks, 1 day ago on The Prosblogion
20. 1972062.898406
In his book Beyond Art, Dominic McIver Lopes presents a multitude of arguments about main problems in the philosophy of art but centred on the problem of defining what is art and what is a work of art . He argues for “buck passing” theory of art . According to this theory, the task is passed to philosophical theories which, first, have to find necessary and sufficient conditions, or at least, some non-trivial conditions what makes each kind of art exactly that kind of art . It has a form: “X is a work of art if and only if X is a work of K, where K is an art” . Lopes offers support for his theory through many arguments around problems of aesthetic appreciation, aesthetic properties, value, appreciative kinds and practices, media used in various artforms, etc . He tries to show that theory is more viable and more informative than its competitors . I present and critically assess some of the claims from the book; it seems that, according to a form of the theory, we can do without the concept of “art” altogether .
Found 3 weeks, 1 day ago on PhilPapers
21. 2012622.898453
This paper employs a case study from the history of neuroscience—brain reward function—to scrutinize the inductive argument for the so-called ‘Heuristic Identity Theory’ (HIT). The case fails to support HIT, illustrating why other case studies previously thought to provide empirical support for HIT also fold under scrutiny. After distinguishing two different ways of understanding the types of identity claims presupposed by HIT and considering other conceptual problems, we conclude that HIT is not an alternative to the traditional identity theory so much as a relabeling of previously discussed strategies for mechanistic discovery.
Found 3 weeks, 2 days ago on PhilSci Archive
22. 2145448.898477
In this paper, I will give an argument for what I call pannormism, the view according to which if x instantiates a metaphysically basic normative property F, then whatever grounds the being of x also instantiates F. In slogan form: if there is normativity, there is normativity all the way down. Such pannormism is in many ways analogous to panpsychism, and my discussion also contains an important lesson for panpsychism, a way to avoid its so-called combination problem. In section 1, I present the argument; in section 2, I discuss its conclusion.
Found 3 weeks, 3 days ago on PhilPapers
23. 2267117.898498
A binary relation R is said to be wellfounded on a domain if and only if every nonempty subset of that domain has an R-minimal element; equivalently, iff every descending R chain is finite. A proper parthood relation can be non-wellfounded in various ways; here are three: (i) gunk: for all x there is a y such that y is a proper part of x; (ii) bounded infinite descent: there is an infinite chain of distinct proper parts ?! > ?2 > ?! > ⋯ which (after infinitely many steps) terminates in a smallest element y; (iii) loops: there are x and y such that x is a proper part of y and y is a proper part of x.
Found 3 weeks, 5 days ago on Aaron J Cotnoir's site
24. 2272487.898519
TRANSITIVITY: Conscious mental states are mental states we are aware of in some way. W-TRANSPARENCY: For at least one conscious state M, it is impossible to: (a) TRANSPARENCY-DIRECT: Stand in a direct awareness relation to M, or; (b) TRANSPARENCY-DE RE: Stand in a de re awareness relation to M, or; (c) TRANSPARENCY-INT: Stand in an introspective awareness relation to M, His basic claim, then, is that there is no way of making precise the statement of transitivity above in such a way as to render it consistent with the weak version of transparency that he thinks should count as a truism or platitude. …
Found 3 weeks, 5 days ago on Richard Brown's blog
25. 2501318.898541
Chapter 6 of Idealism and Christian Theology is “On the Corruption of the Body: A Theological Argument for Metaphysical Idealism” by S. Mark Hamilton. This is easily the best essay in the collection so far, and the most directly focused on the central issues the volume purports to address. …
Found 4 weeks ago on The Prosblogion
26. 2609644.898616
The capacity for “mindreading” is understood in philosophy of mind and cognitive science as the capacity to represent, reason about, and respond to others’ mental states. Essentially the same capacity is also known as “folk psychology”, “Theory of Mind”, and “mentalizing”. An example of everyday mindreading: you notice that Tom’s fright embarrassed Mary and surprised Bill, who had believed that Tom wanted to try everything. Mindreading is of crucial importance for our social life: our ability to predict, explain, and/or coordinate with others’ actions on countless occasions crucially relies on representing their mental states.
Found 1 month ago on Stanford Encyclopedia of Philosophy
27. 2666112.898645
Possible worlds have found many applications in contemporary philosophy: from theories of possibility and necessity, to accounts of conditionals, to theories of mental and linguistic content, to understanding supervenience relationships, to theories of properties and propositions, among many other applications. Almost as soon as possible worlds started to be used in formal theories in logic, philosophy of language, philosophy of mind, metaphysics, and elsewhere, theorists started to wonder whether impossible worlds should be postulated as well. To take just one example, possible-worlds theories of mental content associate sets of worlds with beliefs (or perhaps entire belief systems): the content of a belief is (or is represented by) the set of possible worlds where that belief is true. But what should we say about beliefs that cannot possibly be true: false logical or mathematical beliefs, for example, or beliefs in metaphysical impossibilities? It would be natural to represent these beliefs with sets of impossible worlds. If James thinks that 87 is a prime number, the set of worlds associated with his beliefs includes worlds where 87 is prime, for example. If Jane is undecided about the principle of excluded middle, her belief worlds should include some at which the principle is correct and some where it is incorrect. And so on.
Found 1 month ago on Daniel Nolan's site
28. 2724875.89868
Truth values have been put to quite different uses in philosophy and logic, being characterized, for example, as: primitive abstract objects denoted by sentences in natural and formal languages, abstract entities hypostatized as the equivalence classes of sentences, what is aimed at in judgements, values indicating the degree of truth of sentences, entities that can be used to explain the vagueness of concepts, values that are preserved in valid inferences, values that convey information concerning a given proposition. Depending on their particular use, truth values have been treated as unanalyzed, as defined, as unstructured, or as structured entities.
Found 1 month ago on Stanford Encyclopedia of Philosophy
29. 2788802.898701
Chapter 5 of Idealism and Christian Theology is “Edwardsian Idealism, Imago Dei, and Contemporary Theology” by Joshua Farris. This is an interesting article that directly and constructively addresses one of the central theological issues raised by metaphysical idealism of the Berkeley/Edwards variety. …
Found 1 month ago on The Prosblogion
30. 2957205.898728
This chapter starts from a puzzle. Realism about X is often glossed as the idea that Xs are mind independent: Xs exist, and have their nature, independently of our beliefs, interests, attitudes, or other mental states. Xs are, in a sense ‘out there’, getting on with it independently of our mental life. If this is right, how should we understand realism about cognitive science? Mental processes and states are not mind independent: they don’t take place independently of our beliefs, interests, attitudes, or other mental states. Hence, it seems that we cannot be realists about them. Nevertheless, in line with other areas of philosophy of science, there seems scope for asking the realist question about the posits of cognitive science even if those posits make up our mental life. But unless we state realism differently, there is no way to sensibly ask the realist question about cognitive science. In this paper, I explore the right way to state realism about cognitive science. I introduce three different types of mind dependence and evaluate their merit to stating realism about cognitive science.
Found 1 month ago on Mark Sprevak's site


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시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 128 MB 0 0 0 0.000%

## 문제

In a certain week, a company wants to finish m projects. To this end, the company can employ at most n people from the unemployment agency for a period of one week. Each external employee will cost the company salary euro, unless the project in which he/she is involved is not completed in time. In that case no payment is due.

For each project the company knows from experience the probability that the project will be completed within a week, as a function of the number of employees working on it. These probabilities are given as percentages pij, where i (with 1 ≤ i ≤ m) is the number of the project and j is the number of people working on it. Of course, when nobody is working on a project i, the probability pi0 is zero percent.

If project i is indeed finished within a week, the company earns reward(i) euro; if it is not ready in time, the company has to pay a fine of punishment(i) euro.

Of course the company wants to maximize its total expected profit - Let p (0 < p < 1) be the probability that a job is finished in time, and let E1 be the profit in that case. Furthermore, let E2 be the (negative) profit in case the job is not finished in time. Then the expected profit for this particular job is p⋅E1 + (1 − p)⋅E2 -  at the end of the week by finding the optimal number of external employees to hire, and how to divide them over the projects. The optimal number of employees is the total number of people needed to achieve the maximal expected profit. Your task in this matter is to calculate this optimal number of external employees. Remember that at most n people are available. Furthermore: if a person is employed, he/she works on one and only one project.

## 입력

The first line of the input file contains a single number: the number of test cases to follow. Each test case has the following format:

• One line with one integer m with 1 ≤ m ≤ 100: the number of projects.
• One line with one integer n with 0 ≤ n ≤ 100: the maximal number of available employees.
• One line with one integer salary with 0 ≤ salary ≤ 1,000: the salary of one employee. Remember that the salary is given in euros.
• m lines, each line corresponding to a project i, containing n integers pi1, pi2, …, pin (the percentages, with 0 ≤ pi1, pi2, …, pin ≤ 100), followed by two integers corresponding to the reward and the punishment for project i. All values are separated by single spaces. Both reward and punishment are given in euros and are between 0 and 100,000 (boundaries included).

## 출력

For every test case in the input file, the output should contain two lines.

• The first line contains the maximal expected profit in eurocents.
• The second line contains the total number of external employees that must be hired in order to achieve this maximal expected profit. If the maximal expected profit can be achieved by different (total) numbers of employees, then these different numbers must be given in increasing order. Numbers have to be separated by single spaces.

## 예제 입력

3
1
4
200
90 100 100 100 2000 0
2
2
100
80 80 2100 500
0 100 1700 500
3
4
100
100 80 80 70 1000 100
100 90 80 90 500 50
100 70 60 50 700 100

## 예제 출력

162000
1
100000
1 2
190000
3


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##### What's Hot

We propose a new procedure, TFT$_\pm, to allocate non-unitary topological quantum field theory (TQFT) pairs. [\mathcal{T}_{\rm rank \;0}]$, (2+1)D interaction $\mathcal{N}=4$ superconformal field theory (SCFT) $\mathcal{T}_{\rm rank \;0}$ rank 0, i.e. No and Higgs branches. Topological theory arises from certain degeneracy limits of SCFT. Modular data for non-unitary TQFTs are extracted from degenerate limit supersymmetric partition functions. As an important dictionary, $F = \max_\alpha \left(- \log |S^{(+)}_{0\alpha}| \right) = \max_\alpha \left(- \log |S^ {(-)}_{0\alpha}|\right)$, where $F$ is the circular three-sphere free energy of $\mathcal{T}_{\rm rank \;0 }.$ and $S^{(\pm)}_{0\alpha}$ are the first columns of the modular S matrix of TFT$_\pm$. Derive the lower bound of $F$ from the dictionary. $F \geq -\log \left(\sqrt{\frac{5-\sqrt{5}}{10}} \right) \simeq 0.642965$, which is true for any rank-0 SCFT. This bound is saturated by the minimal $\mathcal{N}=4$ SCFT proposed by Gang-Yamazaki. Gang-Yamazaki’s related topology theory is both Lee-Yang TQFT. Explicitly compute the (Rank 0 SCFT)/(Nonunitary TQFT) correspondence for an infinite number of examples.

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# fitPosterior

Fit posterior probabilities for compact support vector machine (SVM) classifier

## Syntax

``ScoreSVMModel = fitPosterior(SVMModel,TBL,Y)``
``ScoreSVMModel = fitPosterior(SVMModel,X,Y)``
``````[ScoreSVMModel,ScoreTransform] = fitPosterior(___)``````

## Description

````ScoreSVMModel = fitPosterior(SVMModel,TBL,Y)` returns a trained support vector machine (SVM) classifier `ScoreSVMModel` containing the optimal score-to-posterior-probability transformation function for two-class learning. For more details, see Algorithms. If you train `SVMModel` using a table, then you must use a table as input for `fitPosterior`.```

example

````ScoreSVMModel = fitPosterior(SVMModel,X,Y)` returns a trained SVM classifier `ScoreSVMModel` containing the optimal score-to-posterior-probability transformation function for two-class learning. If you train `SVMModel` using a matrix, then you must use a matrix as input for `fitPosterior`.```

example

``````[ScoreSVMModel,ScoreTransform] = fitPosterior(___)``` additionally returns the optimal score-to-posterior-probability transformation function parameters (`ScoreTransform`) for any of the input argument combinations in the previous syntaxes.```

## Examples

collapse all

Load the `ionosphere` data set. Reserve 20 random observations of the data, and consider this set new data.

```load ionosphere n = size(X,1); rng(1); % For reproducibility indx = ~ismember([1:n],randsample(n,20)); % Indices for the training data```

The classes of this data set are inseparable.

Train an SVM classifier using the training data. Standardize the data and specify that `'g'` is the positive class.

```SVMModel = fitcsvm(X(indx,:),Y(indx),'ClassNames',{'b','g'},... 'Standardize',true);```

`SVMModel` is a `ClassificationSVM` classifier.

Use the new data set to estimate the optimal score-to-posterior-probability transformation function for mapping scores to the posterior probability of an observation being classified as `g`. For efficiency, make a compact version of `SVMModel`, and pass it and the new data to `fitPosterior`.

```CompactSVMModel = compact(SVMModel); [ScoreCSVMModel,ScoreParameters] = fitPosterior(CompactSVMModel,... X(~indx,:),Y(~indx)); ScoreTransform = ScoreCSVMModel.ScoreTransform```
```ScoreTransform = '@(S)sigmoid(S,-1.098976e+00,4.520314e-01)' ```
`ScoreParameters`
```ScoreParameters = struct with fields: Type: 'sigmoid' Slope: -1.0990 Intercept: 0.4520 ```

`ScoreTransform` is the optimal score transformation function. `ScoreParameters` is a structure array with three fields: the score transformation function name (`Type`), the sigmoid slope (`Slope`), and the sigmoid intercept estimates (`Intercept`).

Alternatively, you can pass `SVMModel` and the new data to `fitSVMPosterior`, but this process is not as efficient.

Estimate the posterior probabilities that the observations in the new data are in class `g`.

```[labels,postProbs] = predict(ScoreCSVMModel,X(~indx,:)); table(Y(~indx),labels,postProbs(:,2),... 'VariableNames',{'TrueLabel','PredictedLabel','PosteriorProbability'})```
```ans=20×3 table TrueLabel PredictedLabel PosteriorProbability _________ ______________ ____________________ {'g'} {'g'} 0.7844 {'b'} {'b'} 0.024584 {'g'} {'g'} 0.82402 {'b'} {'b'} 0.0061632 {'b'} {'b'} 3.6064e-06 {'b'} {'b'} 0.15688 {'b'} {'g'} 0.96219 {'b'} {'b'} 6.1343e-09 {'b'} {'b'} 0.001964 {'g'} {'g'} 0.72509 {'g'} {'g'} 0.70261 {'b'} {'b'} 0.075298 {'g'} {'g'} 0.90692 {'g'} {'g'} 0.82848 {'b'} {'b'} 0.051175 {'g'} {'g'} 0.95332 ⋮ ```

Load Fisher's iris data set. Use the petal lengths and widths as the predictor data, and remove the virginica species from the data. Reserve 10 random observations of the data, and consider this set new data.

```load fisheriris classKeep = ~strcmp(species,'virginica'); X = meas(classKeep,3:4); Y = species(classKeep); rng(1); % For reproducibility indx1 = 1:numel(species); indx2 = indx1(classKeep); indx = ~ismember(indx2,randsample(indx2,10)); % Indices for the training data gscatter(X(indx,1),X(indx,2),Y(indx)); title('Scatter Diagram of Iris Measurements') xlabel('Petal length') ylabel('Petal width') legend('Setosa','Versicolor')```

The classes are perfectly separable. Therefore, the score-to-posterior-probability transformation function is a step function.

Train an SVM classifier. Standardize the data and specify that `versicolor` is the positive class.

```SVMModel = fitcsvm(X(indx,:),Y(indx),... 'ClassNames',{'setosa','versicolor'},'Standardize',true);```

`SVMModel` is a `ClassificationSVM` classifier.

Use the new data set to estimate the optimal score-to-posterior-probability transformation function for mapping scores to the posterior probability of an observation being classified as `versicolor`. For efficiency, make a compact version `SVMModel`, and pass it and the new data to `fitPosterior`.

```CompactSVMModel = compact(SVMModel); [ScoreCSVMModel,ScoreParameters] = fitPosterior(CompactSVMModel,... X(~indx,:),Y(~indx));```
```Warning: Classes are perfectly separated. The optimal score-to-posterior transformation is a step function. ```
`ScoreTransform = ScoreCSVMModel.ScoreTransform`
```ScoreTransform = '@(S)step(S,-1.338450e+00,2.012495e+00,5.333333e-01)' ```

`fitPosterior` displays a warning whenever the classes are separable, and stores the step function in `ScoreSVMModel.ScoreTransform`.

Display the score function type and its estimated values.

`ScoreParameters`
```ScoreParameters = struct with fields: Type: 'step' LowerBound: -1.3385 UpperBound: 2.0125 PositiveClassProbability: 0.5333 ```

`ScoreParameters` is a structure array with four fields:

• Score transformation function type (`Type`)

• Score corresponding to the negative class boundary (`LowerBound`)

• Score corresponding to the positive class boundary (`UpperBound`)

• Positive class probability (`PositiveClassProbability`)

Alternatively, you can pass `SVMModel` and the new data to `fitSVMPosterior`, but this process is not as efficient.

Estimate the posterior probabilities that the observations in the new data are versicolor irises.

```[labels,postProbs] = predict(ScoreCSVMModel,X(~indx,:)); table(Y(~indx),labels,postProbs(:,2),... 'VariableNames',{'TrueLabel','PredictedLabel','PosteriorProbability'})```
```ans=10×3 table TrueLabel PredictedLabel PosteriorProbability ______________ ______________ ____________________ {'setosa' } {'setosa' } 0 {'setosa' } {'setosa' } 0 {'setosa' } {'setosa' } 0 {'setosa' } {'setosa' } 0 {'setosa' } {'setosa' } 0 {'setosa' } {'setosa' } 0 {'setosa' } {'setosa' } 0 {'setosa' } {'setosa' } 0 {'versicolor'} {'versicolor'} 1 {'versicolor'} {'versicolor'} 1 ```

Because the classes are separable, the step function transforms the positive-class score to:

• `0` if the score is less than `ScoreParameters.LowerBound`

• `1` if the score is greater than `ScoreParameters.UpperBound`

• `ScoreParameters.PositiveClassProbability` if the score is in the interval [ `ScoreParameters.LowerBound` , `ScoreParameters.LowerBound`]

## Input Arguments

collapse all

Trained, compact SVM classifier, specified as a `CompactClassificationSVM` model returned by `compact`.

Sample data, specified as a table. Each row of `TBL` corresponds to one observation, and each column corresponds to one predictor variable. `TBL` must contain all of the predictors used to train `SVMModel`. Optionally, `TBL` can contain an additional column for the response variable. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

If `TBL` contains the response variable used to train `SVMModel`, then you do not need to specify `Y`. If `TBL` does not include the response variable, then the length of `Y` must be equal to the number of rows in `TBL`.

If the sample data used to train `SVMModel` is a `table`, then you must specify the input data for `fitPosterior` as a table.

If you set `'Standardize',true` in `fitcsvm` when training `SVMModel`, then the software fits the transformation function parameter estimates using standardized data.

Data Types: `table`

Predictor data used to estimate the score-to-posterior-probability transformation function, specified as a matrix.

Each row of `X` corresponds to one observation (also known as an instance or example), and each column corresponds to one variable (also known as a feature).

The length of `Y` and the number of rows in `X` must be equal.

If you set `'Standardize',true` in `fitcsvm` when training `SVMModel`, then the software fits the transformation function parameter estimates using standardized data.

Data Types: `double` | `single`

Class labels used to estimate the score-to-posterior-probability transformation function, specified as a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors.

If `Y` is a character array, then each element must correspond to one class label.

The length of `Y` and the number of rows in `X` must be equal.

Data Types: `categorical` | `char` | `string` | `logical` | `single` | `double` | `cell`

## Output Arguments

collapse all

Trained, compact SVM classifier containing the estimated score-to-posterior-probability transformation function, returned as a `CompactClassificationSVM` classifier.

To estimate posterior probabilities for new observations, pass `ScoreSVMModel` and the new observations to `predict`.

Optimal score-to-posterior-probability transformation function parameters, returned as a structure array.

• If the value of the `Type` field of `ScoreTransform` is `sigmoid`, then `ScoreTransform` also has these fields:

• `Slope`: The value of A in the sigmoid function

• `Intercept`: The value of `B` in the sigmoid function

• If the value of the `Type` field of `ScoreTransform` is `step`, then `ScoreTransform` also has these fields:

• `PositiveClassProbability`: The value of π in the step function. This value represents the probability that an observation is in the positive class or the posterior probability that an observation is in the positive class given that its score is in the interval (`LowerBound`,`UpperBound`).

• `LowerBound`: The value $\underset{{y}_{n}=-1}{\mathrm{max}}{s}_{n}$ in the step function. This value represents the lower bound of the score interval that assigns observations with scores in the interval the posterior probability of being in the positive class `PositiveClassProbability`. Any observation with a score less than `LowerBound` has the posterior probability of being in the positive class equal to `0`.

• `UpperBound`: The value $\underset{{y}_{n}=+1}{\mathrm{min}}{s}_{n}$ in the step function. This value represents the upper bound of the score interval that assigns observations with scores in the interval the posterior probability of being in the positive class `PositiveClassProbability`. Any observation with a score greater than `UpperBound` has the posterior probability of being in the positive class equal to `1`.

• If the value of the `Type` field of `ScoreTransform` is `constant`, then `ScoreTransform.PredictedClass` contains the name of the class prediction.

This result is the same as `SVMModel.ClassNames`. The posterior probability of an observation being in `ScoreTransform.PredictedClass` is always `1`.

collapse all

### Sigmoid Function

The sigmoid function that maps score sj corresponding to observation j to the positive class posterior probability is

`$P\left({s}_{j}\right)=\frac{1}{1+\mathrm{exp}\left(A{s}_{j}+B\right)}.$`

If the value of the `Type` field of `ScoreTransform` is `sigmoid`, then parameters A and B correspond to the fields `Scale` and `Intercept` of `ScoreTransform`, respectively.

### Step Function

The step function that maps score sj corresponding to observation j to the positive class posterior probability is

`$P\left({s}_{j}\right)=\left\{\begin{array}{l}\begin{array}{cc}0;& s<\underset{{y}_{k}=-1}{\mathrm{max}}{s}_{k}\end{array}\\ \begin{array}{cc}\pi ;& \underset{{y}_{k}=-1}{\mathrm{max}}{s}_{k}\le {s}_{j}\le \underset{{y}_{k}=+1}{\mathrm{min}}{s}_{k}\end{array}\\ \begin{array}{cc}1;& {s}_{j}>\underset{{y}_{k}=+1}{\mathrm{min}}{s}_{k}\end{array}\end{array},$`

where:

• sj is the score of observation j.

• +1 and –1 denote the positive and negative classes, respectively.

• π is the prior probability that an observation is in the positive class.

If the value of the `Type` field of `ScoreTransform` is `step`, then the quantities $\underset{{y}_{k}=-1}{\mathrm{max}}{s}_{k}$ and $\underset{{y}_{k}=+1}{\mathrm{min}}{s}_{k}$ correspond to the fields `LowerBound` and `UpperBound` of `ScoreTransform`, respectively.

### Constant Function

The constant function maps all scores in a sample to posterior probabilities 1 or 0.

If all observations have posterior probability 1, then they are expected to come from the positive class.

If all observations have posterior probability 0, then they are not expected to come from the positive class.

## Tips

• This process describes one way to predict positive class posterior probabilities.

1. Train an SVM classifier by passing the data to `fitcsvm`. The result is a trained SVM classifier, such as `SVMModel`, that stores the data. The software sets the score transformation function property (`SVMModel.ScoreTransformation`) to `none`.

2. Pass the trained SVM classifier `SVMModel` to `fitSVMPosterior` or `fitPosterior`. The result, such as, `ScoreSVMModel`, is the same trained SVM classifier as `SVMModel`, except the software sets `ScoreSVMModel.ScoreTransformation` to the optimal score transformation function.

3. Pass the predictor data matrix and the trained SVM classifier containing the optimal score transformation function (`ScoreSVMModel`) to `predict`. The second column in the second output argument of `predict` stores the positive class posterior probabilities corresponding to each row of the predictor data matrix.

If you skip step 2, then `predict` returns the positive class score rather than the positive class posterior probability.

• After fitting posterior probabilities, you can generate C/C++ code that predicts labels for new data. Generating C/C++ code requires MATLAB® Coder™. For details, see Introduction to Code Generation.

## Algorithms

The software fits the appropriate score-to-posterior-probability transformation function by using the SVM classifier `SVMModel` and by conducting 10-fold cross-validation using the stored predictor data (`SVMModel.X`) and the class labels (`SVMModel.Y`), as outlined in [1]. The transformation function computes the posterior probability that an observation is classified into the positive class (`SVMModel.Classnames(2)`).

• If the classes are inseparable, then the transformation function is the sigmoid function.

• If the classes are perfectly separable, then the transformation function is the step function.

• In two-class learning, if one of the two classes has a relative frequency of 0, then the transformation function is the constant function. The `fitPosterior` function is not appropriate for one-class learning.

• The software stores the optimal score-to-posterior-probability transformation function in `ScoreSVMModel.ScoreTransform`.

If you re-estimate the score-to-posterior-probability transformation function, that is, if you pass an SVM classifier to `fitPosterior` or `fitSVMPosterior` and its `ScoreTransform` property is not `none`, then the software:

• Displays a warning

• Resets the original transformation function to `'none'` before estimating the new one

## Alternative Functionality

You can also fit the optimal score-to-posterior-probability function by using `fitSVMPosterior`. This function is similar to `fitPosterior`, except it is more broad because it accepts a wider range of SVM classifier types.

## References

[1] Platt, J. “Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods.” Advances in Large Margin Classifiers. Cambridge, MA: The MIT Press, 2000, pp. 61–74.

## Version History

Introduced in R2014a


text:
## ~ Abstracts ~

Real Chevalley involutions and self-dual representations
Jeffrey Adams, University of Maryland, USA

The Chevalley involution of a complex group G takes any algebraic representation to its dual. I'll discuss a real analogue, which has the same property for infinite dimensional representations of a real group G(R). As an application I'll give conditions for every representation of G(R) to be self-dual, and compute the Frobenious Schur (symplectic/orthogonal) indicator for these representations.

« Back...

Some applications of representation theory to estimates of automorphic periods
Joseph Bernstein, Tel Aviv University, Israel

I will talk about new methods that use representation theory of real reductive groups to give some highly non-trivial estimates of periods of automorphic functions.

« Back...

The coherent cohomology of an algebraic group
Michel Brion, Université de Grenoble, France

The coordinate ring of any linear algebraic group G has a structure of Hopf algebra, which uniquely determines G. On the other hand, for an abelian variety G, Serre showed that the cohomology of G with coefficients in its structure sheaf is an exterior algebra on g = dim(G) primitive elements of degree 1. The talk will present a common generalization of these results to an arbitrary algebraic group G.

« Back...

Local theta correspondence and the Gross-Prasad conjecture
Wee Teck Gan, National University of Singapore

I will explain how the Gross-Prasad conjecture for orthogonal and symplectic/metaplectic groups are related via the local theta correspondence.

Given that the GP conjecture for orthogonal groups over p-adic fields are now known (a la Waldspurger and Moeglin-Waldspurger), I will explain what remains to be done to deduce the GP conjecture for the symplectic/metaplectic groups.

« Back...

The arithmetic of elliptic curves
Benedict Gross, Harvard University, USA

I will discuss a problem which has been central in number theory for several centuries - whether an elliptic curve in the plane has infinitely many rational solutions. This led to a precise conjecture by Birch and Swinnerton-Dyer in the 1960s, and to some partial progress in the 1980s. More recently, Manjul Bhargava has introduced a new method to study the average number of solutions. I will survey the progress that has been made in the field.

« Back...

The internal structure of parabolic and parahoric subgroups
Benedict Gross, Harvard University, USA

In the parabolic case, I will discuss the structure of the Levi quotient and its action on the abelian subquotients of the filtration of the unipotent radical. In the parahoric case, I will discuss the reductive quotient and its action on the abelian subquotients of the Moy-Prasad filtration of the pro-unipotent radical. In the first case, the representations have an open orbit. In the second case, they have a polynomial ring of invariants.

« Back...

Hibi algebras and iterated Pieri algebras
Roger Howe, Yale University, USA

In recent years, substantial progress has been made in invariant theory and representation theory by exploiting the idea of toric deformation to semigroup rings. In particular, the special class of semigroup rings constructed by T. Hibi have been found to play a significant role in answering a basic collection of representation-theoretic questions. This talk will focus on the appearance of Hibi rings in describing certain tensor products that generalize the classical Pieri Rule for GL_n.

« Back...

Dirac cohomology. K-characters and branching laws
Jing-Song Huang, Hong Kong University of Science and Technology, Hong Kong

Inspired by work of Enright and Willenbring (Ann. Math. 159), we prove a generalized Littlewood's restriction formula in terms of Dirac cohomology. Our approach is to use a character formula of irreducible unitary lowest weight modules instead of the Bernstein-Gelfand-Gelfand resolution, and the proof is much simpler. We also show that our branching formula is equivalent to the formula of Enright and Willenbring in terms of nilpotent Lie algebra cohomology. This follows from the close relationship between the Dirac cohomology and the corresponding nilpotent Lie algebra cohomology for unitary representations of semisimple Lie groups of Hermitian type, which was established by Huang, Pandizc and Renard. This is a joint work with Fuhai Zhu and Pavle Pandzic to appear in American Journal of Mathematics.

« Back...

Formal degrees and adjoint gamma factors (continued)
Atsuchi Ichino, Kyoto University, Japan

We give an update on the conjecture which relates the formal degree of a discrete series representation of a reductive group over a local field to a certain arithmetic invariant.

« Back...

Stabilization of the trace formula for the covering groups of SL(2) and its application to the theory of Kohnen plus space
Tamotsu Ikeda, Kyoto University, Japan

In this talk, we give a partial stabilization of the trace formula for the metaplectic covering of SL(2). As an application, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half integral weight.

« Back...

Path models for branching to symmetric subgroups
Allen Knutson, Cornell University, USA

Let K be a connected, symmetric subgroup of G, the principal example being G = K x K, and let lambda be a dominant G-weight. I define a polyhedral complex P associated to lambda, and a lattice structure on each face (that may become coarser on smaller faces), together with a linear projection to K's positive Weyl chamber.

Theorem 1. For several pairs (G,K), such as (KxK,K), (GL(2n),Sp(n)), and (SL(3),SO(3)), the multiplicity of the K-weight mu in the finite-dimensional representation V_lambda is the number of lattice points in P lying over mu.

Theorem 2. For every pair (G,K), the corresponding asymptotic statement holds (to leading order in N, where lambda,mu are replaced by N*lambda, N*mu).

The points in P correspond to certain piecewise-linear paths in G's positive Weyl chamber, which in the G = KxK case are closely related to Littelmann's path model for tensor product multiplicities.

Time permitting, I will describe the difficulties in extending these results to infinite-dimensional representations.

« Back...

Finite multiplicity theorems and real spherical varieties
Toshiyuki Kobayashi, University of Tokyo, Japan

I plan to discuss geometric conditions that control the multiplicities of irreducible representations of real reductive groups occurring in branching laws (restriction) and Plancherel formulas (induction).

« Back...

Ubiquity of Schubert varieties
Venkatraman Lakshmibai, Northeastern University, USA

We shall first discuss the classical Schubert varieties in the Grassmannian and the Flag Variety. We shall then show examples of several affine varieties related to Schubert varieties including examples from Classical Invariant Theory. We shall also present results for affine Schubert varieties in the infinite dimensional Flag Variety associated to Kac-Moody Lie alge.

« Back...

On a class of representations of GL(n) over a p-adic field
Erez Lapid, Hebrew University of Jerusalem, Israel

Some time ago Tadic gave a remarkable formula expressing a Speh representation as an alternating sum of standard modules. His proof was subsequently simplified by Chenevier - Renard. I will discuss a different approach which sharpens the statement and works for a wider class of representations. Applications will also be discussed.
Joint work with Alberto Minguez

« Back...

Jian-Shu Li, Hong Kong University of Science and Technology, Hong Kong

This is a report of work in progress with Eischen, Harris and Skinner on the construction of certain p-adic L-functions on unitary groups, with emphasis on those parts related to branching laws.

« Back...

On a pairing of Goldberg-Shahidi for SO(6n)
Wen-Wei Li, Morningside Center of Mathematics, China

In their study of the tempered spectrum of quasisplit even orthogonal groups, Goldberg and Shahidi defined a pairing between supercuspidal matrix coefficients and conjectured that its regular term should be related to twisted endoscopic transfer. These predictions had not been verified except for SO(6), by Shahidi and Spallone. I will discuss the general case in this talk.

« Back...

Derived functor modules, dual pairs and U(g)^K actions
Jia Jun Ma, National University of Singapore

Applying the derived functors on highest weight modules constructs interesting representations. When the highest weight module is in a good range, its derived functor modules are well understood. We consider the singular case. In works of Enright at. el. (1985), Frajria (1991) and Wallach (1994), they established several criteria of irreducibility and unitarity and constructed families of examples for the singular case. Building on the work of Wallach and Zhu, I show these examples are related to theta lifts of characters. This is proved by comparing U(g)^K actions.

Motivated by by the above result and a joint work with Loke, I will also give examples beyond the case of theta lifts of characters, which suggests that derived functor construction is related to dual pair correspondence in a much more general setting.

« Back...

A Fuchsian differential equation with accessory parameters and Zuckemann's tensoring
Tomonori Moriyama, Osaka University, Japan

If we wish to write a "hyperbolic" Fourier expansion of Maass form of weight one, we encounter a second-order Fuchsian differential equation with five singularities, which seems difficult to analyze at a first glance. But we can get a solution of this equation expressed by Gauss's hypergeometric functions by virtue of an interesting relation among differential operators. This is worked out in the master thesis of M. Maeda (Osaka University, 2012). We also explain how the above-mentioned relation can be foreseen by using Zuckemann's tensoring technique.

« Back...

Multiple flag varieties and spherical actions
Kyo Nishiyama, Aoyama Gakuin University, Japan

It is well known that product of two partial flag varieties of a reductive group G have finitely many orbits under the diagonal action of G (the Bruhat decomposition), and Steinberg considered a conormal variety on it. Recently there are several works on triple flag varieties also.

In my talk, I will introduce a double flag variety for a symmetric pair (G, K), which generalize the triple flag varieties above. On this variety K acts diagonally. Sometimes it has finitely many orbits, and they are related to both the Bruhat decompositions and KGB decompositions. It is also related to spherical flag varieties, which can be classified in some sense. We will give an outline of the classification and discuss applications to representation theory/combinatorics.

The talk is based on the joint work with Xuhua He, Hiroyuki Ochiai and Yoshiki Oshima.

« Back...

Buildings by Morse
Gordan Savin, University of Utah, USA

I will apply Bestvina's PL Morse theory to study the affine building for SL(n). Using a PL Morse function I will construct a contraction of the building. Then, following an idea of Opdam and Solleveld, the contraction is used to give a simple proof of a result of Schneider and Stuhler on projective resolutions of smooth representations.

« Back...

Shalika models and cohomological test vectors
Binyong Sun, Chinese Academy of Sciences, China

Let $\pi$ be an irreducible admissible representation of $GL(2n)$ over a local field, with a Shalika model. For arithmetic applications, it is important to find explicit vectors in the Shalika model of $\pi$ whose zeta integral produces the standard L-function of $\pi$. We try to solve this problem when the local field is the field of real numbers, and the representation is essentially tempered and has non-zero cohomology.

« Back...

L-functions and theta correspondence
Shunsuke Yamana, Osaka City University, Japan

The doubling method of Piatetski-Shapiro and Rallis applies in the local situation to define local factors of representations of classical groups. On the one hand, the L-factor is defined as a g.c.d. of the local zeta integrals for all good sections. On the other hand, it is defined from the gamma factor by using the Langlands classification. In this talk I develop a theory of the zeta integral to prove that the two candidates of the L-factor agree. Applications include a characterization of nonvanishing of global theta liftings in terms of the analytic properties of the complete L-functions and the occurrence in the local theta correspondence.

« Back...

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text:
# When is a linear operator invariant for a orthogonal decomposition?

Let $f: V \to V$ be a linear map and $V = U \oplus U^\perp$ be an inner product space.

For its restriction map $f|_U$, is it invariant under $U$? And thus similarly for $f|_{U^\perp}$?

I guess one would try to see where the inner product $( ,)$ would map $f(u)$, but I don't seem to be able to get a good result.

If $f$ is self-adjoint, i.e. $$\langle f(x),y\rangle=\langle x,f(y)\rangle$$ for all $x,y\in V$, then as soon as you have that $f(U)\subseteq U$ then $f(U^\perp)\subseteq U^\perp$ as well.
On the other hand, there's no condition based on the scalar product that will imply that $f(U)\subseteq U$.
When $\langle fu,v\rangle=0$ for all $u\in U$, $v\in U^\perp$ then $fu\in U$ so it leaves $U$ invariant (and similarly for the complement). Does this answer your question?
• The question is when the condition $(fu, v) = 0$ would hold. Aug 24, 2016 at 21:53


text:
## Algebra 1: Common Core (15th Edition)

Published by Prentice Hall

# Skills Handbook - Multiplying and Dividing Fractions - Exercises: 20

26

#### Work Step by Step

We first turn the whole number into fraction form, giving us the following: $\frac{13}{2} \div \frac{1}{4}$ We never can divide by a fraction. Rather, we must multiply the first fraction by the reciprocal of the second fraction. Because the reciprocal of 1/4 is 4/1, we obtain: $\frac{13}{2}\times \frac{4}{1} = \frac{52}{2}=26$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Custom discrete probabilities

Release: 4.6  •  5.0  •  5.1  •  5.2  •  5.3  •  5.4  •  6.0  •  6.1  •  6.2  •  6.3

These functions let you specify a discrete probability distribution using a custom set of values, text (label) values, or numbers.

## ProbTable(): Probability Tables

To describe a probability distribution on a discrete variable whose domain is a list of numbers or list of labels, you use special kind of edit table called a probability table (or probTable), see Arrays and Indexes and ProbTable.

Create a probability table: To define a variable using a probability table:

1. Determine the variable’s domain — number or labels for its possible values.
2. Select the variable and view its definition attribute in the Object window or Attribute panel of the Diagram window.
3. Press the expr menu above the definition field and select Probability Table.
If the variable already has a definition, it confirms that you wish to replace it.
Tip
If the definition of a variable is already a probability table, a ProbTable button appears in the definition. Click it to open the Edit Table window (see Defining a variable as an edit table).
4. The Indexes dialog opens to confirm your choices for the indexes of the table. It already includes the selected variable (Self) among the selected indexes. Other options are variables with a domain that is a list of numbers or list of labels. Add or remove any other variables that you want to condition this variable on.
Tip
Self is required as an index of a probability table. It refers to the domain (possible values) of this variable.
5. Click the OK button. An Edit Table window appears.
6. Enter the possible values for the domain in the left column. As in any edit table, press Enter or down-arrow in the last row to add a row. Select Insert row (Control+i) or Delete row (Control+k) from the Edit menu. If they are numbers, they must be in increasing order.
7. Enter the probability of each possible outcome in the second column. The probabilities should sum to 1. You may enter literal numbers or expressions.

Example: If P is a variable whose value is a probability (between 0 and 1) and the possible weather outcomes are sunny and rainy, you might define a probability table for weather like this.

Expression view of probability table: The Weather probability table when viewed as an expression, looks like this.

ProbTable(Self)(P, (1 - P))

The domain values do not appear in the expression view, and it is not very convenient for defining a probability table. More generally, the expression view of a multidimensional probability table looks like this:

ProbTable(i1, i2, … in) (p1, p2, p3, … pm)

This example is an «n»-dimensional conditional probability table, indexed by the indexes «i1», «i2», … «in». One index must be Self. «p1», «p2», «p3», … «pm» are the probabilities in the array. «m» is the product of the sizes of the indexes «i1», «i2», … «in».

Add a conditioning variable: You might wish to add one or more conditioning variables to a probability table, to create conditional dependency. Each discrete conditioning variable adds a dimension to the table. For example, in the Weather probability table, shown above, the probability of rain might depend on the season. So you might have Season as a conditioning variable, defined as a list of labels:

Variable Season := ['Winter', 'Spring', 'Summer', 'Fall']
1. Open the Edit Table window by clicking the ProbTable button.
2. Click the indexes button to open the Indexes dialog.
3. Click the All Variables checkbox above the left hand list.
4. Move the desired variable, e.g., Season, to add it as an index.
5. Click OK to accept the changes.

The resulting table is indexed by both the domain of your variable and the domains of the conditionally dependent variables. You need to enter a probability for each cell. The probabilities must sum to one over the domain of the variable (sunny and rainy in the example), not over the conditioning index(es).

Tip
You must have already specified the variables as probability tables, before adding them with the Indexes dialog.

## DetermTable(): Deterministic conditional table

DetermTable() defines the value of a variable as a deterministic (not uncertain) function of one or more discrete variables. It gives a value conditional on the value of one or more discrete variables, often including a probabilistic discrete variable and a discrete decision variable defined as a list. DetermTable() is described in Array functions, but we also include it in this section on discrete probability distributions, even though it is not probabilistic, because you usually use it in conjunction with ProbTable and other discrete distributions. It is an editable table, like ProbTable, but with a single (deterministic) value, number, or text, in each cell.

The DetermTable() function looks like an edit table or a probability table, with an index (dimension) from each discrete variable on which it depends. Unlike ProbTable(), it does not need a self index. Its result is probabilistic if any of its conditioning variables are probabilistic.

For the steps to create a determTable, see Creating a DetermTable.

Example: In Create a probability table above, Weather is defined as a probability table. If P, the probability of “sunny” is 0.4, then the probability of “rainy” is 0.6. Party_location is a decision variable with values ['outdoors', 'porch','indoors']. value_to_me is a determTable, containing utility values (or “payoffs”) for each combination of Party_location and Weather.

Evaluating value_to_me gives the value of each party location, considering the uncertain distribution of Weather. The mean value of value_to_me is the expected utility.

## ChanceDist(p, a, i)

Creates a discrete probability distribution, where «a» is an array of outcome values, numbers or text, and «p» is the corresponding array of probabilities. «a» and «p» must both be indexed by «i».

When to use: Use: ChanceDist instead of ProbTable when:

• The array of outcome «a» is multidimensional.
• You want a distribution where each outcome in «a» is equally likely. In this case, you can specify «p» as 1.

or

• You want to use other variables or expressions to define the outcomes or probability arrays.

Example:

Index Index_b :=
 Red White Blue
Variable Array_q :=
Index_b ▶
Red White Blue
0.3 0.2 0.5

The domain of the variable is a list of labels: ['Red', 'White', 'Blue'].

ChanceDist(Array_q, Index_b, Index_b) →


text:
# relation between the Poincaré and Euclidean algebra

Take $d$ a strictly positive integer, and consider the (proper) Euclidean group $E^d$ (the symmetry group of $\mathbf{R}^d$ with the conventional inner product), and the (proper, ortochronous) Poincaré group $P^d$ (symmetry group of the Minkowski space $M^d$).

I need a reference for the following fact.

Take an unit vector $e \in \mathbf{R}^d$ and let $E^{d-1}_e$ be the Euclidean subgroup which fixes that direction. Then the Lie algebra $\mathfrak{e}^d$ of $E^d$ can be decomposed as $\mathfrak{e}^d = \mathfrak{e}^{d-1}_e \oplus \mathfrak{m}^{d-1}_e$ where $\mathfrak{e}^{d-1}_e$ is the Lie algebra of $E^{d-1}_e$ and $\mathfrak{m}^{d-1}_e$ is another appropriate Lie algebra.

Also, the Poincaré algebra can be formally written as $\mathfrak{p}^d = \mathfrak{e}^{d-1}_e \oplus i\mathfrak{m}^{d-1}_e$.

It looks like it has something to do with those Wick rotations, and most of the references I found only treat the Poincaré algebra... Some help would be appreciated!

Thank you.

-

Suppose that you drop one dimension from both groups: time in the Poincaré case, one spatial dimension in the Euclidean (since all directions are the same in that case).

Then you obtain two $(d-1)$-dimensional spaces that have exactly the same metric and, therefore, the same transformation groups.

(By the way, this is the reason why we are so familiar with 3-dimensional rotations.)

If you add back the last dimension, in both cases, what happens?

In the Euclidean case, the added dimension is linked to the previous ones by a rotation. In the Lorentz case, by a boost (since we have said we added time).

As you probably know, at the Lie algebra level the difference between a boost and a rotation is just a $i$ factor. (Think of the difference between $\sin, \cos$ and $\sinh, \cosh$.)

Therefore, whatever was added in the Euclidean case is the same in the Poincaré case, but with a factor of $i$.

(Translations here don't matter at all.)

-


text: As an example, I bet most of your students hated solving the problem $\int e^{-x}cos(x) dx.$ Solve it for them the way they learned it in calculus, by repeated integration by parts and then by $\int e^{-x}cos(x) dx=\Re \int e^{-x(1-i)}dx.$ They should notice how much easier it was to use complex analysis. If you do this enough they might come to appreciate numbers that do not belong to the real world.


text:
## RS Aggarwal Class 9 Solutions Chapter 8 Linear Equations in Two Variables Ex 8A

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Class 9 Solutions Chapter 8 Linear Equations in Two Variables Ex 8A.

Question 1.
Solution:
(i) x = 5 is the line AB parallel to the y-axis at a distance of 5 units.
(ii) y = – 2 is the line CD parallel to x-axis at a distance of – 2 units.
(iii) x + 6 = 0 => x = – 6 is the line EF parallel to y-axis at a distance of – 6 units.
(iv) x + 7 = 0 => x = – 7 is the line PQ parallel to y-axis at a distance of – 7 units.
(v) y = 0 is the equation of x-axis. The graph of y = 0 is the line X’OX
(vi) x = 0 is the equation of y-axis.The graph of x = 0 is the line YOY’
Ans.

Question 2.
Solution:
In the given equation.

y = 3x
Put x = 1, then y = 3 x 1 = 3
Put x = 2, then y = 3 x 2 = 6
Put x = – 1, then y = 3 ( – 1) = – 3
Now, plot the points (1, 3), (2, 6) and ( – 1, – 3) as given the following table

and join them to form a line of the given equation.
Now from x = – 2,
draw a line parallel to y-axis at a distance of x = – 2, meeting the given line at P. From P, draw, a line parallel to x-axis joining y-axis at M, which is y = – 6 Hence, y = – 6 Ans.

Question 3.
Solution:
In the given equation x + 2y – 3 = 0
=> 2y = 3 – x
y = $$\frac { 3-x }{ 2 }$$
put x = 1,then

Question 4.
Solution:
(i) In the equation y = x
When x = 1, then y = 1
when x = 2, then y = 2
and when x = 3, then y = 3

Question 5.
Solution:
In the given equation
2x – 3y = 5

Question 6.
Solution:
In the given equation
2x + y = 6
=> y = 6 – 2x
Put x = 1, then y – 6 – 2 x 1 = 6 – 2 = 4
Put x = 2, then y = 6 – 2 x 2 = 6 – 4 = 2
Put x = 3, then y = 6 – 2 x 3 = 6 – 6 = 0

Question 7.
Solution:
In the given equation
3x + 2y = 6

Hope given RS Aggarwal Class 9 Solutions Chapter 8 Linear Equations in Two Variables Ex 8A are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.


text:
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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki: Year: Volume: Issue: Page: Find

 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2016, Volume 26, Issue 3, Pages 418–438 (Mi vuu549)

MECHANICS

On the periodic motions of a Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a double three-order resonance

A. I. Safonov, O. V. Kholostova

Department of Theoretical Mechanics, Moscow Aviation Institute (National Research University), Volokolamskoe shosse, 4, Moscow, 125993, Russia

Abstract: The paper considers the motion of a near-autonomous time-periodic two-degree-of-freedom Hamiltonian system in a neighborhood of trivial equilibrium being stable in the linear approximation. The third-order double resonance (fundamental and Raman) is assumed to occur in the system, at that Raman resonance can be strong or weak. In both cases the equilibrium considered is unstable in a full nonlinear system. Normalization of Hamiltonians of the perturbed motion is carried out in the terms up to the fourth order with respect to disturbance, taking into account the existing resonances. The problem of the existence and number of equilibrium positions of the corresponding approximate (model) systems is solved, and sufficient and necessary conditions for their stability are obtained. By Poincare's small parameter method, periodic motions of the initial full systems generated from the equilibrium positions of the model systems are constructed. The question of their stability in the linear approximation is solved. In particular, the conditions of the existence of stable (in the linear approximation) periodic motions in a small neighborhood of the unstable trivial equilibrium are obtained.

Keywords: Hamiltonian system, multiple resonance, stability, periodic motion.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00380

DOI: https://doi.org/10.20537/vm160310

Full text: PDF file (398 kB)
References: PDF file   HTML file

Bibliographic databases:

Document Type: Article
UDC: 531.011
MSC: 70H05, 70H14, 70H15, 70K45

Citation: A. I. Safonov, O. V. Kholostova, “On the periodic motions of a Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a double three-order resonance”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 26:3 (2016), 418–438

Citation in format AMSBIB
\Bibitem{SafKho16} \by A.~I.~Safonov, O.~V.~Kholostova \paper On the periodic motions of a~Hamiltonian system in the neighborhood of unstable equilibrium in the presence of a~double three-order resonance \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2016 \vol 26 \issue 3 \pages 418--438 \mathnet{http://mi.mathnet.ru/vuu549} \crossref{https://doi.org/10.20537/vm160310} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3558451} \elib{http://elibrary.ru/item.asp?id=26726587}


text:
Simon’s Graphics Blog

# Representing Rotations in Quaternion Arithmetic

maths

Quaternions crop up a lot in game development, since they are an efficient way to store rotations in 3-space. This article attempts to serve as a mathematical introduction to quaternions, and explains how and why we choose to use them to represent 3D rotations.

## Basic Quaternion Arithmetic

Quaternions form what is called a non-commutative division ring. This space can be visualised in a similar manner to complex numbers. Just as the complex numbers can be represented by a two-dimensional co-ordinate on the complex plane, quaternions can be represented by a co-ordinate in 4-D space. This is written:

$$q=w+x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k} \text{ for } w,x,y,z \in \mathbb{R}$$

The numbers $\boldsymbol{i}$, $\boldsymbol{j}$ and $\boldsymbol{k}$ are related to the real numbers and each other by the following equalities:

$$\begin{array}{c} \boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=-1 \\ \boldsymbol{i}\boldsymbol{j}=\boldsymbol{k} \end{array}$$

Quaternions can be viewed as a vector space over the real numbers. In this way they are given the notation:

$$\begin{array}{c} q=(w,x,y,z) \text{ for } w,x,y,z \in \mathbb{R} \\ \text{or} \\ q=[w,\boldsymbol{v}] \text{ where } \boldsymbol{v}=(x,y,z) \end{array}$$

When adding or subtracting quaternions you perform the same operation as for vectors and matrices: add and subtract component-wise. As with many other vector spaces, the quaternions come equipped with a norm, which is function (usually regarded as a “length”) that returns a single real number given a quaternion. It is usually defined as:

$$\Vert q\Vert=\sqrt{w^2 + x^2 + y^2 + z^2}$$

A non-commutative division ring has more structure than a vector space, namely a non-commutative multiplication operator. Without worrying too much about the exact definitions of these words (those interested can pick up the nearest undergraduate text on algebra) this basically means they form a special kind of number system. The word “non-commutative” means that multiplication is not symmetric, in other words:

$$q_1 q_2 \neq q_2 q_1 \text{ in general}$$

We can define quaternion multiplication using the second vector notation above and the normal vector dot and cross product as:

$$q_1 q_2 = [w_1 w_2 - v_1 \cdot v_2, w_1 v_2 + w_2 v_1 + v_1 \wedge v_2]$$

This is entirely equivalent to multiplying out the various components of i, j and k using the very first notation given earlier. Try it if you don’t believe me! :)

## Conjugates And Inversion

Now that we can add, subtract and multiply quaternions it would be useful to also be able to divide them. As a direct extension of the complex conjugate in complex arithmetic, the quaternion conjugate is defined as:

$$q^\ast=(w,-x,-y,-z)$$

By multiplying this out using the definition of multiplication above, it can be shown that:

$$q^\ast{q}=qq^\ast=\Vert q\Vert^2$$

This equality allows us to construct the inverse of any quaternion:

$$q^{-1}=\frac{q^\ast}{\Vert q\Vert^2}$$

We can use the above definitions to prove that:

$$q^{-1}q=qq^{-1}=1$$

## Exponentials, Logarithms And Powers

Since quaternions are really just numbers you would expect there to be a natural extension of our idea of exponentials and logarithms for real numbers. Well it just so happens that there is, and it is derived in a similar way to the extension for complex numbers. This section is aimed at the mathematically fearless and can be skipped if need be without causing confusion in later sections.

By considering the series expansion of the exponential, it can be shown that:

$$\text{for } q=(w,x,y,z)=[w,\boldsymbol{v}], \text{ let } \boldsymbol{v}=s\boldsymbol{n} \text{ such that }|\boldsymbol{n}|=1 \\ \text{then } e^q=e^w[\cos{s},\boldsymbol{n}\sin{s}]$$

This leads to a very useful factorisation of a quaternion:

$$\text{for } q=re^{[0,\theta\boldsymbol{u}]} \\ \log{q}=\log{r}+[0,\theta\boldsymbol{u}]=[\log{r},\theta\boldsymbol{u}]$$

Using the identity for exponential above, we can write this as:

$$q=r[\cos{\theta},\boldsymbol{u}\sin{\theta}]=re^{[0,\theta\boldsymbol{u}]}$$

Note how similar this is to the $r, \theta$ factorisation of complex numbers. Once we have this factorisation, we can trivially define the quaternion logarithm as:

$$\begin{array}{c} \text{for } q=re^{[0,\theta\boldsymbol{u}]} \\ \log{q}=\log{r}+[0,\theta\boldsymbol{u}]=[\log{r},\theta\boldsymbol{u}] \end{array}$$

We can then define quaternion powers in terms of the logarithm and exponential, namely:

$$q^k=e^{k\log{q}} \text{ for } k\in\mathbb{R}$$

## A Quaternion Rotation

For a rotation of $\phi$ about a unit axis $\boldsymbol{w}$, we will define the representing quaternion $q$ to be:

$$q=[\cos{\frac\phi2},\boldsymbol{w}\sin{\frac\phi2}]$$

Note that using this scheme that the norm of all quaternion rotations is 1 (i.e. they are all unit quaternions). The converse is also true: any unit quaternion can be decomposed into this form (just like we did above for logarithms) and can therefor represent a rotation. It is therefore simple to adjust for numerical drift: simply renormalise the quaternion. This takes far fewer operations than matrix orthonormalisation, which corrects numerical drift in rotation matrices.

To rotate a vector $\boldsymbol{v}$ an angle of $\phi$ around about an arbitrary unit axis $\boldsymbol{w}$, you can use the formula:

$$\boldsymbol{v}^\prime=(\boldsymbol{v}\cdot\boldsymbol{w})\boldsymbol{w}+(\boldsymbol{v}-(\boldsymbol{v}\cdot\boldsymbol{w})\boldsymbol{w})\cos\phi+\boldsymbol{v}\wedge\boldsymbol{w}\sin\phi$$

Representing the same rotation by a quaternion $q$, you can prove (go on, try it!) that this equivalent to:

$$[0,\boldsymbol{v}^\prime]=q[0,\boldsymbol{v}]q^\ast$$

This formula uses quaternion multiplication and conjugation as defined earlier. This equation is one of the reasons for encoding rotations in quaternions, since it greatly simplifies the mathematics of composing rotations. We can also describe composition of rotations:

$$\begin{array}{c} \text{let } r=pq \\ \text{then } r[0,\boldsymbol{v}]r^\ast=pq[0,\boldsymbol{v}](pq)^\ast=pq[0,\boldsymbol{v}]q^\ast{p}^\ast=p(q[0,\boldsymbol{v}]q^\ast)p^\ast \end{array}$$

This shows that multiplication of quaternions is equivalent to composition of rotations. Just like composing rotations, we need to be careful about the order in which we multiply. In the following notation, rotations are applied from right to left.

Similarly, inverting a quaternion (note that for unit quaternions this is just the conjugate directly) produces the inverse rotation. Later, we will show how to use quaternion powers to do partial rotations, or blend between them.

## Quaternion To Matrix Conversion

Although I hate to admit it, it is often necessary to have a quaternion rotation in matrix form, e.g. to load onto graphics hardware for hardware vertex transformations. In this case it useful to have a formula for converting a quaternion quickly into a rotation matrix. Using the scheme of pre-multiplication of matrices given earlier, one such formula for this is:

$$\text{let }q=(w,x,y,z) \\ R=\begin{pmatrix} 1-2y^2-2z^2 & 2xy-2wz & 2zx+2wy \\ 2xy+2wz & 1-2x^2-2z^2 & 2yz-2wx \\ 2zx-2wy & 2yz+2wx & 1-2x^2-2y^2 \end{pmatrix}$$

As I have had recently pointed out to me, applying a quaternion rotation to a vector directly takes a couple of operations more than computing the matrix and applying the matrix directly (41 vs 39 operations). The direct quaternion rotation uses fewer registers for the computation, but once the matrix has been computed only 15 operations are needed to apply it to another vector, so for rotating more than one vector it is much more efficient to use the above matrix form.

## SLERP

SLERP stands for spherical-linear interpolation and is one method of interpolating two rotations smoothly. It is called spherical-linear since the two quaternion rotations are interpolated uniformly along a geodesic in the surface of the 3-sphere.

The general idea behind interpolation is to have a variable t that ranges from 0 to 1, and use this to parameterise a smooth blend between quaternions $a$ and $b$. Spherical linear interpolation uses the following equation:

$$q = (ba^{-1})^{t}a$$

Clearly q is a when t is 0, and b when t is 1. Note how we have been careful with our order of multiplication: quaternions are applied from right to left. General quaternion powers were described earlier, but since t is a real number, and all rotation quaternions are of unit length, we can greatly simplify the power equation:

$$\text{if } q=[\cos\theta,\boldsymbol{u}\sin\theta] \text{ then } q^t=[\cos{t\theta},\boldsymbol{u}\sin{t\theta}]$$

This nicely fits with our intuition that linearly interpolating between the identity transform and a rotation should be a linear change of angle.