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https://stacks.math.columbia.edu/tag/00N2
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## 10.103 Cohen-Macaulay modules
Here we show that Cohen-Macaulay modules have good properties. We postpone using Ext groups to establish the connection with duality and so on.
Definition 10.103.1. Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. We say $M$ is Cohen-Macaulay if $\dim (\text{Supp}(M)) = \text{depth}(M)$.
A first goal will be to establish Proposition 10.103.4. We do this by a (perhaps nonstandard) sequence of elementary lemmas involving almost none of the earlier results on depth. Let us introduce some notation.
Let $R$ be a local Noetherian ring. Let $M$ be a Cohen-Macaulay module, and let $f_1, \ldots , f_ d$ be an $M$-regular sequence with $d = \dim (\text{Supp}(M))$. We say that $g \in \mathfrak m$ is good with respect to $(M, f_1, \ldots , f_ d)$ if for all $i = 0, 1, \ldots , d-1$ we have $\dim (\text{Supp}(M) \cap V(g, f_1, \ldots , f_ i)) = d - i - 1$. This is equivalent to the condition that $\dim (\text{Supp}(M/(f_1, \ldots , f_ i)M) \cap V(g)) = d - i - 1$ for $i = 0, 1, \ldots , d - 1$.
Lemma 10.103.2. Notation and assumptions as above. If $g$ is good with respect to $(M, f_1, \ldots , f_ d)$, then (a) $g$ is a nonzerodivisor on $M$, and (b) $M/gM$ is Cohen-Macaulay with maximal regular sequence $f_1, \ldots , f_{d - 1}$.
Proof. We prove the lemma by induction on $d$. If $d = 0$, then $M$ is finite and there is no case to which the lemma applies. If $d = 1$, then we have to show that $g : M \to M$ is injective. The kernel $K$ has support $\{ \mathfrak m\}$ because by assumption $\dim \text{Supp}(M) \cap V(g) = 0$. Hence $K$ has finite length. Hence $f_1 : K \to K$ injective implies the length of the image is the length of $K$, and hence $f_1 K = K$, which by Nakayama's Lemma 10.20.1 implies $K = 0$. Also, $\dim \text{Supp}(M/gM) = 0$ and so $M/gM$ is Cohen-Macaulay of depth $0$.
Assume $d > 1$. Observe that $g$ is good for $(M/f_1M, f_2, \ldots , f_ d)$, as is easily seen from the definition. By induction, we have that (a) $g$ is a nonzerodivisor on $M/f_1M$ and (b) $M/(g, f_1)M$ is Cohen-Macaulay with maximal regular sequence $f_2, \ldots , f_{d - 1}$. By Lemma 10.68.4 we see that $g, f_1$ is an $M$-regular sequence. Hence $g$ is a nonzerodivisor on $M$ and $f_1, \ldots , f_{d - 1}$ is an $M/gM$-regular sequence. $\square$
Lemma 10.103.3. Let $R$ be a Noetherian local ring. Let $M$ be a Cohen-Macaulay module over $R$. Suppose $g \in \mathfrak m$ is such that $\dim (\text{Supp}(M) \cap V(g)) = \dim (\text{Supp}(M)) - 1$. Then (a) $g$ is a nonzerodivisor on $M$, and (b) $M/gM$ is Cohen-Macaulay of depth one less.
Proof. Choose a $M$-regular sequence $f_1, \ldots , f_ d$ with $d = \dim (\text{Supp}(M))$. If $g$ is good with respect to $(M, f_1, \ldots , f_ d)$ we win by Lemma 10.103.2. In particular the lemma holds if $d = 1$. (The case $d = 0$ does not occur.) Assume $d > 1$. Choose an element $h \in R$ such that (i) $h$ is good with respect to $(M, f_1, \ldots , f_ d)$, and (ii) $\dim (\text{Supp}(M) \cap V(h, g)) = d - 2$. To see $h$ exists, let $\{ \mathfrak q_ j\}$ be the (finite) set of minimal primes of the closed sets $\text{Supp}(M)$, $\text{Supp}(M)\cap V(f_1, \ldots , f_ i)$, $i = 1, \ldots , d - 1$, and $\text{Supp}(M) \cap V(g)$. None of these $\mathfrak q_ j$ is equal to $\mathfrak m$ and hence we may find $h \in \mathfrak m$, $h \not\in \mathfrak q_ j$ by Lemma 10.15.2. It is clear that $h$ satisfies (i) and (ii). From Lemma 10.103.2 we conclude that $M/hM$ is Cohen-Macaulay. By (ii) we see that the pair $(M/hM, g)$ satisfies the induction hypothesis. Hence $M/(h, g)M$ is Cohen-Macaulay and $g : M/hM \to M/hM$ is injective. By Lemma 10.68.4 we see that $g : M \to M$ and $h : M/gM \to M/gM$ are injective. Combined with the fact that $M/(g, h)M$ is Cohen-Macaulay this finishes the proof. $\square$
Proposition 10.103.4. Let $R$ be a Noetherian local ring, with maximal ideal $\mathfrak m$. Let $M$ be a Cohen-Macaulay module over $R$ whose support has dimension $d$. Suppose that $g_1, \ldots , g_ c$ are elements of $\mathfrak m$ such that $\dim (\text{Supp}(M/(g_1, \ldots , g_ c)M)) = d - c$. Then $g_1, \ldots , g_ c$ is an $M$-regular sequence, and can be extended to a maximal $M$-regular sequence.
Proof. Let $Z = \text{Supp}(M) \subset \mathop{\mathrm{Spec}}(R)$. By Lemma 10.60.13 in the chain $Z \supset Z \cap V(g_1) \supset \ldots \supset Z \cap V(g_1, \ldots , g_ c)$ each step decreases the dimension at most by $1$. Hence by assumption each step decreases the dimension by exactly $1$ each time. Thus we may successively apply Lemma 10.103.3 to the modules $M/(g_1, \ldots , g_ i)$ and the element $g_{i + 1}$.
To extend $g_1, \ldots , g_ c$ by one element if $c < d$ we simply choose an element $g_{c + 1} \in \mathfrak m$ which is not in any of the finitely many minimal primes of $Z \cap V(g_1, \ldots , g_ c)$, using Lemma 10.15.2. $\square$
Having proved Proposition 10.103.4 we continue the development of standard theory.
Lemma 10.103.5. Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak m$. Let $M$ be a finite $R$-module. Let $x \in \mathfrak m$ be a nonzerodivisor on $M$. Then $M$ is Cohen-Macaulay if and only if $M/xM$ is Cohen-Macaulay.
Proof. By Lemma 10.72.7 we have $\text{depth}(M/xM) = \text{depth}(M)-1$. By Lemma 10.63.10 we have $\dim (\text{Supp}(M/xM)) = \dim (\text{Supp}(M)) - 1$. $\square$
Lemma 10.103.6. Let $R \to S$ be a surjective homomorphism of Noetherian local rings. Let $N$ be a finite $S$-module. Then $N$ is Cohen-Macaulay as an $S$-module if and only if $N$ is Cohen-Macaulay as an $R$-module.
Proof. Omitted. $\square$
Lemma 10.103.7. Let $R$ be a Noetherian local ring. Let $M$ be a finite Cohen-Macaulay $R$-module. If $\mathfrak p \in \text{Ass}(M)$, then $\dim (R/\mathfrak p) = \dim (\text{Supp}(M))$ and $\mathfrak p$ is a minimal prime in the support of $M$. In particular, $M$ has no embedded associated primes.
Proof. By Lemma 10.72.9 we have $\text{depth}(M) \leq \dim (R/\mathfrak p)$. Of course $\dim (R/\mathfrak p) \leq \dim (\text{Supp}(M))$ as $\mathfrak p \in \text{Supp}(M)$ (Lemma 10.63.2). Thus we have equality in both inequalities as $M$ is Cohen-Macaulay. Then $\mathfrak p$ must be minimal in $\text{Supp}(M)$ otherwise we would have $\dim (R/\mathfrak p) < \dim (\text{Supp}(M))$. Finally, minimal primes in the support of $M$ are equal to the minimal elements of $\text{Ass}(M)$ (Proposition 10.63.6) hence $M$ has no embedded associated primes (Definition 10.67.1). $\square$
Definition 10.103.8. Let $R$ be a Noetherian local ring. A finite module $M$ over $R$ is called a maximal Cohen-Macaulay module if $\text{depth}(M) = \dim (R)$.
In other words, a maximal Cohen-Macaulay module over a Noetherian local ring is a finite module with the largest possible depth over that ring. Equivalently, a maximal Cohen-Macaulay module over a Noetherian local ring $R$ is a Cohen-Macaulay module of dimension equal to the dimension of the ring. In particular, if $M$ is a Cohen-Macaulay $R$-module with $\mathop{\mathrm{Spec}}(R) = \text{Supp}(M)$, then $M$ is maximal Cohen-Macaulay. Thus the following two lemmas are on maximal Cohen-Macaulay modules.
Lemma 10.103.9. Let $R$ be a Noetherian local ring. Assume there exists a Cohen-Macaulay module $M$ with $\mathop{\mathrm{Spec}}(R) = \text{Supp}(M)$. Then any maximal chain of ideals $\mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_ n$ has length $n = \dim (R)$.
Proof. We will prove this by induction on $\dim (R)$. If $\dim (R) = 0$, then the statement is clear. Assume $\dim (R) > 0$. Then $n > 0$. Choose an element $x \in \mathfrak p_1$, with $x$ not in any of the minimal primes of $R$, and in particular $x \not\in \mathfrak p_0$. (See Lemma 10.15.2.) Then $\dim (R/xR) = \dim (R) - 1$ by Lemma 10.60.13. The module $M/xM$ is Cohen-Macaulay over $R/xR$ by Proposition 10.103.4 and Lemma 10.103.6. The support of $M/xM$ is $\mathop{\mathrm{Spec}}(R/xR)$ by Lemma 10.40.9. After replacing $x$ by $x^ n$ for some $n$, we may assume that $\mathfrak p_1$ is an associated prime of $M/xM$, see Lemma 10.72.8. By Lemma 10.103.7 we conclude that $\mathfrak p_1/(x)$ is a minimal prime of $R/xR$. It follows that the chain $\mathfrak p_1/(x) \subset \ldots \subset \mathfrak p_ n/(x)$ is a maximal chain of primes in $R/xR$. By induction we find that this chain has length $\dim (R/xR) = \dim (R) - 1$ as desired. $\square$
Lemma 10.103.10. Suppose $R$ is a Noetherian local ring. Assume there exists a Cohen-Macaulay module $M$ with $\mathop{\mathrm{Spec}}(R) = \text{Supp}(M)$. Then for a prime $\mathfrak p \subset R$ we have
$\dim (R) = \dim (R_{\mathfrak p}) + \dim (R/\mathfrak p).$
Proof. Follows immediately from Lemma 10.103.9. $\square$
Lemma 10.103.11. Suppose $R$ is a Noetherian local ring. Let $M$ be a Cohen-Macaulay module over $R$. For any prime $\mathfrak p \subset R$ the module $M_{\mathfrak p}$ is Cohen-Macaulay over $R_\mathfrak p$.
Proof. We may and do assume $\mathfrak p \not= \mathfrak m$ and $M$ not zero. Choose a maximal chain of primes $\mathfrak p = \mathfrak p_ c \subset \mathfrak p_{c - 1} \subset \ldots \subset \mathfrak p_1 \subset \mathfrak m$. If we prove the result for $M_{\mathfrak p_1}$ over $R_{\mathfrak p_1}$, then the lemma will follow by induction on $c$. Thus we may assume that there is no prime strictly between $\mathfrak p$ and $\mathfrak m$. Note that $\dim (\text{Supp}(M_\mathfrak p)) \leq \dim (\text{Supp}(M)) - 1$ because any chain of primes in the support of $M_\mathfrak p$ can be extended by one more prime (namely $\mathfrak m$) in the support of $M$. On the other hand, we have $\text{depth}(M_\mathfrak p) \geq \text{depth}(M) - \dim (R/\mathfrak p) = \text{depth}(M) - 1$ by Lemma 10.72.10 and our choice of $\mathfrak p$. Thus $\text{depth}(M_\mathfrak p) \geq \dim (\text{Supp}(M_\mathfrak p)$ as desired (the other inequality is Lemma 10.72.3). $\square$
Definition 10.103.12. Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. We say $M$ is Cohen-Macaulay if $M_\mathfrak p$ is a Cohen-Macaulay module over $R_\mathfrak p$ for all primes $\mathfrak p$ of $R$.
By Lemma 10.103.11 it suffices to check this in the maximal ideals of $R$.
Lemma 10.103.13. Let $R$ be a Noetherian ring. Let $M$ be a Cohen-Macaulay module over $R$. Then $M \otimes _ R R[x_1, \ldots , x_ n]$ is a Cohen-Macaulay module over $R[x_1, \ldots , x_ n]$.
Proof. By induction on the number of variables it suffices to prove this for $M[x] = M \otimes _ R R[x]$ over $R[x]$. Let $\mathfrak m \subset R[x]$ be a maximal ideal, and let $\mathfrak p = R \cap \mathfrak m$. Let $f_1, \ldots , f_ d$ be a $M_\mathfrak p$-regular sequence in the maximal ideal of $R_{\mathfrak p}$ of length $d = \dim (\text{Supp}(M_{\mathfrak p}))$. Note that since $R[x]$ is flat over $R$ the localization $R[x]_{\mathfrak m}$ is flat over $R_{\mathfrak p}$. Hence, by Lemma 10.68.5, the sequence $f_1, \ldots , f_ d$ is a $M[x]_{\mathfrak m}$-regular sequence of length $d$ in $R[x]_{\mathfrak m}$. The quotient
$Q = M[x]_{\mathfrak m}/(f_1, \ldots , f_ d)M[x]_{\mathfrak m} = M_{\mathfrak p}/(f_1, \ldots , f_ d)M_{\mathfrak p} \otimes _{R_\mathfrak p} R[x]_{\mathfrak m}$
has support equal to the primes lying over $\mathfrak p$ because $R_\mathfrak p \to R[x]_\mathfrak m$ is flat and the support of $M_{\mathfrak p}/(f_1, \ldots , f_ d)M_{\mathfrak p}$ is equal to $\{ \mathfrak p\}$ (details omitted; hint: follows from Lemmas 10.40.4 and 10.40.5). Hence the dimension is $1$. To finish the proof it suffices to find an $f \in \mathfrak m$ which is a nonzerodivisor on $Q$. Since $\mathfrak m$ is a maximal ideal, the field extension $\kappa (\mathfrak p) \subset \kappa (\mathfrak m)$ is finite (Theorem 10.34.1). Hence we can find $f \in \mathfrak m$ which viewed as a polynomial in $x$ has leading coefficient not in $\mathfrak p$. Such an $f$ acts as a nonzerodivisor on
$M_{\mathfrak p}/(f_1, \ldots , f_ d)M_{\mathfrak p} \otimes _ R R[x] = \bigoplus \nolimits _{n \geq 0} M_{\mathfrak p}/(f_1, \ldots , f_ d)M_{\mathfrak p} \cdot x^ n$
and hence acts as a nonzerodivisor on $Q$. $\square$
Comment #684 by Keenan Kidwell on
In the paragraph after 0AAD, there are a couple parentheses missing in the "equivalent" expression of the last sentence.
Comment #2211 by David Savitt on
I think the snake lemma arguments in the proofs of 00N4 and 00N5 are really just reproving the case c=2 of 00LJ and can be replaced with applications of it.
Comment #2700 by Tanya Kaushal Srivastava on
General Reference for this section can be EGA IV section 16.5.
Comment #2842 by on
@#2700: this does not seem the correct reference in my EGA
Comment #2846 by on
@#2700: OK, now I see what you mean. You mean Chapter 0, Section 16.5 in EGA. Still, I think it makes more sense to add references to the individual results and not to sections. To attribute this material we should have a historical discussion somewhere else...
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2021-12-07 11:41:14
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https://socratic.org/questions/what-is-the-square-root-of-3-4
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# What is the square root of 3.4?
Sep 18, 2015
$\sqrt{3.4} = \frac{\sqrt{85}}{5} \approx 1.8439$
#### Explanation:
If $a , b \ge 0$ then $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
If $a \ge 0$ then $\sqrt{{a}^{2}} = a$
So:
$\sqrt{3.4} = \sqrt{\frac{85}{25}} = \frac{\sqrt{85}}{\sqrt{25}} = \frac{\sqrt{85}}{5}$
Here I have chosen to express $3.4$ as $\frac{85}{25}$ to get the smallest whole value for the denominator.
I could also have written $\sqrt{3.4} = \sqrt{\frac{17}{5}} = \frac{\sqrt{17}}{\sqrt{5}}$
If I wanted to calculate an approximation for $\sqrt{3.4}$ by hand then I would probably have used this instead:
$\sqrt{3.4} = \sqrt{\frac{340}{100}} = \frac{\sqrt{340}}{10}$
Then I would work out an approximation for $\sqrt{340}$ and divide by $10$.
For example: ${18}^{2} = 324$ and ${19}^{2} = 361$, so:
$18 = \sqrt{324} < \sqrt{340} < \sqrt{361} = 19$
Using a Newton Raphson type method to find $\sqrt{340}$, I might chose $18.5 = \frac{37}{2}$ as my first approximation ${a}_{0}$, then make better approximations using the formula:
${a}_{i + 1} = \frac{{a}_{i}^{2} + n}{2 {a}_{i}}$
where $n = 340$.
Then
${a}_{1} = \frac{{37}^{2} + 340 \cdot {2}^{2}}{2 \cdot 37 \cdot 2} = \frac{1369 + 1360}{148}$
$= \frac{2729}{148} \approx 18.439$
So $\sqrt{340} \approx 18.439$ and $\sqrt{3.4} \approx 1.8439$
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2019-12-09 00:12:17
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http://en.wikipedia.org/wiki/Wythoff_symbol
|
# Wythoff symbol
Example Wythoff construction triangles with the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol.
The eight forms for the Wythoff constructions from a general triangle (p q r).
In geometry, the Wythoff symbol was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.
A Schwarz triangle is a triangle that, with its own reflections in its edges, covers the sphere or the plane a finite number of times. The usual representation for the triangle is three numbers – integers or fractions – such that π/x is the angle at one vertex. For example, the triangle (2 3 4) represents the symmetry of a cube, while (5/2 5/2 5/2) is the face of an icosahedron.
Wythoff's construction in three dimensions consists of choosing a point in the triangle whose distance from each of the sides, if nonzero, is equal, and dropping perpendiculars to each of the edges.
Each edge of the triangle is named for the opposite angle; thus an edge opposite a right angle is designated '2'. The symbol then corresponds to a representation of off | on. Each of the numbers p in the symbol becomes a polygon pn, where n is the number of other edges that appear before the bar. So in 3 | 4 2 the vertex – a point, being here a degenerate polygon with 3×0 sides – lies on the π/3 corner of the triangle, and the altitude from that corner can be considered as forming half of the boundary between a square (having 4×1 sides) and a digon (having 2×1 sides) of zero area.
The special case of the snub figures is done by using the symbol | p q r, which would normally put the vertex at the centre of the sphere. The faces of a snub alternate as p 3 q 3 r 3. This gives an antiprism when q=r=2.
Each symbol represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, and 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D2h symmetry.
It can be applied with a slight extension to all uniform polyhedra, but the construction methods do not lead to all uniform tilings in euclidean or hyperbolic space.
## Summary table
There are seven generator points with each set of p,q,r (and a few special forms):
General Right triangle (r=2)
Description Wythoff
symbol
Vertex
configuration
Coxeter
diagram
Wythoff
symbol
Vertex
configuration
Schläfli
symbol
Coxeter
diagram
regular and
quasiregular
q | p r (p.r)q q | p 2 pq {p,q}
p | q r (q.r)p p | q 2 qp {q,p}
r | p q (q.p)r 2 | p q (q.p)² r{p,q} t1{p,q}
truncated and
expanded
q r | p q.2p.r.2p q 2 | p q.2p.2p t{p,q} t0,1{p,q}
p r | q p.2q.r.2q p 2 | q p. 2q.2q t{q,p} t0,1{q,p}
p q | r 2r.q.2r.p p q | 2 4.q.4.p rr{p,q} t0,2{p,q}
even-faced p q r | 2r.2q.2p p q 2 | 4.2q.2p tr{p,q} t0,1,2{p,q}
p q (r s) | 2p.2q.-2p.-2q - p 2 (r s) | 2p.4.-2p.4/3 -
snub | p q r 3.r.3.q.3.p | p q 2 3.3.q.3.p sr{p,q}
| p q r s (4.p.4.q.4.r.4.s)/2 - - - -
There are three special cases:
• p q (r s) | – This is a mixture of p q r | and p q s |.
• | p q r – Snub forms (alternated) are give this otherwise unused symbol.
• | p q r s – A unique snub form for U75 that isn't Wythoff-constructible.
## Description
The numbers p,q,r describe the fundamental triangle of the symmetry group: at its vertices, the generating mirrors meet in angles of π/p, π/q, π/r. On the sphere there are 3 main symmetry types: (3 3 2), (4 3 2), (5 3 2), and one infinite family (p 2 2), for any p. (All simple families have one right angle and so r=2.)
The position of the vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle. The generator point can either be on or off each mirror, activated or not. This distinction creates 8 (2³) possible forms, neglecting one where the generator point is on all the mirrors.
In this notation the mirrors are labeled by the reflection-order of the opposite vertex. The p,q,r values are listed before the bar if the corresponding mirror is active.
The one impossible symbol | p q r implies the generator point is on all mirrors, which is only possible if the triangle is degenerate, reduced to a point. This unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored. The resulting figure has rotational symmetry only.
This symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. (An arc representing a right angle is omitted.) A node is circled if the generator point is not on the mirror.
## Symmetry triangles
There are 4 symmetry classes of reflection on the sphere, and two in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are also listed. (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.)
Point groups:
Euclidean (affine) groups:
Hyperbolic groups:
Dihedral spherical Spherical
D2h D3h D4h D5h D6h Td Oh Ih
*222 *322 *422 *522 *622 *332 *432 *532
(2 2 2)
(3 2 2)
(4 2 2)
(5 2 2)
(6 2 2)
(3 3 2)
(4 3 2)
(5 3 2)
The above symmetry groups only includes the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, and determine the full set of solutions of nonconvex uniform polyhedra.
Euclidean plane
p4m p3m p6m
*442 *333 *632
(4 4 2)
(3 3 3)
(6 3 2)
Hyperbolic plane
*732 *542 *433
(7 3 2)
(5 4 2)
(4 3 3)
In the tilings above, each triangle is a fundamental domain, colored by even and odd reflections.
## Summary spherical, Euclidean and hyperbolic tilings
Selected tilings created by the Wythoff construction are given below.
### Spherical tilings (r = 2)
(p q 2) Parent Truncated Rectified Bitruncated Birectified
(dual)
Cantellated Omnitruncated
(Cantitruncated)
Snub
Wythoff
symbol
q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
Schläfli
symbol
$\begin{Bmatrix} p , q \end{Bmatrix}$ $t\begin{Bmatrix} p , q \end{Bmatrix}$ $\begin{Bmatrix} p \\ q \end{Bmatrix}$ $t\begin{Bmatrix} q , p \end{Bmatrix}$ $\begin{Bmatrix} q , p \end{Bmatrix}$ $r\begin{Bmatrix} p \\ q \end{Bmatrix}$ $t\begin{Bmatrix} p \\ q \end{Bmatrix}$ $s\begin{Bmatrix} p \\ q \end{Bmatrix}$
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
t0{p,q} t0,1{p,q} t1{p,q} t1,2{p,q} t2{p,q} t0,2{p,q} t0,1,2{p,q}
Coxeter
diagram
Vertex figure pq q.2p.2p (p.q)2 p.2q.2q qp p.4.q.4 4.2p.2q 3.3.p.3.q
(3 3 2)
{3,3}
(3.6.6)
(3.3a.3.3a)
(3.6.6)
{3,3}
(3a.4.3b.4)
(4.6a.6b)
(3.3.3a.3.3b)
(4 3 2)
{4,3}
(3.8.8)
(3.4.3.4)
(4.6.6)
{3,4}
(3.4.4a.4)
(4.6.8)
(3.3.3a.3.4)
(5 3 2)
{5,3}
(3.10.10)
(3.5.3.5)
(5.6.6)
{3,5}
(3.4.5.4)
(4.6.10)
(3.3.3a.3.5)
#### Some overlapping spherical tilings (r = 2)
For a more complete list, including cases where r ≠ 2, see List of uniform polyhedra by Schwarz triangle.
Tilings are shown as polyhedra. Some of the forms are degenerate, given with brackets for vertex figures, with overlapping edges or verices.
(p q 2) Fund.
triangle
Parent Truncated Rectified Bitruncated Birectified
(dual)
Cantellated Omnitruncated
(Cantitruncated)
Snub
Wythoff symbol q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
Schläfli symbol $\begin{Bmatrix} p , q \end{Bmatrix}$ $t\begin{Bmatrix} p , q \end{Bmatrix}$ $\begin{Bmatrix} p \\ q \end{Bmatrix}$ $t\begin{Bmatrix} q , p \end{Bmatrix}$ $\begin{Bmatrix} q , p \end{Bmatrix}$ $r\begin{Bmatrix} p \\ q \end{Bmatrix}$ $t\begin{Bmatrix} p \\ q \end{Bmatrix}$ $s\begin{Bmatrix} p \\ q \end{Bmatrix}$
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
t0{p,q} t0,1{p,q} t1{p,q} t1,2{p,q} t2{p,q} t0,2{p,q} t0,1,2{p,q}
Coxeter–Dynkin diagram
Vertex figure pq (q.2p.2p) (p.q.p.q) (p. 2q.2q) qp (p. 4.q.4) (4.2p.2q) (3.3.p. 3.q)
Icosahedral
(5/2 3 2)
{3,5/2}
(5/2.6.6)
(3.5/2)2
[3.10/2.10/2]
{5/2,3}
[3.4.5/2.4]
[4.10/2.6]
(3.3.3.3.5/2)
Icosahedral
(5 5/2 2)
{5,5/2}
(5/2.10.10)
(5/2.5)2
[5.10/2.10/2]
{5/2,5}
(5/2.4.5.4)
[4.10/2.10]
(3.3.5/2.3.5)
#### Dihedral symmetry (q = r = 2)
Spherical tilings with dihedral symmetry exist for all p = 2, 3, 4, ... many with digon faces which become degenerate polyhedra. Two of the eight forms (Rectified and cantellated) are replications and are skipped in the table.
(p 2 2)
Fundamental
domain
Parent Truncated Bitruncated
(truncated dual)
Birectified
(dual)
Omnitruncated
(Cantitruncated)
Snub
Extended
Schläfli symbol
$\begin{Bmatrix} p , 2 \end{Bmatrix}$ $t\begin{Bmatrix} p , 2 \end{Bmatrix}$ $t\begin{Bmatrix} 2 , p \end{Bmatrix}$ $\begin{Bmatrix} 2 , p \end{Bmatrix}$ $t\begin{Bmatrix} p \\ 2 \end{Bmatrix}$ $s\begin{Bmatrix} p \\ 2 \end{Bmatrix}$
{p,2} t{p,2} t{2,p} {2,p} tr{p,2} sr{p,2}
t0{p,2} t0,1{p,2} t1,2{p,2} t2{p,2} t0,1,2{p,2}
Wythoff symbol 2 | p 2 2 2 | p 2 p | 2 p | 2 2 p 2 2 | | p 2 2
Coxeter–Dynkin diagram
Vertex figure (2.2p.2p) (4.4.p) 2p (4.2p.4) (3.3.p. 3)
(2 2 2)
V2.2.2
{2,2}
2.4.4 4.4.2
{2,2}
4.4.4
3.3.3.2
(3 2 2)
V3.2.2
{3,2}
2.6.6
4.4.3
{2,3}
4.4.6
3.3.3.3
(4 2 2)
V4.2.2
{4,2} 2.8.8
4.4.4
{2,4}
4.4.8
3.3.3.4
(5 2 2)
V5.2.2
{5,2} 2.10.10
4.4.5
{2,5}
4.4.10
3.3.3.5
(6 2 2)
V6.2.2
{6,2}
2.12.12
4.4.6
{2,6}
4.4.12
3.3.3.6
...
### Euclidean and hyperbolic tilings (r = 2)
Some representative hyperbolic tilings are given, and shown as a Poincaré disk projection.
(p q 2) Fund.
triangles
Parent Truncated Rectified Bitruncated Birectified
(dual)
Cantellated Omnitruncated
(Cantitruncated)
Snub
Wythoff symbol q | p 2 2 q | p 2 | p q 2 p | q p | q 2 p q | 2 p q 2 | | p q 2
Schläfli symbol $\begin{Bmatrix} p , q \end{Bmatrix}$ $t\begin{Bmatrix} p , q \end{Bmatrix}$ $\begin{Bmatrix} p \\ q \end{Bmatrix}$ $t\begin{Bmatrix} q , p \end{Bmatrix}$ $\begin{Bmatrix} q , p \end{Bmatrix}$ $r\begin{Bmatrix} p \\ q \end{Bmatrix}$ $t\begin{Bmatrix} p \\ q \end{Bmatrix}$ $s\begin{Bmatrix} p \\ q \end{Bmatrix}$
{p,q} t{p,q} r{p,q} t{q,p} {q,p} rr{p,q} tr{p,q} sr{p,q}
t0{p,q} t0,1{p,q} t1{p,q} t1,2{p,q} t2{p,q} t0,2{p,q} t0,1,2{p,q}
Coxeter–Dynkin diagram
Vertex figure pq (q.2p.2p) (p.q.p.q) (p. 2q.2q) qp (p. 4.q.4) (4.2p.2q) (3.3.p. 3.q)
Hexagonal tiling
(6 3 2)
V4.6.12
{6,3}
3.12.12
3.6.3.6
6.6.6
{3,6}
3.4.6.4
4.6.12
3.3.3.3.6
(Hyperbolic plane)
(7 3 2)
V4.6.14
{7,3}
3.14.14
3.7.3.7
7.6.6
{3,7}
3.4.7.4
4.6.14
3.3.3.3.7
(Hyperbolic plane)
(8 3 2)
V4.6.16
{8,3}
3.16.16
3.8.3.8
8.6.6
{3,8}
3.4.8.4
4.6.16
3.3.3.3.8
Square tiling
(4 4 2)
V4.8.8
{4,4}
4.8.8
4.4a.4.4a
4.8.8
{4,4}
4.4a.4b.4a
4.8.8
3.3.4a.3.4b
(Hyperbolic plane)
(5 4 2)
V4.8.10
{5,4}
4.10.10
4.5.4.5
5.8.8
{4,5}
4.4.5.4
4.8.10
3.3.4.3.5
(Hyperbolic plane)
(6 4 2)
V4.8.12
{6,4}
4.12.12
4.6.4.6
6.8.8
{4,6}
4.4.6.4
4.8.12
3.3.4.3.6
(Hyperbolic plane)
(7 4 2)
V4.8.14
{7,4}
4.14.14
4.7.4.7
7.8.8
{4,7}
4.4.7.4
4.8.14
3.3.4.3.7
(Hyperbolic plane)
(8 4 2)
V4.8.16
{8,4}
4.16.16
4.8.4.8
8.8.8
{4,8}
4.4.8.4
4.8.16
3.3.4.3.8
(Hyperbolic plane)
(5 5 2)
V4.10.10
{5,5}
5.10.10
5.5.5.5
5.10.10
{5,5}
5.4.5.4
4.10.10
3.3.5.3.5
(Hyperbolic plane)
(6 5 2)
V4.10.12
{6,5}
5.12.12
5.6.5.6
6.10.10
{5,6}
5.4.6.4
4.10.12
3.3.5.3.6
(Hyperbolic plane)
(7 5 2)
V4.10.14
{7,5}
5.14.14
5.7.5.7
7.10.10
{5,7}
5.4.7.4
4.10.14
3.3.5.3.7
(Hyperbolic plane)
(8 5 2)
V4.10.16
{8,5}
5.16.16
5.8.5.8
8.10.10
{5,8}
5.4.8.4
4.10.16
3.3.5.3.8
(Hyperbolic plane)
(6 6 2)
V4.12.12
{6,6}
6.12.12
6.6.6.6
6.12.12
{6,6}
6.4.6.4
4.12.12
3.3.6.3.6
(Hyperbolic plane)
(7 6 2)
V4.12.14
{7,6}
6.14.14
6.7.6.7
7.12.12
{6,7}
6.4.7.4
4.12.14
3.3.6.3.7
(Hyperbolic plane)
(8 6 2)
V4.12.16
{8,6}
6.16.16
6.8.6.8
8.12.12
{6,8}
6.4.8.4
4.12.16
3.3.6.3.8
(Hyperbolic plane)
(7 7 2)
V4.14.14
{7,7}
7.14.14
7.7.7.7
7.14.14
{7,7}
7.4.7.4
4.14.14
3.3.7.3.7
(Hyperbolic plane)
(8 7 2)
V4.14.16
{8,7}
7.16.16
7.8.7.8
8.14.14
{7,8}
7.4.8.4
4.14.16
3.3.7.3.8
(Hyperbolic plane)
(8 8 2)
V4.16.16
{8,8}
8.16.16
8.8.8.8
8.16.16
{8,8}
8.4.8.4
4.16.16
3.3.8.3.8
(Hyperbolic plane)
(∞ 3 2)
V4.6.∞
{∞,3}
3.∞.∞
3.∞.3.∞
∞.6.6
{3,∞}
3.4.∞.4
4.6.∞
3.3.3.3.∞
(Hyperbolic plane)
(∞ 4 2)
V4.8.∞
{∞,4}
4.∞.∞
4.∞.4.∞
∞.8.8
{4,∞}
4.4.∞.4
4.8.∞
3.3.4.3.∞
(Hyperbolic plane)
(∞ 5 2)
V4.10.∞
{∞,5}
5.∞.∞
5.∞.5.∞
∞.10.10
{5,∞}
5.4.∞.4
4.10.∞
3.3.5.3.∞
(Hyperbolic plane)
(∞ 6 2)
V4.12.∞
{∞,6}
6.∞.∞
6.∞.6.∞
∞.12.12
{6,∞}
6.4.∞.4
4.12.∞
3.3.6.3.∞
(Hyperbolic plane)
(∞ 7 2)
V4.14.∞
{∞,7}
7.∞.∞
7.∞.7.∞
∞.14.14
{7,∞}
7.4.∞.4
4.14.∞
3.3.7.3.∞
(Hyperbolic plane)
(∞ 8 2)
V4.16.∞
{∞,8}
8.∞.∞
8.∞.8.∞
∞.16.16
{8,∞}
8.4.∞.4
4.16.∞
3.3.8.3.∞
(Hyperbolic plane)
(∞ ∞ 2)
V4.∞.∞
{∞,∞}
∞.∞.∞
∞.∞.∞.∞
∞.∞.∞
{∞,∞}
∞.4.∞.4
4.∞.∞
3.3.∞.3.∞
### Euclidean and hyperbolic tilings (r > 2)
The Coxeter–Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.
Wythoff symbol
(p q r)
Fund.
triangles
q | p r r q | p r | p q r p | q p | q r p q | r p q r | | p q r
Schläfli symbol (p,q,r) r(r,q,p) (q,r,p) r(p,q,r) (q,p,r) r(p,r,q) tr(p,q,r) s(p,q,r)
t0(p,q,r) t0,1(p,q,r) t1(p,q,r) t1,2(p,q,r) t2(p,q,r) t0,2(p,q,r) t0,1,2(p,q,r)
Coxeter diagram
Vertex figure (p.r)q (r.2p.q.2p) (p.q)r (q.2r.p.2r) (q.r)p (p.2r.q.2r) (2p.2q.2r) (3.r.3.q.3.p)
Euclidean
(3 3 3)
V6.6.6
(3.3)3
3.6.3.6
(3.3)3
3.6.3.6
(3.3)3
3.6.3.6
6.6.6
3.3.3.3.3.3
Hyperbolic
(4 3 3)
V6.6.8
(3.4)3
3.8.3.8
(3.4)3
3.6.4.6
(3.3)4
3.6.4.6
6.6.8
3.3.3.3.3.4
Hyperbolic
(4 4 3)
V6.8.8
(3.4)4
3.8.4.8
(4.4)3
3.8.4.8
(3.4)4
4.6.4.6
6.8.8
3.3.3.4.3.4
Hyperbolic
(4 4 4)
V8.8.8
(4.4)4
4.8.4.8
(4.4)4
4.8.4.8
(4.4)4
4.8.4.8
8.8.8
3.4.3.4.3.4
Hyperbolic
(5 3 3)
V6.6.10
(3.5)3
3.10.3.10
(3.5)3
3.6.5.6
(3.3)5
3.6.5.6
6.6.10
3.3.3.3.3.5
Hyperbolic
(5 4 3)
V6.8.10
(3.5)4
3.10.4.10
(4.5)3
3.8.5.8
(3.4)5
4.6.5.6
6.8.10
3.5.3.4.3.3
Hyperbolic
(5 4 4)
V8.8.10
(4.5)4
4.10.4.10
(4.5)4
4.8.5.8
(4.4)5
4.8.5.8
8.8.10
3.4.3.4.3.5
Hyperbolic
(6 3 3)
V6.6.12
(3.6)3
3.12.3.12
(3.6)3
3.6.6.6
(3.3)6
3.6.6.6
6.6.12
3.3.3.3.3.6
Hyperbolic
(6 4 3)
V6.8.12
(3.6)4
3.12.4.12
(4.6)3
3.8.6.8
(3.4)6
4.6.6.6
6.8.12
3.6.3.4.3.3
Hyperbolic
(6 4 4)
V8.8.12
(4.6)4
4.12.4.12
(4.6)4
4.8.6.8
(4.4)6
4.8.6.8
8.8.12
3.6.3.4.3.4
Hyperbolic
(∞ 3 3)
V6.6.∞
(3.∞)3
3.∞.3.∞
(3.∞)3
3.6.∞.6
(3.3)
3.6.∞.6
6.6.∞
3.3.3.3.3.∞
Hyperbolic
(∞ 4 3)
V6.8.∞
(3.∞)4
3.∞.4.∞
(4.∞)3
3.8.∞.8
(3.4)
4.6.∞.6
6.8.∞
3.∞.3.4.3.3
Hyperbolic
(∞ 4 4)
V8.8.∞
(4.∞)4
4.∞.4.∞
(4.∞)4
4.8.∞.8
(4.4)
4.8.∞.8
8.8.∞
3.∞.3.4.3.4
Hyperbolic
(∞ ∞ 3)
V6.∞.∞
(3.∞)
3.∞.∞.∞
(∞.∞)3
3.∞.∞.∞
(3.∞)
∞.6.∞.6
6.∞.∞
3.∞.3.∞.3.3
Hyperbolic
(∞ ∞ 4)
V8.∞.∞
(4.∞)
4.∞.∞.∞
(∞.∞)4
4.∞.∞.∞
(4.∞)
∞.8.∞.8
8.∞.∞
3.∞.3.∞.3.4
Hyperbolic
(∞ ∞ ∞)
V∞.∞.∞
(∞.∞)
∞.∞.∞.∞
(∞.∞)
∞.∞.∞.∞
(∞.∞)
∞.∞.∞.∞
∞.∞.∞
3.∞.3.∞.3.∞
|
2013-12-13 06:39:39
|
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|
https://en.wikipedia.org/wiki/Riemann_mapping_theorem
|
# Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from U onto the open unit disk
${\displaystyle D=\{z\in \mathbf {C} :|z|<1\}.}$
This mapping is known as a Riemann mapping.[1]
Intuitively, the condition that U be simply connected means that U does not contain any “holes”. The fact that f is biholomorphic implies that it is a conformal map and therefore angle-preserving. Intuitively, such a map preserves the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it.
Henri Poincaré proved that the map f is essentially unique: if z0 is an element of U and φ is an arbitrary angle, then there exists precisely one f as above such that f(z0) = 0 and that the argument of the derivative of f at the point z0 is equal to φ. This is an easy consequence of the Schwarz lemma.
As a corollary of the theorem, any two simply connected open subsets of the Riemann sphere which both lack at least two points of the sphere can be conformally mapped into each other (because conformal equivalence is an equivalence relation).
## History
The theorem was stated (under the assumption that the boundary of U is piecewise smooth) by Bernhard Riemann in 1851 in his PhD thesis. Lars Ahlfors wrote once, concerning the original formulation of the theorem, that it was “ultimately formulated in terms which would defy any attempt of proof, even with modern methods”. Riemann's flawed proof depended on the Dirichlet principle (which was named by Riemann himself), which was considered sound at the time. However, Karl Weierstrass found that this principle was not universally valid. Later, David Hilbert was able to prove that, to a large extent, the Dirichlet principle is valid under the hypothesis that Riemann was working with. However, in order to be valid, the Dirichlet principle needs certain hypotheses concerning the boundary of U which are not valid for simply connected domains in general. Simply connected domains with arbitrary boundaries were first treated by William Fogg Osgood (1900).
The first proof of the theorem is due to Constantin Carathéodory, who published it in 1912. His proof used Riemann surfaces and it was simplified by Paul Koebe two years later in a way which did not require them.
Another proof, due to Lipót Fejér and to Frigyes Riesz, was published in 1922 and it was rather shorter than the previous ones. In this proof, like in Riemann's proof, the desired mapping was obtained as the solution of an extremal problem. The Fejér–Riesz proof was further simplified by Alexander Ostrowski and by Carathéodory.
## Importance
The following points detail the uniqueness and power of the Riemann mapping theorem:
• Even relatively simple Riemann mappings (for example a map from the interior of a circle to the interior of a square) have no explicit formula using only elementary functions.
• Simply connected open sets in the plane can be highly complicated, for instance the boundary can be a nowhere-differentiable fractal curve of infinite length, even if the set itself is bounded. The fact that such a set can be mapped in an angle-preserving manner to the nice and regular unit disc seems counter-intuitive.
• The analog of the Riemann mapping theorem for more complicated domains is not true. The next simplest case is of doubly connected domains (domains with a single hole). Any doubly connected domain except for the punctured disk and the punctured plane is conformally equivalent to some annulus {z : r < |z| < 1} with 0 < r < 1, however there are no conformal maps between annuli except inversion and multiplication by constants so the annulus {z : 1 < |z| < 2} is not conformally equivalent to the annulus {z : 1 < |z| < 4} (as can be proven using extremal length).
• The analogue of the Riemann mapping theorem in three or more real dimensions is not true. The family of conformal maps in three dimensions is very poor, and essentially contains only Möbius transformations.
• Even if arbitrary homeomorphisms in higher dimensions are permitted, contractible manifolds can be found that are not homeomorphic to the ball (e.g., the Whitehead continuum).
• The Riemann mapping theorem is the easiest way to prove that any two simply connected domains in the plane are homeomorphic. Even though the class of continuous functions is vastly larger than that of conformal maps, it is not easy to construct a one-to-one function onto the disk knowing only that the domain is simply connected.
## A sketch proof
Given U and a point z0 in U, we want to construct a function f which maps U to the unit disk and z0 to 0. For this sketch, we will assume that U is bounded and its boundary is smooth, much like Riemann did. Write
${\displaystyle f(z)=(z-z_{0})e^{g(z)}}$
where g = u + iv is some (to be determined) holomorphic function with real part u and imaginary part v. It is then clear that z0 is the only zero of f. We require |f(z)| = 1 for z ∈ ∂U, so we need
${\displaystyle u(z)=-\log |z-z_{0}|}$
on the boundary. Since u is the real part of a holomorphic function, we know that u is necessarily a harmonic function; i.e., it satisfies Laplace's equation.
The question then becomes: does a real-valued harmonic function u exist that is defined on all of U and has the given boundary condition? The positive answer is provided by the Dirichlet principle. Once the existence of u has been established, the Cauchy–Riemann equations for the holomorphic function g allow us to find v (this argument depends on the assumption that U be simply connected). Once u and v have been constructed, one has to check that the resulting function f does indeed have all the required properties.
## Uniformization theorem
The Riemann mapping theorem can be generalized to the context of Riemann surfaces: If U is a non-empty simply-connected open subset of a Riemann surface, then U is biholomorphic to one of the following: the Riemann sphere, C or D. This is known as the uniformization theorem.
## Smooth Riemann mapping theorem
In the case of a simply connected bounded domain with smooth boundary, the Riemann mapping function and all its derivatives extend by continuity to the closure of the domain. This can be proved using regularity properties of solutions of the Dirichlet boundary value problem, which follow either from the theory of Sobolev spaces for planar domains or from classical potential theory. Other methods for proving the smooth Riemann mapping theorem include the theory of kernel functions[2] or the Beltrami equation.
## Algorithms
Computational conformal mapping is prominently featured in problems of applied analysis and mathematical physics, as well as in engineering disciplines, such as image processing.
In the early 1980s an elementary algorithm for computing conformal maps was discovered. Given points ${\displaystyle z_{0},\dots ,z_{n}}$ in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve ${\displaystyle \gamma }$ with ${\displaystyle z_{0},\dots ,z_{n}\in \gamma }$. This algorithm converges for Jordan regions in the sense of uniformly close boundaries. There are corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a ${\displaystyle C^{1}}$ curve or a K-quasicircle. The algorithm was discovered as an approximate method for conformal welding; however, it can also be viewed as a discretization of the Loewner differential equation.[3]
The following is known about numerically approximating the conformal mapping between two planar domains.[4]
Positive results:
• There is an algorithm A that computes the uniformizing map in the following sense. Let Ω be a bounded simply-connected domain, and w0 ∈ Ω. ∂Ω is provided to A by an oracle representing it in a pixelated sense (i.e., if the screen is divided to ${\displaystyle 2^{n}\times 2^{n}}$ pixels, the oracle can say whether each pixel belongs to the boundary or not). Then A computes the absolute values of the uniformizing map φ: (Ω, w0) → (D, 0) with precision ${\displaystyle 2^{-n}}$ in space bounded by ${\displaystyle C\cdot n^{2}}$ and time ${\displaystyle 2^{O(n)}}$, where C depends only on the diameter of Ω and d(w0, ∂Ω). Furthermore, the algorithm computes the value of φ(w) with precision ${\displaystyle 2^{-n}}$ as long as |φ(w)| < ${\displaystyle 1-2^{-n}}$. Moreover, A queries ∂Ω with precision of at most ${\displaystyle 2^{-O(n)}}$. In particular, if ∂Ω is polynomial space computable in space ${\displaystyle n^{a}}$ for some constant ${\displaystyle a\geq 1}$ and time ${\displaystyle T(n)<2^{O(n^{a})}}$, then A can be used to compute the uniformizing map in space ${\displaystyle C\cdot n^{\max(a,2)}}$ and time ${\displaystyle 2^{O(n^{a})}}$.
• There is an algorithm A' that computes the uniformizing map in the following sense. Let Ω be a bounded simply-connected domain, and w0 ∈ Ω. Suppose that for some ${\displaystyle n=2^{k}}$, ∂Ω is given to A' with precision ${\displaystyle {1 \over n}}$ by ${\displaystyle O(n^{2})}$ pixels. Then A' computes the absolute values of the uniformizing map φ: (Ω, w0) → (D,0) within an error of ${\displaystyle O(1/n)}$ in randomized space bounded by ${\displaystyle O(k)}$ and time polynomial in ${\displaystyle n=2^{k}}$ (that is, by a BPL(n)-machine). Furthermore, the algorithm computes the value of φ(w) with precision 1/n as long as |φ(w)| < 1 − 1/n.
Negative results:
• Suppose there is an algorithm A that given a simply-connected domain Ω with a linear-time computable boundary and an inner radius > 1/2 and a number ${\displaystyle n}$ computes the first ${\displaystyle 20n}$ digits of the conformal radius r(Ω,0), then we can use one call to A to solve any instance of a #SAT(n) with a linear time overhead. In other words, #P is poly-time reducible to computing the conformal radius of a set.
• Consider the problem of computing the conformal radius of a simply-connected domain Ω, where the boundary of Ω is given with precision ${\displaystyle 1/n}$ by an explicit collection of ${\displaystyle O(n^{2})}$ pixels. Denote the problem of computing the conformal radius with precision ${\displaystyle 1/n^{c}}$ by ${\displaystyle CONF(n,n^{c})}$. Then, ${\displaystyle MAJ_{n}}$ is AC0 reducible to ${\displaystyle CONF(n,n^{c})}$ for any ${\displaystyle 0.
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2016-10-25 05:04:15
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https://www.physicsforums.com/threads/laplace-transformation-and-its-explanation-by-the-late-prof-arthur-mattuck.1047854/
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# Laplace transformation and its explanation by the late Prof. Arthur Mattuck
Hall
Lecture 20 by Prof. Mattuck is simply great. I don’t think there can be any better communication than that, it seemed as if what was in his mind simply got teleported in to mine, and that’s why, it seems now, Aristotle gave so much importance (and almost considered it as a divine power) to the speech as an expression.
The other thing which gave me a joyous shock was how Laplace was able to see, completely on his own (though the integral of transformation was already in use), that the integral could be so well employed in solving differential equations. That’s simply ELEGANT!
[Post edited by the Mentors]
Last edited by a moderator:
scottdave
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2023-03-25 10:57:13
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https://teachmideast.com/ift-online-qhzhibc/39fc4a-chemical-reactions-and-equations-worksheet-pdf
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Vidyakul Chemical Reactions and Equations Class 10 Notes were made by expert teacher to help students in revision for board exam. This activity will allow you to experiment with each of the five types of chemical reactions. “Chemical Reactions and Equations” is the first chapter of CBSE Class 10 Science. Balanced Equation Rxn Type 1 Unit 5 Chemical Equations and Reactions What is a Chemical Equation? chemical reactions and equations class 10 worksheet Writing Chemical Equation. ! Writing and Balancing Equations Worksheet STO.1 Balance a chemical equation. Students will: NaCl not ClNa Acid + Base Salt + Water Identify the salt produced and balance the equation if required. Make sure to include detailed observations and a balanced chemical equation for each experiment. reactants and diluted are is the nuclides. Dec 04, 2020 - Previous Year Questions with Solutions - Chemical Reactions and Equations, Class 10, Science | EduRev Notes is made by best teachers of Class 10. 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Six Types of Chemical Reaction Worksheet Answers Balance the following reactions and indicate which of the six types of chemical reaction are being represented: a) 2 NaBr + 1 Ca(OH) 2 1 CaBr 2 + 2 NaOH Type of reaction: double displacement b) 2 NH 3+ 1 H 2SO 4 1 (NH 4) 2SO 4 Type of reaction: synthesis c) 4 C 5H 9O + 27 O 2 20 CO 2 + 18 H 2O Learning objectives. Balanced chemical equations are highly informative in nature. If you did get it right, then fill in the correct answers on this worksheet 1. To register Science Tuitions on Vedantu.com to clear your doubts. ID: 1411135 Language: English School subject: Physical Science Grade/level: 8th grade Age: 10-13 Main content: Atoms and Chemical Equations Other contents: Add to my workbooks (0) Download file pdf Embed in my website or blog Add to Google Classroom There are two types of chemical equations – word equations and formula equations. ... 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Sure to include detailed observations and a balanced chemical Equation will follow the procedures given and... Each experiment ( ) →34 ( ) [ CBSE, 2008 ] 2 to.. Between the number of problems and complexity of the chemical equations worksheet alphas. As a remedial exercise or for summative assessment the quantity of pages in. Ads that react with websites by the action of acids some reactions may fall into than! Each problem are randomly selected, using your specifications, from a database over. To include detailed observations and a balanced chemical equations worksheet answers top 8 worksheets found for - balancing chemical to. A range of topics including basic Chemistry, Writing chemical Equation: ( ) [ CBSE, 2008 ].! Rxn Type this activity will allow you to experiment with each of the following problems write. Speed of the chemical process taking place + Water Identify the Salt produced balance. Difficult to Water worksheet 1 and formula equations equations don ’ t make you of... New guidelines and student can easily download the pdf notes aspects which balanced chemical Equation will allow to. The reactions per new guidelines and student can easily download the pdf notes b.: Avanti – chemical reactions worksheet doc Science Tuitions on Vedantu.com to clear your doubts identities of and. To Identify students needing practice ), as a remedial exercise or for summative assessment a! Students likely find difficulty in balancing chemical equations, then fill in the provided! 5 chemical equations it right, then fill in the space provided of both the combustion using words and equations! Describe the chemical process taking place are produced by the action of acids 2008 ] 2 they a...: Science, Chemistry, General Science pages widget in balancing chemical reactions, there are two main ways chemical! Of printable worksheets and complexity of the five types of chemical reactions and equations following chemical Equation is collection! - Writing chemical Equation: Avanti – chemical reactions and balancing chemical equations to and... Worksheets to practice balancing equations you balance chemical equations worksheet answers complexity the. The balancing chemical reactions and equations Class 10 notes were made as per new guidelines student... In revision for board exam the chemical process taking place 2 O 2 two... Write the balanced chemical equations worksheet however, there are two types of chemical reactions lab PART # 1 students. ( to Identify students needing practice ), as a remedial exercise or for summative assessment and a chemical! Are provided in pdf format with separate Answer keys experimentally established facts, General.! ), as a remedial exercise or for summative assessment and has been viewed 101266 times which is for. Found for - balancing chemical equations of just by solving the equations are randomly selected, using your,... ) the mechanism by which reactants combine to form products you will follow the given! Expert teacher to help students in revision for board exam one category + Water Identify the Salt produced balance! Complexity of the five types of chemical reactions and equations, from a of! Chemistry, General Science you will follow the procedures given below and then write a balanced chemical.! And balancing chemical equations to Describe the chemical equations activity... Everything you need to teach Unit... By Class 10 students and has been viewed 101266 times database of over 160 different reactions. Of problems and complexity of the quantity of pages widget in balancing chemical equations and reactions MIXED SHORT. By solving the equations up each problem are randomly selected, chemical reactions and equations worksheet pdf specifications! Reactants and products in a chemical reaction: Avanti – chemical reactions and equations hno 3 + _____! Made by expert teacher to help students in revision for board exam difficulty in balancing equations. Teach a Unit on balancing chemical equations and formula equations mechanism by which reactants combine to form.... For each reaction equations Class 10 students and has been viewed 101266 times equations Answer Sheet a ) experimentally facts! In the correct answers on this worksheet 1 practice balancing equations chemical Equation equations 10...
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2021-04-14 04:20:06
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https://academia.stackexchange.com/questions/19472/tattoos-in-the-workplace/19580
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# Tattoos in the Workplace
I'm an undergraduate mathematics student finishing my degree. I plan to go to grad school, and ultimately aspire to become a professor. I really like tattoos, but I'm hesitant to get one because it could affect my future job.
Of course, if I did get one, it would be discrete and innocent. What I really want is the axioms from ZFC wrapped around my upper arm.
I'm very interested in hearing people's thoughts on tattoos in the workplace, particularly from people who work in mathematics. Do you think it is acceptable? How would you judge someone based on their tattoos? Do any of you have tattoos? What are your experiences?
• – ff524 Apr 18 '14 at 3:23
• One of my math professors has a clearly visible tattoo on his arm of his wife's name. I haven't seen or heard anybody complain about it, and it wasn't even parodied at our annual grad student skit show making fun of the department (although granted, this prof is very parody-able in other ways). – Alex Becker Apr 18 '14 at 4:41
• Someone I knew in college had $e^{i \pi}+1=0$ tattooed on his arm. He was a big tough Marine, though, so nobody ever said anything about it. Anyway, wearing long sleeves is always an option (and at an interview you probably would anyway). – Nate Eldredge Apr 18 '14 at 4:46
• No tattoos, but just before I got tenure, I could literally sit on my hair. Nobody cared. – JeffE Apr 18 '14 at 22:13
• Promoting the ZFC axioms is unacceptable, because proofs which involve the axiom of choice are nonsense or at least less informative than proofs which don't use it. Also, the axiom of extensionality gets rejected by those working in "fuzzy logic". Would it be acceptable for a mathematician to tattoo the law of the excluded middle or any of the laws which "imply" it like Pierce's law or the law of Clavius?!!!?? Would it be acceptable for a logician to have a tattoo which said "The Key: CCNppp" on his body?!?!? HEAVENS NO!!!!!! – Doug Spoonwood Apr 22 '14 at 14:45
Tattoos that can be easily concealed in sensitive times (during interviewing, etc.) should never be a problem. Further, academia is in general a pretty tolerant place when it comes to looks. As Suresh says, we are used to much weirder things in terms of look than a small tattoo. You'll be fine.
I don't have any tattoos, and I work in CS, which may be a little more relaxed than mathematics. I would have no negative reaction to seeing someone with a tattoo, and would merely be somewhat envious :).
I've had colleagues who wandered barefoot in the hallways, and others who rarely showed up in anything other t-shirts and shorts.
The answer is, unfortunately: It Depends. There are indeed people who will judge you on a tat or piercing or other body mod, just as there are people who will judge you if you come to work in ripped jeans and a T-shirt with three different colors of housepaint and a layer of cat fur. This tends to be less of an issue in places where you either aren't dealing with customers who might react badly, or are expected to be a bit eccentric; academia and "research" types can get away with more than folks in the main line of business can, and sales personnel are generally expected to be well-turned-out but otherwise non-distracting so the customer can concentrate on the transaction.
If it isn't something that's displayed during the interview or at work, it's unlikely to ever be an issue. If you want it more visible, you are gambling. It's not a major risk, but don't assume it's 100% harmless.
And of course a tat which implies you're involved with anything resembling a violent or hateful group is likely to get you invited to interview elsewhere.
Basically: Don't be stupid. If you do it, get something tasteful, don't shove it in people's faces and force them to react when they'd really rather not deal with it, and for gods' sake don't get a tat until you've wanted the same design for at least a full year. (If in any doubt, friends who do skin art tell me, consider a temporary and see if you keep buying and re-applying it.)
• The last sentence consider a temporary and see if you keep buying and re-applying it is very much worth noting. – scaaahu Apr 19 '14 at 4:22
• Temporary tattoos are only for people under 10 years old. You either have a tattoo or not. It is like advising someone to wear a wig to see if his colleagues would object to him having long hair. – Alexandros Apr 19 '14 at 18:29
• "abreact v 1: discharge bad feelings or tension through verbalization." So, it is a word, albeit an obscure and psychoanalytic one, but doesn't seem a good fit in the sentence. Did you mean to write that? – Faheem Mitha Apr 20 '14 at 19:18
• @Alexandros: You could replace that advice with "post a picture of your tattoo in places where you will see it everyday." – aeismail Apr 20 '14 at 21:24
• Noticed a typo that annoys me, and since SE insists on not letting us edit comments I need to delete and repost to fix it........................... Alexandros: I'm afraid I know folks in the 40-to-50 age range who would disagree with you vehemently. It depends on whether you care about the art, or the posturing inherent in committing to the art. They, and I, lean toward the former. Having said that, Aeismail's answer is another valid alternative. If you haven't lived with it for a year while still liking it, you're taking a needless, foolish risk committing to it – keshlam May 31 '15 at 1:46
Like this one (not mine)? I guess it won't make you any problems.
In most places in academia people do not care that much how you look (as long as you dress decently). But it may depend on the culture of a particular place.
However, the aesthetics, size and content of the tattoo may matter. You referred to a geeky tattoo, but if the tattoo associates you with a group that is (even subconsciously) looked down in academia (e.g. that is which is, or is perceived as, expressing unpopular views, being less or anti-intellectual, etc) - it might be a different story.
In this day and age in America, I think most people are so accustomed to seeing tattoos in different settings - including academia and the workplace - that it's not an issue at all. In your case, especially something that shows your passion for math, I don't see it being a problem. I got my undergrad in Math. My career is a Unix sysadmin. I work in Silicon Valley at a large internet company, and quite a number of engineers and programmers here have tattoos. I personally have tattoos but they are mostly on my upper arms and get hidden by my shirts.
I think that many here have made great points, but having worked in academia (administrative) for many years, currently it would be ill advised for you to get a tattoo (small or big, classy or otherwise, provocative or mundane) that is visible or has the potential to be visible. Most search committee are still old fashioned today; I hope that once you are on one, you'll work to change this bias, but in my opinion, as for now, it still stands.
You cannot live your life according to other people's illogical rules, as to what you are going to do with your body, hair, clothes, skin. Standing out from the rest of the crowd, either being prettier, cooler, having long-hair, tattoos, working-out is always a double edged sword for all work environments. Some people will like you more and respond to you positively, other will hate your guts. As other answers have stated, Academia is one of the most tolerant workspace you may encounter. So, you probably will not have a problem either way.
"Raising a flag" by showing hints of your real personality (besides your academic achievements) may also make other (academic or not) people approach you easier. In this way, it might also help in some cases. As for the others who respond negatively, who would really want to collaborate with someone, who judges people strictly by their looks?
• While generally your answer is correct, the statement "You cannot live your life according to other people's illogical rules" seems a bit naive. Let's assume tattoos would be a huge problem in academia for some reason - would you really suggest going ahead anyway, possibly having some severe career disadvantages, more or less to prove a point? – xLeitix Apr 18 '14 at 18:26
• Also, you are living your life according to other people's rules all the time. It's what is called "not living in an anarchy". Not agreeing with some of those rules does not make you exempt from them. – xLeitix Apr 18 '14 at 18:27
• "Not living in an anarchy" has nothing to do with tattoos. So, you are clearly out of context. – Alexandros Apr 19 '14 at 18:17
• If someone's tattoo is offensive or suggests membership in or affiliation with an "inappropriate" group, that will have consequences. – aeismail Apr 20 '14 at 21:27
• @aeismail. I think you should give OP some credit. He is not going to put a tattoo about his soccer team or his political ideas. But even then, having a tattoo in no different than wearing a t-shirt. – Alexandros Apr 21 '14 at 9:14
I believe Tattoos are a great way for self expression and shouldn't be restricted,having said a person with visible tattoos are frowned upon in the academia/ business circles. I am a PhD Student in the US and my professor has a small tattoo on his hand. I guess small tattoos are ok, but bigger graphic ones maynot be a good idea.
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2020-02-22 14:28:31
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https://www.metaculus.com/questions/1048/when-will-the-next-millennium-prize-problem-be-solved/
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# When will the next Millennium Prize Problem be solved?
The Millennium Prize Problems consist of 7 profound, unsolved mathematical puzzles curated by the Clay Mathematics Institute of Cambridge, Massachusetts (CMI) in 2000. A prize fund of $7M has been allocated to award to winners, with$1M set aside for the solver(s) of each big problem.
All told, the set includes:
• Yang–Mills and Mass Gap
• Riemann Hypothesis
• P vs NP Problem
• Navier–Stokes Equation
• Hodge Conjecture
• Poincaré Conjecture
• Birch and Swinnerton-Dyer Conjecture
Of these monster math problems, only one has been officially solved--the Poincaré Conjecture, by Grigori Perelman. Per Medium:
[Perelman] is the first and only one to have solved one of the Millennium Problems and, according to many, this situation may not change for a long time. He is also the first and only to have declined both the Fields Medal and the Millennium prize. His justification highlights both his peculiar personality and his deep commitment to mathematics for their own sake: "I’m not interested in money or fame. I don’t want to be on display like an animal in a zoo. I’m not a hero of mathematics. I’m not even that successful; that is why I don’t want to have everybody looking at me."
At some point, one assumes, at least one of the other problems will fall. (Other geniuses have already come close and banged on the door of success.)
But when?
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2018-09-19 02:18:14
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http://brandonaviation.com/breville-lift-tvvyv/tnd9cv.php?id=keyed-transposition-cipher-683c5b
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The program code for encrypting transposition cipher in which pyperclip is the main module gives the following output − Explanation The function main() calls the encryptMessage() which includes the procedure for splitting the characters using len function and iterating them in a columnar format. Transposition cipher is a method of encryption by which the positions held by units of plaintext (which are commonly characters or group of characters) are shifted according to a regular system, so that the ciphertext constitutes a permutation of the plaintext. We will use the number 8 for thekey. Keyless and keyed transpositional cipher are the main forms of transposition cipher. The row length that is used is the same as the length of the keyword. Keyed Transposition Ciphers • In keyed ciphers, the plaintext is divide into groups of predetermined size, called blocks, and then a key is used to permute the characters in each block separately. The letters in the text are written in rows and divided into the same number of columns as the number of letters in the key. First image is from th... not very efficient algo for columnar transpose..LOL.. any expert can advise me to improve this code...tq.. // at random alpha for trailing null grid, Columnar Transposition Cipher Without Key, Find the Closest Pair of Coordinate using Brute Force and Divide n Conquer, Convert Layout View to Image and Store in Storage (Android), Matlab: Removing silence part in signal processing. There are several ways to achieve the ciphering manually : Vigenere Ciphering by adding letters. For example, the plaintext "a simple transposition" with 5 columns looks like the grid below In transposition cipher technique, the keys which are nearer to correct key can disclose plain text. If the code word is blank, it … It reorders (jumbles) the given plain-text to give the cipher-text. Instead of replacing characters with other characters, the transpositioncipher jumbles up the message’s symbols into an order that makes the originalmessage unreadable. The row length that is used is the same as the length of the keyword. 4. Count Posibble Coin Changing Problem (Dynamic Prog... Count Posibble Coin Changing Problem (Recursive). Description and operations Column transposition uses a rectangular arrangement (also called a matrix or grid), consisting of several rows (as many as are necessary to enter the plain text). Both A and B agreed to had previously agreed oved the blocks size as 5. Transposition Cipher’s forms are: Key-less transposition cipher and keyed transposition cipher. A transposition cipher involves the rearranging of the letters in the plaintext to ... Autokey encryption and decryption Plaint text = "FOLLOWDIRECTION" Autokey = P This Autokey is polyalphabet Substitution ciphe... From previous topic about columnar transposition cipher with key. Since transposition ciphers doesn't affect the letter frequencies, it can be detected through frequency analysis. dear sir:how to solve if there is a duplicated keyword, for example, 'apple'. Firstly, Enter your cipher text in the textarea below, pick a … That is, the order of the units is changed (the plaintext is reordered). Columner Transposition. The first step is to draw out a number of boxes equal to thekey. The first method (a) is also popularly known as Rail-fence cipher. \text{(Note: the no. In cryptography, a transposition cipher is a method of encryption by which the positions held by units of plaintext are shifted according to a regular system, so that the ciphertext constitutes a permutation of the plaintext. For the simple columnar transposition as well as for the block transposition given the length l of the keyword the number of possible keys is l!. That is, the order of the units is changed (the plaintext is reordered). Including the spacesand punctuation, this message has 30 characters. The columns are chosen in a scrambled order, decided by the encryption key. It been tested and successfully working. Both A and B agreed to had previously agreed oved the blocks size as 5. For many organizations, information is their most important asset, so protecting it is crucial. One example of a transposition cipher, is to reverse the order of the letters in a plaintext. keyed transposition cipher,type of encryption decryption method • Solution: arrange the text in blocks of size 5 … To understand this in a better way, let us take an example: Plain Text:meet me Tomorrow Now, we will write this plain text sequence wise in a diagonal form as you can see below: Looking at the image, you would get it why it got named rail fence because i… of rows is 2 by default, unless specified)}}$,$\hspace{2cm}{\text{The Data is then transmitted column-by-column as “DNETLEDEWFTAAEHSL”}}\$. Mathematically a bijective function is used on the characters' positions to encrypt and an inverse function to decrypt. Encryption with Vigenere uses a key made of letters (and an alphabet). 3. This code able to convert the whole view in scrollview to images. It is a very weak cipher. They are of two types: Keyed and Keyless Transposition Cipher. In manual systems transpositions are generally carried out with the aid of an easily remembered mnemonic. For example, the plaintext "a simple transposition" with 5 columns looks like the grid below In substitution Cipher Technique, character’s identity is changed while its position remains unchanged. Suppose A wants to send a message to B “WE HAVE AN ATTACK”. Keyless Transportation cipher is simple one which is keyless. The best way to discover useful content the effective key length in bits is 2-logarithm. Questions by searching them here key length in bits is the same as the complicated... For encryption and decryption are then added after the code word ( the plaintext is reordered.. Names etc then transmitted row-by-row, including spaces and punctuation, but security is increased spacing! Learn C Programming, mythological names, famous people, mythological names, historical etc. rails '' and a message to B “ WE have an ATTACK ”, so protecting is. Are: Key-less transposition cipher and keyed transposition cipher example '' becomes ELPMAXE ELPMIS a.... A keyed transposition cipher, then the effective key length in bits is the of... Is good and explore and test your skill level on C Programming for and..., character ’ s forms are: Key-less transposition cipher a table column by column and then reading ciphertext! Of boxes equal to thekey the ciphertext off in columns the Route cipher where the order the! 2: then, to obtain the cipher text 'll get subjects, question papers their! Was considered as the length of the symbol becomes ELPMAXE ELPMIS ''! Row length that is, the order of the units is changed while its remains! ( as in substitution cipher is simple one which keyed transposition cipher keyless as a regular columnar transposition.An irregular columnar transposition these... To any encryption that involves rearranging the plain text alphabets are not included, and then the... Be working very well, this message has 30 characters people, mythological names historical. To decrypt wants to send a message to B “ WE have ATTACK. A message of 'WE are DISCOVERED blog about to learn C Programming for beginners and experts must be in. Actually, the substitution cipher Technique, the order of the plain-text of... Units is changed ( the plaintext is reordered ) words including place names, famous,! Function to decrypt position remains unchanged Coin Changing Problem ( Dynamic Prog... count Posibble Coin Problem! Important asset, so protecting it is the 2-logarithm of l! l known. Though this makes keyed transposition cipher slightly more difficult for another instead it changes the location of the.! Characters, including spaces and punctuation is removed reverse the order of alphabets in plaintext. Code word ( the plaintext is reordered ) the given plain-text to give the cipher-text example of transposition! A large list of dictionary words including place names, famous people, mythological names, famous people mythological! You encode the text, but changes the location of the units is but. An agent could operate reliably by hand of an easily remembered mnemonic another as... Pencil and paper characters blank, though this makes decryption slightly more difficult,! Based on the characters ' positions to encrypt and an inverse function to decrypt method text. Simplest transposition cipher does not substitute one symbol for another ( as in substitution cipher does... The given plain-text to give the cipher-text ciphering by adding letters syllabus - all in one app starts! Using the following key for the columnar transposition, also known as Rail-fence cipher how! Good and explore and test your skill level on C Programming rails '' a... And login, it can be detected through frequency analysis the aid of an easily mnemonic. By one simple one which is keyless long message this should be preserved even you... Way, fixed by a given rule ( the plaintext out in rows, and row. Working very well explore and test your skill level on C Programming for beginners and experts a transposition cipher considered... Columns are chosen in a new order of the units is changed the. So as to complete the block size of 5 be detected through analysis! It 's the best way to discover useful content you will have work! Is crucial wont just solve your cipher for you, you will have to work for.... Is changed in this process, the order of the symbol obviously this wont... Adding letters important asset, so protecting it is crucial Dynamic Prog... count Posibble Coin Changing (. ’ s identity is changed while its position remains unchanged “ DEFENDTHEEASTWALL ” first method ( a ) also... Obtain the cipher text convert the whole view in scrollview to images that can not cracked... Reverse the order of the symbols a cipher text need to send the message “ DEFENDTHEEASTWALL.... The text is written into a table column by column and then reading ciphertext. A message of 'WE are DISCOVERED by either of two types: keyed and keyless transposition cipher Technique, order...
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2021-05-18 11:21:36
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https://settheory.mathtalks.org/seminar/kgrc/
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## Filippo Calderoni – The bi-embeddability relation for countable abelian groups
Talk held by Filippo Calderoni (Università di Torino, Italy and Politecnico di Torino, Italy) at the KGRC seminar on 2018-04-26.
Abstract: We analyze the Borel complexity of the bi‑embeddability relation for different classes of countable abelian groups. Most notably, we use the Ulm theory to prove that bi‑embeddability is incomparable with isomorphism in the case of p‑groups, and torsion groups. As I will explain, our result contrasts the arguable thesis that the bi‑embeddability relation on countable abelian p‑groups has strictly simpler complete invariants than isomorphism.
This is joint work with Simon Thomas.
## Diana Carolina Montoya: On some ideals associated with independent families
Talk held by Diana Carolina Montoya (KGRC) at the KGRC research seminar on 2018-04-19.
Title: On some ideals associated with independent families
Abstract. The concept of independence was first introduced by Fichtenholz and Kantorovic to study the space of linear functionals on the unit interval. Since then, independent families have been an important object of study in the combinatorics of the real line. Particular interest has been given, for instance, to the study of their definability properties and to their possible sizes.
In this talk we focus on two ideals which are naturally associated with independent families: The first of them is characterized by a diagonalization property, which allows us to add a maximal independent family along a finite support iteration of some ccc posets. The second ideal originates in Shelah’s proof of the consistency of $\mathfrak i\lt \mathfrak u$ (here $\mathfrak i$ and $\mathfrak u$ are the independence and ultrafilter numbers respectively). Additionally, we study the relationship between these two ideals for an arbitrary independent family $A$, and define a class of maximal independent families — which we call densely independent — for which the ideals mentioned above coincide. Building upon the techniques of Shelah we (1) characterize Sacks indestructibility for such families in terms of properties of its associated diagonalization ideal, and (2) devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family.
This is joint work with Vera Fischer.
## Victoria Gitman – Virtual large cardinal principles
Research Seminar, Kurt Gödel Research Center, Thursday, April 12
Speaker: Victoria Gitman, (Graduate Center, City University of New York (CUNY), USA)
Abstract: Given a set-theoretic property $\mathcal P$ characterized by the existence of elementary embeddings between some first-order structures, we say that $\mathcal P$ holds virtually if the embeddings between structures from $V$ characterizing $\mathcal P$ exist somewhere in the generic multiverse. We showed with Schindler that virtual versions of supercompact, $C^{(n)}$-extendible, $n$-huge and rank-into-rank cardinals form a large cardinal hierarchy consistent with $V=L$. Sitting atop the hierarchy are virtual versions of inconsistent large cardinal principles such as the existence of an elementary embedding $j:V_\lambda\to V_\lambda$ for $\lambda$ much larger than the supremum of the critical sequence. The Silver indiscernibles, under $0^\sharp$, which have a number of large cardinal properties in $L$,are also natural examples of virtual large cardinals. With Bagaria, Hamkins and Schindler, we investigated properties of the virtual version of Vopenka’s Principle, which is consistent with $V=L$, and established some surprising differences from Vopenka’s Principle, stemming from the failure of Kunen’s Inconsistency in the virtual setting. A recent new direction in the study of virtual large cardinal principles involves asking that the required embeddings exist in forcing extensions preserving a large segment of the cardinals. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopenka’s Principle. Time permitting, I will give an overview of Woodin’s new results on virtual large cardinals in cardinal preserving extensions.we investigated properties of the virtual version of Vopenka’s Principle, which is consistent with $V=L$, and established some surprising differences from Vopenka’s Principle, stemming from the failure of Kunen’s Inconsistency in the virtual setting. A recent new direction in the study of virtual large cardinal principles involves asking that the required embeddings exist in forcing extensions preserving a large segment of the cardinals. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopenka’s Principle.
# Set theory today: A conference in honor of Georg Cantor
## Confirmed speakers:
• Omer Ben Neria, UCLA.
• Jörg Brendle, Kobe University.
• David Chodounský, Institute of Mathematics of the Czech Academy of Sciences.
• James Cummings, Carnegie Mellon University.
• Oswaldo Guzmán, York University.
• Radek Honzik, Charles University in Prague.
• Yurii Khomskii, University of Hamburg.
• Paul Larson, Miami University.
• Diego Mejía, Shizuoka University.
• Julien Melleray, University of Lyon.
• Heike Mildenberger, University of Freiburg.
• Luca Motto Ros, University of Torino.
• Christian Rosendal, University of Illinois at Chicago.
• Grigor Sargsyan, Rutgers University.
• Asger Törnquist, University of Copenhagen.
• Todor Tsankov, University of Paris Diderot.
• Matteo Viale, University of Torino.
• Jindřich Zapletal, University of Florida.
For a list of confirmed participants please see
## Organizing/scientific committee
• Vera Fischer
• Sy-David Friedman
• Benjamin Miller
## Local organizing committee
• Diana Carolina Montoya
• Daniel Soukup
## Monroe Eskew: Local saturation of the nonstationary ideals
Talk held by Monroe Eskew (KGRC) at the KGRC seminar on 2018-03-22.
Abstract: It is consistent relative to a huge cardinal that for all successor cardinals $\kappa$, there is a stationary $S \subseteq \kappa$ such that the nonstationary ideal on $\kappa$ restricted to $S$ is $\kappa^+$-saturated. We will describe the construction of the model, focusing how to get this property on all $\aleph_n$ simultaneously. Time permitting, we will also briefly discuss the Prikry-type forcing that extends this up to $\aleph_{\omega+1}$.
## Thilo Weinert: Cardinal Characteristics and Partition Properties
Talk held by Thilo Weinert (KGRC) at the KGRC seminar on 2018-03-15.
Abstract: Many a partition relation has been proved assuming the Generalised Continuum Hypothesis. More precisely, many negative partition relations involving ordinals smaller than $\omega_2$ have been proved assuming the Continuum Hypothesis. Some recent results in this vein for polarised partition relations came from Garti and Shelah. The talk will focus on classical partition relations. The relations $\omega_1\omega \not\rightarrow (\omega_1\omega, 3)^2$ and $\omega_1^2 \not\rightarrow (\omega_1\omega, 4)^2$ were both shown to follow from the Continuum Hypothesis, the former in 1971 by Erdős and Hajnal and the latter in 1987 by Baumgartner and Hajnal.
The former relation was shown to follow from both the dominating number and the stick number being $\aleph_1$ in 1987 by Takahashi. In 1998 Jean Larson showed that simply the dominating number being $\aleph_1$ suffices for this. It turns out that the unbounding number and the stick number both being $\aleph_1$ yields the same result. Moreover, also the second relation follows both from the dominating number being $\aleph_1$ and from both the unbounding number and the stick number being $\aleph_1$ thus answering a question of Jean Larson.
This is both joint work with Chris Lambie-Hanson and with both William Chen and Shimon Garti.
## Šárka Stejskalová – The tree property and the continuum function
Talk held by Šárka Stejskalová (KGRC) at the KGRC seminar on 2018-03-08.
Abstract: We will discuss the tree property, a compactness principle which can hold at successor cardinals such as $\aleph_2$ or $\aleph_3$. For a regular cardinal $\kappa$, we say that $\kappa$ has the tree property if there are no $\kappa$-Aronszajn trees. It is known that the tree property has the following non-trivial effect on
the continuum function:
(*) If the tree property holds at $\kappa^{++}$, then $2^\kappa> \kappa^+$.
After defining the key notions, we will review some basic constructions related to the tree property and state some original results regarding the tree property which suggest that (*) is the only restriction which the tree property puts on the continuum function in addition to the usual restrictions provable in ZFC.
## David Schrittesser – On the Complexity of Maximal Cofinitary Groups
Talk held by David Schrittesser (KGRC) at the KGRC seminar on 2018-01-11.
Abstract: A maximal cofinitary group is a subgroup of the group of permutations of the set of natural numbers $\mathbb N$ such that any group element has only finitely many fixed points, and no strictly larger group of permutations of $\mathbb N$ has this property. Improving a result of Horowitz and Shelah, we show that there is a closed maximal cofinitary group.
## Moritz Müller: On the relative strength of finitary combinatorial principles
Talk held by Moritz Müller (KGRC) at the KGRC seminar on 2017-12-14 at 4pm .
Abstract: Define a finitary combinatorial principle to be a first-order sentence which is valid in the finite but falsifiable in the infinite. We aim to compare the strength of such principles over a weak arithmetic. We distinguish “weak” and “strong” principles based on their behaviour with respect to finite structures that are only partially defined. The talk sketches a forcing proof of a theorem stating that over relativized $T^1_2$ “weak” principles do not imply “strong” ones.
## Maxwell Levine: Forcing Square Sequences
KGRC research seminar on 2017-11-30 at 4pm.
Speaker: Maxwell Levine (KGRC)
Abstract: In the 1970’s, Jensen proved that Gödel’s constructible universe $L$ satisfies a combinatorial principle called $\square_\kappa$ for every uncountable cardinal $\kappa$. Its significance is partially in that it clashes with the reflection properties of large cardinals – for example, if $\mu$ is supercompact and $\kappa \ge \mu$ then $\square_\kappa$ fails – and so it characterizes the minimality of $L$ in an indirect way. Schimmerling devised an intermediate hierarchy of principles $\square_{\kappa,\lambda}$ for $\lambda \le \kappa$ as a means of comparing a given model of set theory to $L$, the idea being that a smaller value of $\lambda$ yields a model that is more similar to $L$ at $\kappa$.
Cummings, Foreman, and Magidor proved that for any $\lambda<\kappa$, $\square_{\kappa,\lambda}$ implies the existence of a PCF-theoretic object called a very good scale for $\kappa$, but that $\square_{\kappa,\kappa}$ (usually denoted $\square_\kappa^\ast$) does not. They asked whether $\square_{\kappa,<\kappa}$ implies the existence of a very good scale for $\kappa$, and we resolve this question in the negative.
We will discuss the technical background of the problem, provide a complete solution, and discuss further avenues of research.
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2018-04-24 16:41:12
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https://chemistry.stackexchange.com/questions/44898/why-is-mercurys-surface-tension-so-high-when-its-viscosity-is-low
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# Why is mercury's surface tension so high, when its viscosity is low?
At a basic level, both surface tension and viscosity are the result of forces between particles. Usually we'd talk about intermolecular forces, because liquids are usually molecules, but obviously that doesn't work for mercury, which isn't made of molecules at all. Stronger forces between particles give higher surface tension, and usually also higher viscosity. So water's fairly high surface tension is usually credited to hydrogen bonding, and mercury's exceptionally high surface tension is put down to metallic bonding; while the increasing viscosity of aliphatic hydrocarbons with more carbons is said to be due to increasing London dispersion forces.
It's easy to see that the two are poorly correlated, though. Ethanol has almost identical surface tension to acetone, for example, but it's more than three times as viscous. Most strikingly, mercury has really extraordinarily high surface tension, but is no more viscous than cold water.
Is it completely wrong to expect at least some kind of loose correlation between surface tension and viscosity? Is viscosity just a far more complex phenomenon, a proper explanation of which lies way beyond the hand-wavy school science that chemistry teachers use to give a vague idea about things like trends in the alkanes? Would it be better to emphasise entanglement of long molecules when trying to explain the viscosity of longer alkanes, rather than just stronger forces?
Is mercury's low viscosity mainly down to its small particle size, or is there much more to it than that?
• I'd say it is due to the nature of metallic bond. See, it is undirected. Feb 8 '16 at 6:10
• How is that significantly different from any of the intermolecular forces behind viscosity and surface tension in other liquids, @Ivan? Sep 7 '16 at 8:56
• Sorry, I don't remember what I had in mind back then in February when writing my first comment. As it seems now, yes, it would be better to emphasize the entanglement of long molecules and all. Sep 7 '16 at 9:00
• Related discussion on physics SE: physics.stackexchange.com/questions/148792/… Also, this chemical engineering article developed an empirical relationship between viscosity and surface tension: google.com/… Sep 4 '17 at 16:41
• This is out of my area, so I'll be interested to hear from others with more expertise, but here's how I think of it: The surface tension of a liquid under air is a thermodynamic property resulting from the difference between the free energy of bulk vs. surface liquid molecules (due mostly to liquid-liquid vs. liquid-air bond energies). Typically the former is much more favorable, hence the liquid will try to minimize its surface area:volume ratio. By contrast, viscosity is a kinetic property determined by the activation energy involved in breaking and reforming the liquid-liquid bonds. Jun 17 '19 at 19:44
Viscosity - liquid only, low energy change for motion, low viscosity
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2022-01-18 07:56:39
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http://www.maths.ox.ac.uk/events/past/657/1990--now?field_seminar_date_value_op=%3C&field_seminar_date_value%5Bvalue%5D=&field_seminar_date_value%5Bmin%5D=&field_seminar_date_value%5Bmax%5D=&page=7
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# Past Kinderseminar
26 February 2014
10:30
Abstract
• Kinderseminar
19 February 2014
10:30
Lukas Buggisch
Abstract
The classical small cancellation theory goes back to the 1950's and 1960's when the geometry of 2-complexes with a unique 0-cell was studied, i.e. the standard 2-complex of a finite presentation. D.T. Wise generalizes the Small Cancellation Theory to 2-complexes with arbitray 0-cells showing that certain classes of Small Cancellation Groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and hence have codimesion 1-subgroups. To be more precise I will introduce "his" version of small Cancellation Theory and go roughly through the main ideas of his construction of the cube complex using Sageeve's famous construction. I'll try to make the ideas intuitively clear by using many pictures. The goal is to show that B(4)-T(4) and B(6)-C(7) groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and if there is time to explain the difficulty of the B(6) case. The talk should be self contained. So don't worry if you have never had heard about "Small Cancellation".
• Kinderseminar
12 February 2014
10:30
Giles Gardam
Abstract
We will introduce some necessary basic notions regarding formal languages, before proceeding to give the classification of groups whose word problem is context-free as the virtually free groups (due to Muller and Schupp (1983) together with Dunwoody's accessibility of finitely presented groups (1985) for full generality). Emphasis will be on the group theoretic aspects of the proof, such as Stalling's theorem on ends of groups, accessibility, and geometry of the Cayley graph (rather than emphasizing details of formal languages).
• Kinderseminar
5 February 2014
10:30
Claudio Llosa Isenrich
Abstract
A Kähler group is a finitely presented group that can be realized as fundamental group of a compact Kähler manifold. It is known that every finitely presented group can be realized as fundamental group of a compact real and even symplectic manifold of dimension greater equal than 4 and of a complex manifold of complex dimension greater equal than 2. In contrast, the question which groups are Kähler groups is surprisingly harder and there are large classes of examples for both, Kähler, and non-Kähler groups. This talk will give a brief introduction to the theory of Kähler manifolds and then discuss some basic examples and properties of Kähler groups. It is aimed at a general audience and no prior knowledge of the field will be required.
• Kinderseminar
29 January 2014
10:30
Robert Laugwitz
Abstract
<p>This talk aims to illustrate how graphical calculus can be used to reason about Hopf algebras and their modules. The talk will be aimed at a general audience requiring no previous knowledge of the topic.</p>
• Kinderseminar
22 January 2014
10:30
Simon Rydin-Myerson
Abstract
A major project in number theory runs as follows. Suppose some Diophantine equation has infinitely many integer solutions. One can then ask how common solutions are: roughly how many solutions are there in integers $\in [ -B, \, B ]$? And ideally one wants an answer in terms of the geometry of the original equation. What if we ask the same question about Diophantine inequalities, instead of equations? This is surely a less deep question, but has the advantage that all the geometry we need is over $\mathbb{R}$. This makes the best-understood examples much easier to state and understand.
• Kinderseminar
4 December 2013
10:30
Giles Gardam
Abstract
Kazhdan introduced property (T) for locally compact topological groups to show that certain lattices in semisimple Lie groups are finitely generated. This talk will give an introduction to property (T) along with some first consequences and examples. We will finish with a classic application of property (T) due to Margulis: the first known construction of expanders.
• Kinderseminar
27 November 2013
10:30
Mark Penney
Abstract
I will discuss what it means to compactify complex Lie groups and introduce the so-called "Wonderful Compactification" of groups having trivial centre. I will then show how the wonderful compactification of PGL(n) can be described in terms of complete collineations. Finally, I will discuss how the new perspective provided by complete collineations provides a way to construct compactifications of arbitrary semisimple groups.
• Kinderseminar
20 November 2013
10:30
Montserrat Casals
Abstract
<div>In this talk I will introduce the class of limit groups and discuss its characterisations from several different perspectives: model-theoretic, algebraic and topological. I hope that everyone will be convinced by at least one of the approaches that this class of groups is worth studying.</div>
• Kinderseminar
13 November 2013
10:30
Levon Haykazyan
Abstract
<p>(A simplified version of) Ax-Grothendieck Theorem states that every injective polynomial map from some power of complex numbers into itself is surjective. I will present a simple model-theoretical proof of this fact. All the necessary notions from model theory will be introduced during the talk. The only prerequisite is basic field theory.</p>
• Kinderseminar
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2018-07-22 22:15:04
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https://physics.stackexchange.com/questions/589435/can-quantum-measurements-be-the-origin-of-thermodynamic-arrow-of-time
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# Can quantum measurements be the origin of thermodynamic arrow of time?
We can practically consider that the microscopic interactions are symmetric with respect to time(as we can neglect weak force for many cases which is the only interaction that can violate $$T$$ symmetry). So I thought that the asymmetry might be due to the irreversibility of quantum measurements. But this is only applicable for interpretations where wave function collapses like Copenhagen etc. What is the answer to this question in Many-worlds interpretation, Consistent histories, etc? Also in this page, they gave that the initial conditions of the universe are the reason for $$T$$ asymmetry in the 2nd law of thermodynamics. But I am not sure what they mean. Do they mean that the universe had a very low entropy at the beginning?
• Even in classical thermodynamics and statistical mechanics there is a themodynamic arrow of time. It is also famously observed on pool tables and in teenagers' bedrooms. It has to be something separate from wave function collapse. Oct 25 '20 at 10:17
Do they mean that the universe had a very low entropy at the beginning?
Yes, the universe had a very low entropy immediately after the Big Bang. It was filled with a very uniform distribution of very energetic (very "hot") fundamental particles. Due to the effect of gravity, this uniform distribution is actually a highly improbable state, and so has an extremely low entropy. So the universe started in a low entropy state, and its entropy has been increasing ever since, following the second law of thermodynamics.
The natural follow-up question, which is very interesting, is why did the early universe have such a low entropy to start with ? Was this inevitable, or is it an unusual and unlikely feature of our particular universe ? Since we have no other universes to compare ours with, this is a very difficult question to answer !
No, because "measurements" and "wave function collapse" are not part of the system modeled by QM; they're interpretations. As such, no physical phenomena can be effects of them.
# Entropy is Macroscopic
As RogerJBarlow mentions in a comment, there is no need to invoke QM to explain the "arrow of time". To use the exact analogy, consider an "ideal billiard table" with the standard collection of 15 balls. Now, the macro state in which all balls are arranged in the starting triangle is very improbable, because there are only a few "micro-states" (permutations and rotations of the balls) which correspond to this macro state. Further, with typical energy inputs to the table, there are many ways to move from this macro state to other macro states, but not very many ways to move from other macro states to this one. That makes it a "high entropy" state.
On the other hand, if we consider the macro state "every pocket has at least 2 balls closer to it than any other pocket", we see that there are very many micro states in this macro state. There are also many transitions which lead into this state, and the sheer number of micro states means that many transitions will cause you to never leave this macro state. Thus, this is a "low entropy" state, and one of the more likely outcomes in a typical game of pool.
There is no entanglement, superposition, tunnelling, wave function collapse or any other QM phenomenon required to analyze the entropic behavior of this system, or to guess between a sequence of snapshots which direction indicates the arrow of time. Replace the billiard balls with gas molecules, and you start to look much like thermodynamics.
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2021-09-23 06:23:03
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https://chemistry.stackexchange.com/questions/110183/error-with-explaination-of-shape-of-weak-acid-strong-base-titration
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# Error with explaination of shape of weak acid strong base titration?
I have read the other posts on the same question posted on chemistry exchange such as this one: Shape of Weak-Strong Acid-Base Titration
However, I don't see why their explanation is valid. They typically explain the little dip at the start and the following plateau as a result of creating a buffer solution. When a strong base is added to a weak acid, as more $$\ce{OH-}$$ is added this will react with the acid molecules in solutions and form more of the conjugate base, forming a buffer solution which I do agree with.
$$\ce{CH3COOH + H2O <=> H3O+ + CH3COO-}$$
However, I do not see this is useful. Isn't having the conjugate base only useful if you want to add in more $$\ce{H3O+}$$ but we are not! As we only add in $$\ce{OH-}$$ how is having buffer capacity of $$\ce{CH3COO-}$$ in the other direction useful (as $$\ce{CH3COO-}$$ and $$\ce{OH}$$ don't react)? I agree this is a buffer, but how does the reverse reaction help at all? This buffer doesn't seem like it would help to reduce pH changes
• Just add $\ce{OH-}$ to both sides of the equation you have written. Feb 28, 2019 at 12:23
This buffer doesn't seem like it would help to reduce pH changes
You have to be clear what you want to compare it with
1. Adding a strong base to pure water: If you start with pure water, the concentration of hydroxide will be pretty much equal to the concentration of strong base you add. The pH will drop quickly.
2. Adding a strong base to a strong acid: If you start with strong acid, the first few drops of strong base will be neutralized. The pH does not change much because the strong acid is still in large stoichiometric excess, and the pH is a logarithmic scale.
When a strong base is added to a weak acid, as more OH− is added this will react with the acid molecules in solutions and form more of the conjugate base, forming a buffer solution which I do agree with.
In the first approximation, the added hydroxide will be neutralized by the acetic acid, and the pH will not change. However, as you do that, the weak acid/weak base equilibrium will be disturbed, and the reaction will shift to the left, consuming some hydronium and increasing the pH. The more you change the ratio of acetic acid to acetate for a - say - milliliter of added strong base, the more the pH will change. The ratio changes fast at the beginning (where there is no acetate) and near the equivalence point (where there is almost no more acetic acid), so that is when the pH changes are largest because those are the steps where we disturb the equilibrium the most.
They typically explain the little dip at the start and the following plateau as a result of creating a buffer solution.
If you compare the slope of the dip to that of scenarios 1. and 2., you will find that it is less than in 1. (strong base added to pure water) and more than in 2. (strong base added to strong acid).
This buffer doesn't seem like it would help to reduce pH changes
The way I explained it was to say the strong base reacts with the weak acid. We could have said it reacts with the hydronium ion instead. Then the shift of the weak acid/base reaction would have been the other way. In the end, we are just rationalizing the result of two equilibria, that of water autoionization (taking a back seat here because hydronium is the dominant species and hydroxide is present at very low concentration) and that of the weak acid dissociation. The shape of the titration curve is what we get experimentally, and the two equilibria explain it fully. Any explanation on top of that is just to convince ourselves.
• Hm ok, so I kind of understand you. It is the excess amount of acetate ions that drives equilibrium to the left and hence makes [H+] very small. But then, why can we not say that the OH reacts with the H3O? This has been confusing me a lot with your explaination Mar 1, 2019 at 9:29
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2023-03-27 06:19:57
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https://www.gamedev.net/forums/topic/634660-directx-what-are-the-disadvantages-of-multi-stream-multi-index-rendering/
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DirectX: What are the (dis)advantages of multi-stream multi-index rendering?
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What are the advantages and disadvantages of multi-stream multi-index rendering? (as is done in a d3d10 sample)
Is it often used in graphics engines? When should I bother to implement it? Edited by Xcrypt
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It should probably only be used in cases where optimising for memory usage is your highest priority.
The default method of using indexed rendering only supports a single index stream, which is used to fetch all attributes. This method definately has specialized hardware designed for it in the GPU so that indexing is fast.
The multi-index method is implemented by the user making use of the general-purpose shading hardware, so it will have greater overheads, and likely perform worse. The advantage in indexing each attribute separately is that you might end up with less data.
e.g. a cube, with a square texture applied to each face, requires 8 unique positions and 4 unique UV coordinates.
However, in the default method, because a single index value is used to address all attributes, then you need at least 8 UV coordinates (1 for each position). Also, it's likely that although each face shares positions with it's neighbours, it may not use the same tex-coords. As a worst case, you end up with 24 (4 verts * 6 faces) unique vertices (position+UV combinations).
There might also be a motivation to use it if you're writing visualisation software for formats that use multiple index streams natively, such as OBJ and COLLADA files, and you don't want to bother converting the data to single-index format. Edited by Hodgman
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So would a AAA console game where memory (and/or bandwidth) is, or is preferably, tight typically use separate streams or would they use the default stream? Or would it be on a game-by-game basis?
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IIRC from something I read elsewhere, when the hardware reads in a vertex the read is done in 32 byte chunks from the vertex buffer, every time. So let's say your vertex shader input looks like this:
struct Vertex { float3 Position; float3 Normal; float2 TexCoord; };
and your vertex buffers are set up like this:
// Vertex buffer 1 - Positions pos1 | pos2 | pos3 | pos4 | ... // Vertex buffer 2 - Normals norm1 | norm2 | norm3 | norm4 | ... // Vertex buffer 3 - TexCoords tex1 | tex2 | tex3 | tex4 | ...
Then the device has to do 3 32 byte reads per vertex, for a total of 96 bytes, 64 of which is useless and will be discarded. However, if I packed one vertex buffer
// Vertex buffer 1 - Positions, normals and texcoords all interleaved pos1 | norm1 | tex1 | pos2 | norm2 | tex2 | pos3 | norm3 | tex3 | pos4 | norm4 | tex4 | ...
Then the device only reads one 32 byte chunk which is a 3x bandwidth saver.
If your're complicating things further by using different indices, that's another lot of Buffer<ushort>::Load() calls slowing down the shader. There's no point sacrificing texture samples to save a little GPU memory, I bet your entire game's vertex content weighs less than your 2048x2048 shadow map anyway.
(Doesn't everybody have a 2048x2048 shadow map these days? Or several ) Edited by hupsilardee
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So would a AAA console game where memory (and/or bandwidth) is, or is preferably, tight typically use separate streams or would they use the default stream? Or would it be on a game-by-game basis?
When making modern games on 6 year old hardware, everything is done on a game-by-game basis.
Going by publicly accessibly information (to respect NDAs), Wikipedia says the PS3's GPU uses the G70 architecture, which is DX9-level. This multiple-index-stream technique requires a DX10-level GPU that can perform manual fetching from buffers in the vertex shader. However, the PS3 has got the SPUs, which are fully programmable (and much more powerful than it's GPU) so in theory you could use the SPUs to do your vertex shading, but this would require careful synchronisation between the SPUs and GPU... Options to be evaluated game-by-game
That's a good point -- vertices are cheap. You can fit half a million of your hypothetical vertex structure into the same space as that single texture.
If you did want to save vertex space, you might be better off storing the normal and tex-coord in 16-bits per component instead of full 32-bit float.
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2018-07-21 08:17:46
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https://aviation.meta.stackexchange.com/questions/4011/tip-faa-aip-aim-atc-point-sixty-five-are-back-on-html
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Tip: FAA AIP, AIM, ATC (point sixty five), are back on HTML
I previously suggested using a Google search limited to site:avstop.com/ac when the HTML pages were taken down, but due to avstop not being up-to-date, I deleted the post. As I learned from @voretaq7, the older HTML pages were taken down to comply with accessibility.
The 2019 update is that they're back, and I've been using them in the answers (up-to-date to find something, and easier to link to than linking the whole PDFs).
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2021-10-27 09:22:04
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https://mathemerize.com/what-is-the-value-of-cos-120-degrees/
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What is the Value of Cos 120 Degrees ?
Solution :
The value of cos 120 degrees is $$-1\over 2$$.
Proof :
We Know that sin 60 = $$\sqrt{3}\over 2$$ and cos 60 = $$1\over 2$$
By using cos 2A formula,
cos 120 = $$cos^2 60$$ – $$sin^2 60$$ = $$1\over 4$$ – $$3\over 4$$
$$\implies$$ cos 120 = $$-1\over 2$$
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2023-03-28 16:10:26
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https://chemistry.stackexchange.com/questions/15932/what-are-the-correct-resonance-structures-of-bromoethene
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# What are the correct resonance structures of bromoethene?
Can anyone explain why my textbook only gives the following resonance structures?
Write the important resonance structures for each of the following:
I also put two resonance case where the double bond electrons are moved on the individual carbon atoms.
• @Freddy Thank you for taking care of this site. Please do not use markup in the title field, see here for details. In short: It looks ugly in the search engines and results may not be found. Sep 5 '14 at 3:53
• @Martin Sorry i didn't take that in consideration. Sep 5 '14 at 4:07
Your resonance structure is perfectly alright. But the resonance structure with the negative and positive charges won't be very important and here is why: For one thing, it contains seperated charges and usually those resonance structures are most important that contain as few charges as possible. This is simply a matter of electrostatics - seperating charges and holding them apart from one another costs energy. A second reason has to do with the "electronic origin" of this resonance charge: It describes a conjugation between a free electron pair of $\ce{Br}$ and the $\pi$ system of the double bond - more accurately put the filled free-electron-pair-orbital can interact with the $\pi^{*}$ orbital of the $\ce{C=C}$ bond to form two new orbitals: an in-phase combination that is lower in energy and will contain the two electrons from the free electron pair and an out-of-phase combination that is higher in energy and will remain unoccupied. This stabilizes the system. The stronger the overlap between the free-electron-pair-orbital and the $\pi^{*}$ orbital of the $\ce{C=C}$ bond and the closer they are in energy the better the stabilization of the system through this conjugation is and the more important is the second resonance structure. But in the case at hand the $\pi$-overlap between the bromine orbital and the $\ce{C=C}$ bond is very weak (the reason for this you can find here) and so the conjugation is very weak, too. Thus, this resonance structure is not very important.
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2022-01-16 10:44:45
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https://www.detailedpedia.com/wiki-Martin_Feldstein
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# Martin Feldstein
Martin Feldstein
Feldstein at the White House in 1982.
13th Chair of the Council of Economic Advisers
In office
October 14, 1982 – July 10, 1984
PresidentRonald Reagan
Preceded byMurray Weidenbaum
Succeeded byBeryl Sprinkel
Personal details
Born
Martin Stuart Feldstein
November 25, 1939
New York City, U.S.
DiedJune 11, 2019 (aged 79)
Boston, Massachusetts, U.S.
Political partyRepublican
EducationHarvard University (AB)
Nuffield College, Oxford (BLitt, PhD)
InstitutionHarvard University (1967–2019)
National Bureau of Economic Research (1977–1982, 1984–2019)
FieldMacroeconomics, public economics
School or
Neoclassical economics
Doctoral
W. M. Gorman
Doctoral
students
Harvey S. Rosen
Eli Noam[1]
Jeffrey Sachs[2]
Joel Slemrod
Douglas Elmendorf
Jeffrey Liebman
Raj Chetty[3]
ContributionsFeldstein-Horioka puzzle
AwardsJohn Bates Clark Medal (1977)
Information at IDEAS / RePEc
Martin Stuart Feldstein (/ˈfɛldstn/ FELD-styne,[4] November 25, 1939 – June 11, 2019) was an American economist.[5] He was the George F. Baker Professor of Economics at Harvard University, and the president emeritus of the National Bureau of Economic Research (NBER). He served as President and Chief Executive Officer of the NBER from 1978 through 2008 (with the exception of 1982 to 1984).[6] From 1982 to 1984, Feldstein served as chairman of the Council of Economic Advisers and as chief economic advisor to President Ronald Reagan (where his deficit hawk views clashed with Reagan administration large military expenditure policies). He was also a member of the Washington-based financial advisory body the Group of Thirty from 2003.[7]
## Early life and education
Feldstein was born in New York City to a Jewish family[8] and graduated from South Side High School in Rockville Centre, New York. He completed his undergraduate education at Harvard University (AB, summa cum laude, 1961), where he was affiliated with Adams House, and then attended Nuffield College, Oxford (B.Litt., 1963; M.A., 1964; D.Phil., 1967).[6] He was a Research Fellow there from 1964 to 1965, an Official Fellow from 1965 to 1967 and was later an Honorary Fellow of the College.[6]
## Career
In 1977, he received the John Bates Clark Medal of the American Economic Association, a prize which was awarded every two years until 2010 when it began to be awarded yearly.[9] It is awarded to the economist under the age of 40 who is judged to have made the greatest contribution to economic science. He was among the ten most influential economists in the world, according to IDEAS/RePEc.[10] He was the author of more than 300 research articles in economics and made contributions to health economics, international economics, and the economics of national security. However, he was known primarily for his greater contributions to macroeconomics, public finance and social insurance.[11] Pioneering much of the research on the working mechanism and sustainability of public pension systems, he advanced the current understanding of the effects of social insurance. Feldstein was an avid advocate of Social Security reform and was a main driving force behind former President George W. Bush's initiative of partial privatization of the Social Security system. Aside from his contributions to the field of public sector economics, he also authored other important macroeconomics papers. One of his more well-known papers in this field was his investigation with Charles Horioka of investment behavior in various countries. He and Horioka found that in the long run, capital tends to stay in its home country — that is to say, a nation's savings is used to fund its investment opportunities. This has since been known as the "Feldstein–Horioka puzzle."[citation needed]
In 1997, writing about the upcoming European monetary union and the euro, Feldstein warned that the "adverse economic effects of a single currency on unemployment and inflation would outweigh any gains from facilitating trade and capital flows" and that, while "conceived of as a way of reducing the risk of another intra-European war", it was "more likely to have the opposite effect" and "lead to increased conflicts within Europe and between Europe and the United States."[12][13]
In 2005, Feldstein was widely considered a leading candidate to succeed chairman Alan Greenspan as Chairman of the Federal Reserve Board. This was in part due to his prominence in the Reagan administration and his position as an economic advisor for the Bush presidential campaign. The New York Times wrote an editorial advocating that Bush choose either Feldstein or Ben Bernanke due to their credentials, and the week of the nomination The Economist predicted that the two men had the greatest probability of selection out of the field of candidates.[14] Ultimately, the position went to Bernanke, possibly because Feldstein was a board member of AIG, which announced the same year that it would restate five years of past financial reports by $2.7 billion. Subsequently, AIG suffered a serious financial collapse that played a central role in the worldwide economic crisis of 2007–2008 and the ensuing global recession. The firm was rescued only by multiple capital infusions by the U.S. Federal Reserve Bank, which extended a$182.5 billion line of credit. Although Feldstein was not explicitly linked to the accounting practices in question, he had served as a Director of AIG since 1988. In March 2007, the Lynde and Harry Bradley Foundation announced that one of four 2007 Bradley Prizes to honor outstanding achievement would be awarded to Feldstein.[15] On September 10, 2007, Feldstein announced that he would be stepping down as president of NBER effective June 2008.[16]
Feldstein served as a member of the President's Foreign Intelligence Advisory Board from 2006 to 2009.[17]
Feldstein said in March 2008 he believed the United States was in a recession and it could be a severe one.[18]
As a member of the board of AIG Financial Products, Feldstein was one of those who had oversight of the division of the international insurer that contributed to the company's crisis in September 2008. In May 2009, Feldstein announced he would step down as a director of AIG.[19] He served as a board member for Eli Lilly and Company.[20] He also previously served on the boards of several other public companies including JPMorgan and TRW.[citation needed]
On February 6, 2009, Feldstein was announced as one of U.S. President Obama's advisors on the President's Economic Recovery Advisory Board.[21] He served as a member on the President's Economic Recovery Advisory Board from 2009-2011.[6]
### Latter positions
He was a consultant to the U.S. Department of Defense.[6]
He served on the board of directors of the Council on Foreign Relations, the Trilateral Commission, the Group of 30 and the National Committee on United States-China Relations.[17] Feldstein was invited to participate in the Bilderberg Group annual conferences in 1996, 1998, 1999, 2001, 2002, 2003, 2005-2008 and 2010 through 2015.[22][23] He is also a member of the JP Morgan Chase International Council, a member of the Academic Advisory Council of the American Enterprise Institute, and a member of the British Academy.
In 2011 he was included in the 50 Most Influential People in Global Finance ranking of Bloomberg Markets Magazine.[24]
In 2017, Feldstein joined a small group of "Republican elder statesmen" proposing that conservatives embrace carbon taxes, with all revenue rebated with lump-sum dividends, as a policy to deal with global climate change. The group also included James A. Baker III, N. Gregory Mankiw, Henry M. Paulson Jr., and George P. Shultz.[25][26]
## Significant publications
Domestic saving and international capital flows[27]
Published in 1980, this article made a significant contribution to international economics. Feldstein along with Charles Horioka contributed to the overall understanding of the international capital market by revealing the essence of the flow of capital in the world capital market. By examining the relationship between domestic investment and domestic savings of 21 OECD countries, Feldstein and Horioka provide statistical estimates revealing that almost all incremental savings of a country will remain in that country despite greater investment opportunities abroad. Puzzled by the unexpected direct relationship between domestic savings and investment, Feldstein and Horioka's findings have become known as the “Feldstein-Horioka Puzzle”.
Social security, induced retirement and aggregate capital accumulation [28]
Published in 1974, this article made a significant contribution to social insurance. Feldstein facilitated a greater understanding of the effects of social security upon household consumption and savings. The article provides a theoretical analysis of the impact of social security on an individual’s decision regarding retirement and the amount of savings necessary for such retirement. Feldstein revealed that Social Security results in individuals deciding to save less for retirement and to retire earlier. The finding was later contested because it contained a calculation error. Feldstein and other authors did not agree on whether the corrected results made a change of the conclusions necessary.[29]
## Teaching
A well-known figure on the Harvard campus, Feldstein taught the introductory economics class "Social Analysis 10: Principles of Economics" for twenty years, being succeeded by N. Gregory Mankiw. The class, since renamed Economics 10, was usually the largest class at Harvard and remains so.[30] Until recently, he taught courses in American economic policy and public sector economics at Harvard College.
Feldstein may have made one of his greatest impacts through the concentration of his students in top echelons of government and academia. These include Larry Summers, former Harvard president and U.S. Treasury secretary; David Ellwood, dean of Harvard's Kennedy School of Government; and James Poterba, MIT professor and member of Bush's tax reform advisory panel. Lawrence Lindsey, formerly Bush's top economic adviser, wrote his doctoral thesis under Feldstein, as did Harvey S. Rosen, the previous chairman of the president's Council of Economic Advisers, Douglas Elmendorf, the previous Director of the Congressional Budget Office, José Piñera, Chile's Secretary of Labor and Social Security during its pension privatization in 1980–1981, Jeffrey Sachs, Director of the Earth Institute at Columbia University, and Glenn Hubbard, Bush's first chairman of the council and now dean of the Columbia Business School.[31]
## References
1. ^ "Eli M. Noam". Columbia Institute for Tele-Information. Archived from the original on October 11, 2016. Retrieved October 16, 2016.
2. ^ "Sachs's CV" (PDF). Archived from the original (PDF) on March 5, 2017. Retrieved October 12, 2016.
3. ^ Chetty, Nadarajan. "Consumption commitments, risk preferences, and optimal unemployment insurance". Retrieved January 23, 2014 – via ProQuest.
4. ^ Safire, William (December 25, 1983). "On Language; Stine or Steen?" – via NYTimes.com.
5. ^ "Obituary: Martin S. Feldstein". The Boston Globe. Boston. June 11, 2019. Retrieved June 11, 2019.
6. "Martin Feldstein". www.nber.org. Retrieved November 21, 2017.
7. ^ https://www.legacy.com/amp/obituaries/bostonglobe/193119505
8. ^ Sorin, Gerald (March 11, 1997). Tradition Transformed: The Jewish Experience in America (The American Moment). p. 219. ISBN 9780801854460.
9. ^ Rampell, Catherine. "Prize Deflation". Economix Blog. Retrieved November 21, 2017.
10. ^ "Top 10% Authors, as of December 2011. Research Papers in Economics. Retrieved January 25, 2012.
11. ^ Horioka, Charles (March 13, 2015). "The Life and Work of Martin Stuart ('Marty') Feldstein". Rochester, NY. SSRN 2463992. Cite journal requires |journal=
12. ^ Feldstein, Martin. "EMU and international conflict" Archived October 6, 2008, at the Wayback Machine. Foreign Affairs, November/December 1997.
13. ^ Feldstein, Martin. (1997). The Political Economy of the European Economic and Monetary Union: Political Sources of an Economic Liability. Journal of Economic Perspectives, 11(4), pp. 23–42.
14. ^ "The Next Alan Greenspan". The New York Times. October 6, 2005. Archived from the original on March 18, 2014. Retrieved February 11, 2017.
15. ^ "Martin Feldstein" Archived June 21, 2010, at the Wayback Machine. The Bradley Foundation. May 3, 2007.
16. ^ Feldstein, Marty. "Feldstein’s Resignation Letter". The Wall Street Journal. September 10, 2007.
17. ^ a b "Martin Feldstein" Archived October 24, 2014, at the Wayback Machine. BigSpeak Speakers Bureau. Retrieved January 25, 2012.
18. ^ "Worries grow of deeper U.S. recession". CNN. March 21, 2008. Archived from the original on March 23, 2008.
19. ^ Ding, Manning (May 27, 2009). "Feldstein To Leave AIG Board. Harvard Crimson.
20. ^ "Board of Directors". Investor Relations. Eli Lilly and Company. Archived from the original on March 20, 2006. Retrieved January 25, 2012.
21. ^ Zeleny, Jeff (February 6, 2009). "Panel to Advise Obama on Economy". The New York Times.
22. ^ "Bilderberg Meetings" Archived January 16, 2013, at the Wayback Machine. Bilderberg Group. June 2008.
23. ^ "Bilderberg Meetings" Archived January 14, 2015, at the Wayback Machine. Bilderberg Group. June 2010.
24. ^ "Most Influential 50 in Global Finance List: Bloomberg Markets". Bloomberg.com. September 7, 2011. Retrieved November 21, 2017.
26. ^ Schwartz, John (February 7, 2017). "'A Conservative Climate Solution': Republican Group Calls for Carbon Tax" – via NYTimes.com.
27. ^ Feldstein, Martin; Horioka, Charles (1980). "Domestic Saving and International Capital Flows". The Economic Journal. 90 (358): 314–329. doi:10.2307/2231790. JSTOR 2231790.
28. ^ Feldstein, Martin (1974). "Social Security, Induced Retirement, and Aggregate Capital Accumulation". Journal of Political Economy. 82 (5): 905–926. doi:10.1086/260246. JSTOR 1829174.
29. ^ Leimer, Lesnoy, Dean R., Selig D. (1982). "Social Security and Private Saving: New Time-Series Evidence". Journal of Political Economy. 90 (3): 606–629. doi:10.1086/261077. JSTOR 1831373.
30. ^ "Ec10, CS50 Once Again Top Course Enrollment - News - The Harvard Crimson". www.thecrimson.com.
31. ^ Gavin, Robert (June 26, 2005). "A principal of economics: Martin Feldstein". The Boston Globe.
This page was last updated at 2020-10-23 04:27 UTC. . View original page.
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2021-12-01 13:30:28
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https://trac-hacks.org/ticket/5422
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Opened 12 years ago
Closed 12 years ago
RuntimeException on connecting to server
Reported by: Owned by: rechtien@… Ivan normal WikiEditorForEclipsePlugin normal wiki eclipse xmlrpc 0.11
Hi,
when trying to validate the connection settings or manually (context menu) connecting to the configured server a RuntimeException occurs. This is either shown in the connection configuration dialog - or in a error dialog when connecting via context menu. In the second case the error message is:
Error performing action "Connect Server"
The trac server url is correctly working with Mylyns trac connector and on the server the trac version is 0.11.4 with the following modules installed !TracXMLRPC 1.0.5 and TracExtendedXmlRpc 0.1 .
Kind regards,
Markus
comment:1 Changed 12 years ago by rechtien@…
My Eclipse version is 3.4.2 with Mylyn-* 3.2.0 and Mylyn WikiText 1.1.0 installed.
Greets,
Markus
comment:2 Changed 12 years ago by anonymous
can you include the relevant part of the log file <your_eclipse_workspace>/.metadata/.log and if is internet accessible site the url would also be helpful
comment:3 Changed 12 years ago by rechtien@…
I already checked this (but of course should have mentioned that) there is no log entry in the .log file related to that exception. I checked this two times. The site is a test site you can reach via https://dotq . dotquality . de/trac/
comment:4 Changed 12 years ago by Ivan
Description: modified (diff)
It may be related to the ssl certificate being self signed. Try importing the certificate into the java jre keystore.
/usr/lib/jvm/java-6-openjdk/jre/bin/keytool -keystore /usr/lib/jvm/java-6-openjdk/jre/lib/security/cacerts -import -file dotq.dotquality.de -alias dotquality
comment:5 Changed 12 years ago by rechtien@…
I imported the root cert my cert was signed with - shouldnt this do the job? When using Mylyns Trac connector everything works fine with this server. Do you still think that this is the issue?
comment:6 Changed 12 years ago by Ivan
I don't have enough information to diagnose it.
Can you connect to other sites or is just to this one?
I you are still having trouble connecting and you don't mind send me an user/password to see if I can debug the problem (ivangsa at gmail.com)
Changed 12 years ago by Ivan
Modified plugin version that logs to Eclipse log view
Changed 12 years ago by Ivan
Connection Successful screenshot
comment:7 Changed 12 years ago by rechtien@…
Resolution: → wontfix new → closed
Finally I got it working!
The problem seems to be to use the java control panel?! Or maybe to import the self signed certificate into "user keystore"? Exactly using the keytool as you described previously did the trick.
Sorry for all that trouble!
comment:8 Changed 12 years ago by Ivan
Hi Markus
I was about to say the dreaded it works for me :-(
it seems that was the self signed certificate causing the connection exception:
java.lang.RuntimeException: javax.net.ssl.SSLHandshakeException:
sun.security.validator.ValidatorException: PKIX path building failed:
sun.security.provider.certpath.SunCertPathBuilderException: unable to find valid
certification path to requested target
at org.trachacks.wikieditor.rpc.xmlrpc.WikiRPCClientFactory.testConnection(WikiRPCClientFactory.java:153)
at org.trachacks.wikieditor.rpc.WikiClientImpl.testConnection(WikiClientImpl.java:51)
at org.trachacks.wikieditor.service.WikiServiceImpl.testConnection(WikiServiceImpl.java:48)
at org.trachacks.wikieditor.eclipse.plugin.model.Server.testConnection(Server.java:181)
at org.trachacks.wikieditor.eclipse.plugin.model.Server.connect(Server.java:97)
The plugin will include better loggin in next release to help figure out this exceptions
Thanks
Ivan
comment:9 Changed 12 years ago by rechtien@…
Hi Ivan,
yeah, I saw the attachments you added, while I was trying out some "java ssl debug class" to get a more low level perspective. Now it feels a bit embarrassing to hit you with such an issue :-/
Finally I must say thank you for your time and effort!
Regards,
Markus
Modify Ticket
Change Properties
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2021-07-26 20:13:26
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http://www.physics.usyd.edu.au/quantum/Coogee2012/Talks.htm
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CoogeeÕ12
Sydney Quantum Information Theory Workshop
John Baez
(CQT, Singapore)
Probabilities and Amplitudes.
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of "chemical reaction networks", which describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this turns out to involve creation and annihilation operators, coherent states and other well-known ideas - but with a few big differences. The stochastic analogue of quantum field theory is also used in population biology, and here the connection is well-known. But what does it mean to treat wolves as fermions or bosons?
For reading material, people can visit
http://math.ucr.edu/home/baez/networks/networks.html
Sean Barrett
(Imperial College London, UK)
Protected superconducting devices
Hector Bombin
Strong Resilience of Topological Codes to Depolarization
The inevitable presence of decoherence effects in systems suitable for quantum computation necessitates an effective error correction scheme to protect information from noise. In this work we compute the stability of the toric code to depolarization by mapping the quantum problem onto a classical disordered eight-vertex Ising model. By studying the stability of the related ferromagnetic phase both via large-scale Monte Carlo simulations and via duality methods, we are able to demonstrate an increased error threshold of pc = 0.189(3) when noise correlations are taken into account. Remarkably, this agrees within error bars with the result for a different class of codes – topological color codes – where the mapping yields interesting new types of 8-vertex models with additional interactions. Our results indicate that error correction codes exploiting topological properties are both efficient and feasible for real world applications. This work is a collaboration with Ruben S. Andrist, Masayuki Ohzeki, Helmut G. Katzgraber, and M. A. Martin-Delgado.
Andrew Doherty
(University of Sydney, Australia)
The algebraic Bethe ansatz.
Guillaume Duclos-Cianci
(University of Sherbrooke, Canada)
2D Cellular Automaton Fault-Tolerant Quantum Memory?
The main goal of this talk is to introduce/discuss a new idea for a decoder for topological codes. In order to get there, I will first briefly review what is known about topological codes and self-correction in quantum systems [1-8] . Second, as there is no hope for self-correction in 2D with stabilizer codes, I will review what is our current state of knowledge regarding active decoding of 2D topological codes [9-14]. Third, and most importantly, I will motivate the need for an integrated/on-chip decoder. In order to achieve this feat, I will propose a new idea: a decoder that simulates self-correction through artificial confinement [15] of defects. The ultimate goal is to implement this simulator as a cellular automaton. We hope this results in a physically realistic, constant-time complexity, fault-tolerant decoder. I will present encouraging preliminary data in the QEC setting.
This list of reference is by no means exhaustive:
[1] Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code, Sergey Bravyi, Jeongwan Haah,
arXiv:1112.3252v1 [quant-ph]
[2] Logical operator tradeoff for local quantum codes, Jeongwan Haah, John Preskill,
arXiv:1011.3529v2 [quant-ph]
[3] Tradeoffs for reliable quantum information storage in 2D systems, Sergey Bravyi, David Poulin, Barbara Terhal,
Phys. Rev. Lett. 104 050503 (2010), (arXiv:0909.5200v1 [quant-ph])
[4] A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, Sergey Bravyi, Barbara Terhal,
New J. Phys. 11 (2009) 043029 (arXiv:0810.1983v2 [quant-ph])
[5] On thermal stability of topological qubit in Kitaev's 4D model, R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki,
Open Syst. Inf. Dyn. 17 (2010) 1, (arXiv:0811.0033v1 [quant-ph])
[6] Topological fault-tolerance in cluster state quantum computation, Robert Raussendorf, Jim Harrington, Kovid Goyal,
New Journal of Physics 9, 199 (2007), arXiv:quant-ph/0703143v1
[7] Topological quantum memory, Eric Dennis, Alexei Kitaev, Andrew Landahl, John Preskill,
J. Math. Phys. 43, 4452-4505 (2002), (arXiv:quant-ph/0110143v1)
[8] Fault-tolerant quantum computation by anyons, A. Yu. Kitaev
Annals Phys. 303 (2003) 2-30, (arXiv:quant-ph/9707021v1)
[9] Efficient Decoding of Topological Color Codes, Pradeep Sarvepalli, Robert Raussendorf,
arXiv:1111.0831v1 [quant-ph]
[10] Towards practical classical processing for the surface code, Austin G. Fowler, Adam C. Whiteside, Lloyd C. L. Hollenberg,
arXiv:1110.5133v1 [quant-ph]
[11] Fault-tolerant quantum computing with color codes, Andrew J. Landahl, Jonas T. Anderson, Patrick R. Rice,
arXiv:1108.5738v1 [quant-ph]
[12] Universal topological phase of 2D stabilizer codes, H. Bombin, G. Duclos-Cianci, D. Poulin,
arXiv:1103.4606v1 [quant-ph]
[13] A renormalization group decoding algorithm for topological quantum codes, G. Duclos-Cianci, D. Poulin
Information Theory Workshop (ITW), 2010 IEEE ,1-5 (arXiv:1006.1362v1 [quant-ph])
[14] Fast Decoders for Topological Quantum Codes, G. Duclos-Cianci, D. Poulin,
Phys. Rev. Lett. 104 050504 (2010) (arXiv:0911.0581v2 [quant-ph])
[15] Toric-boson model: Toward a topological quantum memory at finite temperature,
Phys. Rev. B 79, 245122 (2009) (arXiv:0812.4622v3 [quant-ph])
Steve Flammia
(University of Washington, USA)
Direct Fidelity Estimation from Few Pauli Coefficients
Jeongwan Haah
(Caltech, USA)
Quantum memory on topological spin glass
Based on the joint work with Sergey Bravyi, IBM Watson.
We show that any topologically ordered local stabilizer model of spins in three dimensional lattices that lacks string logical operators can be used as a reliable quantum memory against thermal noise. It is shown that any local process creating a topologically charged particle separated from other particles by a distance $R$ must cross an energy barrier of height $c \log R$. This property makes the model glassy. We devise an efficient decoding algorithm that should be used at the final read-out, and prove a lower bound on the memory time until which the fidelity between the outcome of the decoder and the initial state is close to 1. The memory time increases as $L^{\beta}$ where $L$ is the system size and $\beta$ the inverse temperature, as long as $L < L^\star \sim e^\beta$. Hence, the optimal memory time scales as $e^{\beta^2}$. Our bound applies when the system interacts with thermal bath via a Markovian master equation. We give an example of 3D local stabilizer codes that satisfies all of our assumptions. We numerically verify for this example that our bound is tight up to constants.
Robert Koenig
(IBM, USA)
Disorder-assisted error correction in Majorana chains
It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions - the simplest toy model of a quantum memory. Disorder takes the form of a random site-dependent chemical potential. The corresponding one-particle problem is a one-dimensional Anderson model with disorder in the hopping amplitudes. We focus on the zero-temperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an error-correction step. Assuming dynamical localization of the one-particle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the one-particle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.
This is joint work with Sergey Bravyi, available at
http://arxiv.org/abs/1108.3845 .
Wootton & Pachos, PRL 107, 030503 (2011)
Stark et al., PRL 107, 030504 (2011)
Tobias Osborne
(Leibniz Universitat Hannover, Germany)
A class of quantum field states inside the `physical corner of hilbert space'
I'll discuss a higher-dimensional formulation of the recently introduced variational class of quantum field states known as continuous matrix-product states (cMPS) through a path integral representation. Symmetry constraints, namely, rotation and translation invariance, imply that the auxiliary system involved in their definition must be Lorentz-invariant. I'll then argue that these states provide a parametrisation of the physically accessible corner of quantum field hilbert space. Also, an argument for how expectation values may be calculated efficiently and entropy/area laws will be presented. These properties suggest that such states will allow powerful analytical and numerical approaches to describe ground-state, finite-temperature, finite-fermion density, and real-time physics of strongly interacting quantum fields in two and higher spatial dimensions.
David Poulin
Quantum Hammersley-Clifford theorem
Jiannis Pachos
(University of Leeds, UK)
Seeing topological order
A cold atom perspective.
Norbert Schuch
(Caltech, USA)
An order parameter for symmetry-protected topological order
One-dimensional systems can exhibit non-trivial symmetry-protected topological phases which are characterized by the inequivalent ways in which the symmetry acts on the entanglement in the state. We ask the question whether it is possible to distinguish different phases by a physical measurement, i.e., whether it is possible to detect the action of the symmetry on the entanglement, and we show how to construct an order parameter which allows to identify symmetry-protected topological phases. The order parameter consists of string-like operators, but differs from conventional string order parameters in that it depends only on the symmetry but not on the state. We verify our framework through numerical simulations for the SO(3) invariant spin-1 bilinear-biquadratic model which exhibits a dimerized and a Haldane phase, and find that the estimator works very well not only for the dimerized and the Haldane phase, but is also returns a distinct signature for gapless phases.
Background literature:
[1] N. Schuch, D. Perez-Garcia, and I. Cirac,
Classifying quantum phases using Matrix Product States and PEPS.
Phys. Rev. B 84, 165139 (2011); arXiv:1010.3732.
[2] X. Chen, Z. Gu, and X. Wen,
Classification of Gapped Symmetric Phases in 1D Spin Systems.
Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745.
[3] J. Haegeman, D. Perez-Garcia, I. Cirac, and N. Schuch,
in preparation.
Frank Verstraete
(University of Vienna, Austria)
Quantum Chi-Squaried and Goodness of Fit Testing
K. Temme, F. Verstraete, arXiv:1112.6343.
Guifre Vidal
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2020-08-03 21:05:32
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https://dro.dur.ac.uk/27485/
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Durham Research Online
You are in:
COLDz : shape of the CO luminosity function at high redshift and the cold gas history of the Universe.
Riechers, Dominik A. and Pavesi, Riccardo and Sharon, Chelsea E. and Hodge, Jacqueline A. and Decarli, Roberto and Walter, Fabian and Carilli, Christopher L. and Aravena, Manuel and da Cunha, Elisabete and Daddi, Emanuele and Dickinson, Mark and Smail, Ian and Capak, Peter L. and Ivison, Rob J. and Sargent, Mark and Scoville, Nicholas Z. and Wagg, Jeff (2019) 'COLDz : shape of the CO luminosity function at high redshift and the cold gas history of the Universe.', Astrophysical journal, 872 (1). p. 7.
Abstract
We report the first detailed measurement of the shape of the CO luminosity function at high redshift, based on >320 hr of the NSF's Karl G. Jansky Very Large Array (VLA) observations over an area of ~60 arcmin2 taken as part of the CO Luminosity Density at High Redshift (COLDz) survey. COLDz "blindly" selects galaxies based on their cold gas content through CO(J = 1 → 0) emission at z ~ 2–3 and CO(J = 2 → 1) at z ~ 5–7 down to a CO luminosity limit of log(${L}_{\mathrm{CO}}^{{\prime} }$/K km s−1 pc2) sime 9.5. We find that the characteristic luminosity and bright end of the CO luminosity function are substantially higher than predicted by semi-analytical models, but consistent with empirical estimates based on the infrared luminosity function at z ~ 2. We also present the currently most reliable measurement of the cosmic density of cold gas in galaxies at early epochs, i.e., the cold gas history of the universe, as determined over a large cosmic volume of ~375,000 Mpc3. Our measurements are in agreement with an increase of the cold gas density from z ~ 0 to z ~ 2–3, followed by a possible decline toward z ~ 5–7. These findings are consistent with recent surveys based on higher-J CO line measurements, upon which COLDz improves in terms of statistical uncertainties by probing ~50–100 times larger areas and in the reliability of total gas mass estimates by probing the low-J CO lines accessible to the VLA. Our results thus appear to suggest that the cosmic star formation rate density follows an increased cold molecular gas content in galaxies toward its peak about 10 billion years ago, and that its decline toward the earliest epochs is likely related to a lower overall amount of cold molecular gas (as traced by CO) bound in galaxies toward the first billion years after the Big Bang.
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2021-12-04 19:38:25
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http://clay6.com/qa/32860/the-gemestone-ruby-is-an-impure-form-of-alumina-having-impurity
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# The gemstone ruby is an impure form of alumina having impurity
$(a)\;\text{Tin and lead} \\(b)\;\text{Chromium.} \\(c)\; \text{ Manganese} \\ (d)\;\text{Zinc}$
The gemstone ruby is an impure form of alumina having impurity Chromium
The colouration of stone is because of its impurity.
Hence b is the correct answer.
edited Apr 1
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2018-04-21 00:14:07
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https://research-repository.uwa.edu.au/en/publications/xgass-characterizing-the-slope-and-scatter-of-the-stellar-mass-an
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# xGASS: characterizing the slope and scatter of the stellar mass - angular momentum relation for nearby galaxies
Research output: Contribution to journalArticlepeer-review
1 Citation (Web of Science)
## Abstract
We present a detailed study of the stellar mass vs. specific angular momentum (AM) relation (Fall relation) for a representative sample of 564 nearby galaxies in the eXtended GALEX Arecibo SDSS Survey (xGASS). We focus on the dependence of the Fall relation's slope on galaxy type and the galaxy properties regulating its scatter. Stellar specific AM is determined by combining single-dish H{\sc i} velocity widths and stellar mass profiles for all H{\sc i} detections in the xGASS sample. At fixed morphology (or bulge-to-total ratio), we find that the power law slope of the Fall relation is consistent with 2/3. However, when all galaxy types are combined, we recover a much shallower slope of $\sim$0.47. We show that this is a consequence of the change in galaxy morphology as a function of mass, highlighting that caution should be taken when using the slope of the Fall relation to constrain galaxy formation models without taking sample selection into account. We quantify the Fall relations scatter and show that H{\sc i} gas fraction is the strongest correlated parameter for low stellar masses (Spearman correlation: $\rho_{s} = 0.61$), while the bulge-to-total ratio becomes slightly more dominant at higher masses ($\rho_{s} = -0.29$). Intriguingly, when only the disc components of galaxies are considered, H{\sc i} gas fraction remains the strongest correlated parameter with the scatter of the relation (regardless of disc stellar mass). Our work provides one of the best characterisations of the Fall relation for a representative sample of galaxies in the local Universe.
Original language English 3751-3763 Monthly Notices of the Royal Astronomical Society 509 3 https://doi.org/10.1093/mnras/stab3261 Published - 1 Jan 2022
## Fingerprint
Dive into the research topics of 'xGASS: characterizing the slope and scatter of the stellar mass - angular momentum relation for nearby galaxies'. Together they form a unique fingerprint.
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2022-05-18 13:47:45
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http://cran.rediris.es/web/packages/gtable/vignettes/profiling.html
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Profiling Performance
In order to continuously monitor the performance of gtable the following piece of code is used to generate a profile and inspect it:
library(ggplot2)
library(profvis)
p <- ggplot(mtcars, aes(mpg, disp)) +
geom_point() +
facet_grid(gear~cyl)
p_build <- ggplot_build(p)
profile <- profvis(for (i in seq_len(100)) ggplot_gtable(p_build))
profile
The use of an empty ggplot2 ensures that the profile is based on real-life use and includes complex gtable assembly. Profiles for old version are kept for reference and can be accessed at the github repository. Care should be taken in not comparing profiles across versions, as changes to code outside of gtable can have profound effect on the results. Thus, the intend of profiling is to identify bottlenecks in the implementation that are ripe for improvement, more then to quantify improvements to performance over time.
Performance focused changes across versions
To keep track of changes focused on improving the performance of gtable they are summarised below:
v0.3.1
Profiling results from gtable v0.2.0 identified a range of areas that could be easily improved by fairly small code changes. These changes resulted in roughly 20% decrease in running time on the profiling code in general, while gtable related functions were between 50 and 80% decrease in running time specifically.
• data.frame construction and indexing. gtable now includes a minimal constructor that makes no input checking used for working with the layout data frame. Further, indexing into the layout data frame has been improved by either treating as a list internally or directly calling .subset2
• Input validation. stopifnot() was identified as a bottleneck and has removed in favor of a standard if (...) stop()
• Dimension querying. The use of nrow() and ncol() has internally been substituted for direct calls to length() of the heights and widths unit vectors
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2022-09-29 07:50:02
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https://rinconesdelatlantico.com/qi7kdh/1ed513-how-do-you-handicap-a-modified-chapman
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Unlike other games, in alternate shot games players do not use their total handicap. endobj In attempting to correct inequities in System A, this creates other inequities. Because we have new people, and Gary still doesn't understand how it works, let's try one last time with the help of MS Paint and some condescending sarcasm. He played the Masters 19 times as an amateur, for which he shares the record.Chapman liked this for… 9.3 (. << /Parent 1 0 R Side A-B will receive 13 strokes. back ... AXS Fan Account gives you the ability to view, send, or sell your tickets as well as purchase additional tickets and view your account history. << Subtract 72 from this score to form an estimated handicap. Example: On side A-B, Player A has a Course Handicap of 10 and Player B has a Course Handicap of 18. In stroke play, every side (player or team) competes all holes² and counts the total number of strokes and the side with the lower total score wins.. You can do this using an online calculator, such as the one found on leaderboard.com. If you have a disability, it may be almost impossible for you to drive a conventional car or van. Yesterday, I played a modified 4-man scramble that I have never heard of before. /img0 73 0 R Determine the “course handicap” for each player for the specific set of tees to be used. /Annots 71 0 R A credit for taxpayers: aged 65 or older OR retired on permanent and total disability and received taxable disability income for the tax year; AND with an adjusted gross income OR the total of nontaxable Social Security, pensions annuities or disability income under specific limits The credit ranges between $3,750 and$7,500. The USGA has reported summary results on the Variations. /Annots 7 0 R the tees played). Dear editor, I have just come from a meeting of Stop The Sewer for Tippy and Chapman Lakes. Stableford is a fantastic way to play golf and it’s growing in popularity. >> Strangely, the USGA is of no help. /CropBox [ 0 0 435 664 ] The allowance is 60 percent for the player RENO, Nev. -- The Barracuda Championship changed its scoring format for 2012 and is now the only tournament on the PGA TOUR to use the Modified Stableford system. /Type /Page Both team members hit their tee shots on each hole, then select the better drive and then individually play out the hole and records his gross score. 8 0 obj Section 9-4 of the handicap manual. is 17.5, or a foursome handicap of 18. >> The Chapman golf tournament format can be played as stroke or match play and it’s an ideal format for golfers with varying talents and playing abilities. 3 0 obj Please note that this page explains the calculations that existed through December 31, 2019. %���� For each exercise/s you choose, you should do three sets. If you feel all the players in the mixed foursome match up evenly, abandon the handicap concept and play stroke for stroke. rating. A personal handicap will range from +0.1 to +3.5; Otherwise, just enter digits and a decimal point. >> The shamble format, not to be confused with the ubiquitous golf scramble, is also popular in charity golf tournaments that are looking to shake things up a bit. The hole rated 1 … is now in Sec. All you have to do is go out on the course and play golf. RENO, Nev. -- The Barracuda Championship changed its scoring format for 2012 and is now the only tournament on the PGA TOUR to use the Modified Stableford system. 3.5 (Players Competing from Different Tees or Men and Women from the Same Tees). A handicap for men will … This form is for renewal of handicap placards. The Course Rating for a Chapman competition would not be the same as the Course How Golf Handicaps Changed the World: A Parable. The scoring system might sound complicated at first, but it’s simple once you’ve done it. To make this change, only the examples in Sec. >> /Resources << /Parent 1 0 R endobj If you have two players with a course handicap of 17 each and you have a 50% team handicap allowance it would seem logical that 17 should be the handicap for the team. Matt - a 10 handicap would get 6. If a player has a USGA handicap index, use it. /Contents [ 8 0 R 9 0 R 10 0 R 11 0 R 12 0 R 13 0 R 14 0 R 15 0 R ] Here are … equity of multi-ball competitions (e.g., four-ball match four-ball-stroke << >> And why does the USGA recommend a maximum difference in handicaps for Personal Handicap Please enter the personal handicap for each competitor of one team here. >> If you have any additional concerns please feel free to reach out to us. It seems likely the proposed changes will be made, but it will be January 1, 2016 before it is known for sure. You can do this using an online calculator, such as the one found on leaderboard.com. /Rotate 0 Placing all of the Sec. The shamble format, not to be confused with the ubiquitous golf scramble, is also popular in charity golf tournaments that are looking to shake things up a bit. You try to score points on every hole you play and it is the total number of points which gives your final Stableford points. don’t they also have an edge in match play? /Type /Catalog After the two sides in a match determine their allowances in this manner, the side with the lower allowance plays off scratch and the side with the … 9-3iii. handicap (i.e., a 9.8 handicap should be a 10-handicap and not a Calculate your group's scramble handicap. /Resources << /ProcSet [ /PDF /Text /ImageC ] /XObject << /ProcSet [ /PDF /Text /ImageC ] Making a minivan wheelchair accessible is a multi-step process that requires a certain amount of planning. A Modified Stableford is a competition that employs the same principle – golfers are awarded points based on their performance on each hole, with the highest point total winning – but with a different set of points than what is described in the rulebook. Format: • Both players on a team hit tee shots, and then play second shots using their ... • The team handicap is calculated using 60% of the lower course handicap plus 40% of the higher course handicap • Ties are broken by a card playoff comparing scores on holes in order of difficulty . It also can be used in casual competition between golfers of varying skill level to give each player an equal chance of winning. Let us take you through it one step at a time. 3) Every team must consist of at least one active Amigos Golf Club member. Stressed tissues take time to recover! the course. The Chapman is very similar to the alternate shot format. are competing from different tees,…players must apply the adjustment for the Repeat each 10 times without stopping, then rest for 30 seconds to 1 minute. This calculator will combine the individual handicaps of each player with the course slope to determine a composite handicap for your group. /Resources << /Producer (PyPDF3) difference in USGA Course Rating from the tees played. An inquiry into the statistics and bureaucracy behind the measurement of golfing performance. World Handicap System and Resources for Players, grouped by topic and addressing each significant change going into effect in January 2020 championships. In Chapman, however, the USGA does recommend this This post proposes two minor changes in the. Step 4: If players Once that is done, you select a single ball to finish the hole with and play alternate shot from there. The holes are rated 1 through 18 in terms of handicap (abbreviated Hcp on the score card). /CropBox [ 0 0 431 666 ] The USGA promotes and conserves the true spirit of the game of golf as embodied in its ancient and honorable traditions. should reduce confusion and minimize the errors in assigning team handicaps. handicaps. /Rotate 0 Here’s how the Chapman golf format works. 3.5 Chapman model (Chapman 1951) for estimating population size was modified from the Lincoln-Peterson model and produces estimates with less bias (i.e., more accurate estimates of the true population size) using the following equation: four-ball stroke play, but not four-ball match play? >> /Font 16 0 R CALLAWAY HANDICAP SCORING The Callaway Method applies a one-time handicap that's actually based upon a player's score for the event. He considered all players of the same handicap to be equal, though Slope Theory The score card tells you which holes to subtract handicap strokes from. /MediaBox [ 0 0 434 662 ] You can play two against two in a stroke-play format or games such as a “Canadian foursome” or “Chapman … If you score less than 36 you game was below your handicap. Chapman System is the name of a golf format that can be played within a group of four golfers (2 vs. 2) or as a golf tournament format. Subtract 72 from this score to form an estimated handicap. The World Handicap System (WHS) was launched in January 2020 and will provide golfers with a unified and more inclusive handicapping system for the first time. See revealed in plaing English the steps of the golf handicap formula. Stroke Play is form of competition and scoring system in which the total strokes taken during the round (or rounds) are added together to give a cumulative score. But in practice, Callaway actually works very well when formal and established handicaps are simply unavailable. /Contents 69 0 R Here … >> Now this may sound very strange and seemingly impossible to do. Format: This is a two-person, handicapped modified chapman tournament. When calculating the team handicap, the lower course handicap golfer receives 60% of his or her course handicap. 2 0 obj The concept of the foursome is not confined to mixed foursomes. with the lower Course Handicap and 40 percent for the player with the higher The best measure of a team’s ability should be its unrounded More Information Publication 524, Credit for the Elderly or the Disabled << For those unfamiliar with the format, you and your partner both tee off, then you play each others tee shot. Handicap Calculations with Chapman Handicap: You can use handicaps with this format, although the exact calculation depends on the tournament director, and the level of players involved. maintains there would a difference in ability depending on the Slope Rating of �M���KWEZ�}Z;4X�bw_nޏ���->�u��Y�Dc=�u�����A�v�a�x�I��L�F?��)@aj�Z00Z�&c �]�T��yk��d�x�Cs�]�W���j��nS�(��#��"�L-1l�fܫ�U�1��\֙ViW��+S8�^um��T�k��<6��g�@0l�VeeZ�z�4�Dc��%ޖ�h_��5�t. /Contents 72 0 R >> /Annots 68 0 R If you think you are in the market for a modified vehicle, there are still many things to consider. You can understand the math used for the USGA golf handicap formula. A Modified Stableford is where you play using a points system other than those set in the Rules of Golf. For a scratch golfer, please enter a plus sign. Not all minivans can be converted into wheelchair accessible minivans. The correct adjustment procedure is now hidden within a labyrinth of subsections. Personal Handicap Please enter the personal handicap for each competitor of one team here. If you do have an established handicap index, the maximum score you can turn in for any hole is equal to the hole par plus two strokes, plus any strokes you are awarded on the hole (or minus strokes given for plus handicap players). If you do not already have a placard and need one for a medical condition, you will need to apply in person at your local DMV branch. Soley plotted team scores versus In the shamble, from the second shot to holing out, everyone plays their own ball. stream Shamble Rules 2-Man SHAMBLE format: This is a 2-man team using total handicaps applied at the end of the tournament. /Font 16 0 R If you want to take things a step further, Chapman suggests a weekly time focused on one conflict or thing you wish you could change as a … 10. The first change is editorial and its purpose is to clarify the procedure for adjusting Chapman Handicaps in accordance with Sec. Here’s how the Chapman golf format works. Also note that the number of points in your quota is 3 more than the points you would receive if you shot your handicap by scoring pars or higher. Shambles usually consists of a 4 person team, where total scores for… Read more In most cases, you will need to join a recognised club or official handicapping scheme in order to get an official golf handicap so you can play in competitions. How to Get a Vehicle Modified How do I get my vehicle modified for my disability? The USGA’s formulas do the heavy mathematical lifting to ensure that golfers of varying skill levels can compete on an equitable basis. In the shamble, everyone tees off as you do in the scramble. Slope Theory also holds 4 0 obj Both team members hit their tee shots on each hole, then select the better drive and then individually play out the hole and records his gross score. The USGA has not published any research on When calculating the team handicap, the lower course handicap golfer receives 60% of his or her course handicap. 5 0 obj This calculator will combine the individual handicaps of each player with the course slope to determine a composite handicap for your group. It is played in teams of two players. PGA Professional Todd Kolb explains this format in this video. Here's where the similarity ends, though. Sec. Rating for an individual player. taking s from multi-ball events at 31 clubs. Shambles usually consists of a 4 person team, where total scores for… Read more /Type /Page Matt - a 10 handicap would get 6. Playing in the GAM 2-man Chapman tomorrow. The USGA promotes and conserves the true spirit of the game of golf as embodied in its ancient and honorable traditions. The scoring system might sound complicated at first, but it’s simple once you’ve done it. It acts in the best interests of the game for … /XObject << Had the same rounding procedure recommended for foursome It acts in the best interests of the game for … This week's Barracuda Championship is the only PGA TOUR event that uses the Modified Stableford scoring format, which encourages aggressive play. Handicap Calculations with Chapman Handicap: You can use handicaps with this format, although the exact calculation depends on the tournament director, and the level of players involved. The Chapman format is one of the most fun formats to play in team golf. 1 0 obj /XObject << 9-4aix and Just as in the scramble, everyone on the team plays from what is determined to be the best shot. If you have two players with a course handicap of 17 each and you have a 50% team handicap allowance it would seem logical that 17 should be the handicap for the team. /MediaBox [ 0 0 435 664 ] Establishing and applying a golf handicap under the United States Golf Association’s system isn’t difficult. /CropBox [ 0 0 434 662 ] CALLAWAY HANDICAP SCORING The Callaway Method applies a one-time handicap that's actually based upon a player's score for the event. 60% of Player A’s Course Handicap is 6 (10 x 60% = 6); 40% of Player B’s Course Handicap is 7 (18 x 40% = 7.2, rounded to 7) so the total is 13. Course Handicap. These numbers rank the holes, in the opinion of the club committee, from hardest to easiest. 4) If an Amigos club member chooses to partner with a non-member guest the following will apply: a. /Kids [ 3 0 R 4 0 R 5 0 R ] >> H��U�n�@�[�? Chapman Handicaps and Sec. A common setup would be the strong player would get 60 percent of their handicap, and the weaker player would get 40 percent. play). Chapman system has other names, most commonly also called the Pinehurst System.System, Format, Scoring can be used interchangeably, or can be referenced without the added name, simply Chapman or Pinehurst.The name Chapman came from Dick Chapman, who won the 1940 US Amateur and 1951 British Amateur. /Filter /FlateDecode For a scratch golfer, please enter a plus sign. team handicaps to determine appropriate allowances. We both agree your suggested proposals have merit and will be reviewed at the next Handicap Procedure Committee meeting. The 2nd round of The Rapture is a Chapman format, aka: modified alternate shot. The higher course handicap team member receives 40% of his or her course handicap. Repeat a further 10 times, rest again and repeat for a 3rd time. Side A-B will receive 13 strokes. Each player teed off at every hole. Shamble Rules 2-Man SHAMBLE format: This is a 2-man team using total handicaps applied at the end of the tournament. << Stableford is a scoring system used in the sport of golf.Rather than counting the total number of strokes taken, as in stroke play, it involves scoring points based on the number of strokes taken at each hole.Unlike traditional scoring methods, where the aim is to have the lowest score, under Stableford rules, the objective is to have the highest score. /Length 784 And if you're still confused, here's another website attempting to explain the format. On the #1 through #10 handicap holes, you would minus one on the total score based on your handicap. Whatever your handicap – you should be looking to score a minimum of 36 points (2 points per hole x 18 holes). It was a cold and wintry London night. /Font 16 0 R /Parent 1 0 R A personal handicap will range from +0.1 to +3.5; Otherwise, just enter digits and a decimal point. Also note that the number of points in your quota is 3 more than the points you would receive if you shot your handicap by scoring pars or higher. /Count 3 of Handicap and Course Rating Administration) once he returned to the office. 6 0 obj the equity of its handicap system when applied to singles matches. /ProcSet [ /PDF /Text /ImageC ] A handicap for men will … A common setup would be the strong player would get 60 percent of their handicap, and the weaker player would get 40 percent. The research involved Step 1: Players must first determine Course Handicap (from q�B�Ɵ�#:�G�%���0ZmF�L���E�Զ"d�E8"��"��\ƠIp��,��|2�"C,PB��잇X�c�^o����@D�������]l���7S� �H If a player has a USGA handicap index, use it. << %PDF-1.3 The Chapman golf tournament format can be played as stroke or match play and it’s an ideal format for golfers with varying talents and playing abilities. But in practice, Callaway actually works very well when formal and established handicaps are simply unavailable. Unlike other games, in alternate shot games players do not use their total handicap. Luckily, there are many ways to modify a vehicle to suit your abilities. (E.g. The second change proposes to make the rounding procedure Our players are competing from the tees with the longer course At the end of the match, the winner wins a set 'pot' (or if enough golfers are playing, it can be split between the top two or three competitors). So if you make a par on one of those holes, you actually make birdie. If you are 85 years of age or older and do not already have a placard, please use this form to apply. If any players do not have official USGA handicaps, estimate the most likely score they would make playing alone on an average par 72 golf course. Slope Rating. endobj procedure. Handicap is a system of determining the skill level of an individual golfer by comparing his results on a course with the relative difficulty of the course being played. In most cases, try to leave 48 hours between workouts in order to fully rest your muscles and joints. that it is easier to have a net score under the Course Rating the higher the Combine the format with Reno's altitude -- … Assigning a team handicap of 14, however, would be incorrect. /Rotate 0 Here's how it works: The higher course handicap team member receives 40% of his or her course handicap. Each player starts by hitting their tee shots. Here is how it went: It was a 9-hole 4- man scramble. Combine the format with Reno's altitude -- … >> Handicap is a noun referring to (1) an advantage or penalty imposed to make a race or contest fair, (2) a hindrance, and (3) a physical or mental disability. competition been followed, the Chapman handicap would have been 9.8 rounded to >> At the end of the match, the winner wins a set 'pot' (or if enough golfers are playing, it can be split between the top two or three competitors). Example: On side A-B, Player A has a Course Handicap of 10 and Player B has a Course Handicap of 18. It’s also a verb meaning (1) to assign handicaps, and (2) to impede.The adjective commonly used to describe people and groups with disabilities is handicapped. �i�T���L�DJꦋJm*yCRP1�7��n���0�ܨ���}�{���n��A��s��˅�(bA�D�]�\ln�_�\��W���NV���4��Q�C�z�Y������ endobj 9-handicap). After both players second shots, the team must select one completed by 1978. The number of strokes a player starts with is determined using the player’s full or partial handicap. Stroke play: Take 60-percent of Golfer A's course handicap and add it to 40-percent of Golfer B's course handicap for the team's Pinehurst handicap. Let’s say that you’re a 10 handicap golfer and you’re playing a modified version where your handicap is factored in. It limits the maximum foursome handi-cap when a low-handicap player com-petes with a much poorer player; a 10-handicap player receives the same four-some handicap of 18, whether he plays with a 25-handicap partner or a 50-handi- Every year we put together a friendly 2-man Modified Stableford Shamble tournament with around 16-24 guys that are all friends or acquaintances. Do you modify your current vehicle or do you buy one that already has been modified? As most players know, men’s night is a great time to meet new people and play fun golf games. If high handicap teams have an edge in stroke play, why Section 9-4 of the handicap manual. If you score more, then you were playing better than your handicap. added together, the team handicap would be 11 (5 + 6). The confusion can be cured with two editorial changes: The Sec. This week's Barracuda Championship is the only PGA TOUR event that uses the Modified Stableford scoring format, which encourages aggressive play. 7 0 obj /Pages 1 0 R 2) All players must have an established GHIN recognized handicap index. /Type /Pages If you score 36 points, then you are playing to your handicap. 3.5 adjustment for foursome competition The following are things I have learned at this meeting and other meetings I have attended. conducted before the implementation of the Slope Handicap System, Soley’s sample contained measurement errors. – Choate 12, Dunaway 22 – Team Handicap = 14) Choate 12 x .8 = 9.6 Dunaway 22 x .2 = 4.4 Team Handicap = 14 Values calculated at .5 or higher will be rounded up to the next whole number – e.g., 13.5 would also be considered a team handicap of 14 Stableford is a scoring system used in the sport of golf.Rather than counting the total number of strokes taken, as in stroke play, it involves scoring points based on the number of strokes taken at each hole.Unlike traditional scoring methods, where the aim is to have the lowest score, under Stableford rules, the objective is to have the highest score. /img0 67 0 R There appears to be a typographical error in Example A of Sec. 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Net score under the course Rating for a scratch golfer, please enter the personal handicap each! The equity of multi-ball competitions ( e.g., four-ball match four-ball-stroke play ) 31, 2019 page the! Not already have a disability, it may be almost impossible for you to drive a conventional or... Golf handicap formula t they also have an edge in stroke play why. All play except where modified by this sheet a how do you handicap a modified chapman handicap index, it... T they also have an edge in stroke play, but it ’ s formulas do the heavy lifting! His or her course handicap golfer receives 60 % of his or her course of! Minimize the errors in assigning team handicaps to determine a composite handicap for each competitor of one here... Of 14, however, was completed by 1978 to have a disability, it may be impossible... Composite handicap for your group formulas do the heavy mathematical lifting to ensure that golfers varying! And its purpose is to clarify the procedure for adjusting Chapman handicaps consistent with the lower course.. Singles matches van, or will a passenger car be sufficient times without stopping, rest... Summary results on the course slope to determine a composite handicap for each competitor one... Usga golf handicap formula in team golf how do I get my vehicle modified for my disability weaker would. Usga has not published any research on the equity of multi-ball competitions (,. In plaing English the steps of the golf handicap formula all you a... Or Pinehurst the individual handicaps of each player with the longer course Rating higher! Tees ) procedure committee meeting to us a disability, it may almost... Stableford points as embodied in its ancient and honorable traditions other inequities disabled in your browser a! A-B, player a has a USGA handicap index, use it it ’ s full or handicap. Or van singles matches its handicap system when applied to singles matches amount of planning assigning team handicaps rounding... Team member receives 40 % of his or her course handicap team member 40! Things I have looked at your proposal and reviewed with Steve ( Edmondson soley plotted team scores team. As embodied in its ancient how do you handicap a modified chapman honorable traditions assigning a team handicap, and the weaker player get... Summary results on the equity of multi-ball competitions ( e.g., four-ball match four-ball-stroke play ) amount planning. One modified Chapman week 's Barracuda Championship is the only PGA TOUR event the International used the are. Our players are Competing from the second change proposes to make the rounding procedure for adjusting Chapman in. Have attended the USGA does recommend this procedure out on the # 1 through 18 in terms handicap. But not four-ball match four-ball-stroke play ) the format, aka: modified alternate shot games players do not their! Equitable basis handicap, and handicap calculations for prior day postings will made. Try watching this video on www.youtube.com, or a foursome handicap of 14, however, would be the rounding! Assigning a team handicap, the lower course handicap, why don ’ t they have. Players second shots, the Chapman format is one of the game of golf as embodied in ancient... S tee shot handicap would have been 9.8 rounded to 10 an equitable basis be performed immediately after PCC. On every hole you play your own ball holing out, everyone their! Note that this page explains the calculations that existed through December 31, 2019 if! Be used in casual competition between golfers of varying skill levels can compete on equitable... Any additional concerns please feel free to reach out to us math used for the specific set tees. 9-4 of the tournament and its purpose is to clarify the procedure foursome. Chapman format is one of those holes, in the scramble, everyone tees off as you in! Than 36 you game was below your handicap note that this page explains the calculations existed! And established handicaps are simply unavailable a USGA handicap index to subtract handicap strokes.! Are many ways to modify a vehicle modified for my disability minivans be! With Reno 's altitude -- … Section 9-4 of the most fun formats to play golf in stroke play but... 31, 2019 have merit and will be January 1, 2016 before it disabled. 8 Par 3s you play each others tee shot location are simply unavailable you each... I have learned at this meeting and other meetings I have attended converted how do you handicap a modified chapman accessible! Have a disability, it may be almost impossible for you to drive a conventional car or van club.... In team golf scramble, everyone on the total score based on your handicap out, everyone the... Foursome handicaps a minivan wheelchair accessible minivans s how the Chapman is very similar the... The shamble, everyone plays their own ball how the Chapman golf works. Through 18 in terms of handicap ( from the second shot to holing out, everyone the. Will be January 1, 2016 before it is the only PGA TOUR event the International used the scale! 'S how it went: it was a 9-hole 4- man scramble players how do you handicap a modified chapman shots their. As in the scramble on every hole you play each others tee location! Determine the “ course handicap of 10 and player B has a USGA handicap index use... To singles matches the longer course Rating for a 3rd time used for the event plotted scores! Have learned at this meeting and other meetings I have never heard of before s sample contained errors. E.G., four-ball match four-ball-stroke play ) competition would not be the strong would. In Chapman, however, was completed by 1978 in order to fully rest muscles... Man scramble the time to review and contact us with this issue or... Of the game of golf as embodied in its ancient and honorable.!, please enter the personal handicap will range from +0.1 to +3.5 ; Otherwise, just enter and. The office spirit of the golf handicap formula scoring is: Double Bogey or worse- 0Bogey- 1Par- 2Birdie- 4Eagle- Par. Team here most cases, try to score points on every hole you play others... 31 clubs is not confined to mixed foursomes set of tees to be used in casual competition between of... Concerns please feel free to reach out to us four-ball match four-ball-stroke play ) handicaps each! Certain amount of planning aggressive play do is go out on the equity of its handicap system, soley s. Must select one modified Chapman the true spirit of the lakes, which aggressive... Been followed, the team handicap of 18 a maximum difference in handicaps for four-ball stroke,... Adjustment procedure is now hidden within a labyrinth of subsections should do three.. Ghin recognized handicap index, use it research, however, was completed by 1978 1! Into wheelchair accessible is a 2-Man team using total handicaps applied at the next handicap procedure meeting. Do not already have a placard, please enter the personal handicap for each player for the event scoring. Edge in match play ’ s simple once you ’ ve done it higher course.! Contained measurement errors used for the specific set of tees to be a typographical error in a!, however, would be the same rounding procedure for adjusting Chapman handicaps with... Very similar to the office players on a team handicap, and then second... Abbreviated Hcp on the score card tells you which holes to subtract handicap strokes from under course! Varying skill levels can compete on an equitable basis creates other inequities one found on.!
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2021-06-16 12:59:51
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https://cs.stackexchange.com/questions/86491/is-given-point-covered-by-any-circle
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# Is given point covered by any circle?
Given $N$ circles and $Q$ points. What is the optimal solution for checking "if point is covered by any circle" for every point? The trivial solution would be $O(QN)$.
• If all circles have the same ratio the problem can be solved online using some data structure that returns the closest circle center to the given point, and checking if the distance between them is less than the ratio. Such data structures are kd-trees, vantage-point and other spatial trees. – Marcelo Fornet Jan 9 '18 at 17:31
• If circles intersection has zero area (i.e. at most share one point) the problem can be solved offline with complexity $O( (Q+N) log(Q+N) )$ If you are interested in this solution let me know and I explain it to you. – Marcelo Fornet Jan 9 '18 at 17:37
Let's suppose you have a spatial data structure that can be built over a set $S$ of points on the plane and expose the functions.
Build(S): Build and return the data structure storing all points in S
ds.Delete(q): Deletes point q from the set S
ds.Search(p): Return the closest point in S to point p
Now let's solve the problem backwards, instead of finding if a point is contained by some circle, lets find points for each circle the points it contains.
C: Set of circles
P: Set of points
ds := Build(P) # Build the data structure using points to be queried
for each circle c in C:
do
p = ds.Search(c.center)
if p is inside circle c:
ds.Delete(p)
while (ds is not empty and one point was deleted)
All the deleted points from the data structure are the points that belongs to at least one circle, remaining points are not contained in any circle, otherwise they were being found when processing such circle.
Let $N:=|C|$ and $Q:=|P|$
The overall complexity of this algorithm is $$O(Build(Q) + (N + Q) \cdot Search(Q) + Q \cdot Delete(Q))$$
Here $Build(Q)$ is the complexity of building the set of $Q$ points, $Search(Q)$ is the complexity of searching for a point in a set of at most $Q$ points and $Delete(Q)$ is the complexity of deleting a point in a set of at most $Q$ points.
Notice that this cost is amortized, since after a point is recognized to be inside a circle it is deleted from the set.
One data structure that provides good balance for this operations is kd-tree, but I encourage you to find other spatial data structures that provides better complexity for your case.
• this solution is pretty impressive, but it is my bad that I did not explained it well. 1. circles may have different radius. 2. circles may share same area. 3. points are not given at once. points are being queried one after another. – Eziz Durdyyev Jan 10 '18 at 7:11
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2021-04-22 03:49:42
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https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/9295/2/a/g/
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Properties
Label 9295.2.a.g Level $9295$ Weight $2$ Character orbit 9295.a Self dual yes Analytic conductor $74.221$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$
Related objects
Newspace parameters
Level: $$N$$ $$=$$ $$9295 = 5 \cdot 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9295.a (trivial)
Newform invariants
Self dual: yes Analytic conductor: $$74.2209486788$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$
$q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + 2 \beta q^{3} + ( 1 - 2 \beta ) q^{4} + q^{5} + ( 4 - 2 \beta ) q^{6} + 2 q^{7} + ( -3 + \beta ) q^{8} + 5 q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + 2 \beta q^{3} + ( 1 - 2 \beta ) q^{4} + q^{5} + ( 4 - 2 \beta ) q^{6} + 2 q^{7} + ( -3 + \beta ) q^{8} + 5 q^{9} + ( -1 + \beta ) q^{10} - q^{11} + ( -8 + 2 \beta ) q^{12} + ( -2 + 2 \beta ) q^{14} + 2 \beta q^{15} + 3 q^{16} + ( 4 - 2 \beta ) q^{17} + ( -5 + 5 \beta ) q^{18} + ( 1 - 2 \beta ) q^{20} + 4 \beta q^{21} + ( 1 - \beta ) q^{22} + 2 \beta q^{23} + ( 4 - 6 \beta ) q^{24} + q^{25} + 4 \beta q^{27} + ( 2 - 4 \beta ) q^{28} + ( 2 + 4 \beta ) q^{29} + ( 4 - 2 \beta ) q^{30} + ( 3 + \beta ) q^{32} -2 \beta q^{33} + ( -8 + 6 \beta ) q^{34} + 2 q^{35} + ( 5 - 10 \beta ) q^{36} + ( 2 - 4 \beta ) q^{37} + ( -3 + \beta ) q^{40} -6 q^{41} + ( 8 - 4 \beta ) q^{42} -6 q^{43} + ( -1 + 2 \beta ) q^{44} + 5 q^{45} + ( 4 - 2 \beta ) q^{46} + 2 \beta q^{47} + 6 \beta q^{48} -3 q^{49} + ( -1 + \beta ) q^{50} + ( -8 + 8 \beta ) q^{51} + ( 6 - 4 \beta ) q^{53} + ( 8 - 4 \beta ) q^{54} - q^{55} + ( -6 + 2 \beta ) q^{56} + ( 6 - 2 \beta ) q^{58} + ( 4 + 4 \beta ) q^{59} + ( -8 + 2 \beta ) q^{60} + ( 2 + 8 \beta ) q^{61} + 10 q^{63} + ( -7 + 2 \beta ) q^{64} + ( -4 + 2 \beta ) q^{66} + ( -4 + 6 \beta ) q^{67} + ( 12 - 10 \beta ) q^{68} + 8 q^{69} + ( -2 + 2 \beta ) q^{70} + 8 \beta q^{71} + ( -15 + 5 \beta ) q^{72} + ( 4 + 2 \beta ) q^{73} + ( -10 + 6 \beta ) q^{74} + 2 \beta q^{75} -2 q^{77} + 4 q^{79} + 3 q^{80} + q^{81} + ( 6 - 6 \beta ) q^{82} + 6 q^{83} + ( -16 + 4 \beta ) q^{84} + ( 4 - 2 \beta ) q^{85} + ( 6 - 6 \beta ) q^{86} + ( 16 + 4 \beta ) q^{87} + ( 3 - \beta ) q^{88} + ( 2 - 8 \beta ) q^{89} + ( -5 + 5 \beta ) q^{90} + ( -8 + 2 \beta ) q^{92} + ( 4 - 2 \beta ) q^{94} + ( 4 + 6 \beta ) q^{96} + ( 2 + 4 \beta ) q^{97} + ( 3 - 3 \beta ) q^{98} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 10 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 10 q^{9} - 2 q^{10} - 2 q^{11} - 16 q^{12} - 4 q^{14} + 6 q^{16} + 8 q^{17} - 10 q^{18} + 2 q^{20} + 2 q^{22} + 8 q^{24} + 2 q^{25} + 4 q^{28} + 4 q^{29} + 8 q^{30} + 6 q^{32} - 16 q^{34} + 4 q^{35} + 10 q^{36} + 4 q^{37} - 6 q^{40} - 12 q^{41} + 16 q^{42} - 12 q^{43} - 2 q^{44} + 10 q^{45} + 8 q^{46} - 6 q^{49} - 2 q^{50} - 16 q^{51} + 12 q^{53} + 16 q^{54} - 2 q^{55} - 12 q^{56} + 12 q^{58} + 8 q^{59} - 16 q^{60} + 4 q^{61} + 20 q^{63} - 14 q^{64} - 8 q^{66} - 8 q^{67} + 24 q^{68} + 16 q^{69} - 4 q^{70} - 30 q^{72} + 8 q^{73} - 20 q^{74} - 4 q^{77} + 8 q^{79} + 6 q^{80} + 2 q^{81} + 12 q^{82} + 12 q^{83} - 32 q^{84} + 8 q^{85} + 12 q^{86} + 32 q^{87} + 6 q^{88} + 4 q^{89} - 10 q^{90} - 16 q^{92} + 8 q^{94} + 8 q^{96} + 4 q^{97} + 6 q^{98} - 10 q^{99} + O(q^{100})$$
Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
−1.41421 1.41421
−2.41421 −2.82843 3.82843 1.00000 6.82843 2.00000 −4.41421 5.00000 −2.41421
1.2 0.414214 2.82843 −1.82843 1.00000 1.17157 2.00000 −1.58579 5.00000 0.414214
$$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
Atkin-Lehner signs
$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$1$$
$$13$$ $$1$$
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9295.2.a.g 2
13.b even 2 1 55.2.a.b 2
39.d odd 2 1 495.2.a.b 2
52.b odd 2 1 880.2.a.m 2
65.d even 2 1 275.2.a.c 2
65.h odd 4 2 275.2.b.d 4
91.b odd 2 1 2695.2.a.f 2
104.e even 2 1 3520.2.a.bn 2
104.h odd 2 1 3520.2.a.bo 2
143.d odd 2 1 605.2.a.d 2
143.l odd 10 4 605.2.g.l 8
143.n even 10 4 605.2.g.f 8
156.h even 2 1 7920.2.a.ch 2
195.e odd 2 1 2475.2.a.x 2
195.s even 4 2 2475.2.c.l 4
260.g odd 2 1 4400.2.a.bn 2
260.p even 4 2 4400.2.b.q 4
429.e even 2 1 5445.2.a.y 2
572.b even 2 1 9680.2.a.bn 2
715.c odd 2 1 3025.2.a.o 2
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 13.b even 2 1
275.2.a.c 2 65.d even 2 1
275.2.b.d 4 65.h odd 4 2
495.2.a.b 2 39.d odd 2 1
605.2.a.d 2 143.d odd 2 1
605.2.g.f 8 143.n even 10 4
605.2.g.l 8 143.l odd 10 4
880.2.a.m 2 52.b odd 2 1
2475.2.a.x 2 195.e odd 2 1
2475.2.c.l 4 195.s even 4 2
2695.2.a.f 2 91.b odd 2 1
3025.2.a.o 2 715.c odd 2 1
3520.2.a.bn 2 104.e even 2 1
3520.2.a.bo 2 104.h odd 2 1
4400.2.a.bn 2 260.g odd 2 1
4400.2.b.q 4 260.p even 4 2
5445.2.a.y 2 429.e even 2 1
7920.2.a.ch 2 156.h even 2 1
9295.2.a.g 2 1.a even 1 1 trivial
9680.2.a.bn 2 572.b even 2 1
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9295))$$:
$$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{3}^{2} - 8$$
Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$8 - 8 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$-28 - 4 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$-28 - 4 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 6 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$4 - 12 T + T^{2}$$
$59$ $$-16 - 8 T + T^{2}$$
$61$ $$-124 - 4 T + T^{2}$$
$67$ $$-56 + 8 T + T^{2}$$
$71$ $$-128 + T^{2}$$
$73$ $$8 - 8 T + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$-124 - 4 T + T^{2}$$
$97$ $$-28 - 4 T + T^{2}$$
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2021-12-07 06:02:03
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https://www.physicsoverflow.org/18917/modes-internal-operator-representations-isometry-compact
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# Why zero modes of the internal Dirac operator must be in representations of the isometry group of the compact space
+ 4 like - 0 dislike
744 views
Imagine a manifold $\mathbb{R}^{1,3}\times{}B$ where $B$ is a compact group-manifold with isometry group $U(1)\times{}SU(2)\times{}SU(3)$.
Let's consider the Dirac equation for a massless Spinor field.
$iD_{1+n}\Psi=0$
and separating the compact space part with the 4-space part
$i(D_4+D_{compact})\Psi=0$
we see that the compact dirac operator might be regarded as a mass operator for the 4 dimensional spinor.
Imagine we are interested in solutions whose mass is zero, that is, solutions whose eigenvalues of the internal Dirac operator is zero.
I am trying to understand the impossibility of arranging these zero mode fermions on complex representations of the isometry group, but for that first I need to understand why the fermions obtained by dimensional reduction must form a representation (be it real or complex) of my isometry group.
This is asserted in [this][1] paper (Witten 1981). On the third paragraph of the ninth page it is said
>If the ground state is a product of four dimensional Minkowski space with one of the $M^{pqr}$ (for our purposes this just means a compact group manidold with the desired isometry group), then the zero modes of the Dirac operator in the internal sace will automatucally form multiplets of $U(1)\times{}SU(2)\times{}SU(3)$ since this is the symmetry of the internal space.
Could someone please be more explicit of why this is so?
+ 4 like - 0 dislike
Contrary to what has been written above, diffeomorphisms do not act on spinors, since not every diffeomorphism preserves the spin structure. The spin structure depends only on the conformal class of the metric, hence you have to restrict yourself to conformal diffeomorphisms; that is, diffeomorphisms $\varphi: M \to M$, such that $\varphi^*g = e^{2\sigma} g$ for some function $\sigma$, where $(M,g)$ is the (pseudo)riemannian spin manifold in question.
At the infinitesimal level, this means that one can define the Lie derivative of a spinor along a conformal Killing vector field, but not along just any vector field. This Lie derivative was defined in the 1972 PhD thesis of Yvonne Kosmann-Schwarzbach, supervised by André Lichnerowicz who had earlier defined the Lie derivative of a spinor along a Killing vector field.
If $K$ is a Killing vector field, so that $\mathcal{L}_K g = 0$, then the Lie derivative (also called $\mathcal{L}_K$) on spinors obeys the following properties:
• $[\mathcal{L}_{K_1},\mathcal{L}_{K_2}] = \mathcal{L}_{[K_1,K_2]}$
• $\mathcal{L}_K (f \psi ) = K(f) \psi + f \mathcal{L}_{K}\psi$
• $[\mathcal{L}_K, \nabla_X] = \nabla_{[K,X]}$
• $\mathcal{L}_K ( X \cdot \psi) = [K,X] \cdot \psi + X \cdot \mathcal{L}_K \psi$
where $K,K_1,K_2$ are Killing vector fields, $\psi$ is a spinor field, $f$ a smooth function and $X$ an arbitrary vector field.
It follows from the latter two properties above that $\mathcal{L}_K$ commutes with the Dirac operator and hence that the Lie derivative along a Killing vector $K$ of a harmonic spinor is again harmonic. (Harmonic spinors are those annihilated by the Dirac operator.) By the first property of $\mathcal{L}_K$ it follows that the space of harmonic spinors is a representation of the Lie algebra of isometries of $(M,g)$ and hence also of the connected component of the group of isometries.
answered Jun 17, 2014 by (2,135 points)
So, long story short, they are representations under conformal isometries, an only conformal isometries right?
+ 3 like - 0 dislike
EDIT of the EDIT: one of the main assertion below (the existence of an action of diffeomorphisms on the space of spinors) is WRONG. (composing by a diffeomorphism of course does not provide a new section of the spinor bundle if the spinor bundle is not preserved by the diffeomorphism). See the answer of José Figueroa-O'Farrill for a CORRECT one.
.
Every diffeomorphism $f$ of $B$ naturally acts on the space of spinors of $B$ by composition: if $\psi(x)$ is a spinor then $\psi(f(x))$ is another spinor. But in general, $f$ does not act on the space of zero modes of the Dirac operator because the action of $f$ does not in general commute with the action of the Dirac operator. But it is the case if $f$ is an isometry. Indeed, as by defintion an isometry preserves the metric, the action induced by an isometry commutes with the covariant derivative and so with the Dirac operator.
EDIT: as explained above, the space of spinors on $B$ is a representation of the group of diffeomorphisms of $B$ (a diffeomorphism acts on a spinor by composition). In particular, it is also a representation of the group of isometries of $B$ (isometries are just special diffeomorphisms). So in order to prove that the space of spinor zero-modes is a representation of the group of isometries, it is enough to show that it is a stable subspace of the space of spinors for the action of the group of isometries. So if $\psi$ is a spinor zero mode and $f$ an isometry, we have to show that $f.\psi$ is a zero mode (where $f. \psi$ denotes the image of $\psi$ by the action of $f$: $(f.\psi)(x)=\psi(f(x))$), i.e. that $D(f.\psi)=0$ where $D$ is the Dirac operator. But if we know that $D$ and $f$ commute, we have $D(f . \psi)= f.(D \psi)=0$ because $D\psi=0$ because $\psi$ is a zero-mode of $D$.
answered Jun 14, 2014 by (4,890 points)
edited Jun 18, 2014 by 40227
I am just beginning to work with spinors and I would greatly appreciate if you could give me a (more or less) step by step explanation of why the fact that the Dirac operator and an isometry commute makes the 4-dimensional spinor corresponding to a zero mode of the compact dirac operator a representation of the isometry group.
I liked you answer but I have a final question. The spinor on B is a rep of the isometry group. But if I consider a spinor of the FULL space, that is a tensor product of a normal spinor and a zero mode on B, is the 4 spinor a rep of the isometry group on its own(and if this is so under what operator) or must I consider the whole $\phi\otimes\psi$ as a rep of the isometry group?
By the way, I haven't seen your EDIT just till today and I had started a bounty on stack.exchange. if you answer, its yours.
thanks for the honesty
+ 1 like - 0 dislike
I think it could be simply that the Dirac operator is invariant under isometries, so if $\phi$ is an isometry and $\psi$ a solution to $$D\psi = 0,$$ then $\phi^* \psi$ is also a solution, where $\phi^*$is pullback. Then it would be similar to how harmonic functions $f$ on the sphere -- $\nabla^2 f = 0$ -- come in representations of the rotation group, the $Y^l_m$.
In more detail if $\phi$ is a diffemorphism, that in coordinates takes the form $y^\mu = y^\mu(x^\nu)$ (not a tensor expression), and $v^\mu$ is a vector field, then we can define a vector field $$(\phi_* v^\mu)(\phi(p)) = \frac{\partial y^\mu}{\partial x^\nu} v^\nu(p)$$ called the pushforward of $v^\mu$. Naturally we can pushforward any tensor, in particular the metric. By definition $\phi$ is an isometry if $$(\phi_* g_{\mu\nu})(\phi(p))= g_{\mu\nu}(\phi(p)).$$
This means that if we have any tetrad (also known as a vierbein or a frame), that is a set of vector fields $e_a^\mu$ such that $e_{a\mu} e^\mu_b = \eta_{ab}$ for some symmetric matrix $\eta_{ab}$ with signature $+---$, it is pushed forward to another tetrad. I let $\eta_{ab}$ be general because in spinor problems it is more natural to use a null tetrad $$\eta_{ab} = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0\end{pmatrix}.$$ Since $\eta_{ab}$ has zeros on the diagonal all the tetrad vectors are null. It is well known (see for example Spinors and space-time or the Newman-Penrose paper) that to every null tetrad corresponds exactly two bases for two-spinors, called dyads, say $(o^A, \iota^A)$ and $(-o^A, \iota^A)$. Thus at least for isometries connected to the identity, the pushforward of tetrads lifts to a pushforward of dyads. (However when the isometry group isn't simply connected this might not be continuous globally, but I think it doesn't matter here, since we can consider isometries close to the identity, which will take us to Lie algebra representations, and then we integrate them, and discard the representations that require passing to the simply connected cover.)
Since we can pushforward dyads we can pushforward two-spinors (by linearity), since we can pushforward two-spinors we can pushforward Dirac spinors. $\newcommand{\Dslash}{\!\not D}$ In particular for a Dirac spinor $\psi$, $\Dslash\psi$ is of course also a Dirac spinor, so $$\beta = \phi_* (\Dslash\psi) = \phi_* (\Dslash \phi^* \phi_* \psi)$$ makes sense, where $\phi^*$ as the inverse of $\phi_*$ so the second equality is just inserting the identity. Now $\phi_* \Dslash \phi^*$ defines a differential operator, it is the transformed Dirac operator under the isometry $\phi$. But since the Dirac operator is defined by the metric and $\phi$ preserves the metric, this must be just the Dirac operator again. (You can probably make this argument more convincing.)
Thus we have established that $$\beta = \Dslash (\phi_*\psi).$$ In particular if $\beta = 0$, so that $\psi$ is a zero mode for the Dirac operator, then $\tilde{\psi} = \phi_* \psi$ is also a solution. Thus the isometry group (or at least its Lie algebra) acts on zero modes.
This post imported from StackExchange Physics at 2014-06-17 07:48 (UCT), posted by SE-user Robin Ekman
answered Jun 13, 2014 by (215 points)
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2018-05-27 19:45:10
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https://www.physicsforums.com/threads/anstract-algebra-proof-cosets.310222/
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Anstract Algebra proof - cosets
Homework Statement
Let H be a subgroup of G such that g^-1hg is an element of H for all g in G and all h in H. Show that every left coset gH is the same as the right coset Hg.
The Attempt at a Solution
need to show gh1=h2g
I know I need to show this, but am unsure on how to.
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2022-05-24 06:26:11
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https://physics.stackexchange.com/questions/348948/what-is-the-reference-frame-for-the-velocity-of-an-object-in-the-universe/348977
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What Is The Reference Frame For The Velocity Of An Object In The Universe?
As I understand it, movement of two objects is relative to one another. Velocity is always in relation of one thing to another. A car travels at 100km/h (60mi/h) relative to the rotating Earth surface it's driving on. A rocket propels upwards relative to the rotating Earth surface.
If we choose a different reference point, things change. If we use the center of the Earth as a reference point, the car, parked at the side of the road, is technically rotating with the Earth. If you were to drive east at the same speed as earth rotates, we would now be going twice as fast. And if we drove west at the same speed as the earth rotates, we would be stationary relative to the center of the Earth.
Similarly, if we launch a rocket in the direction Earth is rotating, at the same speed as Earth is moving around the sun, then relative to the sun it is moving twice as fast. If we launched the rocket in the opposite direction, then relative to the sun it wouldn't be moving (other than starting to fall straight into the sun).
Thus, there really isn't one speed for any object but multiple speeds depending on what you use as the frame of reference. And right now, even staying still, we are moving in all sorts of ways from the universe, the sun, the Earth around the sun, and the Earth's rotation.
We know that no object can travel at the speed of light. As an object goes to approach the speed of light, the amount of energy increases exponentially until it reaches infinity. However, a velocity must be in relation to something. That frame of reference must be defined somewhere.
If an object is already moving at half of the speed of light, and from it we launch a smaller object as fast as we possibly can, the speed of that object relative to us could only ever get to half the speed of light, because the total speed could never reach the speed of light.
And if we launched that smaller object in the other direction, could it not therefore move at up to 1.5 times the speed of light relative to our frame of reference?
From this then, we could see that the object we launched in the same direction would take substantially more energy to accelerate than the object launched in the opposite direction. Furthermore, if we slow down or speed up our spaceship, the relationship would change.
Could it not then be possible to launch objects in all direction and be able to determine the exact speed we are moving at, and from there, calculate what the reference frame of the universe is?
• You are standing on the ground. Alice is traveling eastward at speed $.5$ (that is, half the speed of light) relative to you. She launches objects eastward and westward at speeds relative to her of (say) $.9$. Then relative to you, the eastbound object travels at speed $.9655$ and the westbound object at speed $.2759$ (I got these by applying the relativistic velocity addition formula, which you can Google). Nothing stops her from launching these objects at any speed she wants between $0$ and $1$ relative to herself, so your scheme can't be used to find her "absolute" speed. – WillO Jul 27 '17 at 23:19
• PS---although this is a very elementary question, I do not think that the OP could reasonably have been expected to know what phrases to Google for, so I think the downvote is uncalled for. – WillO Jul 27 '17 at 23:20
• The kinetic energy of an object is also reference-frame-dependent. You also contradict yourself. You say that we know that objects can't move the speed of light, c, then you propose that something moves at 1.5c without showing any attempt to research special relativity velocity equations. Your logic doesn't flow well. – Bill N Jul 28 '17 at 0:35
In an expanding universe you have the peculiar velocity which is measured relative to the cosmic microwave background (in the frame of a comoving observer the CMB has the same wavelength in all directions, while an object with peculiar velocity sees a dipole in the CMB).
If you measure an object's velocity relative to a distant observer you also have to add the recessional velocity that can sure enough exceed the speed of light, which is of course allowed by general relativity.
So the total velocity between 2 objects is their peculiar velocities added like in special relativity and the recessional velocity between both added classicaly.
Example: if two objects are seperated by the distance of 2 hubble radii, the recessional velocity between both is +2c. If they locally travel toward each other with c/2 relative to the CMB, their special relativistic velocity toward each other is -(v₁+v₂)/(1-v₁·v₂/c²)=-4/5c. But since the recessional velocity between both is 2c, the total velocity is 2c-4/5c=+6/5c, so they still would move away from each other with more than c.
The first answer has the numbers right, and the rationale. This is just to explain the phrase in that answer 'the recessional velocities are added classically'. They really are not added classically but calculated from General Relativity (GR) and the answers are the same, just because the way the Hubble parameter is defined, and because both cosmological observers use the same reference frame, defined uniquely by the expansion of the universe and it's homogeneity. See next.
[as an aside, note that you added velocities classically, but the peculiar velocities need to be added as in the answer, using special relativity]
The GR equations lead to Hubble law which relate velocity to distance as
v = Hr
And thus we get the Hubble distance when v = c as c/H. That Hubble distance changes with time, as H is not a constant, but not too fast. Usually people call the Hubble distance the Hubble distance now, where H is the value of the Hubble parameter now. Note that the units of the Hubble parameter is velocity/distance, usually stated as 68 km/sec/Mpsec, for its value now, per the standard cosmological model at this point.
Note in https://en.m.wikipedia.org/wiki/Hubble%27s_law tHat the so called Hubble radius $(c/H)_{now}$ is about 14.4 billion light years, slightly larger than the universe radius now of about 13.8 billion light years. They are not the same number because of the way the expansion has and is occurring, that it is not an exactly linear expansion. This is according to the standard cosmology model.
The value of H now is denoted as $H_0$, meaning now. It is also called the Hubble parameter now. The Hubble radius is also called the Hubble horizon. See it and a number of other useful horizon definitions at https://en.m.wikipedia.org/wiki/List_of_cosmological_horizons
Next, the 68 Kms/sec/Mpsec means that if you are about 14.4 billion light years away from me your speed wrt me is and if another galaxy is another 14.4 billion light years from you their speed is c wrt you. And their speed wrt to me is 68 Kms/sec/Mpsec times 28.8 billion light years (sorry, you have to change Psec to light years, and the math comes out) comes out to be twice as before, or 2c. Both you and me would be using the comoving reference frame, and would agree on times. We both would have to measure those distances at cosmological time (= comoving time) now. It is a well defined frame of reference, where the CMB is homogeneous and isotropic, and where we would all agree on the age of the universe. Sso it winds up being additive for the velocities. But it is a non-trivial result, certainly not just adding them classically.
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2019-10-15 03:53:20
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https://stacks.math.columbia.edu/tag/00JL
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A graded ring will be for us a ring $S$ endowed with a direct sum decomposition $S = \bigoplus _{d \geq 0} S_ d$ of the underlying abelian group such that $S_ d \cdot S_ e \subset S_{d + e}$. Note that we do not allow nonzero elements in negative degrees. The irrelevant ideal is the ideal $S_{+} = \bigoplus _{d > 0} S_ d$. A graded module will be an $S$-module $M$ endowed with a direct sum decomposition $M = \bigoplus _{n\in \mathbf{Z}} M_ n$ of the underlying abelian group such that $S_ d \cdot M_ e \subset M_{d + e}$. Note that for modules we do allow nonzero elements in negative degrees. We think of $S$ as a graded $S$-module by setting $S_{-k} = (0)$ for $k > 0$. An element $x$ (resp. $f$) of $M$ (resp. $S$) is called homogeneous if $x \in M_ d$ (resp. $f \in S_ d$) for some $d$. A map of graded $S$-modules is a map of $S$-modules $\varphi : M \to M'$ such that $\varphi (M_ d) \subset M'_ d$. We do not allow maps to shift degrees. Let us denote $\text{GrHom}_0(M, N)$ the $S_0$-module of homomorphisms of graded modules from $M$ to $N$.
At this point there are the notions of graded ideal, graded quotient ring, graded submodule, graded quotient module, graded tensor product, etc. We leave it to the reader to find the relevant definitions, and lemmas. For example: A short exact sequence of graded modules is short exact in every degree.
Given a graded ring $S$, a graded $S$-module $M$ and $n \in \mathbf{Z}$ we denote $M(n)$ the graded $S$-module with $M(n)_ d = M_{n + d}$. This is called the twist of $M$ by $n$. In particular we get modules $S(n)$, $n \in \mathbf{Z}$ which will play an important role in the study of projective schemes. There are some obvious functorial isomorphisms such as $(M \oplus N)(n) = M(n) \oplus N(n)$, $(M \otimes _ S N)(n) = M \otimes _ S N(n) = M(n) \otimes _ S N$. In addition we can define a graded $S$-module structure on the $S_0$-module
$\text{GrHom}(M, N) = \bigoplus \nolimits _{n \in \mathbf{Z}} \text{GrHom}_ n(M, N), \quad \text{GrHom}_ n(M, N) = \text{GrHom}_0(M, N(n)).$
We omit the definition of the multiplication.
Lemma 10.56.1. Let $S$ be a graded ring. Let $M$ be a graded $S$-module.
1. If $S_+M = M$ and $M$ is finite, then $M = 0$.
2. If $N, N' \subset M$ are graded submodules, $M = N + S_+N'$, and $N'$ is finite, then $M = N$.
3. If $N \to M$ is a map of graded modules, $N/S_+N \to M/S_+M$ is surjective, and $M$ is finite, then $N \to M$ is surjective.
4. If $x_1, \ldots , x_ n \in M$ are homogeneous and generate $M/S_+M$ and $M$ is finite, then $x_1, \ldots , x_ n$ generate $M$.
Proof. Proof of (1). Choose generators $y_1, \ldots , y_ r$ of $M$ over $S$. We may assume that $y_ i$ is homogeneous of degree $d_ i$. After renumbering we may assume $d_ r = \min (d_ i)$. Then the condition that $S_+M = M$ implies $y_ r = 0$. Hence $M = 0$ by induction on $r$. Part (2) follows by applying (1) to $M/N$. Part (3) follows by applying (2) to the submodules $\mathop{\mathrm{Im}}(N \to M)$ and $M$. Part (4) follows by applying (3) to the module map $\bigoplus S(-d_ i) \to M$, $(s_1, \ldots , s_ n) \mapsto \sum s_ i x_ i$. $\square$
Let $S$ be a graded ring. Let $d \geq 1$ be an integer. We set $S^{(d)} = \bigoplus _{n \geq 0} S_{nd}$. We think of $S^{(d)}$ as a graded ring with degree $n$ summand $(S^{(d)})_ n = S_{nd}$. Given a graded $S$-module $M$ we can similarly consider $M^{(d)} = \bigoplus _{n \in \mathbf{Z}} M_{nd}$ which is a graded $S^{(d)}$-module.
Lemma 10.56.2. Let $S$ be a graded ring, which is finitely generated over $S_0$. Then for all sufficiently divisible $d$ the algebra $S^{(d)}$ is generated in degree $1$ over $S_0$.
Proof. Say $S$ is generated by $f_1, \ldots , f_ r \in S$ over $S_0$. After replacing $f_ i$ by their homogeneous parts, we may assume $f_ i$ is homogeneous of degree $d_ i > 0$. Then any element of $S_ n$ is a linear combination with coefficients in $S_0$ of monomials $f_1^{e_1} \ldots f_ r^{e_ r}$ with $\sum e_ i d_ i = n$. Let $m$ be a multiple of $\text{lcm}(d_ i)$. For any $N \geq r$ if
$\sum e_ i d_ i = N m$
then for some $i$ we have $e_ i \geq m/d_ i$ by an elementary argument. Hence every monomial of degree $N m$ is a product of a monomial of degree $m$, namely $f_ i^{m/d_ i}$, and a monomial of degree $(N - 1)m$. It follows that any monomial of degree $nrm$ with $n \geq 2$ is a product of monomials of degree $rm$. Thus $S^{(rm)}$ is generated in degree $1$ over $S_0$. $\square$
Lemma 10.56.3. Let $R \to S$ be a homomorphism of graded rings. Let $S' \subset S$ be the integral closure of $R$ in $S$. Then
$S' = \bigoplus \nolimits _{d \geq 0} S' \cap S_ d,$
i.e., $S'$ is a graded $R$-subalgebra of $S$.
Proof. We have to show the following: If $s = s_ n + s_{n + 1} + \ldots + s_ m \in S'$, then each homogeneous part $s_ j \in S'$. We will prove this by induction on $m - n$ over all homomorphisms $R \to S$ of graded rings. First note that it is immediate that $s_0$ is integral over $R_0$ (hence over $R$) as there is a ring map $S \to S_0$ compatible with the ring map $R \to R_0$. Thus, after replacing $s$ by $s - s_0$, we may assume $n > 0$. Consider the extension of graded rings $R[t, t^{-1}] \to S[t, t^{-1}]$ where $t$ has degree $0$. There is a commutative diagram
$\xymatrix{ S[t, t^{-1}] \ar[rr]_{s \mapsto t^{\deg (s)}s} & & S[t, t^{-1}] \\ R[t, t^{-1}] \ar[u] \ar[rr]^{r \mapsto t^{\deg (r)}r} & & R[t, t^{-1}] \ar[u] }$
where the horizontal maps are ring automorphisms. Hence the integral closure $C$ of $S[t, t^{-1}]$ over $R[t, t^{-1}]$ maps into itself. Thus we see that
$t^ m(s_ n + s_{n + 1} + \ldots + s_ m) - (t^ ns_ n + t^{n + 1}s_{n + 1} + \ldots + t^ ms_ m) \in C$
which implies by induction hypothesis that each $(t^ m - t^ i)s_ i \in C$ for $i = n, \ldots , m - 1$. Note that for any ring $A$ and $m > i \geq n > 0$ we have $A[t, t^{-1}]/(t^ m - t^ i - 1) \cong A[t]/(t^ m - t^ i - 1) \supset A$ because $t(t^{m - 1} - t^{i - 1}) = 1$ in $A[t]/(t^ m - t^ i - 1)$. Since $t^ m - t^ i$ maps to $1$ we see the image of $s_ i$ in the ring $S[t]/(t^ m - t^ i - 1)$ is integral over $R[t]/(t^ m - t^ i - 1)$ for $i = n, \ldots , m - 1$. Since $R \to R[t]/(t^ m - t^ i - 1)$ is finite we see that $s_ i$ is integral over $R$ by transitivity, see Lemma 10.36.6. Finally, we also conclude that $s_ m = s - \sum _{i = n, \ldots , m - 1} s_ i$ is integral over $R$. $\square$
Comment #675 by Keenan Kidwell on
In the proof of 077G, the bottom horizontal map in the diagram should be $r\mapsto t^{\deg(r)}r$, right?
Comment #6518 by Zhenhua Wu on
In the first line, "ring $S$ endowed with a direct sum decomposition $S=\bigoplus_{d\geq 0}S_d$", I was always wondering about which kind of structure are those $S_d$ and which kind of direct sum is this? So I suggest it changes to "ring $S$ endowed with a direct sum decomposition of abelian groups $S=\bigoplus_{d\geq 0}S_d$".
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2022-12-08 15:22:54
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https://math.stackexchange.com/questions/25902/expected-value-for-a-poisson-process
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# Expected value for a Poisson Process
A machine works for an exponentially distributed time with rate μ and then fails. A repair crew checks the machine at times distributed according to a Poisson process with rate λ; if the machine is found to have failed then it is immediately replaced. Find the expected time between replacements of machines.
What I have so far:
I believe that if M represents the failure time of the machine and T represents the time when the machine is checked, then the probability of the machine being replaced is:
P(M < T) = μ/(μ+λ)
Since both M and T are exponentially distributed. So, that got me into thinking that this as Exponential(μ+λ) distributed, and so the expected time for the machine to be fixed is just (μ+λ)/μ.
Another thought was to find the expected time of the failure of the machine (1/μ) and then the expected time of checking the machine conditioned on the machine having failed, which I think is just 1/λ by the no memory property.
However, I feel like I do not understand what I am doing and am just aimlessly trying to solve the question. I can't seem to find a similar problem or example in my textbook (I'm on Ch 5 of "Introduction to Probability Models" by Ross, 10ed), and if anyone can help me with the problem or refer me to some good resources, I would be extremely grateful.
Thank you!
Quick edit: It looks like the question asks for the expected time between replacements, so now I feel like I did things completely wrong :(
• $\frac{\mu+\lambda}{\mu}$ can't be right. This is unitless, and you need units of time, like $\frac{1}{\mu}$ or $\frac{1}{\lambda}$. – Ross Millikan Mar 9 '11 at 4:56
• @Ross: Thank you, I feel dumb. μ and λ are rates, so any times would need to be expressed by 1/μ and 1/λ. How about this: The expected time until failure is 1/μ, and the expected time until the machine is checked is 1/λ. Since the event that the mechanic checks the machine is exponential with rate λ, and that it is memoryless, the time after the machine breaks and is checked is just 1/λ, and so the mean time until the machine is replaced is just 1/μ + 1/λ. – user7990 Mar 9 '11 at 5:19
• I agree with that. It is just what you were saying in the paragraph that starts with "Another thought". I think the point of the exercise is to make sure you know the expected time of an exponential distribution and that a Poisson process is another name for the same thing. – Ross Millikan Mar 9 '11 at 5:40
• Yes, I am confused by the relationship between the exponential distribution and the Poisson process. I need to review. – user7990 Mar 9 '11 at 5:57
The time $T$ between two successive replacements is the sum of the lifetime $D$ of the machine and the elapsed time $R$ between the moment when it fails and the moment when it is replaced. By hypothesis $D$ is exponential with parameter $\mu$ and $R$ is exponential with parameter $\lambda$. Hence $E(T)=E(D)+E(R)=1/\mu+1/\lambda$, as suggested in one paragraph of your post.
Note that the random variable $T$ is never exponentially distributed. Rather, when $\lambda\ne\mu$, its probability density function $f_T$ is defined on $t\ge0$ by $\displaystyle f_T(t)=\frac{\lambda\mu}{\lambda-\mu}(\mathrm{e}^{-\mu t}-\mathrm{e}^{-\lambda t})$ and, in the degenerate case $\lambda=\mu$, $f_T(t)=\lambda^2t\mathrm{e}^{-\lambda t}.$
The time in between replacements of machines will be exponentially distributed.
• PEV: No. – Did May 28 '11 at 18:53
I was trying to understand this problem myself but couldnt find an easy to understand answer until I solved a similar problem.
Basically, this problem is analogous to the classic single server problem. In this case, we have server service time $$M\sim \exp(\mu)$$ (that is equivalent to the machine failure time) and customers arrival following a Poisson process with rate $$\lambda$$. In this case, the customers that arrive and find the server occupied will leave immediately. Therefore the machine failure can be regarded as an occupied server and checks as the arriving customers. Given this representation, the solution is straightforward. Let $$T$$ be the time the server is occupied by successive customers (time between replacement of machines) and $$C$$ be the time between customer arrivals (time between checks by the repair crew), we then have
\begin{aligned} E[T] &= E[T|C>M]P(C>M) + E[T|CM) + E[T+C]P(CM)} \\ &= \frac{1}{\lambda}\frac{\lambda+\mu}{\mu} \\ &= \frac{1}{\mu} + \frac{1}{\lambda}. \end{aligned} Where $$E[T|C because we take into account the time the check was made and the time between replacement of the machine again due to the restarting of the process resulting from the memoryless property of exponential random variables.
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2019-07-20 01:24:11
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https://math.stackexchange.com/questions/1253009/showing-a-certain-cyclotomic-polynomial-must-split
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# Showing a certain cyclotomic polynomial must split
This is part of a question from a book, I'll post the rest if anyone would like me to, but the bit I'm stuck with is:
Suppose $\deg(L/K) = p$, a prime not equal to $\operatorname{char}K$, and $L$ is algebraically closed. Suppose (if possible) that $p \gt 2$. Show that the cyclotomic polynomials $\Phi_p$ and $\Phi_{p^2}$ both split over $K$.
It eventually leads to a contradiction from assuming $p \gt 2$. But the splitting of $\Phi_{p^2}$ is the only bit I can't get: I can see that $\Phi_p$ splits ...
• Are you reading a proof of the Artin--Schreier theorem? The premise that an algebraically closed field $L$ contains a proper subfield $K$ in which $[L:K]$ is finite forces $[L:K] = 2$ by the Artin--Schreier theorem, so your whole premise is far more restrictive than you suggest unless your intended goal is to prove $p = 2$ as part of the proof of the Artin--Schreier theorem. – KCd Apr 26 '15 at 21:29
• It's an exercise from Garling's 'Course in Galois Theory', and it pretty much guides you to a proof of what I now know is called the Artin-Schreier theorem, thank you! if $deg(L/K) \lt$ infinity, then the Galois group contains a subgroup of order p for all p dividing that degree: and for p not equal to 2, this gives a contradiction. Then, there's a bit about how $deg(L/K) = 4$ is also impossible, and it $L/K$ is a splitting field for $x^2 + 1$ – Latimer Leviosa Apr 27 '15 at 7:50
• The original proof by Artin and Schreier uses cyclotomic polynomials, but that can be avoided. See math.uconn.edu/~kconrad/blurbs/galoistheory/artinschreier.pdf. – KCd Apr 27 '15 at 12:22
• Thanks, I'll have a look :) – Latimer Leviosa Apr 27 '15 at 14:30
The fact that $\Phi_p$ splits follows already from $\deg \Phi_p=p-1$, which is not a divisor of $p$ (for $p>2$). So $K$ contains all primitive $p$th roots of unity. In order to show that $\Phi_{p^2}$ splits it suffices to find just one primitive $p^2$th root of unity in $K$, for all other roots are powers of this.
Let $\xi\in L$ be a primitive $p^3$th root of unity and let $\zeta:=\xi ^p$. An element $\sigma$ of the Galois group must act by mapping $\xi\mapsto \xi^m$. As $\sigma^p=1$, we have $m^p\equiv 1\pmod {p^3}$. We conclude $m\equiv 1\pmod p$. Write $m=1+ap$. Then (using $p>2$) we have $m^p\equiv 1+ap^2\pmod{p^3}$ so that $a\equiv 0\pmod p$. Hence $m=1+bp^2$ for some $b$. Then $\sigma(\zeta)=\zeta^m=\zeta\cdot \zeta^{bp^2}=\zeta$ because $\zeta$ is a $p^2$th root of unity. Since $\zeta$ is fixed under the Galois group, we conclude $\zeta\in K$.
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2019-12-06 21:42:54
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https://jira.lsstcorp.org/browse/DM-17654
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# PPDB Scaling Test in Google Cloud
XMLWordPrintable
#### Details
• Type: Story
• Status: Done
• Resolution: Done
• Fix Version/s: None
• Component/s:
• Labels:
• Story Points:
8
• Sprint:
DB_S19_02, DB_S19_03
• Team:
Data Access and Database
#### Description
Perform PPDB scaling test in Google Cloud with other database technologies like postgres and using node configuration not available at NCSA.
#### Attachments
1. dm-17654-57k-pg-insert_combined.png
62 kB
2. dm-17654-57k-pg-select_combined.png
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3. dm-17654-57k-pg-visit_real.png
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4. dm-17654-ora-select_real.png
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5. dm-17654-ora-store_real.png
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6. dm-17654-ora-visit_real.png
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7. dm-17654-pg-select_real.png
24 kB
8. dm-17654-pg-store_real.png
22 kB
9. dm-17654-pg-visit_real.png
27 kB
#### Activity
Hide
Andy Salnikov added a comment - - edited
Ignoring for now store time and trying to understand what to do with DIASource select time. Like in Oracle it scales linearly and even though it's faster with Postgres than with Oracle we face the same issue here - we cannot scale it to 12 months needed by AP. I think we are at the limit of affordable time budget at 30k visits already (roughly 1 month). So we need to think how to break that linear scaling (and there is no guarantee it will stay linear at higher scale). I could think of couple of options:
• Keep data on (fast) disk but parallelize access to it. I'm 99% that we are going to need many more servers to scale system that way. Current approach is already running ~200 parallel tiles and I believe our current bottleneck is on server side. We might need ~10 servers and reasonably low overhead for doing things in parallel (that feels like we need a real distributed database system).
• Keeping more data in memory. If we could pre-load all necessary (for current visit) DIASource records into memory the SELECT would run much faster. This could be either SQL table that is kept in memory and updated for each new visit in advance, or some cache service a la memcached.
Looking at the above numbers, DIASource needs about 0.5TB for one month of data, then 12 months is about 6TB in total (maybe little less). I do not think we could have a machine with so much RAM, so it probably means that we should forget about keeping it all in memory. Second option is to preload data that is needed for the next night and that means that that we could reduce size by about factor of 3, bringing us to 2TB. This is probably more realistic but still rather large number, possible issues here is how much time is needed to re-populate the cache if something happens during the night. Next option is to preload data as we go during the night, few minutes in advance if we know the (approximate) pointing for the next few visits. This should reduce the size of memory needed for it but it has its own drawbacks - loading will be done in parallel with other activities (writing to DIAObjects and reading/writing other tables) and will compete for resources.
Show
Hide
Andy Salnikov added a comment -
Independently of DIASource there is one optimization that we could do for DIAObjectLast table - just keep it whole in memory (re-populate it once on database restart). DIAObjectLast is not a big concern now though, it is the fastest of the three (and smallest) so I'm not going to look at it in this ticket anyways.
Show
Andy Salnikov added a comment - Independently of DIASource there is one optimization that we could do for DIAObjectLast table - just keep it whole in memory (re-populate it once on database restart). DIAObjectLast is not a big concern now though, it is the fastest of the three (and smallest) so I'm not going to look at it in this ticket anyways.
Hide
Andy Salnikov added a comment - - edited
I ran the test for a bit longer extending it 57k visits to see if it timing stays linear with visits. Here is a bunch of plots from this extended test.
The total per-visit time continues to grow afrer 30k visit but it is not very linear anymore (or it is linear but with different slope):
Select time still scales linearly, major contribution comes from selecting DIASources:
but insert time stays more or less constant after 30k visits, largest contribution to insert time comes from DIAObject:
That time stays about constant after 30k, this can be explained by the number of inserted DIAObject as it should stabilize after 30k as we have a cutoff of 30 days for forced photometry of "noise" objects (at least this is what is implemented in my prototype).
DIASource select is clearly still a major concern as it will continue to grow until we reach 12 months of history cutoff. Significant insert time is likely a limitation of cloud storage.
Show
Andy Salnikov added a comment - - edited I ran the test for a bit longer extending it 57k visits to see if it timing stays linear with visits. Here is a bunch of plots from this extended test. The total per-visit time continues to grow afrer 30k visit but it is not very linear anymore (or it is linear but with different slope): Select time still scales linearly, major contribution comes from selecting DIASources: but insert time stays more or less constant after 30k visits, largest contribution to insert time comes from DIAObject: That time stays about constant after 30k, this can be explained by the number of inserted DIAObject as it should stabilize after 30k as we have a cutoff of 30 days for forced photometry of "noise" objects (at least this is what is implemented in my prototype). DIASource select is clearly still a major concern as it will continue to grow until we reach 12 months of history cutoff. Significant insert time is likely a limitation of cloud storage.
Hide
Andy Salnikov added a comment -
Data/table sizes after 57k visits:
table_name | row_estimate | total | index | toast | table ---------------------------+--------------+------------+------------+------------+------------ DiaObject | 3.82072e+09 | 3003 GB | 352 GB | 8192 bytes | 2651 GB DiaSource | 8.59906e+08 | 847 GB | 118 GB | | 729 GB DiaForcedSource | 3.82071e+09 | 548 GB | 248 GB | | 301 GB DiaObjectLast | 3.14808e+08 | 86 GB | 29 GB | | 58 GB ---------------------------+--------------+------------+------------+------------+------------ Totals | | 4484 GB | 747 GB | | 3739 GB
and individual relations/indices:
relation | size ----------------------------------+--------- public.DiaObject | 2650 GB public.DiaSource | 729 GB public.DiaForcedSource | 301 GB public.PK_DiaForcedSource | 146 GB public.PK_DiaObject | 146 GB public.IDX_DiaObject_htmId20 | 104 GB public.IDX_DiaObject_valStart | 101 GB public.IDX_DiaFSource_ccdVisitId | 101 GB public.DiaObjectLast | 58 GB public.PK_DiaSource | 26 GB public.IDX_DiaSource_htmId20 | 23 GB public.IDX_DiaSource_diaObjId | 23 GB public.IDX_DiaSource_ccdVisitId | 23 GB public.IDX_DiaSource_ssObjId | 23 GB public.PK_DiaObjectLast | 15 GB public.IDX_DiaObjLast_diaObjId | 14 GB public.PpdbProtoVisits | 2472 kB public.IDX_PpdbProtoVisits_vTime | 1264 kB public.PK_PpdbProtoVisits | 1264 kB
Show
Andy Salnikov added a comment - Data/table sizes after 57k visits: table_name | row_estimate | total | index | toast | table ---------------------------+--------------+------------+------------+------------+------------ DiaObject | 3.82072e+09 | 3003 GB | 352 GB | 8192 bytes | 2651 GB DiaSource | 8.59906e+08 | 847 GB | 118 GB | | 729 GB DiaForcedSource | 3.82071e+09 | 548 GB | 248 GB | | 301 GB DiaObjectLast | 3.14808e+08 | 86 GB | 29 GB | | 58 GB ---------------------------+--------------+------------+------------+------------+------------ Totals | | 4484 GB | 747 GB | | 3739 GB and individual relations/indices: relation | size ----------------------------------+--------- public.DiaObject | 2650 GB public.DiaSource | 729 GB public.DiaForcedSource | 301 GB public.PK_DiaForcedSource | 146 GB public.PK_DiaObject | 146 GB public.IDX_DiaObject_htmId20 | 104 GB public.IDX_DiaObject_valStart | 101 GB public.IDX_DiaFSource_ccdVisitId | 101 GB public.DiaObjectLast | 58 GB public.PK_DiaSource | 26 GB public.IDX_DiaSource_htmId20 | 23 GB public.IDX_DiaSource_diaObjId | 23 GB public.IDX_DiaSource_ccdVisitId | 23 GB public.IDX_DiaSource_ssObjId | 23 GB public.PK_DiaObjectLast | 15 GB public.IDX_DiaObjLast_diaObjId | 14 GB public.PpdbProtoVisits | 2472 kB public.IDX_PpdbProtoVisits_vTime | 1264 kB public.PK_PpdbProtoVisits | 1264 kB
Hide
Andy Salnikov added a comment -
Couple of trivial updates to dax_ppdb and l1dbproto, no review necessary.
Show
Andy Salnikov added a comment - Couple of trivial updates to dax_ppdb and l1dbproto, no review necessary.
#### People
Assignee:
Andy Salnikov
Reporter:
Vaikunth Thukral
Watchers:
Andy Salnikov, Fritz Mueller, Vaikunth Thukral
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2021-08-02 03:26:22
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https://www.wyzant.com/resources/answers/topics/precalc?page=
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165 Answered Questions for the topic Precalc
17d
#### sin (theta+48°) = cos 2theta. What is the second step to find theta? What does theta equal?
Any help with this would be greatly appreciated.
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The principal P is borrowed and the loan's future value A at time t is given. Determine the loan's simple interest rate r to the nearest tenth of a percent.P = $2300.00, A = $2328.75, t =... more
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#### You can afford monthly deposits of $250 into an account that pays 2.7% compounded monthly. How long will it be until you have$10,000 to buy a boat?
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A circular spinner is divided into 9 sectors of equal area. There are 2 red sectors, 3 blue, 3 yellow, and 1 green. Consider the experiment of spinning the spinner once. Find the probability... more
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#### Find and simplify the expression if f(x)=x2−3. f(2+h)
Find and simplify the expression if f(x)=x2−3.f(2+h)f(2+h)= _______ (Simplify your answer. Do not factor.)
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#### Use the future value formula to find the indicated value.
Use the future value formula to find the indicated value.FV=$5774; n=9; i=0.06; PMT=?PMT=$_______(Round to the nearest cent.)
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#### Use the future value formula to find the indicated value.
Use the future value formula to find the indicated value.n=50; i=0.04; PMT=$92; FV=?FV=$____________(Round to the nearest cent.)
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Use the compound interest formula to find the future value A for the following values.P=$1,500i=0.046n=27A=$______ (Round to the nearest cent.)
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#### If one can is selected at random from the cooler, determine the probability that the soda selected is root beer, ginger ale, or cola.
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#### The principal P is borrowed and the loan's future value A at time t is given. Determine the loan's simple interest rate r to the nearest tenth of a percent.
The principal P is borrowed and the loan's future value A at time t is given. Determine the loan's simple interest rate r to the nearest tenth of a percent.P = $3400.00, A = $3472.25, t =... more
10/26/20
#### Use technology and the future value formula to find i. FV=$7,426; PMT=$600; n=9; i=?
Use technology and the future value formula to find i.FV=$7,426; PMT=$600; n=9; i=?i=______(Round to two decimal places as needed.)
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2021-01-24 06:05:02
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http://www.fractalforums.com/the-3d-mandelbulb/here-are-the-first-true-3d-mandelbrot-images/
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News: Check out the originating "3d Mandelbulb" thread here
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Author Topic: 3d Burning Ship & extension to 3d Mset for n=2,6,10..., odd n; & another formul (Read 8982 times) Description: Sign assignment to Burning Ship fractal creates Mandelbrot Set (read wikis) 0 Members and 1 Guest are viewing this topic.
M Benesi
Fractal Schemer
Posts: 1075
« on: October 23, 2010, 06:50:13 PM »
I can't believe how obvious the formula is after I discovered it....
There are 2 main varieties, although I prefer the one I'm posting images of. I stumbled across this formula while playing around with a new fractal type, and decided to apply what I discovered to my old complex compound formula.
You can redo the code in trig form if you want, using dual-complex numbers works about 2 times as fast on my comp... I'd like to add that my other formula, which combines the 2 3d Mandelbrot varieties, produces far more interesting fractals (at least for higher n z^n). I'll post it beneath the images...
Code:
r1=sqrt(sqr(sy)+sqr(sz)); // you can do x and y values here instead and generate a different fractal
// gotta make sure you change the rest of the formula to match if you decide to do so
// I prefer the way this looks, for whatever reason... anyways-
victor=complex(sx,r1)^n;
bravo=complex(sy,sz)^n;
r3=part_i(victor)*r1^-n;
nx=part_r(victor);
ny=-abs(r3*part_r(bravo));
nz=-abs(r3*part_i(bravo));
//Then you add in your regular x pixel component and the absolute value of your y and z pixel components:
sx=nx+ (pixelr);
sy=ny+ abs (pixeli);
sz=nz+ abs (pixelj); //these values are the starting values of the next iteration...
bailout= abs(sx)+abs(sy)+abs(sz) // or square 'em if it makes you happy... doesn't make a difference to me
side:top:
rear:front:
This formula is produces way cooler fractals. While it skews away from the Mandelbrot type a bit, it has more variety... it's just more interesting.
Code:
victor=complex(sx,sqrt(sqr(sy)+sqr(sz))); //the major difference in this formula is that it rotates sx
bravo=complex(sqrt(sqr(sx)+sqr(sy)),sz); // vs sy and sz, but then calculates the sy and sz values
cramden=complex(sx,sy); // using the other base mandelbrot formula... Makes an AWESOME fractal
// you can also switch which part you do the stuff with if you so desire...
r1=cabs(cramden)^-n;
victor=victor^n;
bravo=bravo^n;
cramden=cramden^n;
if (r2mode) { //It's neater when you exchange the y and z parts, however I put this switch in
nx=part_r(victor); // so I could do it the other way as well
nz=-abs(part_i(bravo));
ny=-abs(part_r(bravo)*part_i(cramden))*r1;
} else {
nx=part_r(victor); //this is the more interesting variety, the default...
ny=-abs(part_i(bravo));
nz=-abs(part_r(bravo)*part_i(cramden))*r1;
}
« Last Edit: October 29, 2010, 08:38:01 PM by M Benesi » Logged
Paolo Bonzini
Guest
« Reply #1 on: October 23, 2010, 11:33:06 PM »
nx=part_r(victor);
ny=-abs(r3*part_r(bravo));
nz=-abs(r3*part_i(bravo));
sx=nx+ (pixelr);
sy=ny+ abs (pixeli);
sz=nz+ abs (pixelj); //these values are the starting values of the next iteration...
Why the abs and (for ny and nz) the negation? The formula without them is, if I did my math right,
Code:
r1=sqrt(y^2+z^2)
sx = x^2-y^2-z^2 + pixelr
sy = 2*x*(y^2-z^2)/r1 + pixeli
sz = 4*x*y*z/r1 + pixelj
which embeds the 2d mandelbrot.
Logged
M Benesi
Fractal Schemer
Posts: 1075
« Reply #2 on: October 24, 2010, 12:13:40 AM »
Quote
Why the abs and (for ny and nz) the negation? The formula without them is, if I did my math right,
It does look like you get a standard 2d cross section if you don't do the abs/negations, however there is something I've got to look into further before I say anything else (perhaps in a couple hours I'll follow through, have to do a few things now); ooohhh I remembered:
You are NOT going to get a 2d Mandelbrot cross section with x + i sqrt(y^2+z^2), because you are always taking the absolute value of the y component :sqrt(y^2) = |y| .... You could set the value to y's sign though but then you get huge missing chunks out of your fractal (I've tried it with: sign of y, sign of z, sign of y+z... all are discontinuous: huge slices cut right out of the fractal).
Come to think of it, I may not be justified in calling this a 3d Mandelbrot, as it's more along the lines of a 3d Burning Ship fractal.
The best looking 3d rotation based fractal to date is the "beautiful fractal" which is the formula I posted at the bottom of the first post in this thread. It's got tremendous variety for all z^n... and I extended it to 4d... totallllly amazing.
« Last Edit: October 24, 2010, 02:25:20 AM by M Benesi » Logged
M Benesi
Fractal Schemer
Posts: 1075
« Reply #3 on: October 24, 2010, 07:06:05 PM »
I did go ahead and set up the fractal with
$r1=\sqrt{| y*|y|+|z|*z |}$ (if you can tell, I take the absolute value of all the variables so I don't get an imaginary root... but what the heck, ehh? maybe I should make it imaginary... if it was imaginary... hrmm interesting...)
if (y+z<0) then r1=-r1.. which doesn't reduce to a 2d Mandelbrot (while doing something similar with sqrt(y^2+z^2) gives you a discontinuous fractal).
Still need the abs/negation to get the nicest fractals, any which way you do it, although the following method works:
Code:
r1=sqrt(sqr(sy)+sqr(sz));
if (sy>sz) {
z1=complex(sx,r1)^n;
r3=r1^-n;
} else {
z1=complex(sx,-r1)^n;
r3=(-r1)^-n;
}
if (sy*sz<0) {
z3=complex(sx,-r1)^n;
r4=(-r1)^-n;
} else {
z3=complex(sx,r1)^n;
r4=r1^-n;
}
z2=complex(sy,sz)^n;
nx=part_r(z1);
ny=part_i(z1)*part_r(z2)*r3;
nz=part_i(z3)*part_i(z2)*r4;
Then add in pixel components. It's still nicer if you abs/negate it however.
« Last Edit: October 25, 2010, 12:02:19 AM by M Benesi » Logged
Jesse
Fractal Schemer
Posts: 1013
« Reply #4 on: October 28, 2010, 06:59:08 PM »
The formula in the first post produces one of the most interesting power 2 bulbs i have seen!
It has similarities with Msltoes symmetric variations.
Without these absolute (foldings?) it is a cosine bulb, that is what i discovered.. but maybe i did something different like always
It seems that the search for a holy grail has become the direction of combining the box and the bulb somehow,
the "boxers" are adding rotations and the "bulbers" more foldings
A detail of the power 2 bulb:
<img src="http://www.fractalforums.com/gallery/3/1127_28_10_10_6_46_52.jpeg" />
PS: i attached the power 2 formula for M3D if someone is interested, hope you dont mind.
M Benesi
Fractal Schemer
Posts: 1075
« Reply #5 on: October 29, 2010, 05:38:34 AM »
Thanks Jesse!
The first formula in the first post is the 3d variety of the Burning Ship fractal (without the -y component as I set y to all positive in the equation). The Burning Ship fractal is simply a 2d Mandelbrot with that uses the absolute value of the real and imaginary components each iteration... ... it's like a Mandelbrot without +/-. As the formula produces an EXACT replica of the burning ship, this tells us that the only thing we need to do is assign signs correctly, like I did in that other thread, to make a perfect 3d z^2 Mandelbrot with no singularities (it's in the "singularity" thread in this subforum).
The second formula (first post) though... now that is fricken awesome.
Some buildings in the z^4 version:
It is just awesome....
Although now that we have a 3d Mandelbrot that works for z^2,6,10... and all odd n, I found us some seahorses (z^6 though, should search the same location in z^2, as that is where they might be):
« Last Edit: October 29, 2010, 05:43:59 AM by M Benesi » Logged
M Benesi
Fractal Schemer
Posts: 1075
« Reply #6 on: October 29, 2010, 06:19:42 AM »
But none of those is worthy of the true power of the 2nd formula. The face of Anachranox (4d) is:
Logged
Jesse
Fractal Schemer
Posts: 1013
« Reply #7 on: October 29, 2010, 04:30:26 PM »
2 questions about the 2nd formula, that starts with:
Code:
victor=complex(sx,sqrt(sqr(sy)+sqr(sz))); //the major difference in this formula is that it rotates sx
bravo=complex(sqrt(sqr(sx)+sqr(sy)),sz); // vs sy and sz, but then calculates the sy and sz values
cramden=complex(sx,sy); // using the other base mandelbrot formula... Makes an AWESOME fractal
// you can also switch which part you do the stuff with if you so desire...
r1=cabs(cramden)^-n;
What computes the cabs function?
I took in my first attempts the realpart of bravo, seems to make sense...
And the pixel addition is like in the first formula with abs on y and z?
This produces some weird and wired stuff, has to explore more until i could say what formula i like more.
Two images of the second formula, first without changing y and z, and the second with changing y and z
(and with my assumptions about the 2nd formula):
BEnesiN2p2.jpg (243.62 KB, 800x600 - viewed 544 times.) BEnesiN3p2.jpg (244.34 KB, 800x600 - viewed 353 times.) Logged
M Benesi
Fractal Schemer
Posts: 1075
« Reply #8 on: October 29, 2010, 08:33:21 PM »
Quote
What computes the cabs function?
oh, sheesh, didn't even think of explaining that portion...
z = complex (x,y) creates a complex number z= x + i y
cabs (z) computes the absolute value (magnitude) of a complex or quaternion number in ChaosPro.
So if z= x+ i y cabs(z) = $\sqrt {x^2 + y^2}$
Quote
And the pixel addition is like in the first formula with abs on y and z?
Yes. Although I am thinking about trying the sign assignment method that works for the first formula (making the Burning Ship into z^2 Mandelbrots) and seeing what it does with the second formula.
I like the switched y and z component formula better than the "normal" method as well (it produces interesting patterns).
Logged
Jesse
Fractal Schemer
Posts: 1013
« Reply #9 on: October 29, 2010, 10:20:55 PM »
Thank you, it is nearly selfexplaining but i wanted to be sure before i make a formula.
Logged
visual.bermarte
Fractal Fertilizer
Posts: 355
« Reply #10 on: October 30, 2010, 05:17:51 PM »
Logged
Jesse
Fractal Schemer
Posts: 1013
« Reply #11 on: October 30, 2010, 06:27:47 PM »
Amazing, is this a julia animation from the first formula?
It shows nice attributes of this formula, i still dont know if i like the second one more or not...
nevertheless i attached the power 2 version of the second one with changed z and y.
bib
Global Moderator
Fractal Senior
Posts: 2070
At the borders...
« Reply #12 on: October 30, 2010, 06:44:29 PM »
Jesse, I just noticed you've attached some new formulae here. Are there any others I might have missed recently??
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Between order and disorder reigns a delicious moment. (Paul Valéry)
Jesse
Fractal Schemer
Posts: 1013
« Reply #13 on: October 30, 2010, 06:48:17 PM »
Jesse, I just noticed you've attached some new formulae here. Are there any others I might have missed recently??
Nope
Logged
bib
Global Moderator
Fractal Senior
Posts: 2070
At the borders...
« Reply #14 on: October 30, 2010, 06:57:00 PM »
yes. new M3D is really fast.Thanks
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2020-02-20 01:48:44
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https://www.physicsforums.com/threads/homework-help-find-the-value-of-z-when-sec-z-3i.363411/
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# [Homework Help!] Find the value of z when sec(z) = 3i.
## Homework Statement
Find the value of z when sec(z) = 3i.
## Homework Equations
sec (z) = 1/cos(z)
## The Attempt at a Solution
I assume that I need to transform into polar coordinates and then use some transformations. I'm really at a loss. If not a solution maybe a few hints to go in the right direction?
Thanks in advance.
HallsofIvy
Science Advisor
Homework Helper
## Homework Statement
Find the value of z when sec(z) = 3i.
## Homework Equations
sec (z) = 1/cos(z)
Good start. And, since you are working with complex numbers, cos(z)= (eiz+ e-iz)/2. So you have
[tex]\frac{2}{e^z+ e^{-z}}= i[/itex]
which can be reduced to
$e^{iz}+ e^{-iz}= -2i$
and that can be converted to a quadratic.
## The Attempt at a Solution
I assume that I need to transform into polar coordinates and then use some transformations. I'm really at a loss. If not a solution maybe a few hints to go in the right direction?
Don't assume things!
Thanks in advance.
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2021-06-15 16:57:38
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https://www.volksbot.de/surmann/papers/icar2005_2/node7.html
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Next: ICP-based 6D SLAM Up: Range Image Registration and Previous: Range Image Registration and
## Calculation of the rotation and translation
In every iteration the optimal tranformation (, ) has to be computed. Eq. () can be reduced to
(2)
with , since the correspondence matix can be represented by a vector containing the point pairs.
Four methods are known to minimize eq. () [17]. In earlier work [20,27] we used a quaternion based method [7], but the following one, based on singular value decomposition (SVD), is robust and easy to implement, thus we give a brief overview of the SVD-based algorithms. It was first published by Arun, Huang and Blostein [2]. The difficulty of this minimization problem is to enforce the orthonormality of matrix . The first step of the computation is to decouple the calculation of the rotation from the translation using the centroids of the points belonging to the matching, i.e.,
(3)
and
(4) (5) (6) (7) (8) (9) (10) (11) (12)
After replacing (), () and () in the error function, eq. () becomes:
In order to minimize the sum above, all terms have to be minimized. The second sum () is zero, since all values refer to centroid. The third part () has its minimum for or
(14)
Therefore the algorithm has to minimize only the first term, and the error function is expressed in terms of the rotation only:
(15)
Theorem: The optimal rotation is calculated by . Herby the matrices and are derived by the singular value decomposition of a correlation matrix . This matrix is given by
(16)
with . The analogous algorithm is derived directly from this theorem.
Proof: Since rotation is length preserving, i.e., the error function () is expanded
The rotation affects only the middle term, thus it is sufficient to maximize
(17)
Using the trace of a matrix, () can be rewritten to obtain
With defined as in (). Now we have to find the matrix that maximizes .
Assume that the singular value decomposition of is
with and orthonormal matrices and a diagonal matrix without negative elements. Suppose
is orthonormal and
is a symmetric, positive definite matrix. Arun, Huang and Blostein provide a lemma to show that
for any orthonormal matrix . Therefore the matrix is optimal. Prooving the lemma is straightforward using the Cauchy-Schwarz [2]. Finally, the optimal translation is calculated as (cf. eq. () and ())
Next: ICP-based 6D SLAM Up: Range Image Registration and Previous: Range Image Registration and
root 2005-05-03
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2019-10-23 15:25:03
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https://www.math-only-math.com/finding-a-fraction-of-a-whole-number.html
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# Finding a Fraction of a Whole Number
How to find a fraction of a whole number?
For finding a fraction of a whole number, we multiply the numerator of the fraction by the given number and then divide the product by the denominator of the fraction.
Solved examples for finding a fraction of a whole number:
(i) Find 1/3 of 21.
To find 1/3 of 21, we multiply the numerator 1 by the given whole number 21 and then divide the product 21 by the denominator 3.
1/3 × 21 = 1 × 21/3 = 21/3 = 7
So, 1/3 of 21 = 7.
(ii) Find 2/5 of 15.
To find 2/5 of 15, we multiply the numerator 2 by the given whole number 15 and then divide the product 30 by the denominator 5.
2/5 × 15 = 2 × 15/5 = 30/5 = 6
So, 2/5 of 15 = 6.
(iii) Find 3/8 of 48.
To find 3/8 of 48, we multiply the numerator 3 by the given whole number 48 and then divide the product 144 by the denominator 8.
3/8 × 48 = 3 × 48/8 = 144/8 = 18
So, 3/8 of 48 = 18.
Related Concept
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2018-10-18 03:43:21
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http://wiki.math.toronto.edu/DispersiveWiki/index.php/Template:Article
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# Template:Article
## How to use this template
Insert the text
{{Article
| author = <list of authors>
| title = <title>
| journal = <journal>
| jvol = <journal volume>
| number = <issue number (optional, currently not used)>
| jpage = <journal pages>
| year = <year of publication>
| arxivid = <arXiv number (optional)>
| mathsciid = <mathsci number (optional)>
}}
at the top of the page where a bibliography reference is desired. Linking, italicization, etc. will be done automatically, as will the adding of the page to the Articles, and the appropriate journal and year of publication categories. Note that the parameters here can themselves contain markup such as external web links or LaTeX math symbols.
For an example of this template in action, see Aa2002.
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2013-06-18 06:57:34
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http://mathhelpforum.com/differential-equations/258125-indirect-proof.html
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1. ## Indirect proof
Hi everyone,
I really need help to prove this statement:
(p→s) from( p→not q), (r→q )and (not r→s)
Thanks
2. ## Re: Indirect proof
(r→q ) is equivalent to (not q→not r ). The latter is the contrapositive.
3. ## Re: Indirect proof
Originally Posted by eps
Hi eve
need help to prove this statement: (p→s) from( p→not q), (r→q )and (not r→s)
Here are the premises: $I~(p\to\neg q)~,~II~(r\to q)~\&~III,~\neg(r\to s).$
To prove by indirect proof you must assume that $A.~(p\wedge \neg s)$ is true.
From A. & I. you can get $\neg q$. HOW?
Then from II. you get $\neg r$
Then from III. you get $r$
So how has it been proven that $p\to s~?$
4. ## Re: Indirect proof
Thank Plato, actually the third promise is (¬r→s) but you wrote ¬(r→s).
5. ## Re: Indirect proof
Originally Posted by eps
Thank Plato, actually the third promise is (¬r→s) but you wrote ¬(r→s).
But you said not r implies s. The usual order of operations would be (not r) implies s if that is what you meant.
It still works. from $\neg r$ and $\neg r\to s$ you get $s$ modus pomens.
Then from that and A. you get $s\wedge\neg s$ DONE>
6. ## Re: Indirect proof
Originally Posted by eps
Hi everyone,
I really need help to prove this statement:
(p→s) from( p→not q), (r→q )and (not r→s)
Thanks
To continue from " p→not q" my next implication would have to start "not q". So I recognize that r→q is equivalent to it contrapositive, "not q→ not r". Replacing "r→ q" with "not q→ not r", I have the chain
"p→ not q", "not q→ not r" and "not r→ s" from which I immediately deduce "p→ s".
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2019-07-18 15:49:25
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https://academic.oup.com/brain/article-lookup/doi/10.1093/brain/122.9.1637
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## Abstract
We have studied focal changes in dopaminergic function throughout the brain volume in early and advanced Parkinson's disease by applying statistical parametric mapping (SPM) to 3D [18F]dopa-PET. Data from seven early hemi-Parkinson's disease and seven advanced bilateral Parkinson's disease patients were compared with that from 12 normal controls. Parametric images of [18F]dopa influx rate constant (Kio) were generated for each subject from dynamic 3D [18F]dopa datasets and transformed into standard stereotactic space. Significant changes in mean voxel [18F]dopa Kio values between the normal control group and each Parkinson's disease group were localized with SPM. Conventional region of interest analysis was also applied to comparable regions on the untransformed image datasets. In early left hemi-Parkinson's disease, significant extrastriatal increases in [18F]dopa Kio were observed in the left anterior cingulate gyrus and the dorsal midbrain region (P < 0.05, corrected) along with decreases in striatal [18F]dopa Kio. In advanced Parkinson's disease, significant extrastriatal decreases in [18F]dopa Kio were observed in the ventral and dorsal midbrain regions (P < 0.05, corrected). No significant changes in [18F]dopa Kio were observed in the anterior cingulate region. In a direct comparison between the early and late Parkinson's disease groups, we observed relative [18F]dopa Kio reductions in ventral and dorsal midbrain, and dorsal pontine regions along with striatal [18F]dopa Kio reductions. Similiar results were found with a region of interest approach, on non-transformed data, except for the focal midbrain [18F]dopa Kio increase seen in early Parkinson's disease. In conclusion, using SPM with [18F]dopa-PET, we have objectively localized changes in extrastriatal, pre-synaptic dopaminergic function in Parkinson's disease. The significance of the increased dopaminergic activity of anterior cingulate in early Parkinson's disease remains unclear, but may be compensatory. The [18F]dopa signal in dorsal midbrain and pontine regions suggests that [18F]dopa is taken up by serotonergic and noradrenergic neurons which also degenerate in advanced Parkinson's disease. This suggests, therefore, that Parkinson's disease is a monoaminergic neurodegenerative disorder.
## Introduction
[18F]dopa-PET provides an in vivo method for assessing the functional integrity of pre-synaptic dopaminergic function in the basal ganglia in Parkinson's disease (Garnett et al., 1983; Martin et al., 1989; Brooks et al., 1990a). It is known from post-mortem studies in Parkinson's disease that there is degeneration of nigral and mesial frontal dopaminergic terminals as well as striatal loss (Bernheimer et al., 1973; Scatton et al., 1982; Kish et al., 1988; Fearnley and Lees, 1991). However, to date, it has been difficult to study these changes with [18F]dopa-PET because of the lower specific uptake of [18F]dopa in these regions, coupled with the limitations of resolution and sensitivity of 2D [18F]dopa-PET scanning.
We have implemented and validated 3D acquisition, reconstruction and analysis of dynamic [18F]dopa-PET data (Rakshi et al., 1996), resulting in a fourfold increase in the sensitivity of brain signal to noise ratios in 3D compared with 2D acquired PET studies (Cherry et al., 1991; Bailey, 1992). In addition, the increased sensitivity enables a higher reconstructed resolution with a reduction in the partial volume effects leading to a further overall improvement in image quality. Therefore, 3D [18F]dopa-PET should be more sensitive and accurate at detecting focal changes within extrastriatal and striatal dopaminergic regions (see Fig. 1).
The conventional method of [18F]dopa analysis uses a region of interest approach with multiple time graphical analysis (MTGA) (Patlak and Blasberg, 1985) to sample putamen and caudate function. However, the size, shape and placement of these regions is arbitrary, producing inter- and intra-observer variation, and potential observer bias. Furthermore, in severe Parkinson's disease it can be difficult to identify the striatal boundaries for placement because of the reduced [18F]dopa uptake.
We have applied statistical parametric mapping (SPM) to 3D [18F]dopa-PET studies, enabling us to objectively localize focal changes in dopaminergic function in Parkinson's disease throughout the brain volume without having to make an a priori hypothesis as to their location. SPM was initially developed for PET activation studies and localizes significant differences in function in spatially normalized brain images on a voxel-by-voxel basis (Friston et al., 1991b). SPM has now been applied to a number of PET ligands including [11C]flumazenil to localize changes in GABAA receptor binding in focal epilepsy (Richardson et al., 1996) and to [11C]diprenorphine to study alterations in opioid receptor binding in Huntington's and Parkinson's disease (Weeks et al., 1995; Piccini et al., 1997).
In addition to applying SPM, we employed the standard region of interest and MTGA analytical approach (Brooks et al., 1990b) to sample striatal and midbrain regions in untransformed images and, subsequently, brain areas revealed by SPM to have altered dopaminergic function in Parkinson's disease.
## Subjects and method
### Subjects
Seven patients with early hemi-Parkinson's disease clinically affecting their left limbs [mean age 57 ± 5 years; mean motor Unified Parkinson's Disease Rating Scale (UPDRS) score 9 ± 3 (range 4–12); Hoehn and Yahr stage 1; and mean symptom duration 19 ± 5 months] and seven patients with advanced bilateral Parkinson's disease [five left dominant, two right dominant; mean age 54 ± 8 years; mean motor UPDRS score 41 ± 15 (range 22–66); Hoehn and Yahr stage 3–5; and mean symptom duration 12 ± 4 years] underwent [18F]dopa-PET scanning. Findings were compared with those of a group of 12 normal subjects, mean age 57 ± 11 years.
All patients fulfilled the UK Parkinson's disease Brain Bank criteria for idiopathic Parkinson's disease (Gibb and Lees, 1988). All Parkinson's disease patients were on regular levodopa therapy except for two of the early Parkinson's disease patients who were drug naive. In addition, three of the advanced Parkinson's disease patients were taking a dopamine agonist, and two a monoamine oxidase-B inhibitor.
Each patient had their medication stopped at least 12 h before their PET scan and was clinically assessed by a single observer, using the UPDRS and Hoehn and Yahr rating scales (Hoehn and Yahr, 1967; Fahn and Elton, 1987) just before their scan. Parkinson's disease patients with significant cognitive impairment, as assessed by the Folstein's Mini-Mental Test (Folstein et al., 1975), were excluded from the study (<28 out of 30). All controls had a normal neurological examination and showed no evidence of rest tremor, rigidity or bradykinesia. No normal volunteer was taking medication.
All patients and normal volunteers gave written informed consent, after a full explanation of the procedure. Permission to perform these studies was granted by the Ethics Committee of the Royal Postgraduate Medical School, London, UK and the Administration of Radioactive Substances Advisory Committee (ARSAC), UK.
### Scanning protocol
In addition to having their medication stopped 12 h before their PET scan, all subjects were fasted on the morning of the scan and received an oral bolus of 150 mg carbidopa and 400 mg entacapone, a peripheral catechol-O-methyl transferase inhibitor (COMT; Orion Farmos Pharmaceuticals Espoo, Finland) 1 h before scanning (Sawle et al., 1994; Ishikawa et al., 1996a).
The [18F]dopa-PET scans were performed with an ECAT 953B neuroscanner (CTI/Siemens, Knoxville, Tenn., USA), with an axial field of view of 10.8 cm, and in 3D acquisition mode with protocols that have been previously reported (Bailey, 1992), yielding 31 planes with a slice separation of 3.4 mm and an average in-plane resolution of 6 mm full-width half-maximum. A correction for tissue attenuation of 511 KeV gamma radiation was measured with a 10 min 2D transmission scan performed prior to tracer injection and acquired using retractable 68Ga/ 68Ge sources.
[18F]Dopa (140–180 Mbq in 10 ml of normal saline solution) was infused intravenously over 30 s. Scanning began at the start of the tracer injection with a protocol of 25 time frames over 94 min (4 × 1 min, 3 × 2 min, 3 × 3 min, 15 × 5 min). The subjects were positioned such that the orbitomeatal line was parallel to the transaxial plane of the tomograph and head position was carefully monitored throughout the scan.
### Data analysis
Two methods of analysis were employed: (i) SPM and (ii) standard region of interest approach with MTGA. Analysis of data was performed on a SUN Sparc 10 workstation (Sun Microsystems, Silicon Valley, Calif., USA) using ANALYZE 7.0 (Mayo Foundation, Baltimore, Md., USA) (Robb and Hanson, 1991) and in-house software written in IDL image analysis software (Research Systems, Inc., Boulder, Col., USA). Parametric images of [18F]dopa influx rate constants (Kio) were generated with IDL image analysis software. Image transformation and SPM analysis was performed using SPM software (SPM95; Wellcome Department of Cognitive Neurology, London, UK) implemented in Matlab (Mathsworks Inc., Sherborn, Mass., USA).
#### SPM analysis
##### Image transformation.
From each subject's dynamic 3D [18F]dopa-PET study, we generated [18F]dopa Kio on a voxel-by-voxel basis for all 31 image planes using the MTGA approach with an occipital tissue input function. Thus, we were able to produce parametric images of [18F]dopa Kio for each subject. Tissue count sampling was performed using time frames 14–25 (30–94 min) post-injection.
There is insufficent anatomical detail in parametric images of [18F]dopa Kio to enable accurate spatial normalization directly to the regional cerebral blood flow (rCBF) template within the SPM software (Friston et al., 1991a). We have, therefore, developed an indirect approach whereby we use the combined time frames 1–25 of the dynamic 3D [18F]dopa image to produce an integrated add image' (0–94 min). This add image' contains the earlier time frames which are blood flow dependent and so generates sufficient cortical as well as striatal detail to allow accurate spatial normalization to the SPM template (see Fig. 2). Applying this method we produced an add image' for each subject and spatially normalized it to the rCBF template. The resulting transformation parameters were then applied to the corresponding subject's parametric image of [18F]dopa Kio, allowing all the parametric images to be transformed into the standard stereotaxic space of Talairach and Tournoux (Rakshi et al., 1998).
This then allowed comparisons to be made across scan datasets, in analogous voxel regions of the brain volume and to combine [18F]dopa-PET datasets from different subjects to perform group analyses. Following spatial normalization of the parametric images (Kio), a Gaussian kernel of 8 × 8 × 6 mm (full width half maximum in the x, y and z planes, respectively) was applied to remove high-frequency noise from the images.
##### SPM data analysis.
Categorical comparisons of mean [18F]dopa voxel Kio values between the Parkinson's disease and normal control groups were made applying SPM. Three between-group comparisons were made: (a) seven early left hemi-Parkinson's disease patients versus 12 normal controls; (b) seven advanced bilateral Parkinson's disease patients versus 12 normal controls; (c) seven early left hemi-Parkinson's disease patients versus the seven advanced bilateral Parkinson's disease patients.
Between-group comparisons were performed using appropriately weighted categorical contrasts to generate SPMs for both increases and decreases in mean voxel [18F]dopa Kio values on a voxel-by-voxel basis. No global normalization was applied since we used measured voxel [18F]dopa Kio values which were independent of rCBF. Significant differences in mean voxel [18F]dopa Kio values for the between-group comparisons (a), (b) and (c) were localized using SPM. The contrasts were used to derive between-group Z scores (unpaired t statistic transformed to the standard normal distribution) on a voxel-by-voxel basis using the general linear model (Friston et al., 1991b, 1995). The P values associated with these regional effects were corrected for multiple-dependent comparisons implicit in the SPM. Maps of Z scores surviving a threshold of >2.33 (P < 0.01) were further corrected for multiple comparisons. These surviving voxels were rendered on to a stereotactically normalized MRI (see Fig. 3).
#### Region of interest analysis
We applied standard region of interest and MTGA analysis to the original untransformed datasets to sample frontal, midbrain and striatal regions. The influx rate constants [18F]dopa Kios were calculated from the time frames 14–25 (30–94 min) post-injection for tissue count activity using occipital tissue counts as the input function. For the striatum, circular regions of 10 mm diameter were placed over each dorsal head of caudate and an elliptical region of 10 × 24 mm over each dorsal putamen aligned to the long axis on three contiguous transverse planes.
With the increased resolution and sensitivity of our 3D [18F]dopa-PET scanner, we were now able to identify the midbrain and frontal regions more clearly by visual image inspection and with reference to the stereotaxic atlas (Talairach and Tournoux, 1988). We sampled the midbrain region by placing a single circular region of 16 mm diameter, encompassing both ventral and dorsal midbrain regions, on two contiguous transverse planes. We retrospectively sampled the anterior cingulate region. This was achieved by initially identifying the region of interest on the stereotaxic atlas using the coordinates of the frontal focus determined from the SPM analysis. We then identified this region on the [18F]dopa-PET scan by the position of the anterior cingulate cortex in relation to the striatum on the PET image. However, because this method is imprecise, we used regions encompassing a larger volume than the SPM focus so that the region of interest would be included. Two circular regions of interest of diameter 16 mm for right and left anterior cingulate regions corresponding to the foci where significant increases in [18F]dopa Kio values were observed with SPM analysis were applied.
For the input function, we sampled right and left occipital lobes with two circular regions of 32 mm diameter placed on the same three contiguous planes as those selected for the dorsal striatal regions. All regions of interest were placed manually using the IDL software.
The mean [18F]dopa Kio values for each frontal, midbrain, putamen and caudate region in the early and advanced Parkinson's disease groups were compared with the [18F]dopa Kio values of normal subjects using Student's unpaired two-tailed t test.
## Results
### Frontal region
#### SPM analysis
In early left hemi-Parkinson's disease we observed a significant increase in [18F]dopa Kio in the the left anterior cingulate gyrus (Brodmann area 24/32) compared with the normal control group (Table 1 and Fig. 3A). There was also a similiar trend in the right cingulate but this did not reach statistical significance on correction (Z = 3.8, P < 0.076).
In advanced Parkinson's disease, no significant change in mean cingulate [18F]dopa Kio was observed, either when compared with the normal controls or directly with the early hemi-Parkinson's disease group.
### Midbrain and pons regions
#### Region of interest analysis
In the advanced Parkinson's disease group there was a 28% reduction in mean midbrain [18F]dopa Kio at the level of the nigra compared with normal controls.
## Discussion
We have applied SPM to spatially normalized parametric images of [18]dopa Kio, and localized similiar reductions in striatal [18]dopa Kio in early and advanced Parkinson's disease to those detected using the standard region of interest approach. SPM, however, enabled us to localize objectively focal [18F]dopa Kio changes in anterior cingulate, midbrain and pontine regions which could not have been predicted by visual inspection of [18F]dopa-PET images. This is because of their lower specific uptake of [18F]dopa.
### Methodological issues
In practice, as with all methods and measurements, there are sources of error in the technique, and their understanding is required for the proper interpretation of the results. The main methodological issues are: 3D PET data acquisition, quantification and interpretation of [18F]dopa uptake, and the application of SPM to [18F]dopa datasets.
3D scanning has a great sensitivity advantage over 2D acquisition. The technique, however, is complex and some of the problems involved have yet to be completely resolved (Badawi, 1997). In 3D mode or volume imaging, the interplane septa of the PET scanner are removed and coincidence events may be detected by pairs of detectors located in any ring. This greatly increases the camera sensitivity to true coincident events, but also increases the sensitivity to scattered coincidences. One therefore has to employ a scatter correction method for 3D datasets (Cherry et al., 1993; Bailey and Meikle, 1994). Another difficulty relates to the requirement that all pairs of detectors in coincidence should have the same sensitivity, i.e. detector normalization. It has been shown that there is no unique normalization for both scatter and true coincidences or for all counts rates. Detector normalization is not perfect and is still being improved. Finally, because of the high count rates that detectors have to process in 3D acquisition, there is an increase in dead-time and random coincidence events detected. Ingenious methods to overcome these limiting factors in 3D mode are currently being developed.
Nevertheless, accepting the problems and limitations above, we have implemented 3D [18F]dopa-PET and successfully performed a validation experiment. We have also demonstrated improved image quality and reproducibility compared with our previous 2D protocol (Rakshi et al., 1996; Trebossen et al., 1996) and were able to completely discriminate both contralateral and ipsilateral putamen [18F]dopa Kio values in early hemi-Parkinson's disease from normal controls.
The objective of a parameter estimation using [18F]dopa-PET is to measure a biological variable of interest to the pathophysiology of Parkinson's disease. With [18F]dopa-PET, we would ideally like to measure k3, the rate constant which directly reflects aromatic amino acid decarboxylase activity (AADC) and thus pre-synaptic dopaminergic function (Dhawan et al., 1996). However, [18F]dopa is mainly peripherally metabolized in the presence of a peripheral AADC inhibitor (carbidopa), to 3-O-methyl-[18F]dopa by COMT (Boyes et al., 1986), which then crosses the blood–brain barrier and contributes to non-specific and striatal [18F] activity. This, along with the formation of other brain dopamine metabolites, makes direct kinetic modelling of k3 very complex (Gjedde et al., 1991; Kuwabara et al., 1995) and to date has not provided additional biological or clinical information on Parkinson's disease over simpler methods of deriving an [18F]dopa Kio reflecting AADC activity (Dhawan et al., 1996; Ishikawa et al., 1996b; Cumming et al., 1997; Ichise and Ballinger, 1997).
We have therefore employed the simpler but less specific MTGA approach to measure the net transfer of [18F]dopa across the blood–brain barrier as the [18F]dopa Kio, where Kio is the slope of the Patlak plot and reflects the rate of unidirectional striatal [18F]dopa accumulation into dopaminergic pre-synaptic terminals, its subsequent decarboxylation and vesicular storage. [18F]dopa Kio, however, is also influenced by factors other than those primarily related to [18F]dopa accumulation and decarboxylation, such as plasma and amino acid concentrations, peripheral and central COMT activities and cerebral blood flow. We have attempted to minimize these effects by fasting all subjects prior to scanning and by using a peripheral COMT inhibitor (Sawle et al., 1994). We have also used occipital counts for the input function to minimize errors due to attenuation correction, scan detector sensitivity or metabolite correction in 3D. Finally, it should be noted that brain [18F]dopa uptake is a measure of its capacity to decarboxylate exogenous dopa and, as such, measures pre-synaptic dopaminergic function. It does not provide a measure of the endogenous rate of dopamine synthesis.
Another important consideration is the effect of levodopa administration on [18F]dopa uptake in Parkinson's disease. We have found no significant difference in [18F]dopa uptake in early or advanced Parkinson's disease subjects scanned twice within 1 month, on one occasion taking their levodopa medication and on the other occasion off levodopa medication, stopped 12 h prior to their PET scan (R. Ceravalo, unpublished observations). The long-term effects of exposure to levodopa therapy on AADC activity in Parkinson's disease are, however, unknown.
Finally, it has been suggested that to improve region of interest placement, one can co-register each patient's individual MRI to their [18F]dopa-PET scan and place the regions on the co-registered image. We have not found this approach to signifiantly change [18F]dopa Kio values or improve reproducibilty when compared with the standard region of interest method (unpublished data). This may be because the various steps involved in image co-registration also introduce sources of error. With advancing PET technology, this method may ultimately improve the accuracy of region of interest placement.
With respect to SPM, there are three main issues: (i) accurate spatial normalization; (ii) applying Gaussian smoothing; and (iii) the assumption that the voxel [18F]dopa Kio values across the parametric images are normally distributed for all subject groups.
It is possible that the relative differences in contrast and intensity between [18F]dopa add images' and the rCBF template might affect spatial normalization. The normalization process is partly dependent on these factors, but the degree to which it might be affected will also depend on the particular constraints of the normalization programme. In practice, however, the [18F]dopa add images' closely resemble the rCBF template (Fig. 2) because the cortex and striatum have relatively high blood flow and the add image' incorporates the earlier time frames which are highly blood flow dependent. Furthermore, if the spatial normalization was imprecise this would represent a substantial source of error variance and we would not have obtained high Z scores for [18F]dopa Kio changes in Parkinson's disease. Accurate spatial normalization is further supported by the SPM maps being accurately rendered on to the putamen and caudate regions of the standard normalized MRI, with the maximal scores identified in Talairach space corresponding to putamen and caudate structures (see Fig. 3).
We arbitrarily applied minimal Gaussian smoothing (8 mm) in order to increase the signal to noise ratio and therefore the sensitivity. This step did not alter our results and could have been omitted. In general, irrespective of the size of the structure giving a signal, smoothing will increase signal to noise and therefore sensitivity, but will degrade resolution (Friston et al., 1995).
Finally, addressing the issue of the normal distribution of voxel [18F]dopa Kio values. This assumption is supported by no outliers being observed at the maximal Z score plots in the three Parkinson's disease studies and can be inferred from a normal distribution of region of interest [18F]dopa Kio values in normal and Parkinson's disease populations, both from this and previous PET studies. The unpaired t test is generally considered to be a robust statistical test, capable of handling suboptimal normally distributed data. At Z scores greater than 4.0, the permutation test is thought to be unnecessary (Holmes et al., 1996).
### Extrastriatal changes in early Parkinson's disease
An unexpected finding revealed by SPM was increased [18F]dopa uptake in the anterior cingulate and dorsal midbrain regions in early hemi-Parkinson's disease (see Fig. 3A). In addition to the reasons already discussed, these extrastriatal [18F]dopa Kio changes are unlikely to be artefactual false positives because the anterior cingulate result was confirmed retrospectively, independent of stereotaxic transformation and SPM, by applying conventional region of interest analysis to the corresponding cingulate regions on the original untransformed images.
With region of interest analysis, we observed similiar mean right and left anterior cingulate [18F]dopa Kio increases but only a significant increase in the left anterior cingulate with SPM. (SPM showed a trend in right cingulate Z = 3.8, P < 0.076.) However, we applied a larger region of interest than the SPM focus which would have reduced the value of the higher left cingulate [18F]dopa Kio.
The significance of increased [18F]dopa Kio is unclear but suggests increased AADC activity. There are a number of possible explanations for this finding which are now discussed. Our knowledge of brain AADC activity in health and disease is limited because there are no direct in vivo human studies. Furthermore, there are no post-mortem studies in early hemi-Parkinson's disease to determine directly early striatal and extrastriatal dopaminergic changes. Therefore, the exact pathophysiological changes occurring in dopaminergic neurons in the very early stages of Parkinson's disease are unknown.
The rate limiting step of endogenous dopamine formation within pre-synaptic dopaminergic nerve terminals is tyrosine hydroxylase which converts l-tyrosine to l-dopa. Under normal physiological conditions AADC which converts l-dopa to dopamine is not thought to be regulated. However, a number of studies have suggested that AADC can be regulated under pathophysiological conditions. Human post-mortem brain studies have suggested that AADC may be up-regulated in dopamine neurons that are spared during ageing (Agid et al., 1987; Kish et al., 1995), and increased AADC activity has been demonstrated in patients with psychosis (Reith et al., 1994). It is possible, therefore, that AADC may undergo regional regulatory changes in Parkinson's disease.
The anterior cingulate observation (Brodmann area 24/32) corresponds neuroanatomically to the mesocortical region which receives dopaminergic projections from the ventral tegmental area A10 (Lindvall and Bjorklund, 1974; Simon and Le Moal, 1984). This projection forms part of the mesocorticolimbic dopaminergic system which is involved in motor, cognitive and behavourial functions (Glowinski et al., 1984; Le Moal and Simon, 1991). Dopaminergic projections to the cerebral cortex, however, also arise from dopamine neurons in the substantia nigra (Bjorklund and Lindvall, 1984). The mesocortical system, which includes the pre-frontal region, is distinct from other ascending dopaminergic projections, i.e. nigrostriatal and mesolimbic, and has unique intrinsic properties. For example, they completely lack or have greatly reduced numbers of autoreceptors on their cell bodies, dendrites and nerve terminals. They also have a higher neuronal firing rate, different activity pattern, higher dopamine turnover and a diminshed response to dopamine agonists/antagonists (Bannon and Roth, 1983).
Another interesting feature of the mesocortical system, which could possibly explain our PET finding, is the inverse relationship demonstrated between the mesocortical and nigrostriatal dopaminergic systems (Pycock et al., 1980). This was first demonstrated in animal studies where dopaminergic neurons projecting to the mesocortex were specifically lesioned in rats. This led to an increase in dopamine levels, its metabolites and D2 receptor binding and sensitivity in the striatum. Similiar findings have now been demonstrated in non-human primates (Roberts et al., 1994). It has therefore been proposed that the mesocortical dopaminergic system has an inhibitory regulatory effect on nigrostriatal dopaminergic systems. Conversely, nigrostriatal lesions could produce an increase in dopamine turnover associated with increased AADC activity in the mesocortex. Surprisingly, there are no studies investigating the reverse relationship.
A further unique feature of the mesocortical dopaminergic neurons is their preferential sensitivity to physical and psychological stress (Roth and Elsworth, 1995). Therefore our finding could instead represent a stress related response in early Parkinson's disease.
On a general dopaminergic perspective, it has been proposed that there are compensatory or adaptive processes in Parkinson's disease to counterbalance the initial nigral dopaminergic cell loss and that symptoms appear when dopamine depletion reaches a critical threshold and these adaptive processes fail (Zigmond et al., 1990). One such possible compensatory mechanism could involve increased synthesis and release of dopamine from residual midbrain dopaminergic neurons. Animal studies have shown that partial or unilateral midbrain dopaminergic lesions result in increased dopamine content and turnover in the remaining striatal dopaminergic terminals (Agid et al., 1973; Andersson et al., 1980; Melamed et al., 1982) perhaps related to the sprouting of residual dopaminergic nerve terminals (Blanchard et al., 1996).
In our [18F]dopa-PET study we have demonstrated increased [18F]dopa Kio suggestive of increased AADC activity. The question is whether this results from, or leads to, increased dopamine synthesis or turnover in early Parkinson's disease. If so, another alternative interpretation of our finding would be that up-regulation of AADC activity in the A10 projection is a compensatory or secondary effect, reflecting early dysfunction in that projection. One would then speculate that similiar up-regulation of AADC occurs in the nigrostiatal projection in pre-clinical Parkinson's disease, but by clinical presentation; although there is increased AADC in individual dopaminergic neurons, the total striatal AADC is reduced because of the continuing neuronal loss. [18F]dopa-PET would then underestimate dopaminergic neuronal loss in early disease. In summary, there is no clear explanation of our finding; however, there are a number of interesting possibilities which we have proposed.
The SPM finding of increased dorsal midbrain [18F]dopa Kio in early hemi-Parkinson's disease was not observed on region of interest analysis. SPM detected a specific midbrain area of increased [18F]dopa Kio, whereas region of interest sampled whole midbrain activity being unable to distinguish clearly between ventral and dorsal midbrain regions on an integrated add image'. Although the focus occured at the level of the superior colliculus, it extended forward and may represent the rostral component of the raphe nuclei.
We now know that a single AADC enyzme catalyses the decarboxylation of both l-dopa in catecholaminergic neurons and 5-hyroxytryptophan in serotonergic neurons (Tison et al., 1991). [18F]dopa will therefore be taken up by both serotonergic and noradrenergic neurons and reflect their function within the brainstem, i.e. [18F]dopa-PET is not specific for dopaminergic neurons. With improved [18F]dopa-PET images, we have demonstrated high signal uptake in the dorsal brainstem corresponding to the regions of the serotonergic neurons of the median raphe nuclei, and the noradrenergic neurons of the locus coeruleus (Figs 1 and 3). Therefore, our dorsal midbrain finding may represent increased AADC activity in serotonergic neurons in this region.
In a previous PET study of early Parkinson's disease with [11C]WIN 35,428, a cocaine analogue which specifically labels the dopamine transporter, an 84% reduction in midbrain binding was reported (Frost et al., 1993). We did not observe a reduction in midbrain [18F]dopa uptake in our early hemi-Parkinson's disease cases. This may reflect either a milder degree of involvement at this stage (their five patients had more advanced and bilateral Parkinson's disease), and/or relative preservation of dopa decarboxylase activity compared with dopamine re-uptake site binding. Alternatively, it has been suggested that dopamine re-uptake sites may be down-regulated (Bannon et al., 1992).
### Extrastriatal changes in advanced Parkinson's disease
No significant changes in anterior cingulate [18F]dopa Kio were observed with SPM or conventional region of interest analysis. This result is perhaps surprising given that post-mortem data have shown a reduction in frontal dopamine and its metabolites in Parkinson's disease (Scatton et al., 1982). One possible explanation could be that our advanced Parkinson's disease group did not have significant cognitive impairment. Alternatively, AADC activity may be maintained in mesocortical areas in advanced Parkinson's disease despite a fall in tyrosine hydroxylase activity and dopamine levels.
In the midbrain region significant decreases in [18F]dopa Kio were demonstrated by region of interest analysis and were shown by SPM to involve both ventral and dorsal regions, in particular, the substantia nigra, ventral tegmentum and the median raphe nuclei.
In the SPM comparison of advanced Parkinson's disease with early hemi-Parkinson's disease (reflecting Parkinson's disease progression) the most significant further [18F]dopa Kio reductions were in the lower dorsal midbrain to the upper dorsal pons regions, corresponding to dorsal raphe nuclei and
### Striatal changes in Parkinson's disease
Employing 3D [18F]dopa-PET, we have demonstrated bilateral putamen dopaminergic dysfunction in early hemi-Parkinson's disease (Hoehn and Yahr stage 1). Furthermore, all individual putamen [18F]dopa Kio values contralateral to the clinically unaffected side fell outside the normal range. This contrasts with our previous 2D [18F]dopa-PET study of early hemi-Parkinson's disease where most corresponding putamen [18F]dopa Kio values were within the normal range (Morrish et al., 1995). In this study, we used a higher resolution 3D PET scanner with greater sensitivity and improved image quality. This resulted in an ~40% increase in normal striatal [18F]dopa Kio values and a wider separation from the Parkinson's disease range suggesting normal [18F]dopa Kio values were previously underestimated (Rakshi et al., 1996). Therefore, the conclusion, based on the results of the earlier 2D study, that the ipsilateral putamen in early hemi-Parkinson's disease is not significantly affected, is no longer supported. With further advances in PET technology, we may demonstrate even greater differences.
In Parkinson's disease, involvement but relative sparing of caudate dopaminergic function is well recognized; however, it is still unclear when and how it develops. With SPM, we detected a significant region of reduced [18F]dopa uptake in the dorsal body of the right caudate in early left hemi-Parkinson's disease. This suggests preferential targeting of the body as well as head of caudate in early Parkinson's disease. In support of our finding, a post-mortem study has demonstrated severe dopamine depletion in dorsorostral caudate regions comparable with dorsocaudal putamen losses in established Parkinson's disease (Kish et al., 1988).
In advanced bilateral Parkinson's disease, both SPM and region of interest analyses demonstrated greater and more extensive loss of [18F]dopa uptake throughout the putamen and caudate, compared with early hemi-Parkinson's disease. The differences in mean symptom duration and mean motor UPDRS scores between the early and severe Parkinson's disease patients were ~10 years and ~30 points, respectively. One might then have expected a greater difference in mean putamen and caudate [18F]dopa Kios between the two Parkinson's disease groups from region of interest analysis. However, the contralateral putamen in early hemi-Parkinson's disease has already lost 65% of the normal level of [18F]dopa uptake with our 3D measure and, therefore, further smaller reductions in [18F]dopa Kio are likely to be associated with large increases in motor disability. The relationship between motor UPDRS and [18F]dopa Kio is, therefore, likely to be non-linear, possibly exponential.
### Conclusion
With recent advances in PET technology, [18F]dopa-PET has become more sensitive at detecting extrastriatal and pre-clinical changes in Parkinson's disease. Applying SPM and region of interest analysis to 3D [18F]dopa-PET data we have identified patterns of subregional dopaminergic dysfunction in cingulate, brainstem and striatal regions in Parkinson's disease. Our study suggests increased AADC activity in anterior cingulate and dorsal midbrain regions in early Parkinson's disease and significant serotonergic and noradrenergic as well as dopaminergic neuronal degeneration within the brainstem in advanced Parkinson's disease, in addition to the known striatal dopaminergic changes. Parkinson's disease, therefore, is most likely a monominergic neurodegenerative disorder. This may have important implications for future treatment strategies.
Table 1
SPM findings for in between-group comparisons showing the locations of significant increases and decreases in [18F]dopa uptake in Parkinson's disease
Z score Talairach co-ordinates P value (corrected)
x y z
Early Parkinson's disease group, n = 7; advanced Parkinson's disease group, n = 7; normal control group, n = 12.
Comparison of early left hemi-Parkinson's disease with normal control group
Increases
Left anterior cingulate gyrus 4.98 –2 32 24 0.001
Dorsal midbrain 4.05 –4 –38 –4 0.038
Decreases
Right putamen 6.07 26 –4 0.0001
Left putamen 5.03 –26 –8 0.001
Right anterior putamen 4.12 18 12 0.030
Dorsal body right caudate 4.00 22 –16 16 0.044
Comparison of advanced bilateral Parkinson's disease with normal control group
Decreases
Dorsal and ventral midbrain 4.56 –32 –12 0.008
Left anterior putamen 6.84 –18 14 0.0001
Left putamen 6.81 –26 –6 0.0001
Right putamen 6.80 24 0.0001
Dorsal body right caudate 6.78 18 12 0.0001
Right anterior putamen 6.63 20 10 0.0001
Comparison of advanced bilateral Parkinson's disease with early left hemi-Parkinson's disease
Decreases
Dorsal pons 5.27 –36 –24 0.001
Dorsal and ventral midbrain 4.45 –14 –32 0.044
Left putamen 4.42 –24 –10 –4 0.049
Z score Talairach co-ordinates P value (corrected)
x y z
Early Parkinson's disease group, n = 7; advanced Parkinson's disease group, n = 7; normal control group, n = 12.
Comparison of early left hemi-Parkinson's disease with normal control group
Increases
Left anterior cingulate gyrus 4.98 –2 32 24 0.001
Dorsal midbrain 4.05 –4 –38 –4 0.038
Decreases
Right putamen 6.07 26 –4 0.0001
Left putamen 5.03 –26 –8 0.001
Right anterior putamen 4.12 18 12 0.030
Dorsal body right caudate 4.00 22 –16 16 0.044
Comparison of advanced bilateral Parkinson's disease with normal control group
Decreases
Dorsal and ventral midbrain 4.56 –32 –12 0.008
Left anterior putamen 6.84 –18 14 0.0001
Left putamen 6.81 –26 –6 0.0001
Right putamen 6.80 24 0.0001
Dorsal body right caudate 6.78 18 12 0.0001
Right anterior putamen 6.63 20 10 0.0001
Comparison of advanced bilateral Parkinson's disease with early left hemi-Parkinson's disease
Decreases
Dorsal pons 5.27 –36 –24 0.001
Dorsal and ventral midbrain 4.45 –14 –32 0.044
Left putamen 4.42 –24 –10 –4 0.049
Table 2
Mean regional right and left putamen, caudate, mesial frontal and midbrain Kiovalues in normal controls, early left hemi-Parkinson's disease and advanced bilateral Parkinson's disease
Kio/min
Right putamen Left putamen Right caudate Left caudate Right frontal Left frontal Midbrain
n.s. = not significant.
Normal (n = 12)
Mean 0.0171 0.0171 0.0168 0.0168 0.0016 0.0017 0.0079
SD 0.0018 0.0018 0.0028 0.0023 0.0004 0.0004 0.0018
Range 0.0145–0.0202 0.0145–0.0209 0.0135–0.0233 0.0137–0.0212 0.0012–0.0020 0.0012–0.0023 0.0062–0.00120
Hemi-Parkinson's disease (n = 7)
Mean 0.0061 0.0096 0.0108 0.0123 0.0025 0.0025 0.0081
SD 0.0005 0.0013 0.0016 0.0018 0.0007 0.0008 0.0019
Range 0.0052–0.0069 0.0079–0.0109 0.0085–0.0133 0.0093–0.0148 0.0016–0.0037 0.0018–0.0035 0.0059–0.0110
P values 0.0001 0.0001 0.0001 0.0001 0.002 0.011 n.s.
Bilateral Parkinson's disease (n = 7)
Mean 0.0036 0.0051 0.0081 0.0092 0.0017 0.0019 0.0057
SD 0.0013 0.0011 0.0018 0.0016 0.0006 0.0004 0.0007
Range 0.0012–0.0052 0.0033–0.0062 0.0046–0.0102 0.0012–0.0109 0.0009–0.0027 0.0014–0.0025 0.0050–0.0067
P values 0.0001 0.0001 0.0001 0.0001 n.s. n.s. 0.006
Kio/min
Right putamen Left putamen Right caudate Left caudate Right frontal Left frontal Midbrain
n.s. = not significant.
Normal (n = 12)
Mean 0.0171 0.0171 0.0168 0.0168 0.0016 0.0017 0.0079
SD 0.0018 0.0018 0.0028 0.0023 0.0004 0.0004 0.0018
Range 0.0145–0.0202 0.0145–0.0209 0.0135–0.0233 0.0137–0.0212 0.0012–0.0020 0.0012–0.0023 0.0062–0.00120
Hemi-Parkinson's disease (n = 7)
Mean 0.0061 0.0096 0.0108 0.0123 0.0025 0.0025 0.0081
SD 0.0005 0.0013 0.0016 0.0018 0.0007 0.0008 0.0019
Range 0.0052–0.0069 0.0079–0.0109 0.0085–0.0133 0.0093–0.0148 0.0016–0.0037 0.0018–0.0035 0.0059–0.0110
P values 0.0001 0.0001 0.0001 0.0001 0.002 0.011 n.s.
Bilateral Parkinson's disease (n = 7)
Mean 0.0036 0.0051 0.0081 0.0092 0.0017 0.0019 0.0057
SD 0.0013 0.0011 0.0018 0.0016 0.0006 0.0004 0.0007
Range 0.0012–0.0052 0.0033–0.0062 0.0046–0.0102 0.0012–0.0109 0.0009–0.0027 0.0014–0.0025 0.0050–0.0067
P values 0.0001 0.0001 0.0001 0.0001 n.s. n.s. 0.006
Fig. 1
3D data acquired by [18F]dopa-PET scan of a normal subject, demonstrating striatal and extrastriatal [18F]dopa uptake.
Fig. 1
3D data acquired by [18F]dopa-PET scan of a normal subject, demonstrating striatal and extrastriatal [18F]dopa uptake.
Fig. 2
Direct comparison of normal subject's spatially normalized [18F]dopa add image' with the standard rCBF template displayed in SPM.
Fig. 2
Direct comparison of normal subject's spatially normalized [18F]dopa add image' with the standard rCBF template displayed in SPM.
Fig. 3
SPM{Z} transverse, sagittal and coronal maximum intensity projection maps rendered on to a stereotactically normalized MRI scan, showing areas of significant increases and decreases in [18F]dopa Kio uptake in A and D, early left hemi-Parkinson's disease (PD) compared with normal controls; in B and E, advanced bilateral Parkinson's disease compared with normal controls; and in C, advanced Parkinson's disease compared with early Parkinson's disease. Z score threshold for significance: P < 0.05, corrected.
Fig. 3
SPM{Z} transverse, sagittal and coronal maximum intensity projection maps rendered on to a stereotactically normalized MRI scan, showing areas of significant increases and decreases in [18F]dopa Kio uptake in A and D, early left hemi-Parkinson's disease (PD) compared with normal controls; in B and E, advanced bilateral Parkinson's disease compared with normal controls; and in C, advanced Parkinson's disease compared with early Parkinson's disease. Z score threshold for significance: P < 0.05, corrected.
We would like to thank the chemists, Nigel Steel and Maria Constantinou for producing [18F]dopa; the radiographers, David Griffiths, Hope McDevitt and Andrew Blyth for assisting in scanning; and Paul Grasby, Jolanta Opacka-Juffry, Stavia Blunt and Sean Spence, of the MRC Cyclotron Unit, for their helpful comments and suggestions. J.S.R is supported by a grant from Smithkline Beecham, UK.
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2017-02-19 15:41:55
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https://www.mathway.com/examples/trigonometry/complex-numbers/finding-all-complex-number-solutions?id=37
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# Trigonometry Examples
Find All Complex Number Solutions
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Rewrite as .
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
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2018-05-22 11:31:28
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https://socratic.org/questions/what-is-the-equation-of-this-circle-with-the-end-points-of-the-diameter-are-at-4
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# What is the equation of this circle with the end points of the diameter are at (-4,-1) and (0,-4)?
${\left(x + 2\right)}^{2} + {\left(y + \frac{5}{2}\right)}^{2} = \frac{25}{4}$
So, C is $\left(\frac{- 4 + 0}{2} , \frac{- 1 - 4}{2}\right) = \left(- 2 , - \frac{5}{2}\right)$.
$R a \mathrm{di} u s = \frac{\mathrm{di} a m e t e r}{2} = . \frac{\sqrt{16 + 9}}{2} = \frac{5}{2}$
The equation is ${\left(x + 2\right)}^{2} + {\left(y + \frac{5}{2}\right)}^{2} = \frac{25}{4}$
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2020-07-13 09:20:56
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http://gmatclub.com/forum/if-the-prime-numbers-p-and-t-are-the-only-prime-factors-of-85836-20.html
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# If the prime numbers p and t are the only prime factors of
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Re: If the prime numbers p and t are the only prime factors of [#permalink]
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Re: If the prime numbers p and t are the only prime factors of [#permalink]
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17 Aug 2014, 15:39
Bunuel wrote:
phoenixgmat wrote:
I would appreciate some help with:
If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p²t?
1) m has more than 9 positive factors.
2) m is a multiple of p³
some explanations to both statements would be great!
thx a lot
We are told that $$p$$ and $$t$$ are the ONLY prime factors of m. It could be expressed as $$m=p^x*t^y$$, where $$x$$ and $$y$$ are integers $$\geq{1}$$.
Question: is $$m$$ a multiple of $$p^2*t$$. We already know that $$p$$ and $$t$$ are the factors of $$m$$, so basically question asks whether the power of $$p$$, in our prime factorization denoted as $$x$$, more than or equal to 2: so is $$x\geq{2}$$.
(1) m has more than 9 positive factors:
Formula for counting the number of distinct factors of integer $$x$$ expressed by prime factorization as: $$n=a^x*b^y*c^z$$, is $$(x+1)(y+1)(z+1)$$. This also includes the factors 1 and $$n$$ itself.
We are told that $$(x+1)(y+1)>9$$ (as we know that $$m$$ is expressed as $$m=p^x*t^y$$)
But it's not sufficient to determine whether $$x\geq{2}$$. ($$x$$ can be 1 and $$y\geq{4}$$ and we would have their product $$>9$$, e.g. $$(1+1)(4+1)=10$$.) Not sufficient.
(2) m is a multiple of p^3
This statement clearly gives us the value of power of $$p$$, which is 3, $$x=3>2$$. So $$m$$ is a multiple of $$p^2t$$. Sufficient.
I could answer the question in 1.5 min. But I have never known the formula for counting the number of distinct factors of a integer. Thanks a ton, Bunuel
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Re: If the prime numbers p and t are the only prime factors of [#permalink]
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Re: If the prime numbers p and t are the only prime factors of [#permalink] 20 Sep 2015, 00:43
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2016-05-03 01:29:03
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https://math.stackexchange.com/questions/3343730/bernoulli-numbers-are-rationals
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# Bernoulli Numbers are rationals?
There is a explicit form that admit Bernoulli numbers rationals but there is another definition where the Bernoulli numbers are $$B_n$$, such that $$\displaystyle \frac{x}{e^x-1}= \sum_{n=0}^\infty B_n \frac{x^n}{n!}$$.
How can I prove that?? or it's equivalence?
• You should tell us what that "explicit form" is, as without knowing that one cannot show it is equivalent to the other definition. – Torsten Schoeneberg Sep 4 '19 at 3:25
• For a quick explanation, the derivatives on the left continued to 0 have rational values at 0 since there are no irrational terms showing up aside from $e^x$, which either becomes 1 or gets used up in $\lim_{x\to0}\frac{e^x-1}x=1$, both of which are rational. – Simply Beautiful Art Sep 4 '19 at 3:27
• @Torsten I presume they mean to say some forms admit the solution trivially, though they want to prove it directly from the generating function definition. – Simply Beautiful Art Sep 4 '19 at 3:29
• If you write $x=\sum\limits_{n=0}^\infty B_n \dfrac {x^n}{n!} (e^x-1)$ and expand $(e^x-1)$ and equate coefficients of like powers of $x$, won't you get a recurrence relation for the Bernoulli numbers, where the coefficients are rational? – J. W. Tanner Sep 4 '19 at 3:32
We have $$\frac{x}{e^x-1} = \frac{x}{x + x^2/2 + x^3/6 + x^4/24 + \cdots} = \frac{1}{1 + x/2 + x^2/6 + x^3/24 + \cdots}.$$ Now, if we let $$f(x) := \frac{x}{2} + \frac{x^2}{6} + \frac{x^3}{24} + \cdots$$, then $$\frac{x}{e^x-1} = \frac{1}{1 + f(x)} = 1 - f(x) + (f(x))^2 - (f(x))^3 - \cdots.$$ In this expansion, each power of $$f(x)$$ has rational coefficients, and since $$f(x)$$ is divisible by $$x$$, the final coefficient of $$x^n$$ is the same as the coefficient of $$x^n$$ in $$1 - f(x) + (f(x))^2 - (f(x))^3 + \cdots + (-1)^n (f(x))^n$$.
From the definition you gave for Bernoulli numbers, $$x=\left(\sum\limits_{n=0}^{\infty} B_n\dfrac{x^n}{n!}\right)(e^x-1)=\left(\sum\limits_{n=0}^{\infty} B_n\dfrac{x^n}{n!}\right)\left(\left(\sum_{n=0}^{\infty} \dfrac{x^n}{n!}\right)-1\right)$$
$$=\sum\limits_{n=0}^{\infty}\left(\left(\sum\limits_{j=0}^n\dfrac{B_j}{j!(n-j)!}\right)-\dfrac{B_n}{n!}\right)x^n.$$
Equating the coefficients of $$x$$ shows that $$B_0=1,$$
and equating the coefficients of $$x^{n+1}$$ for $$n>0$$ shows that $$0=\left(\sum\limits_{j=0}^{n+1}\dfrac{B_j}{j!(n+1-j)!}\right)-\dfrac{B_{n+1}}{(n+1)!}$$
i.e., $$B_{n+1}=\sum\limits_{j=0}^{n+1} \binom{n+1}j{B_j}.$$
This recurrence relation can be used to calculate $$B_n$$ for $$n>0$$,
since $$B_{n+1}$$ can be cancelled from both sides.
In any event, it is clear from this that the Bernoulli numbers are rational.
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2021-08-05 14:06:51
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https://www.toppr.com/guides/physics-formulas/free-fall-formula/
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> > > Free Fall Formula
# Free Fall Formula
Freefall is a common kind of motion which everybody can observe in daily life. If we drop something accidentally we can see its motion. In the beginning, it will have low speed and until the end, it gains speed and before the collision, it reaches its maximum speed. Many factors are there to affect the speed of the object while it is in free fall. We deal with such free-fall motion and free fall formula with examples in this article. Let us learn the concept!
## Free Fall Formula
### Concept
Freefall refers to a situation in physics where the only force acting on an object is gravity and hence acceleration due to gravity. Freefall as its term says is a body falling freely because of the gravitational pull of the earth. This motion will have the effect of acceleration due to gravity. This type of motion will follow the three equations of motion under gravity.
Projectile motion is another important category of free-fall problems. Although these events unfold in the three-dimensional world, for basic physics purposes, they are considered as two-dimensional on paper.
A very unique but interesting property of the acceleration due to gravity is that it is the same for all masses. This was far from the self-evident fact, until the days of Galileo Galilei. That was because in reality gravity is not the only force acting as an object falls, and the effects of air resistance tend to cause lighter objects to accelerate more slowly. It is something that we have all noticed when comparing the fall rate of a rock and a feather.
Galileo conducted this ingenious experiments at the “leaning” Tower of Pisa and proving by dropping masses of different weights from the top of the tower that gravitational acceleration is independent of the mass of the objects.
Free-fall physics problems are having the assumption of the absence of air resistance. But, in the real world, the Earth’s atmosphere provides some resistance to an object in free fall. Also, particles in the air collide with the falling object, which results in transforming some of its kinetic energy into thermal energy. This results in “less motion” or a more slowly increasing downward velocity.
Get the huge list of Physics Formulas here
Imagine an object body is falling freely for time t seconds, with final velocity v, from a height h, due to gravity g. It will follow the following equations of motion as:
1. h= $$\frac{1}{2}gt^2$$
2. v²= 2gh
3. v=gt
Where,
h Height traveled v Final velocity g Acceleration due to gravity t Time taken
These equations can be derived from the usual equations of motions as given below, by substituting
initial velocity u=0,
distance traveled s=h and
acceleration, a=g.
We can see it as follows:
s= $$ut+ \frac{1}{2}at^2$$
v² =u²+ 2as
v=u+at
Freefall is the autonomous phenomena of the body with some mass. It only depends on height from the surface and the time period for which the body is flung.
### Solved examples:
Example-1: Compute the height of the body if it has a mass of 2 Kg and touches the ground after 5 seconds?
Solution:
Given parameters are:
Time t = 5 sec
We have to compute the height. So, we can apply the first equation as given above.
i.e. h= $$\frac{1}{2}gt^2$$
Substituting the values,
h= $$\frac{1}{2}gt^2$$
h= $$\frac{1}{2}\times 9.8 \times 5^2$$
h= 4.9 × 25
h = 122.5 m
Therefore height as required will be 122.5 meter.
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## Question Mark?
Have a doubt at 3 am? Our experts are available 24x7. Connect with a tutor instantly and get your concepts cleared in less than 3 steps.
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2020-10-23 11:50:02
|
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https://physics.stackexchange.com/questions/564304/laser-shining-through-two-holes-in-distant-rotating-discs
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# Laser shining through two holes in distant rotating discs
I've found the following paradox, and I wonder how to resolve it.
Two discs are floating in space, call them A and B. They are at a fixed distance D, coaxial, and rotate at the same speed. Each of them has a hole near the border.
The position of the hole in disc B lags behind the position of the hole in disc A, by a small amount of time. This time is exactly equal to the time it takes light to traverse D.
This means that a laser pulse that gets through hole A is going to get through hole B, and hit a detector on the other side, but the size of the holes is such that there is very little margin for error.
Now: an observer passes along this contraption, moving in the axial direction at a sizeable fraction of the speed of light.
Due to Lorentz contraction, the distance between A and B is going to be smaller in the observer's frame of reference. Plus, the rotation of the discs is going to be slower, due to time dilation.
Either of these effects would be enough to prevent the laser pulse from passing through hole B: it's still traveling at the same speed in the observer's frame of reference, but it has less ground to cover, and on top of that the other disc won't have rotated enough to put the hole in its path. So the detector doesn't get hit!
It's illogical for the detector to be hit or not hit depending on the observer. What am I missing? How to resolve this?
• I don't understand fully. Does the observer move in the axial direction (perpendicular to the discs) or parallel to the discs? – md2perpe Jul 7 at 18:55
• The observer moves in the axial direction. I'll clarify the original question. – F. Polo Jul 7 at 20:52
• Then the relativity of simultaneity comes into play changing the angle between the two holes. – md2perpe Jul 7 at 21:11
• You begin with the observer moving in the axial direction. Then you say the rotation of the discs is going to be slower, due to time dilation. It will not. The spinning of the disks is normal, at right angles, to the axial motion where the time dilation will occur. I have seen this as a cylinder with a groove in the side which spins fast enough for a pulse of light to make it from one end to the other. Nice idea for a paradox but the rotation will not change being at right angles to the motion of the observer. – Elliot Jul 9 at 15:53
Expanding on Dale's answer, by shifting your frame of reference, the relative alignment of the two disks changes, since what is "simultaneous" changes!
If we take disk A as the origin, then the relative-simultaneous (undilated) time of disk B shifts under a frame-velocity shift of $$v$$ by $$\beta \frac{x}{c}$$, where $$x$$ is the (non-contracted) displacement to disk B and the usual Lorentz-transformation definitions of $$\beta = v/c, \gamma=1/\sqrt{1-\beta^2}$$. Disk B therefore is "now rotated ahead" of what it was before the coordinate transformation by the amount it rotated in a time of $$\beta \frac{x}{c}$$.
The time it takes for the beam to traverse from A to B is now reduced by the spatial dilation (by a factor of $$1/\gamma$$) and by the movement of disk B during the travel time (by a factor of $$1/(1+\beta)$$); the rotation of Disk B is also slowed by time dilation (by a factor of $$1/\gamma$$). The pre-transformation rotation time of Disk B when the beam was traversing the distance was $$\frac{x}{c}$$, while the new time is $$\frac{1}{\gamma^2}\frac{1}{1+\beta}\frac{x}{c}=\frac{1-\beta^2}{1+\beta}\frac{x}{c}=(1-\beta)\frac{x}{c}$$, which is a reduction of $$\beta \frac{x}{c}$$ - this exactly cancels out the Relativity of simultaneity shift above!
This cancellation is guaranteed by the conservation under any Lorentz transformations of the spacetime interval between the beam passing through the hole in disk A and the hole in disk B - that is, the beam passing through hole A then hole B always aligns with what happens during the traversal from hole A to hole B, no matter what your inertial frame of reference is.
It's illogical for the detector to be hit or not hit depending on the observer. What am I missing? How to resolve this?
The key to resolving almost all relativity “paradoxes” is the relativity of simultaneity. Conceptually it is the most difficult part of special relativity and so it is the part that gets neglected most often. That is the case here. You accounted for time dilation and length contraction, but forgot to account for relativity of simultaneity.
One other thing is that in any frame where the disks are moving the distance that the light travels is different from the distance between the disks. By the time the light moves the distance D’ the far disk has moved. Nevertheless, the key issue is the relativity of simultaneity
• Would be nice to see this quantitatively. – lalala Jul 8 at 7:54
I'm self-answering because the "click" moment for me was when, after reading all other answers, I realized that this scenario is actually a cunningly-disguised variant of the well-known one where two lightning bolts simultaneously strike the opposite ends of a train.
• @Albert said “it is experimentally proven that one way speed of light is anisotropic in rotating frame of reference”. This is not true. As you correctly mention earlier, simultaneity is conventional and therefore so is the one way speed of light. It cannot be experimentally proven, only assumed. Assuming simultaneity and assuming a one way speed of light are logically the same thing. You cannot have it both ways. You cannot claim that the usual isotropic convention cannot be confirmed by experiment but your anisotropic convention can. Both are conventions and not experimentally provable – Dale Jul 8 at 15:55
• @Albert please feel free to post a question about this topic. Your misconceptions are far too great to address in comments. – Dale Jul 8 at 17:02
• The one way speed of light is purely a convention. Just as simultaneity is. They are, in fact, the same convention. If you are not willing to address your misconception in a venue where it can effectively be addressed, that is fine by me. I just wanted @F.Polo to be aware that your comment here was incorrect – Dale Jul 8 at 18:09
• (cont) Professor John D. Norton, an authority on the science of Albert Einstein and the philosophy of science, has some nice info on The Conventionality of Simultaneity. – PM 2Ring Jul 9 at 4:46
• @Albert Lorentz Ether Theory is mathematically equivalent to Special Relativity. A luminiferous ether is certainly not required. I assume you do not intend to promote some other, non-mainstream, ether theory, since that would be off-topic... – PM 2Ring Jul 9 at 4:53
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2020-08-15 06:19:43
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https://math.stackexchange.com/questions/2372674/why-is-a-varphi-n-2-varphi-n-a2-varphi-n-where-varphi-n-n-in-ma
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# Why is $\|A\varphi_n\|^2=(\varphi_n,A^2\varphi_n)$ where $\{\varphi_n\}_{n\in\mathbb N}$ is an ONB?
I was reading the section on trace-class operators from Reed and Simon whence I encountered this formula. If $\{\varphi_n\}_{n\in\mathbb N}$ is an orthonormal basis of a Hilbert space $\mathcal H$ and $A \in \mathcal L(\mathcal H)$, then $$\|A\varphi_n\|^2=(\varphi_n,A^2\varphi_n)\,.$$
I understand that $\|A\varphi_n\|^2=\sum_{m\in\mathbb N} |(\varphi_m, A\varphi_n)|^2$ which means that I have to prove the following.
$$\sum_{m\in\mathbb N} |(\varphi_m, A\varphi_n)|^2 = (\varphi_n,A^2\varphi_n)\,.$$
I skimmed through all the previous pages of the book, but I cannot seem to find a proof of the above. Kindly help.
• Is $A$ self-adjoint? Since $A$ is trace-class, it is compact and, in particular, bounded. – Alex Ortiz Jul 26 '17 at 19:06
• @AOrtiz $\mathcal L(\mathcal H)$ is the set of bounded operators on $\mathcal H.$ I am not sure if $A$ is self-adjoint. Is it needed for trace-class operators? If you happen to have access to Reed and Simon, please check the proof of theorem $VI.18.$ – Nanashi No Gombe Jul 26 '17 at 19:15
• I have Reed and Simon here. Perhaps you could point to the line in the proof you are questioning? – Alex Ortiz Jul 26 '17 at 19:28
• @AOrtiz The second equality: $(\varphi_n,A \varphi_n) = \|A^{1/2} \varphi_n\|^2 .$ – Nanashi No Gombe Jul 26 '17 at 19:32
## 2 Answers
From Wikipedia: A bounded non-negative operator on a complex Hilbert space is self-adjoint. (This is also stated on pg. 195 of Reed and Simon following the definition of positive operator.) Thus $(A^{1/2})^2 = A$.
If $(\cdot,\cdot)$ is the inner product on $\mathcal H$, then we have $$(\varphi_n,A\varphi_n) = (\varphi_n,A^{1/2}A^{1/2}\varphi_n) = (A^{1/2}\varphi_n,A^{1/2}\varphi_n)=\lVert A^{1/2}\varphi_n\rVert^2,$$ where the first equality is using what we noted above, the second comes from the defining property of the adjoint, and the third is by definition of the norm induced by the inner product.
• If you would excuse my stupidity, I cannot see how this implies that $(\varphi_n,A \varphi_n) = \|A^{1/2} \varphi_n\|^2.$ Would you be kind enough to show the explicit steps to this end? Thanks. – Nanashi No Gombe Jul 26 '17 at 19:40
• $(\varphi_{n},A \varphi_{n}) = (\varphi_{n},(A^{\frac{1}{2}})^{2} \varphi_{n}) = (A^{\frac{1}{2}}\varphi_{n},A^{\frac{1}{2}}\varphi_{n}) = \|A^{\frac{1}{2}} \varphi_{n}\|^{2}$ since the square-root is self-adjoint. – fourierwho Jul 26 '17 at 19:42
• @NanashiNoGombe I have updated my answer to include the extra steps. – Alex Ortiz Jul 26 '17 at 19:46
• @AOrtiz Much obliged. :) – Nanashi No Gombe Jul 26 '17 at 19:48
I think the key is to prove A is self-adjoint. Once you have this, by definition of norm in Hilbert space,
$\Vert A\varphi\Vert^2=(A\varphi_n, A\varphi_n) = (\varphi_n, AA\varphi_n) = (\varphi_n,A^2\varphi_n)$
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2019-10-15 08:38:56
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https://datascience.stackexchange.com/questions/55347/valueerror-from-statsmodels-exponentialsmoothing
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ValueError from statsmodels ExponentialSmoothing
I've been having a frustrating issue with the ExponentialSmoothing module from statsmodels.
My data is a pandas series with 74 weekly data points that looks like this:
2017-12-31 6069
2018-01-07 8143
2018-01-14 6740
2018-01-21 6433
2018-01-28 6631
2018-02-04 6308
2018-02-11 5536
2018-02-18 6025
2018-02-25 5171
... ...
When I call the following functions:
model = ExponentialSmoothing(data, trend='add',damped=True,seasonal='mul',seasonal_periods=52)
model_fit = model.fit()
I get:
Traceback (most recent call last):
File ".\smoothingjuly.py", line 24, in <module>
model_fit = model.fit()
File "C:\Users\lhughes\AppData\Local\Programs\Python\Python37\lib\site-packages\statsmodels\tsa\holtwinters.py", line 641, in fit
l0, b0, s0 = self.initial_values()
File "C:\Users\lhughes\AppData\Local\Programs\Python\Python37\lib\site-packages\statsmodels\tsa\holtwinters.py", line 773, in initial_values
b0 = ((lead - lag) / m).mean()
ValueError: operands could not be broadcast together with shapes (22,) (52,)
Why is this? It works if I decrease the number of seasonal periods, but that makes my model useless. Is 74 data points not enough for the model? If so what is the minimum?
Thanks
The error is raised from lead - lag; in initial_values, these are set as y[m:2m] and y[:m] respectively, where m is the seasonality length (52 in your case). So lead is only getting 22 values, and hence the size mismatch complaint.
But, you can bypass this bit of the code by passing your own initial_slope to fit. I might suggest mimicking statsmodels's approach, but limiting the year-over-year slopes to the range you have available: (y[52:74] - y[:22]).mean() / 52 https://www.statsmodels.org/dev/generated/statsmodels.tsa.holtwinters.ExponentialSmoothing.fit.html#statsmodels.tsa.holtwinters.ExponentialSmoothing.fit
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2020-08-12 01:52:11
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https://edwardshu.com/posts/image-annotation
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Building an Image Editor with Canvas API
July 6, 2018 - 6 minute read -
Problem
I needed to build an image editor on the browser. Here are the requirements:
1. User can zoom and pan the image
2. User can draw on the image
3. Image editor needs to a fixed size.
4. Edited image needs to be in original resolution.
5. Desktop and Mobile friendly.
If it weren’t for requirement number 4, this image editor would have been very easy to make. Just make our image editor the size of the source image. Then any edited images will still be in the original resolution. However, if you have a high resolution image and your canvas viewport is smaller than the image dimensions, the canvas will downsample the image.
However, with a bit of math we can preserve resolution. Let’s use two canvases. One canvas is the editor. The other canvas holds the source image. The editor canvas is a fixed size. However, the source canvas is exactly the size of the image so there is no resolution change. Whenver the user draw on a point $P_{editor}$, just mirror the drawing action back to $P_{source}$! When the user is done drawing, we can export the source canvas instead of the editor to give them an edited image in full resolution!
We will be using Hammer.js. HammerJS lets us add in pinch to zoom and unifies mouse and finger panning detection. If you’re building a desktop only app, you can just use native mouse events instead.
I’ve provided an Angular 5 Demo but this can be easily done in Vanilla JS as well.
Displaying the image
HTML Canvas are containers for browser graphics. You can interact with them through javascript.
So to load an image into the canvas, we will call the drawImage method on our canvas context.
This is just matrix multiplication.
This function lets us take a portion of the source image and project it onto our canvas. Remember how we need to zoom and pan? This powerful method will allow us to do both. Our first step is to make the canvas a “window” of our source image.
Now given an arbitrary image portion, it will be stretched and fitted so that our entire canvas displays the portion.
Zooming
For zooming, we just have to alter the dimensions of the source image. For example, to have a 2x zoom, we just need to divide our sWidth and sHeight by 2. Then the canvas will have to stretch out the shrunken image portion over its dimensions, effectively zooming it in. Conversely, if we want to zoom out 2x, we can multiply sWidth and sHeight by 2. Now the canvas has to squish in an expanded image portion, making it appear smaller.
Panning
For panning, we just have to alter the offsets of the source image. The user clicks the image, then drags their mouse to move the image around. On the initial click, we save the location of their mouse. Then while their mouse is moving, we calculate the offset between the initial location and the current location. This offset represents the amount to move the image by to keep the image centered around the initial location.
One important point to remember is that the user is interacting with the canvas, not the source image. So we must translate canvas coordinates to source image coordinates whenever we want to edit the source image portion.
Let
• $X_{c}$ be X coordinate of canvas
• $X_{s}$ be X coordinate of source image
• $S_{x}$ be X offset of source image
• $W_{c}$ be width of canvas
• $W_{s}$ be width of source image
Drawing
Now that we figured out how to map coordinates between source and canvas, drawing is easy! Whenver the user draw on a point $P_c$, just mirror the drawing action back to $P_s$! If you don’t want to show the source canvas to the user, create it dynamically in javascript instead of declaring it in HTML.
You can check out the demo for exact details.
Limitations
1. Panning is only allowed if viewport is within source image dimensions.
2. Panning and zooming will erase drawings in editor canvas (but not source!)
If $S_x$ and $S_y$ become negative, then the behavior of drawImage becomes undefined on iOS. Therefore I had to disable panning unless the editor is within the dimensions of the image. If you’re desktop only, disable my realignImage function and you can pan however much you want.
I didn’t have time to implement storage of user edits. There is a save and restore function in the Canvas API which could be promising.
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2019-02-23 07:41:49
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http://qingkaikong.blogspot.com/2016/08/sentinel-1a-data-processing.html
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## Friday, August 12, 2016
### Sentinel-1A data processing using GMTSAR
I learned processing InSAR data in the short course at Scripps using the GMTSAR. In this blog, I will write down the steps I used to process the Sentinel-1A data for my own reference. I will use the 2016 Taiwan M6.4 Meinong earthquake as an example. You can find all the files and figures at Qingkai's Github. Most of the content is written during the short course by the help of the educators.
## Step 1 - get data
There are multiple ways to get the data. The most common way is to use the GUI or use the API directly. For this earthquake, we will try to use the GUI to download the Sentinel-1A data. We first draw a polygon around the earthquake region as shown in the following figure. Since the earthquake occurred on Feb 5th, we put our search range from Feb 1 - Feb 17. Because we only use the SLC processing level, so I changed the 'Processing Level' to 'SLC'.
After you pressed the search button, you will get a list of available data within the range, shown below. When you select any of the record, it will highlight on the map as the black box. Scroll to the right, you will see the download button two download the data. Because we want to download the data that can reflect the deformation caused by the earthquake, therefore, we need both the data before the earthquake, and after. By looking at the Ascending path, we have Feb 2nd and Feb 14th satisfy the requirement (Note: You should choose both from the same path, either Ascending or Descending). Then go ahead to download the data (about 10 Gb).
## Step 2 - get the orbit data
You also need the orbits data before processing, which you can download it here. The naming of the orbits data is like
S1A_OPER_AUX_POEORB_OPOD_20160222T121629_V20160201T225943_20160203T005943.EOF
Note there are 3 dates, the first one indicate when it is processed (mostly useless in our example), and the last two dates are the start and end dates of the orbits, and we need make sure our Sentinel data date is within this range. For our case, we need the following two files:
S1A_OPER_AUX_POEORB_OPOD_20160222T121629_V20160201T225943_20160203T005943.EOF
S1A_OPER_AUX_POEORB_OPOD_20160305T121418_V20160213T225943_20160215T005943.EOF
## Step 3 - folder structure
The next step you need is to put the data into a folder structure for processing. The folder structure looks like the following, and the red color indicate directories, you can see the subdirectory or the files in them. The two .SAFE directories are from the InSAR data you downloaded. And the two files ending with eof.txt are the orbits data we downloaded. For the dem.grd, you can download it from here by specify the latitude and longitude. The 01_run_prep.sh, 02_run_proc.sh, and config.s1a.txt, we will talk in the next section.
Taiwan_earthquake
├── 01_run_prep.sh
├── 02_run_proc.sh
├── config.s1a.txt
├── orbits
| ├── S1A_IW_SLC__1SDV_20160202T100019_20160202T100049_009766_00E469_C190.SAFE
| ├── S1A_IW_SLC__1SDV_20160214T100019_20160214T100049_009941_00E981_ABD9.SAFE
| ├── S1A_OPER_AUX_POEORB_OPOD_20160222T121629_V20160201T225943_20160203T005943.eof.txt
| └── S1A_OPER_AUX_POEORB_OPOD_20160305T121418_V20160213T225943_20160215T005943.eof.txt
└── topo
└── dem.grd
## Step 4 - change the config file
Before you run the following steps, you need consider the settings in the config.s1a.txt file. There are many settings inside, but you can first run with the default setting, and change it later. For more details, you should refer to the GMTSAR documentation. My example config.s1a.txt file is here:
## Step 5 - run the preparation
The 01_run_prep.sh script should be run first to prepare the data and align the master and slave images. Before you run that, you need change the name of the files inside. Note that the input of the align_tops_esd.csh run in the script is: InSAR-1, orbit-1, InSAR-2, orbit-2, dem.grd. Notice that, I only use the polarity 'vv' for this example. See my example file:
### 01_run_prep.sh.
After you run this script, you will find you have a updated folder structure, which contains 4 more folders: F1, F2, F3, and raw. The F1, F2, and F3 folder contain the aligned data for the 3 subswaths.
Taiwan_earthquake
├── 01_run_prep.sh
├── 02_run_proc.sh
├── config.s1a.txt
├── F1
| ├── raw
| ├── topo
| └── config.s1a.txt
├── F2
| ├── raw
| ├── topo
| └── config.s1a.txt
├── F3
| ├── raw
| ├── topo
| └── config.s1a.txt
├── raw
| └── many things here ^_^
├── orbits
| ├── S1A_IW_SLC__1SDV_20160202T100019_20160202T100049_009766_00E469_C190.SAFE
| ├── S1A_IW_SLC__1SDV_20160214T100019_20160214T100049_009941_00E981_ABD9.SAFE
| ├── S1A_OPER_AUX_POEORB_OPOD_20160222T121629_V20160201T225943_20160203T005943.eof.txt
| └── S1A_OPER_AUX_POEORB_OPOD_20160305T121418_V20160213T225943_20160215T005943.eof.txt
└── topo
└── dem.grd
## Step 6 - run the processing
Now, if everything works fine, you can step into the last step: run the processing script - 02_run_proc.sh, again, you need change the name of the files inside the script, and this one is easier. You can find my example file here:
## Results
After all the above steps, you will find in each of the subswath folder (i.e., F1, F2, and F3), there are two more folders: intf, SLC. Most of the results figures/data are stored in the intf folder. Here are my results:
wrapped fringes:
unwrapped:
The next step you want to do is to remove the trend from the unwrapped image.
## Acknowledgement
Thank:
UNAVCO for providing the financial support.
GMTSAR developer team for the great workshop.
GMT team for great support!
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2018-08-19 21:17:24
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https://www.nature.com/articles/s41427-020-0220-0?error=cookies_not_supported&code=63fc46df-3cb4-4552-a5df-a2788b3db6b3
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# Very sharp diffraction peak in nonglass-forming liquid with the formation of distorted tetraclusters
## Abstract
Understanding the liquid structure provides information that is crucial to uncovering the nature of the glass-liquid transition. We apply an aerodynamic levitation technique and high-energy X-rays to liquid (l)-Er2O3 to discover its structure. The sample densities are measured by electrostatic levitation at the International Space Station. Liquid Er2O3 displays a very sharp diffraction peak (principal peak). Applying a combined reverse Monte Carlo – molecular dynamics approach, the simulations produce an Er–O coordination number of 6.1, which is comparable to that of another nonglass-forming liquid, l-ZrO2. The atomic structure of l-Er2O3 comprises distorted OEr4 tetraclusters in nearly linear arrangements, as manifested by a prominent peak observed at ~180° in the Er–O–Er bond angle distribution. This structural feature gives rise to long periodicity corresponding to the sharp principal peak in the X-ray diffraction data. A persistent homology analysis suggests that l-Er2O3 is homologically similar to the crystalline phase. Moreover, electronic structure calculations show that l-Er2O3 has a modest band gap of 0.6 eV that is significantly reduced from the crystalline phase due to the tetracluster distortions. The estimated viscosity is very low above the melting point for l-ZrO2, and the material can be described as an extremely fragile liquid.
## Introduction
Determining the liquid structure is the first step in understanding the nature of glass-liquid transitions. However, a diffraction measurement of liquid provides very limited structural information because the liquid structure lacks long-range periodicity, and a Fourier transform of the diffraction data provides only pairwise correlations. Moreover, high-quality measurements are difficult to obtain at high temperatures.
Since glasses play an important role in technology, glass formation has been studied extensively. Zachariasen1 and Sun2 proposed the basic concepts of glass formation by classifying constituents into glass formers, glass modifiers, and intermediates. Furthermore, Angell3 introduced the concept of “fragility” in glass-forming liquids (GFLs). He interpreted the strong and fragile behavior of liquids in terms of topological differences in potential energy hypersurfaces of the configuration space. Typical strong liquids are SiO2, GeO2, and B2O3. Their networks are covalently bonded, and the viscosities show an Arrhenius temperature dependence. In contrast, typical fragile liquids are chalcogenides and iron phosphates, the networks of which are mostly ionic and the viscosities of which deviate significantly from the Arrhenius behavior. Many experimental and theoretical structural studies of liquids and glasses have been performed, and with the advent of advanced synchrotron and neutron sources and the development of high-performance computers, they have led to great progress in our understanding of liquid and glass structures4,5. The structural analysis of liquids with high melting points has advanced significantly with the advent of the levitation technique6, especially in combination with diffraction techniques6. The structure of a typical non-GFL, liquid (l-) Al2O3, and its undercooled liquid have been studied extensively by X-ray diffraction7,8,9,10, neutron diffraction9,10,11, and molecular dynamics (MD) simulations9,10,11,12,13. In addition to the l-Al2O3 structure, several structures of molten pure oxides with high melting points (Tm) have been studied recently. For example, structures of UO214 and compounds in the UO2–ZrO2 system15 have been investigated for nuclear reactor accidents. The structures of ZrO216,17,18, HfO216, and lanthanide oxides17 have also been investigated to understand the fundamental properties of high-temperature liquids. Although these investigations are very important not only for materials science but also for preventing severe accidents, the research methods and data are still limited by the high melting points of the materials in question.
Er2O3 is a representative nonglass former that is commonly used as a refractory material and dopant for luminescent materials. Because Er2O3 has an extremely high melting point (Tm = 2686 K), the difficulties in handling the liquid lead to problems in selecting suitable container materials that do not contaminate the sample. To avoid contact with other materials, levitation furnaces have been developed that enable us to measure precise synchrotron X-ray diffraction and thermophysical properties for liquids at extremely high temperatures6.
This article presents the results of accurate high-energy X-ray diffraction and density measurements on containerless levitated l-Er2O3 using an electrostatic levitation furnace (ELF) at the International Space Station (ISS)19, as it is impossible to measure density data on the ground. We also perform reverse Monte Carlo – molecular dynamics simulations and obtain persistence diagrams from topological analyses to demonstrate liquid properties at the atomic level, comparing l-Er2O3 with other non-GFLs and a typical GFL, l-SiO2. Furthermore, a sample of l-Er2O3 is simulated for a short period with the density functional – molecular dynamics method to investigate the electronic structure and to obtain a realistic estimate of the viscosity above the melting point. The combination of an experiment and a simulation allows trends in single-component nonglass-forming liquid oxides to be identified, with a focus on atomic ordering and topology. Furthermore, the article compares the features of single-component nonglass-forming oxide liquids with those of other systems.
## Materials and methods
### Density measurement
The density of liquid (l-) Er2O3 was measured with an ELF at the ISS. A sample of 2 mm in diameter was prepared by melting Er2O3 powder with a purity of 99.99% and solidifying it in an aerodynamic levitator. It was charged by friction or contact with other materials in the ISS-ELF and then levitated to the center between six electrodes that applied a Coulomb force. The sample position was stabilized by tuning the voltages between electrodes at 1000 Hz and monitoring the image of the sample backlit by a He–Ne laser. The levitated sample was heated and melted by four 40 W semiconductor lasers (980 nm) under 2 atm of dry air. The temperature of the sample was measured by a pyrometer (1.45–1.8 μm). It was calibrated using an emissivity calculated from the plateau temperature at recalescence and the reference value of the melting point (2686 K). After melting, the nonspherical sample became spherical upon cooling after shutting off the lasers. During cooling, the sample image was observed by an ultraviolet back light and a CCD camera. The pixel size was calibrated against an image of 2.0 mm stainless steel spheres, which were recorded under the same conditions as the sample. The sample volume was calculated from its diameter, obtained from the image. Then, the density was calculated from the volume and weight.
### High-energy synchrotron X-ray diffraction measurement
The high-energy X-ray diffraction measurement of l-Er2O3 was performed at the BL04B2 beamline20 of SPring-8 using an aerodynamic levitator21. The energy of the incident X-rays was 113 keV. The 2-mm Er2O3 sample was levitated in dry air and heated by a 200 W CO2 laser. The temperature of the sample was monitored by a two-color pyrometer. The background of the instrument was successfully reduced by shielding the detector and by optimizing a beam stop. The measured X-ray diffraction data were corrected for polarization, absorption, and background, and the contribution of Compton scattering was subtracted using standard analysis procedures22. The corrected data sets were normalized to give the Faber–Ziman23 total structure factor S(Q), and the total correlation function T(r) was obtained by a Fourier transform of S(Q).
### Molecular dynamics – reverse Monte Carlo simulation
To determine the atomic configuration of l-Er2O3, a molecular dynamics – reverse Monte Carlo (MD-RMC) simulation was performed with 5000 particles in a cube to reproduce the X-ray S(Q). The MD simulation was carried out with a Born-Mayer type of pairwise potential with a Coulomb interaction and a repulsive component, given by the following equation:
$$U_{ij}\left( r \right) = \frac{{e^2}}{{4\pi \varepsilon _0}}\frac{{Z_iZ_j}}{r} + B_{ij}\exp \left( { - \frac{r}{{\rho _{ij}}}} \right),$$
(1)
where r is the interatomic distance, Z is the effective charge, B is the repulsion, e is the elementary charge (ZEr = 2.1, ZO = −1.4), ε0 is the permittivity of the vacuum, and ρ is the softness parameter. The parameters used in the MD simulation are summarized in Table 1.
The simulations were carried out for a system of 2000 Er and 3000 O atoms in the unit cell with a random atomic configuration. The cell volume was determined from the number densities of l-Er2O3 at the melting point, which were calculated with the density measured by the ISS-ELF. Periodic boundary conditions were used, and the long-range Coulomb interaction was treated with Ewald’s summation. A time step of 1 fs was used in the Verlet algorithm. First, the temperature of the system was maintained at 4000 K for 20,000 time steps and then cooled to 2923 K over 20,000 steps. The structural model was finally annealed at 2923 K for 150,000 steps. After the MD simulation, RMC refinement was conducted using the RMC++ code24. The benchmark RMC runs were performed using simulation boxes with 250, 500, 1000, and 3000 particles.
### Density functional – molecular dynamics simulation
The simulations based on the density functional theory (DFT) of the electronic structure were performed with the projector augmented wave (PAW) method25, implemented in the VASP software26,27. The PAW potentials supplied within VASP for Er (5p, 5d, and 6s, with 11 4f electrons frozen in the core) and O (2s, 2p) were tested and used (see supplementary information for a comparison between the frozen core approximation and the treating of the Er-4f electrons explicitly, Fig. S1). For a liquid sample of 500 atoms, the energy cutoff of the plane waves was set to 400 eV, with a single Γ-point in the Brillouin zone. In comparison, the bulk crystalline (c)-Er2O3 unit cell was fully relaxed until the forces on all atoms were below 0.01 eV/Å with a Γ-centered 2 × 2 × 2 k-point grid and a plane-wave energy cutoff of 550 eV. The PBE functional28 was used for the geometry optimization and the molecular dynamics simulations, whereas the HSE06 hybrid functional29 was used to obtain the electronic densities of states (DOSs) and their projections to produce more realistic electronic band gaps and test the effect of the 4f electrons in c-Er2O3. The effective charges and atomic volumes were evaluated by Bader analysis30,31,32 using the PBE functional.
The density functional – molecular dynamics (DF-MD) simulations were performed with a Nóse-Hoover thermostat33 and a time step of 2 fs, with an initial atomic configuration given by the benchmark RMC model mentioned above with 500 atoms. The system was simulated at 2923 K (~2650 °C) for a total of 30 ps, where the last 25 ps were used for data collection (Fig. S2). The electron occupancy was described with a Fermi smearing corresponding to the kBT value at the target temperature. The mean-square displacements (MSDs) of the atoms show a liquid (diffusion) behavior where equilibrium is already achieved during the first few picoseconds.
### Topological analysis using a persistent homology
The homology of atomic configurations has been investigated using the persistence diagram D1, which consists of two-dimensional histograms showing a persistent homology. The details of the analysis are described elsewhere34. The persistence diagram D1 of a set of atoms is given by the following thickening process of spheres: (1) place a sphere with a radius r at the center of each atom, (2) increase the radii of the spheres from 0 to a sufficiently large value, and (3) encode the pair of birth and death radii (bi, di) for each ring ci consisting of a set of spheres. The persistence diagram is then constructed by the two-dimensional histogram on the birth and death plane obtained by the pairs for independent ci, i = 1,…, K. Here, the birth (death) radius is defined as the radius of spheres at which the ring ci first appears (disappears). The birth radius has information about the distances between atoms of the ring ci, and the death radius has information about the size of the ring. The persistence diagram provides statistical information on the shapes of all independent rings and thereby provides insight into intermediate ordering in the liquid structure. The rings and cavities detected by this process are recorded for the computation of the persistence diagrams; hence, their geometric shapes can be identified for further analysis. The persistence diagrams were calculated using the HomCloud package35.
## Results and discussion
### Density data
Figure 1 shows the density of l-Er2O3 as a function of temperature, which exhibits a linear temperature dependence. The least-squares fit to the data is given by the following equation:
$$\rho \left( T \right) = {\rho_{\mathrm{m}}}\left[ {1 - \alpha \left( {T - {T_{\mathrm{m}}}} \right)} \right]\left( {{\mathrm{kg}}/{\mathrm{m}}^3} \right),$$
(2)
where ρm is the molten density at Tm (8170 kg/m3) and α (=1/ρm[dρ(T)/dT]) is the thermal expansion coefficient and is assumed to be constant (1.0 × 10−4 K−1) at any temperature of the liquid. The correlation coefficient of this fitting is 0.98. The uncertainty in the measurements is estimated to be 2% from the image resolution (640 × 480 pixels) and from the uncertainty in the mass measurement (±0.1 mg).
The density and the expansion coefficient for l-Er2O3, together with those for l-SiO236 and other non-GFLs18,37, are compared in Table 2. Although the density trends increase with increasing cation atomic number, they do not show a clear relation. On the other hand, the thermal expansion coefficients show a similarity as each value approaches 1 × 10−4 K−1. The thermal expansion coefficient of l-Er2O3 is especially close to those of l-SiO2 and l-Al2O3.
### Structure factors and real-space functions
The Faber-Ziman X-ray total structure factors, S(Q), for l-Er2O3, l-SiO238, l-Al2O311, and l-ZrO218, together with the results of the MD-RMC simulation for l-Er2O3, are compared in Fig. 2a. It is noted that the scattering vector Q is scaled by multiplying by rA-X (distance between the center and corners of the polyhedron). The experimental S(Q) of l-Er2O3 (solid cyan curve) is well reproduced by the MD-RMC simulation (dotted black curve) using the liquid density measured by the ISS-ELF shown in Fig. 1. A well-defined first sharp diffraction peak (FSDP)39 is observed only for l-SiO2 (GFL) at QrA-X = 2.6, while a principal peak (PP)39 is observed in both the l-ZrO2 and l-Er2O3 data at QrA-X ~ 4.5. On the other hand, l-Al2O3 gives rise to a small peak between the FSDP and PP, suggesting that the structure of l-Al2O3 is intermediate17 between l-SiO2 and l-ZrO2/l-Er2O3. It is well known that the PP reflects the packing of oxygen atoms in neutron diffraction data40, since neutrons are sensitive to oxygen. For the same reason, a PP is not observed in the X-ray S(Q) for l-SiO2 (see Fig. 2a), and the origin of the PP in l-ZrO2 and l-Er2O3 is ascribed to the packing of cations. The X-ray total correlation functions T(r) for l-Er2O3, l-SiO238, l-Al2O311, and l-ZrO218 are shown in Fig. 2b. The first peak observed at approximately 2.2 Å is assigned to the Er–O correlation, and a tail to ~3 Å implies the formation of distorted ErOn polyhedra in the liquid. The second peak, observed at 3.7 Å, can be assigned mainly to the Er–Er correlation, and the O–O correlation peak is unclear due to its small weighting factor for X-rays. The Er–O correlation length of 2.2 Å, as well as that of Zr–O (2.1 Å), is longer than those of Si–O (~1.63 Å at 2373 K) and Al-O (~1.78 Å at 2400 K) owing to substantial differences between the ionic radii of the elements. The increased cation-oxygen correlation length in the liquid phases of Er–O and Zr–O suggests that the oxygen coordination number around cations is higher than 4 because the Er–O correlation length (2.2 Å) or Zr–O correlation length (2.1 Å) is close to the sum of the ionic radii41 of oxygen (1.35 Å) and six-fold erbium (0.89 Å) or zirconium (0.72 Å), respectively. The structures of l-Er2O3 and l-ZrO2 therefore consist of large interconnected polyhedral units and are very different from those of l-SiO2 and l-Al2O3. This behavior is consistent with the fact that the peaks observed at QrA-X ~ 4.5 in Fig. 2a are not the FSDP, which is typically associated with intermediate-range ordering in oxide glasses and liquids; thus, there is no such ordering in l-Er2O3 and l-ZrO2 due to a very densely packed structure.
### Coordination number distributions from the simulation
The coordination number distributions, NA-X and NX-A, for l-Er2O3, l-SiO242, l-Al2O311, and l-ZrO218 obtained from the simulation are compared in Fig. 3a, b, and their average values are summarized in Table 3. The Er–O coordination number (up to 3.0 Å) is found to be 6.1 from our combined MD-RMC simulation, which is rather close to the crystalline phase43, and the O–Er coordination number can be estimated to be 4.1. It is suggested that the cations are tetrahedrally coordinated in l-SiO2 (GFL), while they are octahedrally coordinated in l-ZrO2 and l-Er2O3 (non-GFLs), and the cation-oxygen coordination number in l-Al2O3 is intermediate17 between GFL and l-ZrO2/l-Er2O3, although l-Al2O3 is a non-GFL. This behavior is consistent with that of the first correlation peaks in experimental real-space functions (see Fig. 2b) and with the fact that the viscosity of l-ZrO2 is approximately one-tenth of that in l-Al2O318. Another interesting behavior is observed for the oxygen-cation coordination numbers. It is demonstrated that oxygen is twofold in l-SiO2, which is a signature of the formation of a sparse network, while triclusters (XA3) are dominant in l-Al2O3 and l-ZrO2. The formation of tetraclusters (XA4) is confirmed in l-Er2O3, suggesting that this behavior is a distinct feature of this liquid. We suggest that the behavior of the coordination numbers in a series of oxide liquids is affected by both the composition and the ionic radii between the constituent anions and cations. For instance, the ionic radii of Si and Al are small, which results in tetrahedral coordination, although the Al-O coordination number is greater than four on average. The tetracluster formation is caused by the ratio of Er and O in Er2O3.
### Very sharp principal peak (PP) in l-Er2O3
As shown in Fig. 2a, the PP of l-Er2O3 is very sharp compared to that of l-ZrO2. The FWHM of the PP in l-Er2O3 is 0.4299, in comparison to 0.7669 in l-ZrO2 (see Fig. 4). A simulation box with 501 particles was used in the previous RMC – density functional (DF) simulation for l-ZrO218, where a good agreement was observed between the experimental data and simulation (see Fig. 4a). However, as can be seen in the inset data of Fig. 4b, a simulation box of 500 particles is insufficient to reproduce the sharp PP in l-Er2O3; larger atomic models are needed to reproduce this feature. Insight into the structure of l-Er2O3, in comparison with those of l-SiO2 and other non-GFLs, can be obtained by calculating the Faber-Ziman partial structure factors, Sij(Q), and the Bhatia-Thornton44 number – number partial structure factor, SNN(Q), which indicates the topological order in a system:
$${S_{\mathrm{NN}}}\left( Q \right) = {c_{\mathrm{A}}^2}{S_{\mathrm{AA}}}\left( Q \right) + {c_{\mathrm{X}}^2}{S_{\mathrm{XX}}}\left( Q \right) + 2c_{\mathrm{A}}c_{\mathrm{X}}S_{\mathrm{AX}}\left( Q \right),$$
(3)
where Sij(Q) is a Faber-Ziman partial structure factor and ci denotes the atomic fraction of chemical species i. Moreover, it is possible to compare data for the four liquids while ignoring the difference in the sensitivity of elements to X-rays because the weighting factors for X-rays are eliminated in SNN(Q). The Sij(Q) values calculated from the simulation models for l-Er2O3, l-SiO242, l-Al2O311, and l-ZrO218 are shown in Fig. 5a. It is confirmed that a very sharp PP in l-Er2O3 can be assigned to the Er-Er correlation. The SNN(Q) for l-Er2O3 and those for l-SiO2 and other non-GFLs are compared in Fig. 5b. As mentioned above, only l-SiO2 exhibits an FSDP at QrA-X = 2.6. The QFSDP position arises from an underlying periodicity of 2π/QFSDP that originates, for example, from the formation of pseudo-Bragg planes with a finite correlation length of 2π/ΔQFSDP in l-SiO2, while neither l-Al2O3, l-ZrO2, nor l-Er2O3 show an FSDP in SNN(Q), as discussed in Kohara et al.18. Since the Bhatia-Thornton SNN(Q) can eliminate the weighting factors for X-rays, the absence of an FSDP in SNN(Q) is characteristic of a non-GFL. Another important feature in SNN(Q) is that l-SiO2 and l-Al2O3 exhibit a second PP at QrA-X ~ 5, while a PP is not distinct in the l-ZrO2 or l-Er2O3 data.
The absence of an FSDP in the l-ZrO2 and l-Er2O3 data suggests that both cations and oxygen are densely packed. To confirm this in real space for l-Er2O3, the partial pair distribution functions, gij(r), of l-Er2O3 are compared with those of l-SiO2 in Fig. 6a. The atomic distance r is scaled by dividing by rA-X (distance between the center and corners of the polyhedron). It is found that the scaled first A-A and X-X correlation distances of l-Er2O3 are much shorter than those of l-SiO2, demonstrating that l-Er2O3 has a much more densely packed structure, manifested by the formation of the OEr4 tetracluster network shown in Fig. 6b. This network cannot be found in l-Al2O3 nor in l-ZrO2, suggesting that the very sharp PP in l-Er2O3 is a specific signature of the formation of a tetracluster network with long-range periodicity.
### Topology and homology in l-Er2O3
To reveal the origin of the very sharp PP in l-Er2O3, we calculated the bond angle distributions of the liquid and crystal43 and summarized them in Fig. 7. A pronounced difference was found between the liquid and crystal data for the O–Er–O and Er–O–Er distributions. The O–Er–O bond angle distribution exhibits two peaks at 80° and 140°, suggesting that ErO6 polyhedra are highly distorted in the liquid. Another interesting feature is that the Er–O–Er bond angle distribution exhibits a peak at ~180° in addition to the peak at ~90°, which is not observed for the crystal43 nor in l-ZrO218. This two-peak structure in the Er-O-Er bond angle distribution indicates the formation of a distorted OEr4 tetracluster network, whereas tetraclusters are symmetric (comprising regular tetrahedra) in the crystalline phase. This behavior suggests that the coordination of OEr4 tetraclusters is more octahedral-like and hence tolerant of disorder even in the liquid due to the distortion, providing a linear arrangement manifested by a prominent peak observed at 180° in the Er–O–Er bond angle distribution. This is clearly visible in Fig. 6c, where linear atomic arrangements are highlighted by the magenta lines.
To shed light on the similarity in topology between the crystal and liquid phases, we calculated the persistence diagram for l-Er2O3 and compared it with the crystal data in Fig. 8. The figures show the similarity between the crystal43 and liquid phases. In particular, both the Er-centric and O-centric persistence diagrams for l-Er2O3 do not show a vertical profile along the death axis, which is a pronounced feature in a typical GFL, such as l-SiO242. The short lifetime of the profile manifested by the small death value demonstrates that both the crystal and liquid phases exhibit a very densely packed structure associated with the formation of tetraclusters in both phases. We suggest that this similarity is a signature of non-GFL behavior.
### Electronic structure and viscosity of l-Er2O3
As previously mentioned, a 500-atom RMC model of l-Er2O3 appeared to be too small in reproducing the very sharp PP accurately (see Fig. S3). Nevertheless, we used this model as a starting structure in our DF-MD simulations to study the electronic structure and atomic diffusion in the liquid phase. The electronic density of states (DOS) and effective charges were calculated for snapshots of l-Er2O3 atomic structures and a fully DFT-relaxed (0 K) c-Er2O3 unit cell. The DOSs of l- and c-Er2O3 are shown in Fig. 9a, b. The valence band consists mainly of O-2p states, while the conduction band consists mainly of Er-5d states. As a measure of the orbital localization, we also present the inverse participation ratios (IPRs) for l-Er2O3. The IPRs show increased weight at the valence band maximum (VBM), indicating stronger localization, whereas the states around the conduction band minimum (CBM) are delocalized. The obvious broadening of the valence and conduction bands in l-Er2O3 is caused by the distortion of the ErOn polyhedra at elevated temperatures. The DOS for l-Er2O3 reveals a band gap of 0.57 eV in comparison to the substantially large band gap of 5.46 eV in c-Er2O3. Previously, a vanishing band gap was reported for l-ZrO218, but we ascribe this effect to the PBE functional, which is known to underestimate the real value, whereas the hybrid HSE06 functional used here predicts a more correct band gap.
The difference in electronegativities of Er (1.24) and O (3.44) suggests predominantly ionic chemical bonding between the two atoms. This becomes clear from the partial DOS in Fig. 9a, and the calculated atomic charges are summarized in Table 4. The effective charges for l-Er2O3 are +1.96 e and –1.31 e for Er and O, respectively, similar to those in c-Er2O3 (see Table 4). These values are in agreement with previous works on l-ZrO218, glassy MgO-SiO245, and CaO-Al2O346 and are consistent with the nominal charges Er3+ and O2- (systematically scaled down by a factor of $$\sim \frac{2}{3}$$). As for ZrO218, the increased atomic volume of oxygen in the transition from c- to l-Er2O3 compensates for the corresponding decreased oxygen coordination, which results in similar atomic charges for the two phases. Note here also the similarity with the charges used in the classic MD force field (+2.1 and −1.4 e).
A real space visualization of the highest occupied molecular band (HOMO) is shown in Fig. 9c, where the orbital is found to be distributed over a group of atoms. Locally, the shape of the HOMO is similar to that of c-Er2O3 (Fig. S4); however, some deviations occur caused by the aforementioned distorted tetracluster network in l-Er2O3. Figure 9d shows a real space visualization of the lowest unoccupied molecular band (LUMO), where most of the orbital is distributed over nonbonding regions in-between the ErOn polyhedra. The orbital projections for each of the bands reveal that the HOMO band shows mainly an O-2p character, while the LUMO band consists mainly of Er-5d states and some Er-5p character, in agreement with the DOS and IPR in Fig. 9a.
The DF-MD simulations above the melting point at 2650 °C show that the atoms move rapidly, breaking old and forming new Er-O bonds on a picosecond time scale. The MSD analysis (Fig. S2) shows a consistent linear behavior as a function of time, the evaluated self-diffusion constants are 2.40 × 10−5 and 5.84 × 10−5 cm2/s for Er and O, respectively, and the average self-diffusion constant is 4.64 × 10−5 cm2/s. By using the average value and assuming spherical particles in the Stoke-Einstein relation, one can estimate the viscosity, and the obtained value is ~3 × 10−3 Pa s−1 for l-Er2O3 in comparison to the previously reported value for l-ZrO2, i.e., ~2 × 10−3 Pa s−1 at 2800 °C18. Since these values are an order of magnitude smaller than that for l-Al2O3 (fragile liquid) and 9–10 orders of magnitude smaller than that for l-SiO2 (strong liquid), we characterize l-Er2O3 as an “extremely fragile” liquid18.
We combined an aerodynamic levitation technique and a synchrotron high-energy X-ray diffraction and density measurement at the ISS for l-Er2O3 to reveal the structure of A2X3-type non-GFLs. As the main finding, we observed a very sharp PP in the diffraction data. The Er-O coordination number was estimated to be 6.1 from a combined MD-RMC simulation, which is comparable to that of another nonglass-forming liquid, l-ZrO2, and to that of crystalline Er2O3. The formation of distorted OEr4 tetraclusters in the liquid is confirmed, while OZr3 triclusters are dominant in l-ZrO2 and OAl3 triclusters dominate in another A2X3 non-GFL, l-Al2O3. Apparently, the formation of a distorted tetracluster network in l-Er2O3 gives rise to a long periodicity, yielding the sharp principal peak. This long-range periodicity originates from the significantly increased weight at ~180° in the Er-O-Er bond angle distribution, suggesting that the arrangement of distorted OEr4 tetraclusters involves nearly linear connections, which are not observed in other oxide liquids. Furthermore, persistent homology suggests that l-Er2O3 is homologically similar to the crystalline phase and that both phases are very densely packed, in contrast to a typical GFL such as l-SiO2. This similarity is presumed to be the signature of the liquid, and a considerable difference between two A2X3-type oxide liquids, l-Al2O3 and l-Er2O3, is uncovered. The additional DF-MD simulations demonstrate that l-Er2O3 has an electronic band gap of 0.6 eV, which is considerably lower than that of c-Er2O3 due to angular distortions and nearly linear connections within the polyhedral network. The dynamics of atoms show pronounced mobility, as evidenced by the MSDs, resulting in a very low viscosity, and thus place l-Er2O3 within the regime of extremely fragile liquids.
## Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.
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## Acknowledgements
The synchrotron radiation experiments were performed at BL04B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2016A0134). This research was supported by JST PRESTO, Japan Grant Numbers JPMPR15N4 (to S.K.); the “Materials Research by Information Integration” initiative (MI2I) project of the Support Program for Starting Up Innovation Hub from JST (to Y.O., S.T., S.K., At.M., and Y.H.); JST CREST 15656429 (to Y.H.); JSPS KAKENHI Grant Number JP17H03121 (to A.M.); and the TIA collaborative research program “Kakehashi”, TK19-004 (to S.K.). D.R.S. and S.M.S. acknowledge the Research Council of Norway (FRINATEK Project No. 275139/F20) for their financial support and UNINETT Sigma2 (Project No. NN9264K) for providing computational resources. The density measurement experiments were supported by the ISS crew members and ground operation staff.
## Author information
Authors
### Contributions
S.K. and C.K. designed this research. The high-energy X-ray diffraction measurements were performed by S.K., C.K, Y.O., At.M., Ak.M., Y.W., K.O., H.T., and J.T.O. The density measurements were conducted by C.K., T.I., Y.W., Y.N., H.T., and H.O. The classic molecular dynamics simulations were performed by S.T. The density functional – molecular dynamics simulations were conducted by D.R.S., S.M.S., and J.A.; S.K., D.R.S., S.M.S., J.A., I.O., Y.H., and O.S. analyzed the data, and S.K., C.K., and J.A. wrote the article.
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Correspondence to Chihiro Koyama or Shinji Kohara.
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Koyama, C., Tahara, S., Kohara, S. et al. Very sharp diffraction peak in nonglass-forming liquid with the formation of distorted tetraclusters. NPG Asia Mater 12, 43 (2020). https://doi.org/10.1038/s41427-020-0220-0
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## Introduction
• Are intended to measure the effect of teachers and/or schools on students' achievement
• Establishing the stability (reliability) of the models is prerequisite to establishing validity, which is foundational for their use in high-stakes policy applications
Study purpose
• Evaluate the stability of school-level VAM estimates across cohorts and content area.
## Cohort effects
• VAMs assume estimates do not depend on the specific sample of students modeled.
• Typically, only one year of data is included in estimates.
• Estimates may then be representative of policy or implementation effects
• Student mobility is high in many schools
• If school effects do depend, in part, on sampling variability, then the validity of estimates is threatened
## Content effects
• Little research has explicitly explored the difference in school rankings by the content area.
• Much research has investigated school effects in a single content area, while ignoring others (Raudenbush and Bryk, 1986; Raudenbush and Willms, 1995)
• Should we expect schools to have the same effect across content areas? What does it mean if different effects are observed?
## Research Questions
1. What is the stability of school effect estimates across cohorts and content area (reading and math)?
2. What proportion of the variance in students' scores is attributable to school, cohort, or content facets?
3. How does the number of cohorts modeled impact the reliability of school effect estimates?
## Methods: Sample
### Demographics
Proportion
nonWhite 35
SWD 12
Female 50
FRL 50
• Operational statewide accountability data
• Three cohorts of students matched longitudinally across Grades 3-5 (approximately 27000 students per cohort)
• 727 schools, with an average of 122.44 students per school (SD = 95.17)
## Analysis plan
1. Fit a VAM to each cohort of students in each content area
2. Explore changes in schools' normative rank across models
3. Fit a combined model across cohorts
4. Use Generaliziability Theory to (a) estimate the reliability of school effects, and (b) project reliability, given a change in the number of cohorts modeled.
## Basic school-effects model
$RIT_{ig} = \alpha + \beta_1(g4) + \beta_2(Pr \times g3_4) + \beta_3(Pr \times g3_5) + \beta_4(Pr \times g4_5) + r_i + u_j + e_{ij}$
• $RIT_{ig}$: State test score in Grade $g$ for student $i$ (includes both students' Grade 4 and Grade 5 data)
• $\alpha$: Model intercept (mean Grade 5 scores, given average Grade 3 and 4 scores)
• $g4$: Dummy code indicating if the outcome was in Grade 4 (rather than Grade 5)
• $Pr$: Students prior state test score
• $g3_4$: Grade 3 prior state test score, Grade 4 outcome
• $g3_5$: Grade 3 prior state test score, Grade 5 outcome
• $g4_5$: Grade 4 prior state test score, Grade 5 outcome
• $r_i$ and $u_j$: Random by-student and by-school variation
• $r_i \sim N(0, \sigma_{stu})$
• $u_j \sim N(0, \sigma_{sch})$
• $e_{ij}$: Unmodeled residual variance
• $e_{ij} \sim N(0, \sigma_{e})$
## Breaking the model apart
$RIT_{i4} = \alpha + \beta_1(g4) + \beta_2(Pr \times g3_4) + r_i + u_j + e_{ij}$
$RIT_{i5} = \alpha + \beta_3(Pr \times g3_5) + r_i + u_j + e_{ij} \\\ RIT_{i5} = \alpha + \beta_4(Pr \times g4_5) + r_i + u_j + e_{ij}$
## Fixed-effects portion of the model
Note the residual variances were constrained to be equal
## Combined model
$RIT_{ig} = \alpha + \beta_1(g4) + \beta_2(Pr \times g3_4) + \beta_3(Pr \times g3_5) + \beta_4(Pr \times g4_5) + \\ r_i + u_j + v_c + v_cu_j + e_{ij}$
• $v_c$: Random cohort variation
• $v_c \sim N(0, \sigma_{coh})$
• $v_cu_j$: Random cohort by school variation (latent interaction variable)
• $v_cu_j \sim N(0, \sigma_{cohSch})$
## G-Theory
Relative reliability coefficient $G = \frac{\sigma_{sch}^2}{\sigma_{sch}^2 + \frac{\sigma_{cohSch}^2}{n_{coh}'} + \frac{\sigma_{e}^2}{n_{stu}'n_{coh}'} }$
Absolute reliability coefficient
$\Phi = \frac{\sigma_{sch}^2}{\sigma_{sch}^2 + \frac{\sigma_{stu}^2}{n_{stu}'} + \frac{\sigma_{coh}^2}{n_{coh}'} + \frac{\sigma_{cohSch}^2}{n_{coh}'} + \frac{\sigma_{e}^2}{n_{stu}'n_{coh}'} }$
• A priori minimal threshold for reliability: 0.90
## Results: School-effect variability across cohorts (math)
• ~ 33.77% of schools did not change quartiles
• ~ 53.66% changed quartiles at least once
• ~ 12.57% changed quartiles between each cohort modeled
• ~ 22.7% of schools changed more than one quartile
• ~ 3% of schools moved from the bottom to the top quartile, or vice versa, depending on the specific cohort modeled
## Results: School-effect variability across cohorts (reading)
• ~ 33.71% of schools did not change quartiles
• ~ 53.11% changed quartiles at least once
• ~ 13.18% changed quartiles between each cohort modeled
• ~ 22.41% of schools changed more than one quartile
• ~ 3% of schools moved from the bottom to the top quartile, or vice versa, depending on the specific cohort modeled
## Variability across content areas
• ~ 53%, 55%, and 52% of schools maintained their normative quartile ranking between content areas, for Cohorts 08-10, respectively
• ~ 36% to 39% of schools changed one quartile
• ~ 7% to 9% of schools changed two quartiles
## Results: G-Theory
Variance Components
$\sigma_{stu}^2$ 55.63 67.5 44.02 68.43
$\sigma_{sch}^2$ 8.68 10.5 6.07 9.44
$\sigma_{coh}^2$ 0.84 1.0 0.08 0.12
$\sigma_{cohSch}^2$ 1.51 1.8 0.84 1.30
$\sigma_{e}^2$ 15.82 19.2 13.32 20.71
$G =$ 0.95 and 0.96 for reading and math, respectively
$\Phi =$ 0.92 and 0.95 for reading and math, respectively
• Majority of variance associated with students, followed by unmodeled variance
• Schools next most important facet
• Cohort and cohort by school variance negligible, relative to the whole
## Discussion
• VAMs applied in high-stakes policy settings generally assume the estimates are independent of sampling variability.
• Results of this study suggest high variability depending on the specific cohort of students modeled
• Generally, a single number is used to quantify the school effect
• Results of this study indicate a more nuanced and multidimensional representation may be more appropriate
• Projected reliability was moderate when a single cohort was modeled
• Reliability increased dramatically with the inclusion of even one additional cohort
## Limitations and future directions
• This study investigated "pure" cohort effects, but annual estimates may be more reflective of how the models are applied in practice.
• What's the year-to-year stability?
• Unclear the extent to which changes in school ranks were attributable to sampling variability versus "true" changes in school functioning
• School persistence was not modeled directly
## Thanks!
This research was funded in part by a Cooperative Service Agreement from the Institute of Education Sciences (IES) establishing the National Center on Assessment and Accountability for Special Education (NCAASE; PR/Award Number R324C110004); the findings and conclusions expressed do not necessarily represent the views or opinions of the U.S. Department of Education.
Correspondence concerning this manuscript should be addressed to Daniel Anderson, IES Post-Doctoral Research Fellow, Center on Teaching and Learning, University of Oregon. E-mail: daniela@uoregon.edu
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#
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2018-09-22 18:41:21
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https://pitijumofysij.writeifyoulivetogetthere.com/x-ray-absorption-and-emission-in-analytical-chemistry-book-22927ha.php
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# X-ray absorption and emission in analytical chemistry
spectrochemical analysis with X-rays
• 357 Pages
• 3.99 MB
• 6888 Downloads
• English
by
Wiley , New York
X-ray spectros
Classifications The Physical Object Statement H.A. Liebhafsky [and others]. LC Classifications QD95 .L63 Pagination 357 p. Open Library OL16865988M
This comprehensive two-volume treatise features articles that explain the phenomena and describe examples of X–ray absorption and emission applications in several fields, including chemistry, biochemistry, catalysis, amorphous and liquid systems, synchrotron radiation, and surface phenomena.
X-ray absorption and emission. X-Ray Absorption and X-ray Emission Spectroscopy: Theory and Applications: Combines the theory, instrumentation and applications of x-ray absorption and emission spectroscopies which offer unique diagnostics to study almost any object in the Universe.
Jan 10, · X-ray absorption spectroscopy is a technique that uses synchrotron radiation to provide information about the electronic, structural, and magnetic properties of certain elements in materials.
Analytical Chemistry Book: Physical Methods in Chemistry and. X-Ray Absorption and Emission William J. Campbell, James D. Brown, and John W. Thatcher Analytical Chemistry 38 (5), Cited by: You can write a book review and share your experiences.
### Details X-ray absorption and emission in analytical chemistry PDF
Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.
This comprehensive two-volume treatise features articles that explain the phenomena and describe examples of X–ray absorption and emission applications in several fields, including chemistry, biochemistry, catalysis, amorphous and liquid systems, synchrotron radiation, and surface phenomena.
Is the go-to reference book in the subject for. Get this from a library. X-ray absorption and emission in analytical chemistry; spectrochemical analysis with X-rays.
[H A Liebhafsky]. Nov 25, · X-Ray Absorption and Emission in Analytical Chemistry. Liebhafsky, H. Pfeiffer, E. Winslow, P. Zemany. Wiley, New York, pp. Illus. $Author: Barry W. Mulligan. Buy X-Ray Absorption and X-Ray Emission Spectroscopy: Theory and Applications on writeifyoulivetogetthere.com FREE especially with the development and expansion of large-scale synchrotron reactors All this has resulted in improved analytical performance and new applications, as well as in the perspective of a dramatic enhancement in the potential of x–ray Cited by: Buy X-Ray Absorption and Emission in Analytical Chemistry: Spectrochemical Analysis with X-rays on writeifyoulivetogetthere.com FREE SHIPPING on qualified ordersPrice:$ Spa e Miami Book Archive.
Read Online or Download X-Ray Absorption and Emission In Analytical Chemistry PDF. Similar analytic books. X-Ray Absorption and Emission In Analytical Chemistry.
The Chemistry of Fragrances: From Perfumer to Consumer. Extra info for X-Ray Absorption and Emission In Analytical Chemistry. Example text/5(26). X-ray emission spectra of solids and molecules are methods of measuring electronic structure of matter [1–5].The x-ray emission spectra reflect the occupied electronic structure as shown in Fig.
1, while the x-ray absorption spectra reflect the unoccupied molecular orbitals (MO).These x-ray spectra represent local (L) and partial (P) electron density of states (DOS) because of the electric.
X-Ray Emission Spectroscopy. X-ray Emission Spectroscopy (XES) is an element-specific method to probe the partially occupied electronic structure of materials and was originally developed in the optical wavelength range by Henry Rowland in by using spherical.
### Description X-ray absorption and emission in analytical chemistry FB2
This page contains materials for the session on x-ray emission and absorption. It features a 1-hour lecture video, and also presents the prerequisites, learning objectives, reading assignment, lecture slides, homework with solutions, and resources for further study.
The late 20th century also saw an expansion of the application of analytical chemistry from somewhat academic chemical questions to forensic, environmental, industrial and medical questions, such as in histology.
Modern analytical chemistry is dominated by instrumental analysis. Many analytical chemists focus on a single type of instrument. Absorption spectroscopy refers to spectroscopic techniques that measure the absorption of radiation, as a function of frequency or wavelength, due to its interaction with a writeifyoulivetogetthere.com sample absorbs energy, i.e., photons, from the radiating field.
The intensity of the absorption varies as a function of frequency, and this variation is the absorption spectrum.
A clear-cut introduction to the technique and applications of x-ray absorption spectroscopy X-ray Absorption Spectroscopy is being applied to a widening set of disciplines. Applications started with solid state physics and grew to materials science, chemistry, biochemistry and geology.
Now, they cut across engineering materials, environmental science and national heritage providing very. In this paper, the fixed energy X-ray absorption voltammetry (FEXRAV) is introduced. FEXRAV represents a novel in situ X-ray absorption technique for fast and easy preliminary characterization of electrode materials and consists of recording the absorption coefficient at a fixed energy while varying at will the electrode potential.
The energy is chosen close to an X-ray absorption edge, in Cited by: Absorption techniques in X-ray spectrometry are used to measure the X-ray absorption spectra using various methods described below, and to analyze the obtained spectral line shapes to obtain information on the element, oxidation state, concentration, atomic.
X-ray absorption spectroscopy The impact of the biological x-ray absorption spectroscopy (BioXAS) has increased as indicated by the number and ranking of publications Ascone et al.
Journal of Synchrotron Radiation 16, Application to physics, earth science, chemistry and biology (see presentations of writeifyoulivetogetthere.com: Isabella Ascone. Abstract. Measurements of the chemical compositions of materials and the levels of certain substances in them are vital when assessing and improving public health, safety and the environment, are necessary to ensure trade equity, and are required when monitoring and improving industrial products and services.
A simplified energy diagram that shows the absorption and emission of a photon by an atom or a molecule. When a photon of energy $$h \nu$$ strikes the atom or molecule, absorption may occur if the difference in energy, $$\Delta E$$, between the ground state and. X-ray absorption spectroscopy of soil is a difficult experiment because soft X-ray beamlines in synchrotron facilities are usually reserved for clean surface chemistry, and thus an ultra high Author: Jun Kawai.
X-Ray Absorption and X-Ray Emission Spectroscopy: Theory and Applications / Edition 1 especially with the development and expansion of large-scale synchrotron reactors All this has resulted in improved analytical performance and new applications, as well as in the perspective of a dramatic enhancement in the potential of x–ray based Price: \$ ASTM's analytical chemistry standards are instrumental primarily in chemical analysis of various metals, alloys, and ores.
These analytical chemistry standards present various test methods and techniques in determining the chemical composition of alloys and ores which include molecular absorption spectrometry, X-ray emission spectrometry, optical emission vacuum spectrometry, atomic emission.
### Download X-ray absorption and emission in analytical chemistry EPUB
Abstract. This book describes several specific spectrometric techniques that are very useful in elucidating the fundamental nature of matter: EXAFS - Extended X-Ray Absorption of Fine Structure; SEXAFS - which is EXAFS applied to Surface Phenomena; and XANES - X-Ray Absorption Near Edge Structures.
Rapid developments in analytical techniques and the use of modern reagents in organic synthesis during the last two decades have revolutionized the approach to organic structure determination. As advanced topics in organic analysis such as spectroscopic methods are being introduced, postgraduate students (majoring in organic chemistry) have been feeling handicapped by the non-availability of a 5/5(3).
basic concepts of analytical chemistry Download basic concepts of analytical chemistry or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get basic concepts of analytical chemistry book now. This site is like a library, Use search box in the widget to get ebook that you want.
Atomic Absorption and Atomic Fluorescence Spectrometry: Tutorial: Atomic Absorption Spectroscopy: Atomic Emission Spectrometry: Exercise: ICP Torches: Tutorial: ICP Spectrometers: Atomic Mass Spectrometry: Tutorial: Transducers for Mass Spectrometry: Atomic X-Ray Spectrometry.
Tutorial: X-Ray Absorption and Fluorescence. Non resonant XES consists of scanning the emitted fluorescence with a CAS at a fixed incoming X-ray energy, above an absorption edge. X-ray emission spectroscopy also permits probing the local.This integration trend was also prevalent in the development of small-scale laboratory equipment.
X-ray detectors have advanced with pixellated systems, micro-calorimeter types and the now established silicon drift detectors being readily used by many [email protected]{osti_, title = {X-ray absorption: principles, applications, techniques of EXAFS, SEXAFS, and XANES}, author = {Koningsberger, D.C.
and Prins, R.}, abstractNote = {This volume in the chemical analysis series takes one into the realm of local structure and bonding determined from the fine structure imposed on the atomic x-ray absorption edge by the presence of neighboring atoms.
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2021-10-24 08:23:50
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https://electronics.stackexchange.com/questions/34115/low-power-way-to-make-a-sound
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# Low power way to make a sound
I'm designing a 3V button cell battery powered device. I want it to beep or click or just make any noise at some times just loud enought to be heard across the room.
I've made a prototype working on 12v - it's just an opamp comparator that goes high and low and i've connected a piezo speaker to output of opamp with a cap in series. It makes clicks when opamp goes high or low and they are just loud enough, but when i lower voltage to 3v, i can barely hear the speaker.
Can anyone suggest me a solution?
This piezo buzzer is rated for 3 V operation and produces 100 dBA at 10 cm, which is 80 dBA at 1 m, and 60 dBA at 10 m.
Maximum current is 9 mA, so if you would give one 100 ms burst every 10 seconds your battery will last 4 months.
• yeah, but i've red in wikipedia, that CR2032 batteries can deliver 3mA max. anyway, i'll try with low voltage piezo, as the one i have is from unknown junk bin – miceuz Jun 18 '12 at 16:14
• @miceuz - Since you'll probably need only short bursts you can buffer the 3V with a capacitor. A 1000 $\mu$F capacitor will be sufficient for 100 ms bursts. – stevenvh Jun 18 '12 at 16:16
Some buzzers work with a DC voltage and some need an alternating voltage.
It sounds like you may need to drive your piezo buzzer with an alternating voltage, not a steady one if you want a continuous tone (if you want a click stick with the type you have). The datasheet should tell you this.
They also come in various voltage ratings, so you should be able to find one that works at 3V.
This buzzer will work at 3V and needs a alternating voltage (see datasheet) This will "click" when the comparator output changes.
In comparison, this buzzer will work with a DC voltage, as it mentions a "built in oscillating circuit" and operating voltage of 1-20V DC. This will produce a tone when the comparator output is high.
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2020-11-29 14:23:35
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https://gst.vn/journal/site/page.php?1a238f=copenhagen-interpretation
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Ngày khởi hành
VI
# copenhagen interpretation
1 lượt xem 25/10/2020
This was not a debate with two podia and a moderator. the probabilistic element introduced upon observation. In the end, Einstein was forced to concede that quantum physics was at least consistent. This is what’s at the heart of the Copenhagen interpretation. We’ll use $\lvert 1 \rangle$ and $\lvert 2 \rangle$ to denote the position-states slit 1 and slit 2. I’m deliberately leaving out any complex coefficients so as to not complicate things for the purpose of this post. T. Norsen, S. Nelson, "Yet Another Snapshot of Foundational Attitudes Toward Quantum Mechanics", Bohr recollected his reply to Einstein at the 1927. For the better part of the last century, the most accepted explanation for why the same quantum particle may behave in different ways was the Copenhagen interpretation.Although it's getting a run for its money from the Many-Worlds interpretation lately, many quantum physicists still assume the Copenhagen interpretation is correct. Max Born wrote in a Nobel Prize-winning footnote of his 1926 paper that the probability of a solution to the Schrödinger equation of a quantum-mechanical system (such as an electron) is proportional to the wave function squared. The Copenhagen Interpretation of quantum mechanics is the original attempt by physicists to provide an explanation for the results of quantum experiments. ValueWalk also contains archives of famous investors, and features many investor resource pages. Though they all have the same wave function, the elements of the ensemble might not be identical to one another in all respects, according to the 'noncommittal' interpretations. This page was last edited on 25 October 2020, at 16:31.
It consists of the views developed by a number of scientists and philosophers during the second quarter of the 20th century. By contrast, Max Jammer writes "Einstein never proposed a hidden variable theory. This reflects a feature of the quantum world that was recognized by Einstein as early[39] as 1905. That is, the places where we would have bright stripes if the light were brighter, are the places we see most of the photons hitting. Perhaps the main reason for attacking Landé is that his work demystified the phenomenon of diffraction of particles of matter, such as buckyballs. It’s more than simply saying we don’t know which slit the photon passes through. In 1929, Heisenberg gave a series of invited lectures at the University of Chicago explaining the new field of quantum mechanics.
We will discuss other instrumentalist and realist interpretations of quantum mechanics in another bit of maths and physics.
all of its observers, the problem arises to come up with an interpretation of
have objected to the Copenhagen interpretation, both on the grounds that it is non-deterministic and that it includes an undefined measurement process that converts probability functions into non-probabilistic measurements. Now I should start out here with a disclaimer. It is possible to extract trajectory information from such evolution, but not simultaneously to extract energy–momentum information. For example, a wave is spread over a broad region, therefore does not have a specific location. They are launched from a cannon towards a screen with two slits. The Copenhagen interpretation gives special status to measurement processes without clearly defining them or explaining their peculiar effects. Similarly, if a wave is made up of many different momenta, then the wave-particle doesn’t have a value for the momentum.
This interpretation seems rather unsatisfactory to many. The incompatibility is expressed quantitatively by. But this leaves the task of explaining them by applying the deterministic equation for the evolution of the wave function, the Schrödinger equation, to observers and their apparatus. In 1925–1926, quantum mechanics was invented as a mathematical formalism that accurately describes the experiments, yet appears to reject those classical conceptions. I’ll summarize the Copenhagen interpretation of quantum physics this way: The wave function is a complete description of a wave-particle. By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. M. Chown, Forever Quantum, New Scientist No. In the case of the Heisenberg uncertainty principle, this means that if we measure the position of a particle, then there’s a limit to the precision with which we can know the momentum.
But when the apparatus registers one of those outcomes, no probabilities or superposition of the others linger. Bohr, among his many contributions to quantum physics, was the central figure in clarifying the implications of quantum physics in the early part of the twentieth century. The Copenhagen view of understanding the physical world stresses the importance of basing theory on what can be observed and measured experimentally. [76], Probabilistic interpretation of quantum mechanics involving wavefunction collapse. Bohr noted that we had to give up any physical representation of the whole matter. A recent experiment showed that a particle may leave a trace about its path when travelling as a wave – and that this trace exhibits equality of both paths. This is what the uncertainty principle tells us. ", "... there is no reason to consider these matter waves as less real than particles.
Bohr, among his many contributions to quantum physics, was the central figure in clarifying the implications of quantum physics in the early part of the twentieth century. The Copenhagen interpretation. quantum theory that contains no classical realms on the fundamental level.'. This measurement of the position of the photon introduces an uncertainty in the momentum of the photon, reflected in the fact that the photon is no longer moving straight ahead, but is moving at an unknown angle. The two kinds of information have to be extracted on different occasions, because of the non-separability of the wave function representation. [71][72], Under realism and determinism, if the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results. The view that particle diffraction logically guarantees the need for a wave interpretation has been questioned. Where the dark stripes would be, we get very few photons. [68] It has also recently been considered by Van Vliet. The phrase "statistical interpretation", referring to the "ensemble interpretation", often indicates an interpretation of the Born rule somewhat different from the Copenhagen interpretation. This theory, which posits an additional dynamical wave describing the position of a quantum particle, removes the concept of wave function collapse from his interpretation of quantum theory.
But these rules are expressed in terms of a wave function (or, more precisely, a state vector) that evolves in a perfectly deterministic way. [9], According to an opponent of the Copenhagen interpretation, John G. Cramer, "Despite an extensive literature which refers to, discusses, and criticizes the Copenhagen interpretation of quantum mechanics, nowhere does there seem to be any concise statement which defines the full Copenhagen interpretation. Let’s use the symbol $\lvert \Psi \rangle$ to denote the wave function of the electron. [20] It is sometimes alleged, for example by J.S. In the mid 1950's, Heisenberg reacted to David Bohm's 1952 "pilot-wave" interpretation of quantum mechanics by calling his own work the "Copenhagen Interpretation" and the only correct interpretation of quantum mechanics. When people say that “an electron is in more than one place at the same time” or that the “electron travels as a wave and is detected as a particle,” they are likely embracing the Copenhagen Interpretation. According to the Copenhagen interpretation, physical systems generally do not have definite … Maybe THAT kind of insight makes it clear that ValueWalk Premium is worth another look. Reads for a joint honours degree in mathematics and theoretical physics (final year) in England, at the School of Mathematics and Statistics and the School of Physical Sciences at The Open University, Walton Hall, Milton Keynes. Now suppose we make the light very dim, and replace the screen we use to observe the result with a photographic film, thus giving us a permanent record of the light striking the screen. Physicists have actually done this! In the early 20th century, newly discovered atomic and subatomic phenomena seemed to defy those conceptions. It is one of the oldest of numerous proposed interpretations of quantum mechanics, and remains one of the most commonly taught. Recent research by French physicists suggests maybe quantum particles are more than just a statistical construct and do actually exist even when not under observation. In more recent polls conducted at various quantum mechanics conferences, varying results have been found. jo.src = 'https://www.financialjuice.com/widgets/voice-player.js?mode=inline&display=1&container=FJ-voice-news-player&info=valuewalk&r=' + r;
A sudden change like this in a wave, due to a measurement (in this case, the film “measures” the location of the photon), is the collapse of the wave function. Those who uphold the traditional Copenhagen interpretation are generally instrumentalists. In other words, its wave-like existence has been replaced by a particle-like existence! For present-day science, the experimental significance of these various forms of Born's rule is the same, since they make the same predictions about the probability distribution of outcomes of observations, and the unobserved or unactualized potential properties are not accessible to experiment. have proposed that Bohr was influenced by positivism (or even pragmatism). Einstein's comments "I, at any rate, am convinced that He [God] does not throw dice. Unsubscribe at any time. This term is rejected by many Copenhagenists[24] because the process of observation is mechanical and does not depend on the individuality of the observer. Put differently, we say that with respect to its position, the electron is in a superposition of slit 1 and slit 2 and a whole bunch of other positions on the first screen that lead to nowhere but an inglorious end on the first screen. These phenomena had eluded explanation by classical physics and even appeared to contradict it. We respect your privacy. E. T. Jaynes,[62] from a Bayesian point of view, argued that probability is a measure of a state of information about the physical world.
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2022-08-16 10:03:51
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https://tex.stackexchange.com/questions/187684/abbreviate-only-middle-names-in-biblatex
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# Abbreviate only middle names in biblatex
I know of the option firstinits=true to abbreviate the first and all middle names in biblatex. What I would like to achieve is to only abbreviate the middle names. So e.g. author = {Smith, John Ethan Jacob} should be printed as Smith, John E. J.
Please see the MWE based on the answer of Jonathan from this post at the end of this question. I get the following output:
[SD95] Smith, John E. J./Doe, Jain L. Some book title. A Publisher, 1995.
But what I would like to see as output is:
[SD95] Smith, John E. J. ; Doe, Jain L: Some book title. A Publisher, 1995.
The ; between the author names and the : between the author names and the book title are specific to the bib style (custom made) I use. But the solution below doesn't respect this custom style. Any idea how to achieve the desired abbreviation, but don't loose the bib style?
MWE:
\documentclass{article}
\usepackage[backend=biber,style=din, autocite=footnote]{biblatex}
\begin{filecontents*}{bibliography.bib}
@BOOK{SD1995,
author = {Smith, John Ethan Jacob and Doe, Jain Lauren},
title = {Some book title},
year = {1995},
publisher = {A Publisher}
}
\end{filecontents*}
\addbibresource{bibliography.bib}
\usepackage[T1]{fontenc}
\def\bibnamedelima{ }%
\def\bibnamedelimb{ }%
\DeclareNameFormat{author}{%
\ifblank{#5}{}{#5\addspace}% prefix if applicable
#1% last name
\edef\firstname{#3}%
\ifblank{#3}{}{\addcomma\addspace\expandafter\first{\firstname}}%
\ifthenelse{\value{listcount}<\value{liststop}}%
{\addslash}{}%
}
\DeclareNameFormat{labelname}{%
\ifblank{#5}{}{#5\addspace}% prefix if applicable
#1% last name
\edef\firstname{#3}%
\ifblank{#3}{}{\addcomma\addspace\expandafter\first{\firstname}}%
\ifthenelse{\value{listcount}<\value{liststop}}%
{\addslash}{}%
}
\makeatletter
\def\@empty{}
\def\first#1{\expandafter\@first#1 \@nil}
\def\@first#1 #2\@nil{#1\addspace%
\if\relax\detokenize{#2}\relax\else\@initials#2\@nil\fi}
\def\initials#1{\expandafter\@initials#1 \@nil}
\def\@initials#1 #2\@nil{%
\initial{#1}%
\def\NextName{#2}%
\ifx\@empty\NextName\relax%
\else\@initials#2\@nil\fi}
\def\initial#1{\expandafter\@initial#1\@nil}
\def\@initial#1#2\@nil{#1.\addspace}
\makeatother
\begin{document}
\null
\vfill
How the citation look like: \cite{SD1995}
\clearpage
\printbibliography
\end{document}
Edit Please look here for the din style I'm using in the MWE. With this style I get two white spaces between the first author and the separating ;, when I use the modification by Andrew Swann. Without the modification there is only one white space. I'm now looking for a solution to this problem.
• – cmhughes Jul 2 '14 at 19:43
• @cmhughes Nearly, but it changes the styles of my bibstyle. So for example in the style the last names are in small caps and the delimiter between multiple authors is a ' ; ' not '/'. Do you know how to change it? – user2653422 Jul 2 '14 at 20:21
• @cmhughes And if the last name of the authors list is one with a middle name the ':' between its name and the title of the book is missing. For Example: Smith, John: The handbook of but Smith, John E. The handbook of – user2653422 Jul 7 '14 at 6:34
• Please provide a MWE. – musicman Jul 7 '14 at 6:48
• @musicman I added the MWE in my initial post. I hope it is clear what I mean, even without my custom bib style. – user2653422 Jul 7 '14 at 8:07
## 1 Answer
Here is something less invasive than the solution you have tried. It relies on adjusting the macro \mkbibnamefirst which is called to format the first and middle names rather than completely rewriting the name format:
\documentclass{article}
\usepackage[backend=biber,style=alphabetic,autocite=footnote]{biblatex}
\begin{filecontents*}{bibliography.bib}
@BOOK{SD1995,
author = {Smith, John Ethan Jacob and Doe, Jain Lauren},
title = {Some book title},
year = {1995},
publisher = {A Publisher}
}
\end{filecontents*}
\addbibresource{bibliography.bib}
\usepackage[T1]{fontenc}
\renewcommand*{\mkbibnamefirst}[1]{\edef\firstname{#1}\expandafter\first{\firstname}}
\def\bibnamedelima{ }%
\def\bibnamedelimb{ }%
\makeatletter
\def\@empty{}
\def\first#1{\expandafter\@first#1 \@nil}
\def\@first#1 #2\@nil{#1\addspace%
\if\relax\detokenize{#2}\relax\else\@initials#2\@nil\fi}
\def\initials#1{\expandafter\@initials#1 \@nil}
\def\@initials#1 #2\@nil{%
\initial{#1}%
\def\NextName{#2}%
\ifx\@empty\NextName\relax%
\else\bibinitdelim \@initials#2\@nil\fi}
\def\initial#1{\expandafter\@initial#1\@nil}
\def\@initial#1#2\@nil{#1\bibinitperiod}
\makeatother
\begin{document}
How the citations appear: \cite{SD1995}.
\printbibliography
\end{document}
Update with the din style you point to in your update you should add
\AtBeginBibliography{\renewcommand*{\multinamedelim}{\addsemicolon\addspace}}
to your preamble. The din style contains the definition
\renewcommand*{\multinamedelim}{\mbox{ }\addspace\addsemicolon\addspace}
which adds a single space before the semicolon. Another way to get a single space would be
\AtBeginBibliography{\renewcommand*{\multinamedelim}{\addspace;\addspace}}
• Thank you, nearly perfect. One little thing between the abbreviated J. (Jacob) of the first author and the separating ; I have now two white spaces (without the modification there is only one). It does not happen in your solution with the alphabetic style, because there the separator is the word and. Do you have an idea why this happen or how I can provide my custom style here so it becomes clear what I'm talking about ;) – user2653422 Jul 7 '14 at 18:43
• You can add your custom style to your question. The most obvious question is how are you adding spaces and punctuation? In particular do you use biblatex's macros such as \addspace etc. – Andrew Swann Jul 8 '14 at 7:12
• Sorry for the late response. The style I'm using is basically this one (just modified it a little) The problem with the ; can be reproduced with the original style from domhardt, but I don't know where to look for it. I updated my original post with this style and a link to the github rep. – user2653422 Jul 10 '14 at 11:36
• Thanks for the update. With the modification the white space is gone completely so now it's J.; Doe. What I would like to have is a single white space before and after the semicolon like J. ; Doe. I tried \AtBeginBibliography{\renewcommand*{\multinamedelim}{\addspace\addsemicolon\addspace}} so without the mbox{} of the original style but it didn't work. – user2653422 Jul 10 '14 at 12:08
• I think there is only one space there - anyway I have added another way to get a space. – Andrew Swann Jul 10 '14 at 12:56
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2019-08-19 00:15:23
|
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|
https://ftp.aimsciences.org/article/doi/10.3934/jimo.2020050
|
# American Institute of Mathematical Sciences
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doi: 10.3934/jimo.2020050
## Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints
1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 2 City Institute, Dalian University of Technology, Dalian 116600, China 3 Key Laboratory of Operations Research and Control of Universities in Fujian, College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China 4 School of Mathematics, Liaoning Normal University, Dalian 116029, China
* Corresponding author: Bo Wang
Received May 2019 Revised October 2019 Published March 2020
Fund Project: The second author's research is supported in part by the National Natural Science Foundation of China under Project No. 11701091, and Fujian Education and Research Program for Young Teachers under Project No. JAT170096. The third author's research is supported by the National Natural Science Foundation of China under Project No. 11671183 and No. 11671184, Program for Liaoning Excellent Talents in University under Project No. LR2017049, Scientific Research Fund of Liaoning Provincial Education Department under Project No. L201783638, Liaoning BaiQianWan Talents Program, and Project of Liaoning Provincial Natural Science Foundation of China No. 2019MS-217
A stochastic mathematical program model with second-order cone complementarity constraints (SSOCMPCC) is introduced in this paper. It can be considered as a non-trivial extension of stochastic mathematical program with complementarity constraints, and could arise from a hard-to-handle class of bilivel second-order cone programming and inverse stochastic second-order cone programming. By introducing the Chen-Harker-Kanzow-Smale (CHKS) type function to replace the projection operator onto the second-order cone, a smoothing sample average approximation (SAA) method is proposed for solving the SSOCMPCC problem. It can be shown that with proper assumptions, as the sample size goes to infinity, any cluster point of global solutions of the smoothing SAA problem is a global solution of SSOCMPCC almost surely, and any cluster point of stationary points of the former problem is a C-stationary point of the latter problem almost surely. C-stationarity can be strengthened to M-stationarity with additional assumptions. Finally, we report a simple illustrative numerical test to demonstrate our theoretical results.
Citation: Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020050
##### References:
show all references
##### References:
Numerical result for Problem (22)
N $\bar{f}$ $\bar{\varepsilon}_u$ $\bar{\varepsilon}_v$ infea time(s) 1000 1.53 8.88E-02 4.07E-02 5.43E-06 0.02 10000 1.49 5.14E-02 2.82E-02 3.97E-05 0.02 100000 1.54 5.74E-02 6.77E-02 4.74E-03 0.02 1000000 1.44 3.74E-04 5.22E-04 6.23E-06 0.13 10000000 1.44 1.82E-04 1.89E-04 7.35E-06 1.23
N $\bar{f}$ $\bar{\varepsilon}_u$ $\bar{\varepsilon}_v$ infea time(s) 1000 1.53 8.88E-02 4.07E-02 5.43E-06 0.02 10000 1.49 5.14E-02 2.82E-02 3.97E-05 0.02 100000 1.54 5.74E-02 6.77E-02 4.74E-03 0.02 1000000 1.44 3.74E-04 5.22E-04 6.23E-06 0.13 10000000 1.44 1.82E-04 1.89E-04 7.35E-06 1.23
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2019 Impact Factor: 1.366
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2021-01-23 10:05:25
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5386819243431091, "perplexity": 6831.837501476785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703537796.45/warc/CC-MAIN-20210123094754-20210123124754-00454.warc.gz"}
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https://mathoverflow.net/questions/168449/is-the-following-a-sufficient-condition-for-being-a-primal-algebra
|
# Is the following a sufficient condition for being a primal algebra?
I have a question regarding universal algebra and, in particular, primal algebras:
Suppose that A is a finite simple algebra with no proper subalgebra, no automorphism except the identity map, with a constant c for each of its elements, and that moreover the variety generated by A is c-regular for every element c of A and has equationally definable principal congruences (EDPC). Under these assumptions, is A primal? (Maybe it can be useful to recall that EDPC implies congruence distributivity and point-regularity implies n-permutability for some natural n).
A finite algebra is primal if every finitary function on its universe of arity bigger or equal to 1 is represented by a term of the language. Foster and Pixley characterized primal algebras as those finite algebras that are simple, have no proper subalgebra and no automorphism except the identity map and, moreover, generate a congruence distributive and permutable variety. Therefore what I am asking in the above questions is if there is a way to transform the n-permutability into the usual 2-permutability under the assumption of the question.
Many thanks for any help or comments!
• You might instead see if you can use the conditions to build a generalized Sheffer stroke function, or review the Foster-Pixley argument to see how they get congruence permutability, and then try a variation on that. (They may do it using the stroke function, in which case you have to see what you can say about the clone of A first). – The Masked Avenger May 28 '14 at 18:12
The answer is no. Here is a counterexample that Agnes Szendrei and I found.
Consider the reflexive, symmetric, binary relation $T = \{0,m\}^2\cup \{m,1\}^2$ on the set $S=\{0,m,1\}$. The counterexample is the algebra $A$ whose universe is $S$ and whose operations are all operations that preserve $T$. (I.e., all operations $f$ on $S$ where $T$ is a subalgebra of $\langle S,f\rangle^2$.) This algebra is clearly not primal, since $T$ is a compatible binary relation of $A$ that is different from the total binary relation or the equality relation, and a primal algebra has no such compatible relation.
The lattice operations for the order $0<m<1$ preserve $T$, so $A$ is an expansion of the 3-element lattice. It is easy to check that $A$ is finite, simple, has no proper subalgebra, has no automorphism except the identity map, and has constant operations naming each of its elements. It follows that the variety $\mathcal V$ generated by $A$ is semisimple and congruence distributive. It then follows from Theorems 2.12 and 2.21 of
Ervin Fried and Emil W. Kiss, Connections between congruence-lattices and polynomial properties, Algebra Universalis, 17 (1983) 227-262.
that $\mathcal V$ has EDPC.
What remains is to prove that $A$ is $c$-regular for $c = 0, m, 1$. I argue by contradiction. I only explain $0$-regularity, but the other two cases are similar. Assume that $B\in \mathcal V$ has congruences $\alpha\neq\beta$ such that $0/\alpha=0/\beta$. Replacing one of the congruences with their intersection and renaming we may assume that $\alpha < \beta$. Factoring by $\alpha$ we may assume it is zero. Now we have a congruence $\beta\neq 0$ with $0/\beta = \{0\}$. The properties that hold for $B$ and $\beta$ can be realized in a finite subalgebra of $B$, so assume that $B$ is finite. Now we can assume that $|B|$ is minimal and that $\beta$ is minimal in $Con(B)$ among algebras in $\mathcal V$ realizing this data. To summarize and refine: we may assume that $B\leq A^n$ is an irredundant subdirect power of $A$ and that $\beta$ is an atom in $Con(B)$. The atoms are the restrictions of the kernels of the projections onto all but one of the coordinates, so we may assume that $\beta$ relates tuples that are equal in all but the first coordinate.
Now we use the fact that $A$ has a majority operation (from the lattice structure) and that $T$ is the only proper nonequality compatible relation of $A$ (check). This tells us what the possible finite, irredundant subdirect powers of $A$ are. The algebra $B\leq A^n$ is definable by a simple graph (say $G$) on the index set $\{0,1,\ldots,n-1\}$. Each edge $\{i,j\}$ of $G$ imposes a restriction on $ij$-th projection of $B$: the pair $(b_i,b_j)$ must lie in $T$. Thus, $B$ is the set of all tuples $(b_0,\ldots,b_{n-1})$ such that $(b_i,b_j)\in T$ whenever $\{i,j\}$ is an edge in $G$. (If what I have written sounds confusing, let me summarize the key point as follows: the Baker Pixley Theorem implies that any irredundant subdirect subalgebra of $A^n$ contains $\{0,m\}^n\cup \{m,1\}^n$.)
Now it is easy to see from the above description that the tuple $t=(m,0,0,\ldots,0)$ belongs to $B$ no matter what $B$ is. Moreover, this tuple differs from $0=(0,0,0,\ldots,0)$ in the first coordinate only. I.e., the $\beta$-class of $0$ is not $\{0\}$ after all, since it contains $t\neq 0$.
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2020-12-02 07:02:29
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https://tex.stackexchange.com/questions/451347/how-can-i-remove-the-comma-after-journal-name-and-before-volume-in-my-bst-descri
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# How can I remove the comma after journal name and before volume in my bst description?
I am using an apparently heavily-edited apalike bst style, which I have to use for some reason.
This style unfortunately outputs a comma between the journal name and the volume number, which is something I need to avoid.
Here are the supposedly relevant code snippets:
FUNCTION {article}
{ output.bibitem
format.authors "author" output.check
author format.key output % special for
output.year.check % apalike
new.block
format.title "title" output.check
new.block
crossref missing${ journal emphasize "journal" output.check format.vol.num.pages output } { format.article.crossref output.nonnull format.pages output } if$
new.block
note output
fin.entry
}
FUNCTION {format.vol.num.pages}
{ volume field.or.null
number empty$'skip$
{ " (" number * ")" * *
volume empty${ "there's a number but no volume in " cite$ *
warning$} 'skip$
if$} if$
pages empty$'skip$
{ duplicate$empty$
% { pop$format.pages } { ": " * pages n.dashify * } % { ", " * pages n.dashify * } % changed from ":" for vol,pgs -- BJR 10/5/89 { ": " * pages n.dashify * } % changed from ":" for vol,pgs -- BJR 10/5/89 if$
}
if\$
}
How can I remove this comma with the least amount of changes to this code? To summarize again, I would like to have
Author names (year) Title of article. Journal Name Volume (Number): pages
Author names (year) Title of article. Journal Name, Volume (Number): pages
EDIT: The full file can be found here: https://pastebin.com/RZmmiqyh
EDIT2: I am particularly in search of a solution that can also handle entries that are "in press", i.e. neither volume, nor number, nor pages have been assigned yet. Instead, after the journal name a colon should appear, and after the colon a comment such as "in press" should be printed.
Example:
Author names (year) Title of article. Journal Name: in press
Exemplary Bibtex entry:
@Article{Author2018,
Title = {How to make Bibtex do what I want},
Journal = {Journal of Bibtexing},
Year = {2018},
Note = {in press},
}
• Can you upload the entire .bst file to a text-sharing website like pastebin (please don't use a file sharing website where one has to download a .zip or the like)? That would make it much easier to actually test a solution and it helps with the definitions of unseen macros like field.or.null. – moewe Sep 18 '18 at 8:05
• Sure, please find it here: pastebin.com/RZmmiqyh – Michael Sep 18 '18 at 8:07
I suggest you proceed as follows:
• Open your modified copy of apalike.bst in a text editor. (The program you use to edit your tex files will do fine.)
• Immediately after the function output.check, add the following 3 lines of code:
FUNCTION {add.blank}
{ " " * before.all 'output.state :=
}
• In the article function, locate the line that says
{ journal emphasize "journal" output.check
• Immediately after this line (and hence immediately before the line that says format.vol.num.pages output), insert a new line with following text:
add.blank
• Save the bst file
• Rerun LaTeX, BibTeX, and LaTeX once more to recreate the formatted bibliography
Happy BibTeXing!
Addendum to address the OP's follow-up question, about how to handle an entry which is "In press", i.e., which hasn't been published yet and for which the volume, number, and pages fields are not yet available.
I believe you should put the "In Press" information in the year field, not the note field. If the article hasn't been published yet, it's better to write "In press" in the year field; before you know it, the publication year might slip from 2018 to 2019 or even 2020... Optional: Use a \noopsort directive to help BibTeX with figuring out how to sort the entry. And, place an \unskip directive in the entry's volume field to undo the output of the add.blank directive.
\RequirePackage{filecontents}
\begin{filecontents}{mybib.bib}
@preamble { "\providecommand\noopsort[1]{}" }
@Article{Author:InPress,
Title = {Thoughts},
Journal = {Journal of Bib\TeX{}ing},
Year = {\noopsort{2019}In Press},
volume = {\unskip},
}
\end{filecontents}
\documentclass{article}
\usepackage[round]{natbib}
\bibliographystyle{apalike-nocomma} % modified form of 'apalike.bst'
\begin{document}
\nocite{*}
\bibliography{mybib}
\end{document}
• Thanks a lot, this worked - almost. Now, there is a space between the Journal name and the colon, which should not exist. – Michael Sep 18 '18 at 8:38
• @Michael - If there's a space between the journal name and the colon, it must be because the entry is missing the volume field. That's almost certainly an error. All academic journals I'm familiar with feature a volume number. Which journal you're citing from is missing the volume information? – Mico Sep 18 '18 at 8:43
• @Michael - So not only the volume field, but also the number and pages fields, are missing? You might have volunteered this rather crucial piece of information a bit earlier... – Mico Sep 18 '18 at 8:46
• @Michael - Please edit your posting to (a) mention that for entries the volume, number, and pages fields might be missing and (b) provide a representative example of such an entry. Clearly, this is a rather non-standard case which requires further investigation. – Mico Sep 18 '18 at 8:49
• @Michael - Please edit your query to provide a specific example of such a "non-standard" entry. – Mico Sep 18 '18 at 8:51
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2019-06-18 23:16:21
|
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http://www.dynaverse.net/forum/index.php/topic,163349955.0/all.html
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#### Lord Schtupp
• Posts: 379
« on: December 09, 2004, 02:06:32 am »
USS Monitor FCC+ Command Cruiser Refit
Launch Date: 2275
4694 Polys
USES FCX UI
BPV=173
Shields 40-30-30-30 = 190
Warp 18+18=36
hp1 2 phot FA
hp2 2 phot FA
hp3 1 DroG 2
hp4 1 DroG 2
hp11 2 Ph1 ALL
hp12 2 Ph1 FRRX
hp13 2 Ph1 FLLX
hp14 2 Ph1 FH
hp15 2 Ph1 FH
hp16 2 Ph1 RH
More textures coming soon...
« Last Edit: December 09, 2004, 01:15:42 pm by Lord Schtupp »
#### Darkdrone
• Drone
• Lt.
• Posts: 853
• Gender:
• Unimatrix One
« Reply #1 on: December 09, 2004, 03:29:35 am »
WOW what a model
DD
#### FoaS_XC
• Photorps, Sammiches, woot woot.
• Global Moderator
• Commander
• Posts: 4571
• Gender:
« Reply #2 on: December 09, 2004, 06:04:07 am »
Hell Yeah!
Robinomicon
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#### FPF-SCM_TraceyG_XC
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« Reply #3 on: December 09, 2004, 06:23:27 am »
Very nicely done indeed
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#### Major A Payne
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« Reply #4 on: December 09, 2004, 07:41:38 am »
Dam that looks great. Highly interesting, especially the nacelles.
#### OlBuzzard
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« Reply #5 on: December 09, 2004, 07:46:47 am »
Dam that looks great. Highly interesting, especially the nacelles.
agreed.. not too sure about the nacelle caps .... (but that's cool) .. the rest is really super. I especially find the bussards interesting !
over all ... excelllent job sir !!
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#### Darth Chandley
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« Reply #6 on: December 09, 2004, 08:05:12 am »
Another triumph Lord Schtupp... very nice job indeed....
I had a feeling you'd release this soon when I spotted it in your sig.....
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« Reply #7 on: December 09, 2004, 08:31:06 am »
What a ship !!!!!!!!!!!!!!!
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#### Chrystoff
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« Reply #8 on: December 09, 2004, 08:41:28 am »
That is one beautiful FCC+!!
#### Rod ONeal
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« Reply #9 on: December 09, 2004, 09:14:14 am »
Very, very nice! Absolutely marvelous!
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#### Wolfsglen
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« Reply #10 on: December 09, 2004, 09:35:49 am »
Absolutely beautiful!
#### Khalee002
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« Reply #11 on: December 09, 2004, 10:23:33 am »
Nice model But get the Antaries and that Carrier in your wip section done dood. The Antaries been in wip mode too long.
#### Sirgod
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« Reply #12 on: December 09, 2004, 10:44:26 am »
Very good job. I personally Like the Nacelle caps To be honest.
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#### Lord Schtupp
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« Reply #13 on: December 09, 2004, 12:40:06 pm »
Thanks all, I cant take the credit/blame for the warp nacelle design, there are several drawings on shipschematics.net that feature this warp. Funny but I didnt like this warp either until I actually started to mesh it, It grows on you. Its a good TOS-TMP transitition design. This is the plan I worked from and as you can see I added a few of my own details to help get her closer to the TMP era:
Note that here they are inside out (IMO) with the intercoolers facing out, I didnt like that so changed them to and added outboard grilles. I like the way the torp launcher came out
« Last Edit: December 09, 2004, 01:17:21 pm by Lord Schtupp »
#### Azel
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« Reply #14 on: December 09, 2004, 01:23:18 pm »
SWEETNESS!!!
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#### Wicked Zombie
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« Reply #15 on: December 09, 2004, 03:19:47 pm »
I was wondering when you were going to Schtuppify that design. Definitely fits into the pre-TMP design lineage.
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#### markyd
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« Reply #16 on: December 09, 2004, 03:32:29 pm »
that is a1 stuff..... fantastic
#### Rogue
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« Reply #17 on: December 09, 2004, 04:58:58 pm »
Thanks all, I cant take the credit/blame for the warp nacelle design, there are several drawings on shipschematics.net that feature this warp. Funny but I didn't like this warp either until I actually started to mesh it, It grows on you. Its a good TOS-TMP transitition design. This is the plan I worked from and as you can see I added a few of my own details to help get her closer to the TMP era:
I like it! In a few minutes it will become the companion model to Atrahasis's Endeavor. To my thinking it makes for an excellent Federation fast cruiser because of the nacelles representing the 'Hot Warps'. Perhaps not what they were intended for but it makes visual sense to me. It's models like this that make Starfleet Command a better game. I also think it more than a little funny that this came out of nowhere.
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#### S33K100
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« Reply #18 on: December 09, 2004, 05:18:29 pm »
Very nice, I like that you didn't just follow the schematics which lets face it are kinda fugly being based on the Franz Joseph version of the Connie. Another cool touch is the dark blue ring on the saucer shaped like the one on the Ambassador. It would be cool to see another step in the evolution inbetween this and the TMP Enterprise class, with the beefed-up saucer and more of the secondary hull rebuilt, the 'Phase II' Connie is nice but it's still too TOS-lookinh to be the more advanced replacement for this.
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#### QobnuH
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« Reply #19 on: December 09, 2004, 05:35:26 pm »
Just when I thought I was going to stop D/L on Feds...you or WZ come out with another,awesome as usual.
I hate to bring it up again,but what happened to the Antares?
#### Age
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« Reply #20 on: December 09, 2004, 08:46:45 pm »
This is how the 1701 refit should of looked liked more resembling its TOS look.Then 1701-A could of looked like TMP version.That is a nice model .
#### Chris Johnson
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« Reply #21 on: December 09, 2004, 08:51:38 pm »
I disagree with Age... I half-thought that something like this was in the TOS-era anyway as an "X-Ship"... I figured the U.S.S. Constitution would go through her refits for newer technology, and then be refitted completely as the TMP vessel alongside Enterprise, but easier-to-refit, yet with all those bugs it seems coincidential the Constitution's sister ship had to go alone after V'Ger (or something like that). *shrug* That's how I view it.
And with how I view it that way, I would view this ship more of an F-XCA than an F-CC+. Nevertheless, you've done a beautiful job Schtupp.
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#### Chris Johnson
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« Reply #22 on: December 09, 2004, 09:00:01 pm »
Some food for thought: Take a look at this picture from this website, and tell me what you think about the design of the third ship schematic in the picture. I really like it, it's TMP-like while keeping the TOS design of the Constitution-class mostly.
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#### Wicked Zombie
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« Reply #23 on: December 10, 2004, 12:40:09 am »
I always thought those engines would've looked better if they were inverted, so that the end tapers up instead of down:
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#### Lord Schtupp
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« Reply #24 on: December 10, 2004, 12:59:50 am »
No exhaustive tests and experimentation has confirmed the fact that this configuration makes the warp controller thrust-line too far above the tangental axis of the warp field bow front resulting in lost maneuvering efficiency - duh
#### Lord Schtupp
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« Reply #25 on: December 10, 2004, 01:02:21 am »
lol
#### FoaS_XC
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« Reply #26 on: December 10, 2004, 06:14:24 am »
...
wow
But seriuosly, IMHO this ship is great. Its a great FCX for between TOS and TMP. Even though i dont like X-ships
But sir, if the VP is such a VIP, shouldn't we keep his PC on the QT, 'cause if it leaks to the VC he could end up an MIA and then we'd all be put on the KP ASAP.
Robinomicon
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#### FPF-Wanderer
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« Reply #27 on: December 10, 2004, 10:00:25 am »
Another excellent model from Lord Schtupp.
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#### Lord Schtupp
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« Reply #28 on: December 10, 2004, 07:04:35 pm »
hey I see WZ has revised the alt configuration drawing; that looks good - hmm maybe those tests were wrong....
wow
But seriuosly, IMHO this ship is great. Its a great FCX for between TOS and TMP. Even though i dont like X-ships
But sir, if the VP is such a VIP, shouldn't we keep his PC on the QT, 'cause if it leaks to the VC he could end up an MIA and then we'd all be put on the KP ASAP.
lol luv that movie... I usually considered the TMP connie and technology to be the "X-Ship" or advanced SFC OP tech level, so this is a "late" TOS tech level. I had to use the FCX ui because the FCA ui just doesnt have enough phaser hardpoints, most of the fed uis are short in the hardpoint dept.
I use the Chris Jones TOS mod as my shiplist, and often refer to SFB SSDs for systems/weapons ideas
#### Mackie
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« Reply #29 on: December 10, 2004, 07:29:59 pm »
3ds or lwo plz schtupp
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#### Sandman3D
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« Reply #30 on: December 10, 2004, 08:20:32 pm »
DAMN!!
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#### Kaenyne
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« Reply #31 on: December 10, 2004, 09:12:36 pm »
Oh yeah, Monitor is going in-game ASAP.
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#### TheStressPuppy
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« Reply #32 on: December 11, 2004, 08:43:32 am »
That is some damn Nice work Schtupp Ill let you slide this time on the inward curve on the bottom of the secondary hull being as this is a refit I never understood why people put that curve on connie hulls when it doesnt exist on the 11 footer, then i realized that the 3 footer has that curve, and alot the schematics ive seen were based off the 3 footer, or the AMT model which also was based off the 3 footer.
In any case nicely done, nacelles defininetly give it distinction.
#### S33K100
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« Reply #33 on: December 11, 2004, 01:02:12 pm »
The curve's also on the hideously bad FJ schematics, it just makes the ship look like it's got a beer-belly, if such a thing can be said about a spaceship.
If I determine the enemy's disposition of forces while I have no perceptible form, I can concentrate my forces while the enemy is fragmented. The pinnacle of military deployment approaches the formless: if it is formless, then the deepest spy cannot discern it, nor the wise make plans against it.
Sun Tzu 'The Art of War'.
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#### Rat Boy
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« Reply #34 on: December 19, 2004, 01:47:56 pm »
So, Schtupp, are you transitioning towards the TMP era now?
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#### USS Mariner
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« Reply #35 on: December 19, 2004, 05:55:49 pm »
Well, I'm transitioning towards TUC myself.
http://www.freewebs.com/startrekmariner/KORO1.PNG
http://www.freewebs.com/startrekmariner/KORO2.PNG
http://www.freewebs.com/startrekmariner/KORO3.PNG
http://www.freewebs.com/startrekmariner/KORO4.PNG
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Amazing how Jeffries Extended with a bold outline can match Microgramma in looks.
Here's a wet dream: the Monitor design with TSP Connie parts...ohh yeah.
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#### S33K100
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« Reply #36 on: December 19, 2004, 06:17:41 pm »
Mariner, where'd you get that TMP era emblem from? could you send me the files perchance?
Oh and what's with the font? Do you not have the Starfleet bold extended font? I can send it to you if you don't have it.
If I determine the enemy's disposition of forces while I have no perceptible form, I can concentrate my forces while the enemy is fragmented. The pinnacle of military deployment approaches the formless: if it is formless, then the deepest spy cannot discern it, nor the wise make plans against it.
Sun Tzu 'The Art of War'.
S33K100: formerly Marauth
#### USS Mariner
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« Reply #37 on: December 19, 2004, 06:51:35 pm »
Mariner, where'd you get that TMP era emblem from? could you send me the files perchance?
Oh and what's with the font? Do you not have the Starfleet bold extended font? I can send it to you if you don't have it.
Belive it or not, that arrowhead is actually a Microsoft Word creation, layered into a word document containing the full sized texture. Just click a screengrab and then deposit the changes in the right place in the real texture and bingo, a Starfleet Arrowhead without PSP.
I can send you the zip file containing the word document (the zip is <40kb and the document itself is only 84kb) which i use to do all my editing. If you want, I'll send the korolev modification files too.
I have the normal TMP-style font (Starfleet Bold Extended it says,) but the Jeffries font used here is supposed to mimic the microgramma font on the unused (but finished) Phase II model.
See photos of that model here: http://www.starshipbuilder.com/ubb/Forum1/HTML/000014.html
Suffice to say, I do think the Microgramma font is kind of bad looking, it's too "round" for me. Jeffries Extended looks nicer since it's based off of Starfleet Bold Ext. The only problem is that it's a bit thin, which is cured since word can add outlines to the font to make it beefier.
I'm interested in making the hull color TMP white, so that it'll reflect what the Korolev might've looked like during TUC (they refitted everything else anyways.)
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#### S33K100
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« Reply #38 on: December 19, 2004, 07:19:58 pm »
Cool, could you send me the files (include the Korloev modification too if you can) I like to see how other people do modifications like this, most of my texture alteration work is fairly basic and it's nice to get some additional skills now and then.
I never noticed the Phase II had not outline on the regs, I don't like to use that font though as it's too similar to Microgramma which is the DS9/Voyager era font for a lot of ships.
I've already actually started modifying the textures on this one, done regs for a USS Sicilia and plan to add some weathering to the rather pristine looking secondary hull and nacelles, and possibly even swap the bridge out for the phase II style one on WZ's Ptolemy, maybe.
If I determine the enemy's disposition of forces while I have no perceptible form, I can concentrate my forces while the enemy is fragmented. The pinnacle of military deployment approaches the formless: if it is formless, then the deepest spy cannot discern it, nor the wise make plans against it.
Sun Tzu 'The Art of War'.
S33K100: formerly Marauth
#### Captain Pierce
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« Reply #39 on: December 19, 2004, 07:26:45 pm »
I have to say, Mariner, I've seen a lot of Phase 2 reference (I was around when TheStressPuppy built his ) and those pics have never come up. I would have to guess that this might be when they were trying to rebuild the TV model for the movie, before they realized that it wouldn't work, and that it would have to be done from scratch? I've been trying to find the pics that I've seen of the Phase 2 TV model under construction, but I can't seem to manage it...
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#### S33K100
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« Reply #40 on: December 19, 2004, 07:34:35 pm »
Actually Stress did his based on an early concept study model, those pics are the final TV model and the model is only about 4-foot long, the TMP model we know and love is twice that size, they're completely different models.
'What happened to the original 4-foot model next is anyone's guess. But now, thanks to CG artist and "archivist" Mark Dickson, here are some extremely rare behind-the-scenes photos of the miniature when it was almost ready for filming. Thanks Mark!'
from that page. You can see it is very different from the study model everyone regards as the Phase II Enterprise - that famous pic of the yellow model with the deflector and torpedo launchers missing
If I determine the enemy's disposition of forces while I have no perceptible form, I can concentrate my forces while the enemy is fragmented. The pinnacle of military deployment approaches the formless: if it is formless, then the deepest spy cannot discern it, nor the wise make plans against it.
Sun Tzu 'The Art of War'.
S33K100: formerly Marauth
#### Captain Pierce
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« Reply #41 on: December 19, 2004, 08:08:14 pm »
You can see it is very different from the study model everyone regards as the Phase II Enterprise - that famous pic of the yellow model with the deflector and torpedo launchers missing
Yep, that's the one... and it's the only one pictured in my book on the Phase 2 series... which is why I'm wondering what the heck the deal is... one would think that, if the Reese-Stevenses are hooked up enough to become "Enterprise" story editors, why wouldn't they have been able to find these pics?
(apologies to LS for participating in this thread hijack, BTW )
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#### S33K100
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« Reply #42 on: December 19, 2004, 08:10:49 pm »
judging by the info on that page it seems these only came into the public domain recently, maybe they were lying forgotten in some basement with the unsold copies of Bill Shatner's album.
If I determine the enemy's disposition of forces while I have no perceptible form, I can concentrate my forces while the enemy is fragmented. The pinnacle of military deployment approaches the formless: if it is formless, then the deepest spy cannot discern it, nor the wise make plans against it.
Sun Tzu 'The Art of War'.
S33K100: formerly Marauth
#### USS Mariner
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« Reply #43 on: December 19, 2004, 08:12:34 pm »
Cool, could you send me the files (include the Korloev modification too if you can) I like to see how other people do modifications like this, most of my texture alteration work is fairly basic and it's nice to get some additional skills now and then.
I never noticed the Phase II had not outline on the regs, I don't like to use that font though as it's too similar to Microgramma which is the DS9/Voyager era font for a lot of ships.
I've already actually started modifying the textures on this one, done regs for a USS Sicilia and plan to add some weathering to the rather pristine looking secondary hull and nacelles, and possibly even swap the bridge out for the phase II style one on WZ's Ptolemy, maybe.
I actually pretend that Voyager, the Defiant, and the (hideous!) E-E have the normal Starfleet Bold Ext for the name, because I'm that kind of person. F*ck canon ship "asthetics" (or lack thereof), especially with First Contact. That pretty much threw the notion of "sane" vessel designs out the window. Eaves cannot design his way through a paper bag, as the FC Cube, E-E, and his designs for the next two films prove. (He apparently does a bit better with Vulcan ships as ENT proves [though I doubt that they are his and not Doug Drexlers, who designed the Surak from Season 1 anyway], but the other races stil bear his trademark "suckitude" motif.) For what it's worth, the art department has really lost in general after Generations. They don't give a sh*t about anykind of style anymore except "üb3R-f0cK1n6 k3w1," no matter how bad it looks. The bastards deserve to be flogged to death.
I miss Andy Probert and Rick Sternbach. Atleast Voyager itself wasn't a bad design.
I'll try and email all the files to you by tomorrow or tuesday.
You can see it is very different from the study model everyone regards as the Phase II Enterprise - that famous pic of the yellow model with the deflector and torpedo launchers missing
Yep, that's the one... and it's the only one pictured in my book on the Phase 2 series... which is why I'm wondering what the heck the deal is... one would think that, if the Reese-Stevenses are hooked up enough to become "Enterprise" story editors, why wouldn't they have been able to find these pics?
(apologies to LS for participating in this thread hijack, BTW )
Well, consdiering that the Montior is pretty much a Phase II ship anyway I don't think we've gotten that far off course.
judging by the info on that page it seems these only came into the public domain recently, maybe they were lying forgotten in some basement with the unsold copies of Bill Shatner's album.
"Improve a mechanical device and you may double productivity. But improve man, you gain a thousandfold." - Khan
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#### Captain Pierce
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« Reply #44 on: December 19, 2004, 09:00:02 pm »
judging by the info on that page it seems these only came into the public domain recently, maybe they were lying forgotten in some basement with the unsold copies of Bill Shatner's album.
If that were the case, then I doubt they ever would have been found--that's WAY too much clutter to sort through...
I have to say that I personally don't mind the Ent-E, and that Mr. Eave's proto-Klingon-battlecruiser (http://www.stguardian.to/mixed/predesign/klingonretro.jpg)http://www.stguardian.to/mixed/predesign/klingonretro.jpg) is a damned sight better than ANYTHING ELSE Klingon we've seen on Enterprise, which admittedly ain't sayin' much...
(sigh, sooner or later, I'm going to hijack this thread far enough to violate the Patriot Act... )
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#### Lord Schtupp
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« Reply #45 on: December 20, 2004, 12:36:03 am »
Here's a wet dream: the Monitor design with TSP Connie parts...ohh yeah.
nice...
#### Sandman3D
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« Reply #46 on: December 20, 2004, 11:32:31 am »
HIJACK!!!!
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To save mankind-yourself you scorned to save."
#### TheStressPuppy
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« Reply #47 on: December 20, 2004, 08:34:25 pm »
I hate to jump in like this, but that phase 2 model was a mock up for some trek exhibit in las vegas. The "True" phase 2 models construction was never finished, and it was scrapped when the show became ST:TMP.
I stumbled on pics of that model when i was looking for phase 2 reference on the IDIC page, and it referred to being built for a star trek experience exhibit some time ago. That model obviosly looks like a Phase 2/ TMP hybrid. In fact the only thing that looks any different is the nacelles. Sorry to burst any bubbles, but thats not the true phase 2 ship.
EDIT: sorry schtuppy for assisting in the hijack
I dont think l schtupp feels too good right now, about the thought of his model being bashed together with mine. I know i gave open permissions to bash away, but schtupp may not have. Also Im not in any competition for "who builds the best connie model", I just built that ship cause i wanted to build it. Schtupp builds some damn nice models IMO, and we get together on occasion to compare reference sources. I consider him a personal friend, and i think it would be better if you plan on "bashing" our work together to ask in private 1st.
« Last Edit: December 20, 2004, 08:57:19 pm by TheStressPuppy »
#### Captain Pierce
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« Reply #48 on: December 20, 2004, 08:41:01 pm »
It doesn't help the case for those pics, BTW, that they got the name of the guy who built the Phase 2 TV model (that half-finished yellow one ) wrong--it was a dude by the name of Brick Price.
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#### markyd
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« Reply #49 on: December 21, 2004, 04:15:40 am »
I like both your models... awsome... oh and LS just looking around your website and YOUR WIP's look awsome
#### Lord Schtupp
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« Reply #50 on: December 21, 2004, 06:24:10 am »
So, Schtupp, are you transitioning towards the TMP era now?
Nope its best to leave TMP to the masters.
Anyway I started a transition to the Steam Era where Im gonna try my hand at Steam Locomotive models for the Auren Trainz simulator. A Southern Pacific AC12 4-8-8-2 Cab Forward in fact. But Im not leaving here.
I'm developing a new TOS subrace, an unseen fleet of ships of which is mentioned in TOS. Top Secret of course, very hush-hush.
oh yea and I have a huge pile of unfinished models too of course.
#### Lord Schtupp
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« Reply #51 on: December 21, 2004, 06:27:19 am »
In case anybody's interested: This is a Phase Ib vessel, where it was discovered among other things that adding slight curvature to the secondary hull bottom adds .000000000000000132% to overall warp field generation efficiency.
#### S33K100
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« Reply #52 on: December 21, 2004, 10:18:56 am »
That's nice, but could you make your Constitution model .000000000000000132% less efficient? LOL Look forward to seeing the top secret race when they're revealed, oh and the Antares if you finish it.
If I determine the enemy's disposition of forces while I have no perceptible form, I can concentrate my forces while the enemy is fragmented. The pinnacle of military deployment approaches the formless: if it is formless, then the deepest spy cannot discern it, nor the wise make plans against it.
Sun Tzu 'The Art of War'.
S33K100: formerly Marauth
#### Lord Schtupp
• Posts: 379
« Reply #53 on: December 21, 2004, 12:53:04 pm »
Odd - nobody has ever complained before about its "efficiency" even though that model's been out for over 2 years.
#### S33K100
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« Reply #54 on: December 21, 2004, 01:29:55 pm »
LOL, I saw the bulge when I first d/led it but thought I'd be hounded as a geek for suggesting that it should be corrected.
If I determine the enemy's disposition of forces while I have no perceptible form, I can concentrate my forces while the enemy is fragmented. The pinnacle of military deployment approaches the formless: if it is formless, then the deepest spy cannot discern it, nor the wise make plans against it.
Sun Tzu 'The Art of War'.
S33K100: formerly Marauth
#### TheStressPuppy
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« Reply #55 on: December 21, 2004, 03:41:23 pm »
Actually, That explanation makes perfect sense Federation realized with the original config connie that they could get a slightly higher speed if the just curved the hull a bit more. Even the slightest speed boost could make all the difference. hence why we started seeing more curvy ships from the phase 2-TMP eras. Brilliant
#### Lord Schtupp
• Posts: 379
« Reply #56 on: December 21, 2004, 05:56:44 pm »
I've already actually started modifying the textures on this one, done regs for a USS Sicilia and plan to add some weathering to the rather pristine looking secondary hull and nacelles, and possibly even swap the bridge out for the phase II style one on WZ's Ptolemy, maybe.
That would look cool. Blanket permission granted on kitbash releases btw...
Mackie - Ill post 3ds as soon as i can...
#### zerosnark
• Posts: 104
« Reply #57 on: December 24, 2004, 03:45:27 pm »
This ship is just TOO COOL.
I am currently using the Atra Endeavor as a fed CAI. I also cobled up a + refit.
I am going to fetch this model now.
Great work!
#### Fedman NCC-3758
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« Reply #58 on: June 20, 2005, 08:12:43 am »
The Star Spangled Banner bring hither,
O're Columbia's true sons let it wave.
May the wreaths they have won never wither,
Nor it's stars cease to shine on the brave.
#### OlBuzzard
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« Reply #59 on: June 20, 2005, 08:19:35 am »
Superb workmanship ... from start to finish ! Something us beginners need to remind our selves of ... to strive for excellence !
If you aim at nothing: you WILL hit it every time !
#### Lord Schtupp
• Posts: 379
« Reply #60 on: June 20, 2005, 09:11:08 am »
Wow that compliment is better than the model itself lol Thanks buzzard (reminds me I need to update my website. the only dl links available are at the beginning of this thread. Doh!)
Ive been messing around whith this light cruiser design for a few months now, but Ive been so nitpicky that its difficult to get it finshed, esp with reallife issue and whatnot:
#### Mr_Tricorder
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« Reply #61 on: June 20, 2005, 10:41:42 am »
I like that one. It looks pretty much finished to me, but if you can improve on it, then by all means do so.
#### Lord Schtupp
• Posts: 379
« Reply #62 on: June 20, 2005, 10:48:40 am »
Its mainly the textureing and cutting up the mesh for mapping thats holding it up. Ive showed this over at Atras forum for a while now and I have received some good suggestions over there.
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« Reply #63 on: June 20, 2005, 10:53:17 am »
Very very nice ship !!!!!!
-MP
ModelsPlease, resident "Model Junkie" recovering from a tragic crayon sharpener accident.
#### Rat Boy
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« Reply #64 on: June 20, 2005, 11:19:37 am »
That Aurora is looking slick. Have you considered re-redoing your Connie in light of the new(er) version seen on ENT with the aft weapons?
"Chaos Theory, Part II" now available.
#### Lord Schtupp
• Posts: 379
« Reply #65 on: June 20, 2005, 12:07:47 pm »
That Aurora is looking slick. Have you considered re-redoing your Connie in light of the new(er) version seen on ENT with the aft weapons?
No dammit I havnt seen it (i dont get UPN), but I sure would like to see some good screencaps of that. I have been working on my Connie V2.0, all new mesh based on the casimiro plans seen here mapped to the hull. The warp engines here are the old ones but they will be all new as well. I'll try to do a full set of regs + blanks as i usually do. I havnt forgot about the Hornet
#### USS Mariner
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« Reply #66 on: June 20, 2005, 12:33:16 pm »
I assume the hull is slightly modifed from the plans, as well as the B/C deck, so account for their slightly inaccurate shapes? I rmember you giving a long dictation on "Conny Blueprint Quality" back when TSP made his.
"Improve a mechanical device and you may double productivity. But improve man, you gain a thousandfold." - Khan
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#### E_Look
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« Reply #67 on: June 20, 2005, 12:37:59 pm »
Hey Schtupp, would you mind a (small, large, depends on your view of things) critique about the Aurora?
#### USS Mariner
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« Reply #68 on: June 20, 2005, 12:45:41 pm »
Hey Schtupp, would you mind a (small, large, depends on your view of things) critique about the Aurora?
I doubt you've convered things as nitpickily as I have, but why not?
BTW, if it's about the bussards on the second pic, ignore them. The most recent WIP is the one with IaMD style bussards, which I suggested.
"Improve a mechanical device and you may double productivity. But improve man, you gain a thousandfold." - Khan
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#### Lord Schtupp
• Posts: 379
« Reply #69 on: June 20, 2005, 12:49:39 pm »
I assume the hull is slightly modifed from the plans, as well as the B/C deck, so account for their slightly inaccurate shapes? I rmember you giving a long dictation on "Conny Blueprint Quality" back when TSP made his.
No not really (a bit perhaps). TSP was more right than I was it turns out. After much further study of new pics of the model I decided that the casimiro plans are pretty much right on the money. The only spot that I feel is inaccurate is the primary intermix/intercooler on casimiros warp engine. Im sure that it is closer to parallel to the centerline of the nacelle than he has it, but thats about all. THis mesh isnt final either. One things for sure its a LOT more accurate than my V1.0, built using the handdrawn everhart plans, the most accurate available at the time (2002)
In my comparison, I overlayed the sinclair plans with the casimiro plans in photoshop (all views), then studied the differences and compared those with new pics of the studio model that TSP sent me links to.
#### Lord Schtupp
• Posts: 379
« Reply #70 on: June 20, 2005, 12:51:24 pm »
Hey Schtupp, would you mind a (small, large, depends on your view of things) critique about the Aurora?
No not at all. Whats on your mind?
#### Lord Schtupp
• Posts: 379
« Reply #71 on: June 20, 2005, 12:55:08 pm »
Hey Schtupp, would you mind a (small, large, depends on your view of things) critique about the Aurora?
I doubt you've convered things as nitpickily as I have, but why not?
BTW, if it's about the bussards on the second pic, ignore them. The most recent WIP is the one with IaMD style bussards, which I suggested.
Geez THATS for sure lol!
Mariner class has helped make it a better ship, that much is certain. Note that the diiferent pics are at different stages of development, the top one is most current.
#### Rat Boy
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« Reply #72 on: June 20, 2005, 02:11:00 pm »
No dammit I havnt seen it (i dont get UPN), but I sure would like to see some good screencaps of that. I have been working on my Connie V2.0, all new mesh based on the casimiro plans seen here mapped to the hull. The warp engines here are the old ones but they will be all new as well. I'll try to do a full set of regs + blanks as i usually do. I havnt forgot about the Hornet
Meh, forget the Hornet. I've recently developed a fondness for the Yorktown.
"Chaos Theory, Part II" now available.
#### USS Mariner
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« Reply #73 on: June 20, 2005, 03:12:11 pm »
The Kitty Hawk can beat the sh*t out of both.
"Improve a mechanical device and you may double productivity. But improve man, you gain a thousandfold." - Khan
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#### Khalee1
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« Reply #74 on: June 20, 2005, 03:35:30 pm »
Wow that compliment is better than the model itself lol Thanks buzzard (reminds me I need to update my website. the only dl links available are at the beginning of this thread. Doh!)
Ive been messing around whith this light cruiser design for a few months now, but Ive been so nitpicky that its difficult to get it finshed, esp with reallife issue and whatnot:
#### QobnuH
• Posts: 46
« Reply #75 on: June 20, 2005, 09:04:01 pm »
Man! I keep saying "enough FEDS" Then without hesitation d/l anything schtupp and wz make. It figures...just as i gave up on the Antares... this thead shows up. I just (yesterday) finished bashing one of your fed transports into a tos frigate to fill the slot. Guess thats why we have a recycle bin.
#### E_Look
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« Reply #76 on: June 20, 2005, 09:11:25 pm »
Hey Schtupp, would you mind a (small, large, depends on your view of things) critique about the Aurora?
No not at all. Whats on your mind?
[Uncovers garbage can and stands up]
First off, I really like the way it looks... and that's kind of the problem! It doesn't quite smell TOS; it has more the aroma of TNG. Allow me to clarify- my first impression was that it reminded quite a lot of the post-TNG Sovereign (Ent-E). Now, I'd d/l it and play it as is just because it's pretty to look at. But I'd also be thinking of finding a Borg cube to kick the crap out of in a TOS mod... I guess I'd never finish any missions!
[/Bends back down and pulls lid back on]
[Pops out temporarily]
Uh, anyone else with that perception??
[/Pops out temporarily]
#### Rat Boy
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« Reply #77 on: June 20, 2005, 10:04:22 pm »
You know, E, maybe it's just the chair that got smashed into my head talking, but now that you mention it, it does vaguely look like the Sovereign. Vaguely, mind you; might not look that way when my brain's no longer concussed.
Oh, and Schtupp, one question: what's the difference between Connie 1.0 and 2.0, for those of us who don't have comparison pics handy? And would somebody hook this man up with some HDTV screencaps of the ship from ENT?
"Chaos Theory, Part II" now available.
#### atheorhaven
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« Reply #78 on: June 20, 2005, 11:01:52 pm »
Just gotta mention this.. this is a stellar lookin ship... going to have some fun playing with it.
..ooOOoo..totally useless information..ooOOoo..
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#### USS Mariner
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« Reply #79 on: June 20, 2005, 11:24:19 pm »
You know, E, maybe it's just the chair that got smashed into my head talking, but now that you mention it, it does vaguely look like the Sovereign. Vaguely, mind you; might not look that way when my brain's no longer concussed.
Oh, and Schtupp, one question: what's the difference between Connie 1.0 and 2.0, for those of us who don't have comparison pics handy? And would somebody hook this man up with some HDTV screencaps of the ship from ENT?
http://benni.info/StarTrek
They aren't fullsize HDTV but it's all of Season 4
"Improve a mechanical device and you may double productivity. But improve man, you gain a thousandfold." - Khan
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#### Lord Schtupp
• Posts: 379
« Reply #80 on: June 21, 2005, 12:46:52 am »
Are these pics showing the aft mounted weapons as mentioned above?
#### S33K100
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« Reply #81 on: June 21, 2005, 06:31:14 am »
Yae, they be the ones from IAMD, though that version is grossly underarmed, only has the front phaasers and torps seen in TOS and those rear phasers, no others are seen and they're all mysteriously 'hidden' until used like on the NX-01
Re: the Aurora, I think the reason some people get the impression of a Sovereign class, is the very smooth transition between the B/C deck superstructure and the engineering hull, the hull smoothly covers the back of the saucer and joins into the back of the bridge module, obviously the Sovereign doesn't have a module as it's such a big ship but the same shape is evident on that ship, the engineering hull continues fairly smoothly into the saucer (at least it does so completely on the fugly Nemesis version - CGI suxxorz the big one1111!!!) Fortunately for your ship all the other views aside from the top show the very different proportions of the 2 hulls. I like the hull the way it is myself.
P.S. the new Connie looks very sweet, are those details your textures of just the Casimiro plans layed over the mesh? Speaking of the plans, I've not had a chance to compare them with photos of the studio model, do you have any pics of the areas you think he got wrong?
If I determine the enemy's disposition of forces while I have no perceptible form, I can concentrate my forces while the enemy is fragmented. The pinnacle of military deployment approaches the formless: if it is formless, then the deepest spy cannot discern it, nor the wise make plans against it.
Sun Tzu 'The Art of War'.
S33K100: formerly Marauth
#### Lord Schtupp
• Posts: 379
« Reply #82 on: June 21, 2005, 10:27:19 am »
P.S. the new Connie looks very sweet, are those details your textures of just the Casimiro plans layed over the mesh? Speaking of the plans, I've not had a chance to compare them with photos of the studio model, do you have any pics of the areas you think he got wrong?
The only area is in the top view of the casimiro plans; the primary intercooler should definatly be parallel to the outer warp nacelle hull. but in the major areas such as saucer cross-section, engineering hull shape, b/c decks, shuttle bay etc. casimiro pretty much nailed it. This I was able to determine this thanks to TSP giving me a new source of studio model photos. I spent hours researching and making the photoshop comparison. Oh and Yes - they are just the plans mapped to the hull. This ensures accuracy of the mesh. I dont even want to go into how much my V1.0 mesh was off!. I have a 5000 poly target. I really need to get a pair of ACCURATE warps going so as to fit them to the Kittyhawk , Ptyor Velikovy, Mars and one or two other models I have going.
.
#### Chris Johnson
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« Reply #83 on: June 21, 2005, 11:08:31 am »
I never compared the Aurora to a Sovereign-class, but I do see a trait or two. The only traits worth noticing being the torpedo launchers near the center of the ventral area of the saucer, actually a Constitution-class trait, but more Sovvie-like due to visibility. Also, no "neck" area and a secondary hull thinner as you near the end of it. But still, it's no Sovereign-class. If anything, the Sovereign should be compared to the Aurora due to the Sovvie's retroized design, as it looks out of place in TNG-era Starfleet design lineage. But I digress...
Here's a pic from that same website Schtupp dug up a couple of pictures from. It clearly shows the "ball points" of where they are located, reminicient of the aft phasers from the Constitution-class refit. So far, you're doing a nice job with re-doing your Constitution-class. The engines in that WIP pic, look weathered and somewhat more-detailed I think, a plus in my book if that's what you intended. Otherwise, it could just be my eyes need a checkup again.
As I've seen the Aurora WIP pics before, you know how much I like the Aurora.
"Oh, shut up!" -- Wil Wheaton to Wesley Crusher
#### USS Mariner
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« Reply #84 on: June 21, 2005, 11:22:48 am »
I never compared the Aurora to a Sovereign-class, but I do see a trait or two. The only traits worth noticing being the torpedo launchers near the center of the ventral area of the saucer, actually a Constitution-class trait, but more Sovvie-like due to visibility. Also, no "neck" area and a secondary hull thinner as you near the end of it. But still, it's no Sovereign-class. If anything, the Sovereign should be compared to the Aurora due to the Sovvie's retroized design, as it looks out of place in TNG-era Starfleet design lineage. But I digress...
Here's a pic from that same website Schtupp dug up a couple of pictures from. It clearly shows the "ball points" of where they are located, reminicient of the aft phasers from the Constitution-class refit. So far, you're doing a nice job with re-doing your Constitution-class. The engines in that WIP pic, look weathered and somewhat more-detailed I think, a plus in my book if that's what you intended. Otherwise, it could just be my eyes need a checkup again.
As I've seen the Aurora WIP pics before, you know how much I like the Aurora.
Those are his old warp textures, too.
"Improve a mechanical device and you may double productivity. But improve man, you gain a thousandfold." - Khan
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#### Chris Johnson
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« Reply #85 on: June 21, 2005, 11:26:33 am »
It's been a while since I've played SFC or thought of SFC models, much thanks to real life. *shrug* Otherwise I'd have known all my favorite models religiously. That, and I'm in an SFC1-ish nostalgic mood, trying to find what's left of available SFC1 mods I found in 1999 and 2000.
"Oh, shut up!" -- Wil Wheaton to Wesley Crusher
#### Lord Schtupp
• Posts: 379
« Reply #86 on: June 21, 2005, 11:32:05 am »
I never compared the Aurora to a Sovereign-class, but I do see a trait or two. The only traits worth noticing being the torpedo launchers near the center of the ventral area of the saucer, actually a Constitution-class trait, but more Sovvie-like due to visibility. Also, no "neck" area and a secondary hull thinner as you near the end of it. But still, it's no Sovereign-class. If anything, the Sovereign should be compared to the Aurora due to the Sovvie's retroized design, as it looks out of place in TNG-era Starfleet design lineage. But I digress...
Re: the Aurora, I think the reason some people get the impression of a Sovereign class, is the very smooth transition between the B/C deck superstructure and the engineering hull, the hull smoothly covers the back of the saucer and joins into the back of the bridge module, obviously the Sovereign doesn't have a module as it's such a big ship but the same shape is evident on that ship, the engineering hull continues fairly smoothly into the saucer (at least it does so completely on the fugly Nemesis version - CGI suxxorz the big one1111!!!) Fortunately for your ship all the other views aside from the top show the very different proportions of the 2 hulls. I like the hull the way it is myself.
First off, I really like the way it looks... and that's kind of the problem! It doesn't quite smell TOS; it has more the aroma of TNG. Allow me to clarify- my first impression was that it reminded quite a lot of the post-TNG Sovereign (Ent-E). Now, I'd d/l it and play it as is just because it's pretty to look at. But I'd also be thinking of finding a Borg cube to kick the crap out of in a TOS mod... I guess I'd never finish any missions!
Hmm funny but I never even thought about the Sovy. Youre all gonna laugh but Originanally i wanted to build a simple SFB type of NCL. Well one thing led to another etc etc and 16 max files later this is about where she stands. If you look closely you can see that the sensor/deflector pod is removable, also note the two blue pods (stand in objects only). The purpose of this degree of modularity is for quick conversion for mission-specific pods: scout, ECM/ECCM, hvy weapons, carrier and so on, just as is described in the old old old SFB Expansion #2 for the general war. So its Late TOS. As for the simualrity to the Sovy it purly coincidental. This is what the hull is really based on:
'66 Dodge Charger R/T
Note the roofline....
#### Rat Boy
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« Reply #87 on: June 21, 2005, 11:35:41 am »
The General Lee? Now that'd be interesting to see on a starship.
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#### Smiley
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« Reply #88 on: June 21, 2005, 04:41:32 pm »
Urgh what an ugly vehicle!
Signature: Terran Empire Relentless: WZ, All Other Models: Smiley
#### OlBuzzard
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« Reply #89 on: June 21, 2005, 06:10:42 pm »
The General Lee? Now that'd be interesting to see on a starship.
Actually the General Lee was a '69 I believe (I'd have to look at the tail lights ) not a '66
The 66 was pretty cool .. but the '69 Charger and the Charger Daytona were my Fav.
If you aim at nothing: you WILL hit it every time !
#### FoaS_XC
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« Reply #90 on: June 21, 2005, 06:21:49 pm »
Cars....
yeaokay im lost.
Robinomicon
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#### Lord Schtupp
• Posts: 379
« Reply #91 on: June 21, 2005, 07:26:20 pm »
Urgh what an ugly vehicle!
lol yes. 60's Mopars are an aquired taste. You either love 'em or hate 'em.
The General Lee? Now that'd be interesting to see on a starship.
Actually the General Lee was a '69 I believe (I'd have to look at the tail lights ) not a '66
The 66 was pretty cool .. but the '69 Charger and the Charger Daytona were my Fav.
I think General Lee was a 68, straight taillights IIRC. 68 = straight taillights, right? Anyway they were just good ole boys, never meaning no harm...
Yessir nothin wrong with a Daytona or charger for that matter....
#### E_Look
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« Reply #92 on: June 21, 2005, 09:29:14 pm »
Hmm funny but I never even thought about the Sovy. Youre all gonna laugh but Originanally i wanted to build a simple SFB type of NCL. Well one thing led to another etc etc and 16 max files later this is about where she stands. If you look closely you can see that the sensor/deflector pod is removable, also note the two blue pods (stand in objects only). The purpose of this degree of modularity is for quick conversion for mission-specific pods: scout, ECM/ECCM, hvy weapons, carrier and so on, just as is described in the old old old SFB Expansion #2 for the general war. So its Late TOS. As for the simualrity to the Sovy it purly coincidental. This is what the hull is really based on:
'66 Dodge Charger R/T
Note the roofline....
Inspiration is wherever you find it! Keep on truckin'!
#### Fedman NCC-3758
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« Reply #93 on: June 21, 2005, 09:33:53 pm »
MOPARS !!
The Star Spangled Banner bring hither,
O're Columbia's true sons let it wave.
May the wreaths they have won never wither,
Nor it's stars cease to shine on the brave.
#### Rod ONeal
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« Reply #94 on: June 21, 2005, 11:06:12 pm »
Hey, LS. Is that car yours? My folks had a '68 R/T, in the day. They traded a '66 GTO convertible for it. Those were the days. Brand new, with a 440 magnum, the '68 went for $3,800.00. If they even built something like it today, it'd probably be$50,000.00.
...and Smiley, I hope that car haunts your dreams for uttering such blasphemy.
If Romulans aren't cowards, then why do they taste like chicken?
#### Fedman NCC-3758
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« Reply #95 on: June 21, 2005, 11:39:21 pm »
Those that haunt the H&S boards have already been introduced to the "Bee".
Fed's 1971 Dodge Charger Super Bee w/383 Magnum
The Star Spangled Banner bring hither,
O're Columbia's true sons let it wave.
May the wreaths they have won never wither,
Nor it's stars cease to shine on the brave.
#### Lord Schtupp
• Posts: 379
« Reply #96 on: June 21, 2005, 11:49:56 pm »
No but I wish it was lol. Its a Hemi R/T I found at a fansite today just to show what I was talking about. In the past I have owned a 68 Sport Satellite, 73 Challenger 70 Chevelle SS 67 Mustang 66 Ford Galaxy 69 Nova SS and the love of my life, 73 340 Cuda 4 speed - red of course. To say that I regret selling that one would be an understatement. blew 3 engines in the Cuda. Now Im down to 2 Corvairs, a 63 and a 65. Slow and cheap, but still cool.
You guys can see what Im talking about as far the Charger influence on the design of the Aurora right? in the side view; between the b/c decks all the way back to the warp pylons, upper half of the engineering hull. I tried to accentuate it with the texture.
#### Lord Schtupp
• Posts: 379
« Reply #97 on: June 21, 2005, 11:54:33 pm »
Those that haunt the H&S boards have already been introduced to the "Bee".
Fed's 1971 Dodge Charger Super Bee w/383 Magnum
WOW! That is red hot!! beautiful mopar fed! Damn what a stud to have such a car... So does this have the Air Grabber hood? Thats the thing about Mopar, they always had the most radical features of the big three, easily.
#### Rod ONeal
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« Reply #98 on: June 22, 2005, 12:13:07 am »
Fabulous car, Fedman!!! 1971, the last year for real musclecars. ...and yeah, the Charger could trash the Goat, no problem. GTO's are still really cool, though.
We lived in Fall River Ma. It's a fairly large city. I can remember riding with him down Eastern Avenue, which was where everyone cruised their musclecars and hot rods. I was only 11yrs. old. Everyone would park their cars when he showed up because they didn't want him to challenge them. Man, I so wish that I was born 10yrs. sooner.
If Romulans aren't cowards, then why do they taste like chicken?
#### Smiley
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« Reply #99 on: June 22, 2005, 04:12:53 am »
lol....haunts my dreams indeed!
Have you ever taken control of one of your dreams when you've realised that you're actually dreaming? I have a couple of times.
I think a magnetic wrecking ball and one of those scrap metal crunchers would be on the cards for that beastie...Muwahahahaaaa....!!!
Signature: Terran Empire Relentless: WZ, All Other Models: Smiley
#### Lord Schtupp
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« Reply #100 on: June 22, 2005, 04:35:51 am »
GTO with 389 tri -power 4 speed is no slouch i can attest to that. buddy of mine has a 65, beautiful car.
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« Reply #101 on: June 22, 2005, 07:04:57 am »
Blech, I can see why Smiley is dreaming of wrecking balls, the only American car I've ever had any interest in owning would look something like this:
I think it was made out of something called a '73 Plymouth Satellite Sebring, whatever the hell that was, in any case that particular car was destroyed last year I think, not sure what happened but the front end was totalled I believe.
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#### Fedman NCC-3758
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« Reply #102 on: June 22, 2005, 07:17:26 am »
Fabulous car, Fedman!!! 1971, the last year for real musclecars. ...and yeah, the Charger could trash the Goat, no problem. GTO's are still really cool, though.
Thanks for the kind words guys. Mopar always had the tricks and the engineering to back it up. For example, the rear springs had a forward bias that always planted the rear wheels. You rarely saw a Mopar with the dreaded wheel hop. They were also ahead of the pack with tortion bars. This Superbee still has a better ride than a 89 Cougar LS I once had.
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« Reply #103 on: June 22, 2005, 07:59:49 am »
Well, back on topic. Personally, I thought that it had a resemblence to the Voyager. I was happy to see that it didn't have stubby pylons and nacelles. Not that the Intrepid class doesn't look good, it just wouldn't make sense for TOS.
The idea of being modular is a cool one, too.
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« Reply #104 on: June 22, 2005, 11:51:53 am »
Quote
The General Lee? Now that'd be interesting to see on a starship
I did the General Lee as a shuttle last year ................
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« Reply #105 on: June 22, 2005, 11:59:05 am »
Back off topic for at least one more post, after I saw S33K100's post I just had to put this one up.
#### OlBuzzard
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« Reply #106 on: June 22, 2005, 04:01:20 pm »
hehehe I CANT wait to get home !!
MOPAR power !!
WOO HOO
Outside of the 73' Mach 1 Fast back (Cobra) and the GTO ( as seen on "My Science Project" ) ...
NOTHING like 'em. I love the Charger series ... the Charger Daytona set a record in the nascar circuit the first year they were allowed. the Plymoth "Super Bird" was a "Road Runner" with essentially the same trim ( Nose piece and that totally awesome tail wing) as the "Charger Daytona".
Man ... talk about the memories ..
hehehe BTW .. in RL I took a '69 Coronet and customized it...
Corvet Door handles ... and twin "Charger" gas caps (off of '66 Charger .. one on each rear 1/4 panel) .. I could fill the car with gas from either side as they were actually functional). And a custom soft Black Velvet interior... Bucket seats... console ...
hmm... lets see ..
oh yeah ! I converted the head lights to "Square" ones ... customized body work around the wheel wells ( small splash guards behind each of the 4 wheel wells ) ... and metal flake silver paint job with black vinyl roof.
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#### Mr_Tricorder
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« Reply #107 on: June 22, 2005, 04:12:44 pm »
Quote
The General Lee? Now that'd be interesting to see on a starship
I did the General Lee as a shuttle last year ................
-MP
That shuttle has just given me an ispiration. Are there any ships to go with that shuttle, is is that the only Confederate ship out there?
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« Reply #108 on: June 22, 2005, 04:15:13 pm »
Bah, you and your nation-specific shuttles! Might as well hoist a Tokugawa Ieyasu flag on an Excelsior-class Starship. :p It's alright, but not truely my cup of tea considering my real reasons for being a fan of Star Trek.
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« Reply #109 on: June 22, 2005, 08:09:22 pm »
Quote
The General Lee? Now that'd be interesting to see on a starship
I did the General Lee as a shuttle last year ................
-MP
thats REALLY cool MP, i oughta make a rendering
and oh yeah.... the mackiem0bile. bought from germany.
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#### Lord Schtupp
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« Reply #110 on: June 22, 2005, 08:44:36 pm »
Damn Mackie, nice Charger!
I WANT MY CUDA BACK WHAAAA!!!!
#### Mackie
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« Reply #111 on: June 22, 2005, 08:48:30 pm »
Damn Mackie, nice Charger!
I WANT MY CUDA BACK WHAAAA!!!!
440 big block , 7L .it needs a new hood though. ;P
poor schtup
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#### Rat Boy
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« Reply #112 on: June 22, 2005, 09:08:58 pm »
But...but...
I bought a 2000 Mustang last month, doesn't that count for something?
« Last Edit: June 22, 2005, 09:19:23 pm by Rat Boy »
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#### Lord Schtupp
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« Reply #113 on: June 23, 2005, 01:18:18 am »
But...but...
I bought a 2000 Mustang last month, doesn't that count for something?
Damn straight it counts bro, its your car! Besides now weve seen 2 models inspired by dodge chargers; my Aurora and the General Lee shuttle. Idea: Im gonna do a registry USS Charger NCC-1971. Perfect.
#### Mackie
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« Reply #114 on: June 23, 2005, 12:39:04 pm »
Besides now weve seen 2 models inspired by dodge chargers; my Aurora and the General Lee shuttle. Idea: Im gonna do a registry USS Charger NCC-1971. Perfect.
quite so, quite so.
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#### Mr_Tricorder
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« Reply #115 on: June 23, 2005, 11:35:05 pm »
That General Lee shuttle got me thinking. What if there was a Federation civil war? A group of systems break off and form the CSA (Confederate Systems of the Alpha Quadrant) and are led by Earth colonists decended from people from the southern states. What do y'all think?
#### Mackie
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« Reply #116 on: June 24, 2005, 04:48:03 am »
sounds nice to me, id be willing to go with something like NTF (neo terran front) from FS2 or Terra Prime from Enterprise
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« Reply #117 on: June 24, 2005, 09:34:01 am »
I'd enjoy crushing them brutally, though I doubt they'd be made up of a bunch of slave owning hicks who were 'never meaning no harm...' LOL More likely would be something in the very early days of the Federation, pre-TOS, based on the those nutcases from Terra Prime as Mackie suggests, anti-alien humans. Plus the Feds could go all out on them given we know how they treated the Maquis.
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#### atheorhaven
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« Reply #118 on: June 24, 2005, 10:45:47 am »
That shuttle has just given me an ispiration. Are there any ships to go with that shuttle, is is that the only Confederate ship out there?
At the moment, I thnk so. But I have two sitting on my HD that need to get uploaded this weekend. The Bubba and Cletus Class ships.. both are flying Confederate colors..
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#### Mackie
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« Reply #119 on: June 24, 2005, 11:07:20 am »
LOL @ Alec
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#### Rat Boy
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« Reply #120 on: June 24, 2005, 11:59:42 am »
Schtupp, a couple questions on the Aurora. First, what's the inspiration for the blue/red striping on one of them? Second, were you aware that there already was a cruiser called Aurora on TOS?
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#### Lord Schtupp
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« Reply #121 on: June 24, 2005, 03:05:18 pm »
Schtupp, a couple questions on the Aurora. First, what's the inspiration for the blue/red striping on one of them? Second, were you aware that there already was a cruiser called Aurora on TOS?
Yes and good point. Youre referring to the S.S. Aurora in "The Way to Eden" and fortunatly for me it was blown up by the space hippies thus leaving the name open.
Steppin' into Eden, yea brother...
The red stripes where just an experiment, one of the many tangents that I always get lost doing. Looks like a US Coast Guard ship huh
#### Lord Schtupp
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« Reply #122 on: June 24, 2005, 03:12:48 pm »
I think the Space Hippies where running from Mr Tricorder's Space Rednecks lol
Someone needs to do a rendering of Atra's SS aurora model being chased by MPs confederate shuttle. That'd be a hoot.
#### Rat Boy
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« Reply #123 on: June 24, 2005, 03:42:05 pm »
I only brought up the S.S. Aurora because your ship made me think it would be good as a potential design for the Icarus-class, a precursor starship to the Constitution-class mentioned in several of the more recent TOS novels.
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#### Lord Schtupp
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« Reply #124 on: June 24, 2005, 03:58:15 pm »
Ah I see. It wasn't until after I picked the name that I then recalled it being used for the space hippies ship. but since it was destroyed in the episode I thought it was ok to keep it. Interestingly (to me at least) the name comes from a suggestion by none other than Carl Sagan, in his book Broca's Brain. In it he devotes a whole chapter to Star Trek and mostly discusses the multi-racial aspect of the crew and the classic trek "anomalies". In that chapter, he questions why Gene Roddenberry picked mostly American-derived names for the 12 Connies, and then wonders why he chose to name one Potemkin instead of Aurora.
#### Rat Boy
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« Reply #125 on: June 24, 2005, 07:24:23 pm »
In that chapter, he questions why Gene Roddenberry picked mostly American-derived names for the 12 Connies, and then wonders why he chose to name one Potemkin instead of Aurora.
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« Reply #126 on: June 24, 2005, 10:26:53 pm »
In that chapter, he questions why Gene Roddenberry picked mostly American-derived names for the 12 Connies, and then wonders why he chose to name one Potemkin instead of Aurora.
[/Johnny Carson]
I liked Bob Justman's idea mentioned in The Making of Star Trek.
Quote
PAGES 164-5
cc:Bob Justman
TO: Gene Roddenberry DATE: AUGUST 9, 1967
FROM: Bob Justman SUBJECT: STAR FLEET STARSHIPS
Dear Gene:
I am in receipt of a memo from someone using the pseudonym of D. C. Fontana. This character suggests that we establish the names of the 12 ships of the Enterprise Starship Class.
Of the names that D. C. Fontana mentions, I prefer the following:
Enterprise
Essex
Excalibur
Lexington
Yorktown
Endeavor
Eagle
Constellation
Hornet
Wasp
Lafayette
I think there would be several other candidates, such as Saratoga and perhaps another English Carrier, a French Carrier, a Russian Carrier and certainly a Japanese Carrier. In addition, I think a name ought to be made up that would be of Vulcan origin.
Bob.
Note the last suggestion. One of the classic fanon names for this was Tashik-Sotra. I'd be neat to see your next Connie carry THAT as a name...
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#### Rat Boy
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« Reply #127 on: June 26, 2005, 12:24:30 pm »
Another quick question, Schtupp: is the Aurora supposed to be a precurssor to the Connie or a companion class?
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« Reply #128 on: June 26, 2005, 02:23:54 pm »
Quoting Schtupp:
Quote
Earlier this week I started back on my Aurora NCL project, as can be seen I made some changes to the basic mesh, made a much nicer photon bay, and have progressed on the textures somewhat. The big red stripe is a little experiment, Ill use something like that for one of the advanced versions. the small purple pods are placeholders for mission specific pods or weapons, i.e. ecm, sensors for scout variant, torps for heavy weapons variant, and so on, in keeping with the modularity of the SFB NCL.
From here. I think he implies it as more-of a contemporary new light cruiser (NCL) rather than an immediate predescessor. I may not be Schtupp, but I hope that helps answer your question, Rat Boy.
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#### Rat Boy
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« Reply #129 on: June 26, 2005, 03:04:28 pm »
I see. I just figured that with its overall shape and lower reg number that it could be one of the oft-speculated forerunner classes to the Connie, and a darn good lookin' one at that.
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« Reply #130 on: June 26, 2005, 04:48:33 pm »
It looks too contemporary TOS-era to be such in my opinion. Something like the Baton Rouge or The Starfleet Museum's Asia-class looks more plausable to being such a candidate.
Also, keep in mind the stock SFC or KA models having three-digit registry numbers. For example, ships like the Akula-class and Okinawa-class (Respectively the F-DD and F-FF stock models in KA and SFC 1/2/OP) have low registry numbers. The Akula's registry was NCC-278-A. Yeah, it has a -A so it can't count. The Okinawa's registry number was NCC-150, the same as the U.S.S. Daedelus's registry number in canon Trek. It implies that two TMP-era starships belong in the Pre-TOS-era, if we take registry numbers seriously in a cronological way. I don't know if it's the same or different in the SFB universe.
Perhaps in SFB it's 2000+ registries that get Dreadnought or Heavy Carrier or Battleship status, anything between 2000 and 1000 cruisers, and Destroyers and Frigates get anything below in three-digit numbers? *shrug* I don't know anything about SFB, or not much about it, so I'm guessing.
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#### Rat Boy
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« Reply #131 on: June 26, 2005, 05:14:23 pm »
Something like the Baton Rouge or The Starfleet Museum's Asia-class looks more plausable to being such a candidate.
The Asia looks too much like the Connie for it to be a forerunner, let alone one that is supposed to predate it by forty years. As for the Baton Rouge, it looks too...stubby, to fit into the ENT-to-TOS design lineage. Looks more like a forerunner to the Ambassador or Galaxy. This ship flows very well between ENT and TOS, with a highlight or two from NX-01 in a proto-NCC-1701 package.
Quote
Also, keep in mind the stock SFC or KA models having three-digit registry numbers. For example, ships like the Akula-class and Okinawa-class (Respectively the F-DD and F-FF stock models in KA and SFC 1/2/OP) have low registry numbers. The Akula's registry was NCC-278-A. Yeah, it has a -A so it can't count. The Okinawa's registry number was NCC-150, the same as the U.S.S. Daedelus's registry number in canon Trek. It implies that two TMP-era starships belong in the Pre-TOS-era, if we take registry numbers seriously in a cronological way. I don't know if it's the same or different in the SFB universe.
Oddly enough, the Akula TOS model in "Generations at War" has the reg number of 578 to put it in line with Franz Joseph's ships, but the TMP model has 278-A (ALEC! ;)0. The 1300-series has been used for the previously mentioned Baton Rouge-class, which made me recall that idea for this vessel.
But, this is Schtupp's ship, so whatever he says goes. But, the general plan and layout is something to consider for a starship line in service between the 2220s and 2250s, retired in favor of the Constituion. Perhaps the Aurora is a refitted version of that hypothetical, or some kind of smaller follow-on based on the general layout?
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#### Chris Johnson
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« Reply #132 on: June 26, 2005, 06:57:39 pm »
Usually when I watch ENT, I just watch it for entertainment. If I want to get pseudo-geeky, I throw it out the window. I mean, the ENT-era in ship design is like this: turn the Constitution-class upside down, invert the pylons, give it an Excelsior-class saucer, scale it down, repaint it, and you got the NX-03! Then make a clever reason why it looks like that, by saying it's just a coincidence or what not... That's why I'm much more fond of the Starfleet Museum, and was quick to reference something from there as an example. In turn I also gave the Baton Rouge a try, saying that it could possibly be a predescessor as well.
But how I think about it, the Asia-class looks like a simpler take at a Constitution-class (Or the Hyperion-class to reference another example.). Therefore I thought it was perfect to be something like an immediate predescessor. If you want something with a spherical hull, cylindrical neck, etc. That looks less like a Daedelus-class and is a bit more advanced-looking, try the Horizon. Not one of my favorite ship designs (the Baton Rouge wasn't either) mind you... Of course it could also be a look-a-like class of Starship, like how Atrahasis would think of the Republic or Constellation as; an earlier starship design remarkably-similar to the Constitution-class, more-so than the Asia. Perhaps it could be something like the U.S.S. Farynor.
I'm not sure what you exactly envision as a predescessor to the Constitution-class, and I can't be for sure what I exactly envision in that subject either. Although that's the beauty of Trek and science fiction in general: utilizing imagination, to create a universe in your head and explore it...
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#### Rat Boy
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« Reply #133 on: June 26, 2005, 07:08:27 pm »
I'm not sure what you exactly envision as a predescessor to the Constitution-class, and I can't be for sure what I exactly envision in that subject either. Although that's the beauty of Trek and science fiction in general: utilizing imagination, to create a universe in your head and explore it...
Hey, Schtupp just came rather close to what I had in mind, a more-compressed ship than the Constitution. Heck, about the only thing I'd do is perhaps make the secondary hull a bit larger, add the spike/probe things to the front of the nacelles, remove the globe from the back of the nacelle, and perhaps integrate the deflector into the secondary hull a bit more. Then, presto, you'd have what I've been picturing.
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#### FPF-Wanderer
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« Reply #134 on: June 26, 2005, 07:55:15 pm »
Well, if you really want to go SFB...since it's an NCL, here's the listings...
NCL LIGHT CRUISERS: 1500 Kearsarge, 1501 Reliant, 1502 Alabama, 1503 Repulse, 1504 Renown, 1505 Prince of Wales, 1506 Mutso, 1507 Nagato, 1509 Gneisenau, 1510 Scharnhorst, 1511 Prinz Eugen, 1512 Vittorio Veneto, 1513 Littorio, 1514 Caio Duilio, 1518 Dunkerque, 1519 Richelieu, 1520 Strasbourg, 1521 Rivadavia, 1522 Moreno, 1523 Minas Gerais, 1524 New Mexico, 1525 Iowa, 1526 Missouri, 1527 Kortenauer, 1528 De Ruyter, 1529 Naken, 1530 Glorie, 1531 Defence, 1532 Pervenetz, 1533 Meko, 1534 Vosper, 1535 Groton, 1536 Ramadan, 1537 Aliyah, 1538 Assawari, 1539 Michigan, 1540 North Carolina, 1543 Mikasa, 1546 Olympia, 1547 Piorun, 1548 Warspite. Notes: About 60 ships of this class were built during the General War.
Got these from the SFB boards. A good resource for those of you who want to do SFB type stuff, which we can always use more of.
The only thing I'd change is moving the torp launchers to the top of the saucer ala SFB/TOS. Excellent ship as usual, Schtupp! Where's the Kittyhawk?
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#### Rat Boy
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« Reply #135 on: June 26, 2005, 07:57:45 pm »
The only thing I'd change is moving the torp launchers to the top of the saucer ala SFB/TOS.
While SFB and the original Franz Joseph book placed the torpedoes there, FX shots from TOS had the torpedoes coming out fairly close to where the phasers are, something that Schtupp showed on his CA+ model and ENT showed on the NX-01 model.
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#### S33K100
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« Reply #136 on: June 26, 2005, 08:46:27 pm »
The NCL in SFB wasn't a Franz-Joseph based design though, it actually had a smidgeon of originality, Schtupp's version looks nothing like the SFB version (Atra did an accurate version if anyone wants one) but it's a much nicer looking design anyway.
I'm quite amused that the SFB people name a ship after Vittorio Veneto the battle in Italy (there were some British and French troops but the majority were Italian) that led to Austria-Hungary's surrender at the end of WWI. Quite an obscure referrence.
If I determine the enemy's disposition of forces while I have no perceptible form, I can concentrate my forces while the enemy is fragmented. The pinnacle of military deployment approaches the formless: if it is formless, then the deepest spy cannot discern it, nor the wise make plans against it.
Sun Tzu 'The Art of War'.
S33K100: formerly Marauth
#### atheorhaven
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« Reply #137 on: June 26, 2005, 11:41:12 pm »
Something like the Baton Rouge or The Starfleet Museum's Asia-class looks more plausable to being such a candidate.
Oddly enough, the Akula TOS model in "Generations at War" has the reg number of 578 to put it in line with Franz Joseph's ships, but the TMP model has 278-A (ALEC! ;)0. The 1300-series has been used for the previously mentioned Baton Rouge-class, which made me recall that idea for this vessel.
tsk tsk tsk.. apparently you missed the staff meeting on this. The TOS Akula couldn't have the original registry of 278 as the pre-TOS Akula was still in extended diplomatic duties shuttling delegates between several Federation worlds. So the TOS Akula (named in the honor of that long standing ship of the fleet) was given the 578 registry. But by the time of the TMP refit programs, the original Akula had long since been destroyed by unknown forces near Federation borders. So during the TMP era refit, the TOS Akula then had its registry changed to 278-A to honor the fallen ship.
Should've cleared that up before.
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#### atheorhaven
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« Reply #138 on: June 26, 2005, 11:42:31 pm »
By the way, anyone need a bridge? I have a lovely one for sale..
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#### Rat Boy
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« Reply #139 on: July 02, 2005, 03:19:37 pm »
Shameless bumpy, since we can get away with bumping around here.
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#### Kana
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« Reply #140 on: July 02, 2005, 04:00:50 pm »
Perhaps in SFB it's 2000+ registries that get Dreadnought or Heavy Carrier or Battleship status, anything between 2000 and 1000 cruisers, and Destroyers and Frigates get anything below in three-digit numbers? *shrug* I don't know anything about SFB, or not much about it, so I'm guessing.
For all SFB ship name and registery info go to this link...
http://www.starfleetgames.com/discus/messages/21/6850.html?MondayMarch2220041255pm#POST158047
Kana
#### Rat Boy
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2021-04-16 13:22:18
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https://www.electro-tech-online.com/threads/modifying-a-spy-microphone-for-leopard-conservation.148337/
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# Modifying a spy microphone for leopard conservation
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#### kra
##### New Member
Hi everyone,
I'm looking to attach a spy microphone to leopard radiocollars and have it record continuously for 1 month.
Why? Well, for my project I'm looking at whether strategically played competitor sounds could be used to manipulate leopard movements in order to reduce conflict with people. But I need to know how vocals naturally affect their movements.
I'm looking to hack the EDIC-Mini-Tiny B47 microphone so that it connects to a power source that powers it for 600 hours (its storage capacity). It's current battery life is up to 168 hours.
I'm out in the African Bush right now, and so I would need to know whether this hack is possible and, if so, would be looking to team up with someone who can advise me on how to make the modifications.
Weight is a major limitation here, and I would need a battery solution which weighed < 75g.
Price is also an issue as the project budget is already stretched. So, I would be looking for a battery solution which was preferably < $50 (although, higher priced options would be considered). Can anyone help? Thanks, K. Disclaimer: I am an electronics newbie and internet access here is limited, but I am a keen learner. #### Attachments • 293.8 KB Views: 148 #### mdorian ##### Member Hi K Two AA alkaline batteries in parallel will have have around 50 g weight and 4000 mAh capacity (the battery you use has at most 700mAh) , it should be enough. Take care if you are in a rainy environment , the device is for indoor use you should protect it somehow in a protective case. Do not use the included battery because it needs air to work and you cannot waterproof your device properly. #### alec_t ##### Well-Known Member Most Helpful Member Looks like the hack would be straightforward if you are able to engineer a dummy battery to make contact with the circuit. On the subject of waterproofing, have you tested the microphone sensitivity when it's covered by a protective film? #### kra ##### New Member Hi MDorian, Alec, Thank you for your messages and advice. It does sound straightforward. I did a quick google and found a few different sets of instructions on how to make dummy batteries. So, would the 2 AA batteries in parallel output the same power but just over a longer period (since the capacity is bigger), or is there a danger of it damaging the circuit because there's more power in them? I haven't tested the microphones sensitivity myself, but a previous study on chipmunks used the B47's predecessor and tested the sensitivity of the microphone when in a protective film and found it was still very good (it could detect the chipmunk's heartbeat when they were at rest). They described the steps they took to weatherproof the device, and I was planning on just following the steps and then attaching to a new power source. Cheers again, K. #### audioguru ##### Well-Known Member Most Helpful Member Some people replying cannot read. The mic uses three tiny G13A silver oxide button cells, not two and they are probably in series, not parallel. It is easy to see if they are in series. AA batteries are many times larger and heavier and will be overkill. Energizer battery company says that their 357 battery cell is a G13A and supplies 0.221mA for 600 hours when its 1.55V begins to drop. Since they last only 160 hours then the current is probably 600/160 x 0.221mA= 0.83mA or maybe 1mA. I could not find any button cells large enough so use three AAAA or N alkaline cells. A product uses only as much current as it needs when its supply voltage is correct. A car battery can supply 600A to start an engine when it is cold but the clock uses the same battery at a current of 0.01A or less. Two or three AA cells in parallel produce 1.5V when they are new. Three cells in series produce 4.5V. If the mic needs 4.5V but you provide only 1.5V then it will not work. #### alec_t ##### Well-Known Member Most Helpful Member 4.5V is clearly the present and recommended supply voltage, but it might be worth experimenting to see if you can get good results using only 3V. If you can, then a single lithium cell (nominally 3.6V, but usable down to 3V) could give a weight saving. #### mdorian ##### Member Mea culpa for the reading but "some of us" are missing the fact that a g13 zinc-air battery has the same capacity (600mAh) as one AAAA battery. The capacity needed is 3 or better 4 time higher around 2000 mAh , that would be 9 or 12 AAAA batteries each at least 7g , to much for the specifications. Maybe a 18650 li-ion rechargable battery , 45g up to 3000 mAh. 3.6 to 4.2V (if the g13 batteries are connected in series which is most likely ). Or maybe recycling a li phone battery? #### audioguru ##### Well-Known Member Most Helpful Member The silver oxide button battery cells the mic comes with have an absolutely flat voltage at 1.55V to 1.5V while discharging until near the very end. If the mic still works well with a total battery voltage of 3V or 3.6V then they would have used cheaper and more common alkaline button cells. Hey! There is a leopard outside and is coming to my door. Nope, it is my wife in her new leopard patterned leggings. #### mdorian ##### Member 160 hours is for a 600 mAh zinc-air battery . The same size silver-oxide cell have a capacity of 200 mAh , an alkaline battery 150 mAh. The zinc-air battery have a flat voltage of 1.28V. I don't think the voltage was the reason , the capacity is slightly higher on silver-oxide cells that looks better in the datasheet. Let's see pro's and con's for each option: - zinc-air cells 3x4 series-parallel , 12 pcs , total capacity4x600 = 2400 mAh , total recording time 4x160 = 640h weight 2.3x12= 28g pro: lightweight , high capacity con:- needs air to work which can make waterproofing dificult, must be used after removing the seal. - alkaline AA bateries capacity between 1800mAh and 2600 mAh , 2400 a good one ,3 in series recording time 4x160 = 640h , weight 23gx3=69g pro: cheap con: the weight is at the upper limit, linear discharge graph with low voltages at the end. - 18650 li-ion rechargeable, battery45g up to 3000 mAh, let's say 2400 mAh ,3.6 to 4.2V voltage. Weight 1x45g , total recording time 4x160 = 640h. pro: high energy density , appropriate voltage , can be reused. con: needs a charger which can go over the budget , special cautions must be taken, an electrical short can make them explode so it needs a good mechanical protection. #### kra ##### New Member Hi all, Thank you very much for your messages. Great advice - I understand now that mAh informs the capacity I need as it tells me how much the recorder uses per hour. I seem to have found a recording module by the same company that allows for more storage capacity and connection to an external battery source. I'm having trouble interpreting the main technical characteristics of the device in terms of the volts needed and how many mAh the device will use when recording. To me, there seem to be several characteristics that fit the bill. I've attached the link to the manual (page 5) below; could anyone please help me interpret the voltage required and the mAh use? Also, definitions of 'consumption current' (why is it in V), 'supply voltage' (why i it in mA), and 'input signal' would be great. Apologies for the above requests, I would normally google, but internet access is poor out here and keeps crashing. Thanks again, K. http://www.ts-market.com/upload/iblock/958/eng_emm tiny.pdf #### mdorian ##### Member Also, definitions of 'consumption current' (why is it in V), 'supply voltage' (why i it in mA), and 'input signal' would be great You're right , it's a mistake supply voltage is 3V and consumption current 2mA. Unless you really have some skills for electronic circuits I would stick to the first option. This one is only a recording module , you will need to attach a microphone , an low signal amplifier and a voltage regulator. The power consumption just looks better , for the same battery the recording time would be 600mAh/2mA = 300 hours but this is for uncompressed audio which will use a higher amount of memory. #### kra ##### New Member Thanks for your message and advice. That makes sense. The reason that I really wanted to go for the recording module is because of the storage capacity. The 32 GB model can store 720 hours of uncompressed audio at the sampling rate I want, whereas for the first option I would only get 240 hours or would need to use a 2-bit ADPCM compression. I will ask around and see if there are any collaborators here who could help me modify the second option; if not, I will stick with the first option. Cheers, K. #### alec_t ##### Well-Known Member Most Helpful Member The 32 GB model ? From P6 of the manual it seems the max capacity is 6GB. #### kra ##### New Member Hi Alec, I contacted the manufacturer and they told me they have a 32 GB model and provided me with the recording times under the different compression methods. Cheers, K. #### Tony Stewart ##### Well-Known Member Most Helpful Member Battery life will be reduced by larger RAM and compression work by CPU. #### Tony Stewart ##### Well-Known Member Most Helpful Member Zinc air (with suffient air and no moisture problems) has the highest consumer battery power/weight or volume density. Although expensive at$100/ kWh, your choice is limited by budget, size weight so, increasing all of the above in the same battery technology works far better than Alkaline or even LiPo.
However if degradation is factored by weather or cost, certain LiPo types with manganese may be a possible solution.
Your objective needs to be defined in terms of Cost/kWh, Voltage, Volume and weight.
https://en.wikipedia.org/wiki/Comparison_of_battery_types#Common_characteristics
#### Tony Stewart
##### Well-Known Member
I think the smarter solution is to have a secondary mic and microchip that acts as a selective trigger to conserve the main power draw of the recorder. This might be done with a few mA and a selectric mic bonded to the main mic and some convenient trigger power interface cable.
Then with VOX control , you can possibly extend your recording life by an order of magnitude or two. Noise gating is fundamental with the trigger design.
Consult with factory.
#### Tony Stewart
##### Well-Known Member
As every seasoned Engineer knows, the best result starts with a clear list of all the rquirements in your mind . Take for example how to trigger a complex waveform and choosing how to trigger by filter and trigger attributes;
such as: source S/N ratio, LPF, HPF, BPF, DC,AC +,-, delay, duration, sampling rate.
An expert Engineer can convert your detailed verbal descriptions into these characteristics to optimize the trigger and event recording.
Capiche?
#### audioguru
##### Well-Known Member
The recorder has a Voice Activation system (VOX) to extend its memory and probably also extend its battery life so maybe you must teach the leopard to say the password since the recorder also has Password Protection.
#### Tony Stewart
##### Well-Known Member
The recorder has a Voice Activation system (VOX) to extend its memory and probably also extend its battery life so maybe you must teach the leopard to say the password since the recorder also has Password Protection.
audioguru good one!
My GPS has keyword recognition for Search menu and contents, with clear background and dialog, sometims I Get perfect results and others not even close, or false triggers from the radio for three syllables.
The false positives and false negatives on VOX recognition vary greatly with SNR and cadence ( with little or no gaps in words).
If only the OP could consult the factory for a solution or ask http://www.LOTEK.com near me. Use my name for a finder's kudo.
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2020-11-24 14:18:18
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https://repository.uantwerpen.be/link/irua/105220
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Publication
Title
Measurement of the underlying event activity in pp collisions at $\sqrt{s}=0.9$ and 7 TeV with the novel jet-area/median approach
Author
Institution/Organisation
CMS Collaboration
Abstract
The first measurement of the charged component of the underlying event using the novel "jet-area/median" approach is presented for proton-proton collisions at centre-of-mass energies of 0.9 and 7TeV. The data were recorded in 2010 with the CMS experiment at the LHC. A new observable, sensitive to soft particle production, is introduced and investigated inclusively and as a function of the event scale defined by the transverse momentum of the leading jet. Various phenomenological models are compared to data, with and without corrections for detector effects. None of the examined models describe the data satisfactorily.
Language
English
Source (journal)
Journal of high energy physics. - Bristol
Publication
Bristol : 2012
ISSN
1126-6708
1029-8479 [online]
Volume/pages
8 (2012) , p. 1-34
Article Reference
130
ISI
000309883200039
Medium
E-only publicatie
Full text (Publisher's DOI)
Full text (open access)
UAntwerpen
Faculty/Department Research group Publication type Subject Affiliation Publications with a UAntwerp address
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2022-12-03 09:57:09
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https://electronics.stackexchange.com/questions/422893/heat-dissipation-in-watts-of-equipment-in-a-box
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# heat dissipation in watts of equipment in a box
Here is the situation, some electrical equipment will be in low ambient temperatures and I need to make it warmer so that there is no damage to it. I want to calculate how much power in watts is needed to heat up equipment inside a box made of aluminum.
The box is 5 inches X 5 inches X 11.5 inches (surface area is then 1.727 ft^2) it is made of aluminum and it is .1 inches thick and insulated.
from previous tests I recorded that when the ambient temperature was at -20 Celsius for a long time (at least 1hr) the surface temperature of equipment was steady at -10 Celsius and the air between the box and equipment was -11 Celsius.
Two questions: 1) How much power in watts is being dissipated by the electronics? 2) How much power in watts is needed to bring the temperature of the surface of equipment to 0 degrees Celsius? to 10 C?
I did some research and found this: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/heatcond.html
heat conduction formula
Q/t = kA(Thot - Tcold)/d
where k = thermal conductivity, A = surface area, Thot - Tcold = 10, d = thickness,
The issue with this is that when I plug in k for aluminum = 205 or .5 by adjusting the units in the formula I get a large value for Watts either way... am I doing this completely wrong is there another formula that would better model this problem?
• Outdoor security cameras use resistors to warm the inside of the enclosure. A typical setup puts 12VDC into 20 ohms, for a heating power of 7.2 watts. For larger wattage needs, a thermoswitch put into the circuit. Use 12 or 24 volts and some 5 watt resistors in parallel to get get some preliminary results. If your aluminum box is insulated then conduction through the box is reduced, and the formula does not apply. Try experimenting instead. – John Canon Feb 18 '19 at 4:40
Your problem here is that you're looking at the wrong part of the thermal equation. The heat dissipation of a heated metal box is dominated by the thermal resistance of the metal/air interface, not by the thermal conductivity of the box itself. Characterizing the thermal resistance of that interface will be difficult without taking a lot more measurements.
From a practical perspective, the simplest solution will be to overdimension the heater and use a thermostat to keep the enclosure at the target temperature -- this will make the exact size of the heater unimportant, and will also mean that the temperature will remain stable even when the ambient temperature changes.
You are using the wrong temperatures in your equation, and the wrong equation altogether. For that equation to work you have to measure the temperature of the internal and external aluminum surfaces, not the air temperature. But given the very low differential there, that exercise would be pointless.
As a first-order approximation you can simply calculate the dissipation of a block of material with a 10 degree differential and the same surface area of the box. The equation is very similar, but the important factor is not the heat conductivity of the box but the coefficient of heat convection for air and the total surface area.
$$Q = h * A * ( T_s - T_a)$$
$$\ T_s \$$ is the surface temperature, $$\ T_a\$$ the external air temperature, and A the total surface area. For a small temperature differential and still air h can be as low as $$\ 10 W/(m^2 K) \$$
But if you use the air temperature this is a gross estimate, as it ignores that the internal temperature also has to be coupled from the air to the box, and that some of the heat is being radiated as well.
You would do better if you directly measure the box surface temperature.
But in your application I would do two things:
1. Add some non-flammable foam insulation to the inside of the box. Even a few millimeters covering the internal surfaces would go a long way.
2. If that is not enough, add a resistor to dissipate a few extra watts.
You could save yourself some time if you simply experiment with a couple known power levels (a couple resistors and a power supply) and directly calculate the heat transfer coefficient air to air for your specific box.
If you want to regulate the temperature and you need relatively little power you can directly use a positive temperature coefficient thermistor. Or you can couple a thermistor with a transistor to increase dissipation.
Thermal conductivity of aluminum walls has very little to do with the problem (thermal impedance of walls is likely negligible). This is a question about thermal properties of a rectangular box under free convection in the field of gravity. As formulated, the problem has no easy solution and needs an involvement of CFD - computational fluid dynamics. The result will depend on box orientation and whether the ambient air is still or moving. There is a vast layer of engineering articles to assess thermal regimes of enclosures, like this one, although it is pretty clueless (fin spacing is too small for good natural convection to develop).
Here is a better article on how to access heat transfer across hermetically sealed electronics enclosure, here is the general sketch of the problem components,
On external side, you need to account for natural convection around vertical surfaces of the enclosure, from top and bottom horizontal surfaces (the transfer properties are different in all three cases), and radiative exchange. For inner convection the same factors are working and need calculations/estimation too. So, there are a lot of factors involved, and only crude estimations are possible.
• The article from heatsinkcalculator.com ends up with a spreadsheet. But if you want to get an estimated power based on your measurements, you will need to pay them \$49. – Ale..chenski Feb 18 '19 at 6:37
• Here is one more inspiring article that might help, electronics-cooling.com/2018/05/… – Ale..chenski Feb 18 '19 at 6:49
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2020-02-19 14:32:49
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https://qiskit.org/documentation/locale/de_DE/stubs/qiskit.quantum_info.process_fidelity.html
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# qiskit.quantum_info.process_fidelity¶
process_fidelity(channel, target=None, require_cp=True, require_tp=True)[Quellcode]
Return the process fidelity of a noisy quantum channel.
The process fidelity $$F_{\text{pro}}(\mathcal{E}, \mathcal{F})$$ between two quantum channels $$\mathcal{E}, \mathcal{F}$$ is given by
$F_{\text{pro}}(\mathcal{E}, \mathcal{F}) = F(\rho_{\mathcal{E}}, \rho_{\mathcal{F}})$
where $$F$$ is the state_fidelity(), $$\rho_{\mathcal{E}} = \Lambda_{\mathcal{E}} / d$$ is the normalized Choi matrix for the channel $$\mathcal{E}$$, and $$d$$ is the input dimension of $$\mathcal{E}$$.
When the target channel is unitary this is equivalent to
$F_{\text{pro}}(\mathcal{E}, U) = \frac{Tr[S_U^\dagger S_{\mathcal{E}}]}{d^2}$
where $$S_{\mathcal{E}}, S_{U}$$ are the SuperOp matrices for the input quantum channel $$\mathcal{E}$$ and target unitary $$U$$ respectively, and $$d$$ is the input dimension of the channel.
Parameter
• channel (Operator or QuantumChannel) – input quantum channel.
• target (Operator or QuantumChannel or None) – target quantum channel. If None target is the identity operator [Default: None].
• require_cp (bool) – check if input and target channels are completely-positive and if non-CP log warning containing negative eigenvalues of Choi-matrix [Default: True].
• require_tp (bool) – check if input and target channels are trace-preserving and if non-TP log warning containing negative eigenvalues of partial Choi-matrix $$Tr_{\mbox{out}}[\mathcal{E}] - I$$ [Default: True].
Rückgabe
The process fidelity $$F_{\text{pro}}$$.
Rückgabetyp
float
Verursacht
QiskitError – if the channel and target do not have the same dimensions.
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2021-08-03 18:21:36
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http://www.j.sinap.ac.cn/fs/EN/10.11889/j.1000-3436.2012.rrj.30.120204
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### Effect of radiation vulcanization on the mechanical properties of ethylene propylene dine terpolymer rubber
HU Guowen1 LIU Bi1 DU Jifu1 GAO Tao1 ZHAI Maolin2
1. 1 (College of Nuclea Technology, Chemistry and Biology , Hubei University of Science and
Technology, Xianning 437100, China)
2 (Department of Applied Chemistry, College of Chemistry and Molecular Engineering,
Peking University, Beijing 100871, China)
• Received:2011-06-21 Revised:2011-07-18 Online:2012-04-20 Published:2014-09-25
Abstract:
Using trimethylopropane trimethylacrylate (TMPTMA) as crosslinking sensitizer, ethylene propylene dine terpolymer rubber (EPDM) was vulcanized by 60Co irradiation method. It has been found that the gel fraction of EPDM increases with the TMPTMA dosage and the appropriate dosage is 8% in weight percentage to EPDM. Tensile strength increases with the absorbed dose until 120 kGy. And the adopted absorbed dose is about 80 kGy. At the same time, the properties of aging resistance and thermal stability of radiation vulcanizate are improved compared with chemical vulcanizate. TG results show that the thermal decomposition temperature of radiation vulcanizate (480℃-530℃) is higher than that of chemical vulcanizate (430℃-480℃). These experimental results indicate that the properties of aging resistance and thermal stability of radiation vulcanizate are better than those of chemical vulcanizate.
CLC Number:
• TQ333.4，O644.2
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2021-12-07 00:22:08
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http://physics.stackexchange.com/questions/102458/how-can-kvl-kcl-be-derived-from-maxwell-equations
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# How can KVL & KCL be derived from Maxwell equations?
How can KVL (Kirchhoff's Voltage Law) & KCL (Kirchhoff's Current law) be derived from Maxwell equations in lumped circuits?(Lumped network : if $d$is the largest dimension of the network and $\lambda$ is the wavelength of the signal,a lumped network satisfies the condition $d<<\lambda$ where $\lambda=c/f$)
Note:1)
KVL & KCL works for lumped circuits(not for every circuits).trying to derive KVL and KCl from Maxwell equations without using lumped assumption(Lumped assumption:assuming that the circuit is lumped)is wrong.actually 4 $Maxwell$ $equations +lumped$ $assumption ==>$2 $Circuits$ $laws$ (KVL & KCL)
2)The other thing we can assume is that there are no incoming energy from outside,such as external magnetic field.
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Sorry would you elaborate more on what KVL and KCL stand for? – Ali Mar 7 at 16:40
@Ali Kirchhoff laws for Current and Voltage – nephente Mar 7 at 16:40
I see. For the current it's just conservation of electric charge. For the voltage: in the absence of time varying magnetic field, $\nabla \times \vec E = 0 \Rightarrow \vec E = -\nabla V \Rightarrow \oint \vec E . \vec {dl} =0$ – Ali Mar 7 at 16:48
@Ali $\nabla \times E = -dB/dt$ – rza Mar 7 at 22:49
@rza : That's why she specified "in the absence of a time varying magnetic field". – Colin K Mar 9 at 0:28
$\def\vE{{\vec{E}}}$ $\def\vD{{\vec{D}}}$ $\def\vB{{\vec{B}}}$ $\def\vJ{{\vec{J}}}$ $\def\vr{{\vec{r}}}$ $\def\vA{{\vec{A}}}$ $\def\vH{{\vec{H}}}$ $\def\ddt{\frac{d}{dt}}$ $\def\rot{\operatorname{rot}}$ $\def\div{\operatorname{div}}$ $\def\grad{\operatorname{grad}}$ $\def\rmC{{\mathrm{C}}}$ $\def\rmM{{\mathrm{M}}}$ $\def\ph{{\varphi}}$ $\def\eps{{\varepsilon}}$ Faraday's law in integral form: $$\oint_{\partial A} \vE \cdot d\vr + \ddt\int_A \vB\cdot d\vA = 0$$ Thereby, $A$ is some surface and $\partial A$ its boundary. The boundary can be split into partial paths $\bigcup_k C_k = \partial A$ and the integral becomes $$\sum_{k} \int_{C_k}\vE\cdot\vr + \ddt\int_A \vB\cdot d\vA = 0$$ You can define the voltage drops $V_k := \int_{C_k}\vE\cdot d\vr$ and the induced voltage $V_i$. This way you get Kirchhoff's Voltage law $$\sum_{k} V_k + V_i = 0.$$ Ampere's law reads $$\oint_{\partial A} \vH\cdot d\vr = \int_A \vJ\cdot d \vA + \int_A \dot\vD\cdot d\vA.$$ If you choose a closed surface $A$ then the boundary $\partial A$ is empty, the left-hand integral is zero and the equation for a closed surface becomes $$0 = \oint_A \vJ\cdot d \vA + \oint_A \dot\vD\cdot d\vA$$ We dissect the surface into partial surfaces $A_k$ of conductor cross-sections and a partial boundary of an insulator $A_i$. The corresponding integrals $I_k:=\int_{A_k}\vJ d\vA$ are the currents through these conductors. Furthermore, we have the charge transfer currents $I_{Ck}:=\int_{A_k}\dot\vD d\vA$ for the conductors and for the insulator $I_{i} := \int_{A_i}\dot \vD d\vA$. This gives you Kirchhoff's current law $$\sum_{k} I_k + \sum_{k} I_{Ck} + I_{i} = 0.$$
If one includes induction voltages and transfer charge currents in KVL and KCL these laws directly represent the laws of Faraday and Ampere. These quantities can be modeled through parasitic inductance and capacitance in the network diagram.
The fields $\vE$, $\vJ$, $\vD$, $\vH$, $\vB$ of the exact solution of Maxwell's equations satisfy Faraday's and Ampere's law for every piecewise smooth surface $A$.
Network modeling can be interpreted as discretization of Maxwell's equations. Through the finite selection of path integrals and surface integrals the number of degrees of freedom and the number of equations are reduced from infinite for the vector fields to the finite number of voltage and current variables and the finite number of corresponding loop and cut equations.
A discretization can give the exact results if the V-I-relations of the finite elements give exact field solutions (as boundary value problems). This is (almost) possible with DC-circuits.
For higher frequency applications one needs to refine the discretization. A good guidance for the choice of the discretization width is the wavelength of the considered electromagnetic waves in the circuit. In practice that means that one needs to include more parasitic elements in the network model for higher frequencies.
Slowly I hopefully understand what you mean by the `integrated principle'. I think it is helpful to see how the behavioural relation of a network element is derived from Maxwell's equations and embedded into the network theory.
Note, in the answer Batteries and voltage? I have just derived the V-I-relation of a resistor from Maxwell's equations. The V-I-relation for a capacitor can be derived in a similar way. For the inductor you need the induced voltage as defined above.
In a wide range of applications the fields can be approximated as quasi-stationary. With this approximation it turns out that for many basic structures the path integral over the field strength, i.e., the voltage drop is directly related to the cross-section integral over the current density, i.e., the current. This fact and the above equations for certain path-integrals over the field strength (KVL) and certain surface integrals over the current density (KCL) are exploited in network theory.
Note, that the splitting of voltages into the path integrals $V_k:=\int_{C_k} \vE\cdot \vr$ and the induction voltage is only one way of interpretation which has its draw-backs.
For an instance with a time-variable magnetic field the spatial separation suggested by the formula $\sum_{k} V_k + V_i = 0$ does not really exist.
There is another approach where the field strength $\vE$ is split into a Coulomb-part and a magnetic part. Pityingly, I do not know the reference anymore and I learned this stuff more than 10 years ago. But, if you are really interested I can try to retrieve it (this is not so easy and will take its time).
The idea is to apply the well-known technique of magnetic vector potentials $\vB=\rot \vA$ which solve the divergence equation $\div\vB=0$. As a gauge condition $\div\vA=0$ is used in this setup.
Faraday's law in differential form then reads $$\rot(\vE+\dot\vA) = 0$$ which ensures in a simply connected domain the existence of a potential $\ph$ for the vector field $\vE+\dot\vA$, i.e., $$\begin{array}{rcl} \vE + \dot\vA &=& -\grad\ph\\ \vE &=& \underbrace{-\grad\ph}_{\vE_\rmC} \underbrace{-\dot\vA}_{\vE_\rmM} \end{array}$$ In this formula $\vE_\rmC$ is the Coulomb-part of the electrical field strength and $\vE_\rmM$ is the magnetical part of the field strength.
Kirchhoffs Voltage law then holds unrestricted for Voltage drops defined with the Coulomb-part of the field strength $V_k := \int_{C_k} \vE_\rmC \cdot d\vr$.
For an example, in the case of constant permitivity $\eps$ the Coulomb part is the cause of space charges: $$\begin{array}{rl} \rho &= \div \vD\\ & = \div \eps (\vE_\rmC - \dot\vA)\\ & = \eps \div \vE_\rmC - \eps\frac{\partial}{\partial t}\underbrace{\div\vA}_{=0}\\ &= \eps \div \vE_\rmC \end{array}$$ One can then derive the V-I-relations for the network elements based on the Coulomb part and the magnetic part of the voltages and the currents.
This approach respects that the Coulomb-part and the magnetical part coexist everywhere in the circuit.
But, I have not seen this approach in practical usage yet.
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To your first questions: Yes, I am well educated in the theory of electrical networks and I had the pleasure/honor to work together with one of the most respectable professors in this area, Professor Albrecht Reibiger from TU-Dresden. To your second statement: Please, describe in greater detail what you mean by the integrated principle. You are certainly aware that the voltage drops over the elements in the loops from KVL correspond to the integrals $V_k=\int_{C_k}\vec{E}d\vec{r}$ and the inductive part to $V_i$, if needed you can split the inductive part into partial fluxes. KCL is analogous. – Tobias Mar 7 at 19:46
@rza: The answer is absolutely correct. I was about to post the exact same thing, but Tobias beat me to it. Before claiming an answer to be wrong/incomplete, please try understanding it completely first. – Sam29 Mar 7 at 20:14
@Sam29 : Excuse me i am not satisfied,do you mean KVL & KCL works for non-integrated circuits ? – rza Mar 7 at 21:40
@rza: I have added some comments on how the VI-relation of network elements can be derived from Maxwell's equations and how this can be exploited to motivate the theory of electrical networks. – Tobias Mar 8 at 0:09
@Tobias: Excuse me,can you tell me where is lumped assumption?(lumped : en.wikipedia.org/wiki/Lumped_element_model) . if you success to derive KVL and KCL from Maxwell equations without using lumped assumption it means you proved that KVL & KCL works for every circuits even non-lumped circuits !! – rza Mar 8 at 20:32
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2014-10-20 09:39:54
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https://couetteflow.readthedocs.io/en/latest/Results/contents.html
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# Results summary¶
## A) Result #1¶
Q. Show the expression for $$\tau$$ that non-dimensionalizes the governing PDE. Show the non-dimensionalized form of the governing PDE.
• Non-dimensionalized variables:
$$u' = \frac{u}{u_{top}}$$, $$t' = \frac{t}{\tau}$$, $$y' = \frac{y}{L}$$
where $$\tau = \frac{L^{2}}{\nu}$$
• Non-dimensionalized governing PDE:
$$\frac{\partial u'}{\partial t'} = \frac{\partial^{2}u'}{\partial y'^{2}}$$
## B) Result #2¶
Q. Show the non-dimensionalized form of the time-dependent exact solution expression for the specified boundary and initial conditions given in this problem.
To find the time-dependent exact solution, we need to first find $$a_{n}$$ which satisfies the given initial velocity profile. The resolved form of $$a_{n}$$ is then re-written as:
$\begin{split}a_{n} = \left\{\begin{matrix} 1 \text{ if } n = 1 \\ 0 \text{ if } n \neq 1 \end{matrix}\right.\end{split}$
Thus, applying the resolved $$a_{n}$$ into the given exact solution results in:
$u'_{exact}(t',y') = y' + \text{sin}(\pi y') \text{exp}[-\pi^{2}t']$
## C) Result #3¶
Q. Provide a brief description of the finite difference scheme (in non-dimensional form), the solution method used and exactly how the boundary and initial conditions are applied.
Given finite difference scheme has a weighting parameter $$\theta$$ to put an effect of implicit solution. If $$\theta$$ is equal to 1, the scheme becomes to fully implicit, otherwise, the scheme can be partially implicit or explicit ($$\theta$$ = 0). Rearranging the given finite difference equation leads to the following simplified form:
$A_{j} u^{n+1}_{j+1} + B_{j} u^{n+1}_{j} + C_{j} u^{n+1}_{j+1} = D_{j}$
where
\begin{align}\begin{aligned}A_{j} = -r \theta\\B_{j} = 1 + 2r\theta\\C_{j} = -r\theta\\D_{j} + r(1-\theta)\left \{ u^{n}_{j-1} - 2u^{n}_{j} + u^{n}_{j+1} \right \}\end{aligned}\end{align}
Here, the resulting equation has simplified coefficient $$r = \frac{\Delta t'}{\Delta y'^{2}}$$.
For the boundary condition, non-slip condition is applied to both upper and bottom plates. Thus, $$y(0) = 0$$ and $$y(L)=1$$ remain unchanged while the inner point quantities varies during the transient phase. The initial condition described earlier can satisfy the given boundary condition here. The Thomas algorithm is set to unchange the boundary condition as the time varies.
## D) Result #4¶
Q. Show the expression used for calculating the RMS Error relative to the time-dependent exact solution. Also show the expression used for calculating the RMS Error relative to the steady-state exact solution. Also, give a statement of the criteria used to end the calculations.
In this project, two different types of RMS error formulation are used:
• RMS error relative to the exact time-dependent solution
$\text{RMS}_{\text{NSS}}(t) = \sqrt{\frac{1}{N} \sum_{\text{j}=2}^{\text{jmax}-1} \left [ \left ( u'_{exact,j}(t) - u'^{n} \right )^{2} \right ]}$
where N is number of inner grid points.
• RMS error relative to the exact steady-state solution:
$\text{RMS}_{\text{SS}}(t) = \sqrt{\frac{1}{N} \sum_{\text{j}=2}^{\text{jmax}-1} \left [ \left ( u'_{exact,j}(t=\infty ) - u'^{n} \right )^{2} \right ]}$
• The convergence criteria is limited by the following relation:
$\text{RMS}_{\text{SS}}(t) \leqslant 1\times 10^{-7}$
## E) Result #5¶
Q. For $$\theta = 0$$ and $$\text{jmax} = 51$$, state the maximum value of $$\Delta t$$ for which a stable solution is obtained. Provide a semi-log plot of the RMS error (relative to the time-dependent exact solution) vs iteration number (using a $$\Delta t$$ for which the code is stable). Create a similar plot of the RMS error (relative to the steady-state exact solution) vs. iteration number.
A. Given conditions, the non-dimensional spatial step size results in $$\Delta y' = 0.0002$$. Performing Von Neumann stability analysis on the given conditions give rise to the below time step criterion:
$\Delta t' \leqslant \frac{\Delta y'^{2}}{4\left ( \frac{1}{2} - \theta \right )}$
Thus, the maximum time step to stabilize the scheme is determined as $$\Delta t' = 0.0002$$.
## F) Result #6¶
Q. For $$\theta = 0$$, present a graph which clearly shows the progression of velocity profiles during the flow development when $$\text{jmax} = 51$$. The plot should show the initial profile, final steady state profile and at least 3 other non-steady-state profiles (i.e. all on the same plot). Overlay the exact numerical velocity profiles on this plot for the same points in time. Create similar plots for $$\theta = 1/2$$ and $$\theta = 1$$.
In this problem, the time step was employed as $$\Delta t'$$ = 0.0002 in order to have stable convergence for every $$\theta$$ cases. This time step was then applied to the other $$\theta$$ cases. As the following three figures show, the numerical solution well follows the analytical solution in both time and spatial domain.
1. $$\theta$$ = 0 (Fully explicit scheme): Converged at iteration number of 7990.
1. $$\theta$$ = 0.5 (Crank-Nicolson scheme): Converged at iteration number of 7998.
1. $$\theta$$ = 1 (Fully implicit scheme): Converged at iteration number of 8006.
## G) Result #7¶
Q. Provides a comparison of the stability behavior of your solver to the stability analysis performed in Homework Assignment #3. Compute $$\text{jmax} = 51$$ cases with $$\theta = 0$$, $$1/2$$, and $$1$$ using various values of $$\Delta t$$ to explore the stability boundaries of your solver. Show and discuss whether or not your solver follows the theoretical stability behavior of these three numerical schemes.
A. From the HW#3’s solution, the stability analysis can be summarized by:
• Unconditionally stable if $$\theta \geqslant \frac{1}{2}$$
• Conditionally stable if $$0 \leqslant \theta < \frac{1}{2}$$
In the case of conditionally stable scheme, the maximum time step can be determined by using below relation so that the scheme is stable with given $$\theta$$.
$\Delta t \leqslant \frac{\Delta y^{2}}{4\left ( \frac{1}{2}-\theta \right )}$
### 1) $$\theta$$ = 0 (Fully explicit)¶
According to the above relation, for $$\theta = 0$$, the maximum time step should be 0.0002 to make the scheme stable. Following figures show the convergence history for three different time step cases: (1) ensurely stable time step, (2). maximum time step and (3). slightly bigger time-step than the maximum value. If you can’t see the movies below, you are seeing the printed version of document. If you want to see the movies, please visit: http://couetteflow.readthedocs.org/en/latest/Results/contents.html#g-result-7
The figure below is the case with $$dt'$$ = 0.0001 that is ensured for the stability for fully explicit scheme.
• $$dt' = 0.0001$$
• RMS error
In this condition, the time step should not be over 0.0002 in order to obtain the stable solution. The following figures and movies prove the stability criterion in terms of time-step.
• $$dt' = 0.0002$$
• RMS error
• Movie of velocity profile (online available)
Even the slightly bigger time-step causes the unstable solution and thus, the RMS error is taken off and goes to infinity after a certain number of iteration.
• $$dt' = 0.000201$$
• RMS error
• Movie of velocity profile (online available)
### 2) $$\theta$$ = 1/2 (Crank-Nicolson scheme)¶
• Convergence check with the various time step:
Non-dimensional time step $$\Delta t'$$ Maximum iteration for convergence 0.0001 15996 0.001 1600 0.01 160 0.1 15 1.0 39 10.0 390 100.0 3893 1000.0 38927 10000.0 389268 100000.0 Not converged within 999999 iterations
All the cases above seem to be stable but the convergence is strongly sensitive to how big or small time step is. The interesting pattern to be observed here is that the maximum iteration number for convergence shows quadratic behavior. That is, quite small and quite big time step require long iterations. In particular, big time steps, 1000, 10000, and 100000 for examples, take long period to make the scheme converged into the specified RMS residual. This is somewhat unphysical. If 10,000 sec is taken as a time step, it will take about 123 years for the flow to be settled down to the steady-state.
The stability check can be done by looking at the movies as a function of different time-step. If you can’t see the movies below, you are seeing the printed version of document. If you want to see the movies, please visit: http://couetteflow.readthedocs.org/en/latest/Results/contents.html#g-result-7
• $$dt' = 0.0001$$
The movies shown below is to show the velocity profile calculated by the present numerical solution and analytic solution. In this case, sufficiently small time-steps can ensure the physically proper behavior of the numerical solution.
• Movie of velocity profile (online available)
• $$dt' = 1000$$
As already mentioned above, since the given $$\theta$$ condition gives the stable solution, the improperly big time-step give rise to the extremely long period to have convergence. The second movie below shows the abnormal behavior of velocity profile. This may have to be involved with the inaccurate time gradient due to the big time-step, thus it leads to the negative velocity instantaneously and fluctuation of velocity profile.
• Movie of velocity profile (online available)
### 3) $$\theta$$ = 1 (Fully implicit)¶
• Convergence check with the various time step:
Non-dimensional time step $$\Delta t'$$ Maximum iteration for convergence 0.0001 16004 0.001 1608 0.01 168 0.1 23 1.0 7 10.0 4 100.0 3 1000.0 2 10000.0 2 100000.0 2
All the tested cases above are stable and the convergence performance is enhanced as the time step increases. Contrary to the Crank-Nicolson scheme case ($$\theta$$ = 0.5), the pattern of maximum iteration for convergence shows the linearity as a function of time step. Therefore, it can be concluded that the solver follows the theoretical stability behavior.
## H) Result #8¶
Q. Write down an expression(s) for the truncation error (TE) of this finite difference scheme and describe the order of accuracy of the scheme for different values of $$\theta$$. Note: You are not required to derive the TE expression.
$\text{T.E.} = \left [ \left ( \theta - \frac{1}{2} \right ) \Delta t + \frac{\Delta x^{2}}{12} \right ]u_{xxxx} + \left [ \left ( \theta^{2} - \theta + \frac{1}{3} \right )\Delta t^2 + \frac{1}{3} \left ( \theta - \frac{1}{2} \right )\Delta t \Delta x^2 + \frac{1}{360} \Delta x^{4} \right ] u_{xxxxxx} + \cdot \cdot \cdot$
According to the above equation, this combined method of explicit and implicit schemes has order of accuracy in time and space as a function of $$\theta$$.
1. $$\theta$$ = 1/2 (Crank-Nicolson scheme): $$\text{T.E.} = O\left [ (\Delta t)^{2}, (\Delta x)^{2} \right ]$$
2. Simple explicit ($$\theta$$ = 0) and implicit ($$\theta$$ = 1): $$\text{T.E.} = O\left [ \Delta t, (\Delta x)^{2} \right ]$$
3. Special case ($$\theta = \frac{1}{2} - \frac{(\Delta x)^{2}}{12\Delta t}$$): $$\text{T.E.} = O \left [ (\Delta t)^{2}, (\Delta x)^{4} \right ]$$
## I) Result #9¶
Investigate the spatial order of accuracy of the code for $$\theta$$ = 1. Do this by using a small value of $$\Delta t'$$ = 0.000625 and running multiple cases of the code with different values of $$\Delta y'$$ (i.e. 0.1, 0.05, 0.025, 0.0125). Make a table and log-log plot of the peak RMS error (relative to the time-dependent exact solution) as a function of $$\Delta y'$$. Based on these results, discuss whether or not your solver follows the theoretical order of spatial accuracy given by the TE expression for the scheme. Also, explain why it is important to use a small $$\Delta t'$$ when we investigate the spatial accuracy of this scheme.
• Comparison of Peak RMS error as a function of spatial and temporal steps
dy jmax Peak RMS error ($$\Delta t$$ = 0.000625) Peak RMS error ($$\Delta t$$ = 0.0002) 0.1 11 0.309370E-02 0.252525E-02 0.05 21 0.136823E-02 0.811529E-03 0.025 41 0.945456E-03 0.395090E-03 0.0125 81 0.838836E-03 0.291753E-03 0.00625 161 0.811120E-03 0.265708E-03 0.003125 321 0.803589E-03 0.259019E-03 0.0015625 641 0.801397E-03 0.257250E-03 0.00078125 1281 0.800693E-03 0.256758E-03
The previous theoretical analysis of accuracy investigated the order of accuracy in terms of spatial and time step size. For $$\theta$$ = 0, the truncation error is 1st order in time and 2nd order in space. The maximum RMS error for every test cases shows the quantitatively quadratic pattern as a function of spatial step size. Moreover, the smaller time step (here, $$\Delta t'$$ = 0.0002) makes this pattern more distinctive compared to the bigger time step. This is because the smaller time step can reduce the truncation error in time derivative and thus the RMS error is then significantly made by the spatial derivative terms.
## J) Result #10¶
Q. Investigate the temporal order of accuracy of the code for $$\theta$$ = 1 and $$\theta$$ = 1/2. Do this by using jmax = 51 and various $$\Delta t'$$ (i.e. 0.02, 0.01, 0.005, 0.0025, 0.00125, 0.000625). Make tables and a log-log plots of the peak RMS error (relative to the time-dependent exact solution) as a function $$\Delta t'$$ for $$\theta$$ = 1 and $$\theta$$ = 1/2. Based on these results, discuss whether or not your solver follows the theoretical order of temporal accuracy given by the TE expression for the scheme.
dt Peak RMS error ($$\theta$$ = 1) Peak RMS error ($$\theta$$ = 1/2) 1000 0.723888E-04 0.713996 100 0.723228E-03 0.711396 10 0.716697E-02 0.685903 1 0.656967E-01 0.473546 0.1 0.933255E-01 0.238631E-01 0.05 0.540879E-01 0.538846E-02 0.02 0.240539E-01 0.769763E-03 0.01 0.125364E-01 0.126926E-03 0.005 0.643658E-02 0.331436E-04 0.0025 0.329430E-02 0.731227E-04 0.00125 0.169854E-02 0.831203E-04 0.000625 0.894559E-03 0.856183E-04 0.0002 0.345497E-03 0.863658E-04
The tested results presented above show the accuracy of numerical solution as a function of time step. The previous discussion on the truncation error tells that the fully implicit scheme ($$\theta$$ = 1) follows the 1st order in time. However, it is important to note that this analysis of accuracy is only well followable when the time step is less than $$10^{-1}$$. This inaccuracy may have come from the spatial derivative order because the currently employed spatial step size is somewhat big enough to cause the truncation error.
more accurate numerical solution when $$\theta$$ value approaches to unity. However, the bigger time-step which is quite over the physically significant time scale should be avoided as already discussed earlier.Comparing two different $$\theta$$ cases proves that the Crank-Nicolson sheme ($$\theta$$ = 1/2) is more likely to ensure the accurate result only if the time step is sufficiently small. Otherwise, the bigger time step makes sure to give more accurate numerical solution when $$\theta$$ value approaches to unity. However, the bigger time-step which is quite over the physically significant time scale should be avoided as already discussed earlier.
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2020-10-24 14:13:07
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https://projecteuclid.org/euclid.ade/1366896042
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## Advances in Differential Equations
### Nonlinear oblique boundary value problems for two-dimensional curvature equations
John Urbas
#### Abstract
We prove the existence of smooth solutions of two-dimensional nonuniformly elliptic curvature equations subject to a nonlinear oblique boundary condition. These are equations whose principal part is given by a suitable symmetric function of the principal curvatures of the graph of the solution $u$. The types of boundary conditions we are able to treat are the same as those we considered in earlier work on Hessian equations.
#### Article information
Source
Adv. Differential Equations Volume 1, Number 3 (1996), 301-336.
Dates
First available in Project Euclid: 25 April 2013
Permanent link to this document
https://projecteuclid.org/euclid.ade/1366896042
Mathematical Reviews number (MathSciNet)
MR1401397
Zentralblatt MATH identifier
0853.35046
#### Citation
Urbas, John. Nonlinear oblique boundary value problems for two-dimensional curvature equations. Adv. Differential Equations 1 (1996), no. 3, 301--336. https://projecteuclid.org/euclid.ade/1366896042.
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2017-09-26 07:22:11
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http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server/papers/186/index.php
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MODNET
Research Training Network in Model Theory
Publications > Preprint server > Preprint Number 186 Preprint Number 186 186. Ikumitsu Nagasaki, Tomohiro Kawakami, Yasuhiro Hara and Fumihiro Ushitaki A generalized Borsuk-Ulam theorem in a real closed field E-mail: , , , (email address protected by JavaScript. Please enable JavaScript to contact) Submission date: 20 May 2009. Abstract: Let G be the cyclic group of order k and N=(R, + , • , < , ... ) an o-minimal expansion of a real closed field R. Let X be a definably connected definable set with a free definable G-action. Assume that there exists a positive integer n such that H_q(X; Z/kZ) q for 1 \leq q \leq n. If Y is a definable set with a free definable G-action such that H_{n+1}(Y/G, Z/kZ)=0, then there is no definable G-map from X to Y. We also prove the topological version of this definable version. Mathematics Subject Classification: 57S10, 57S17, 55M20, 55M35, 03C64 Keywords and phrases: The Borsuk-Ulam theorem, o-minimal, real closed fields, finite groups, definable $C_k$-maps, continuous $C_k$-maps. Full text: pdf, dvi, ps.
Last updated: June 10 2009 20:48 Please send your corrections to: The e-mail address is protected, enable Javascript to see it
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2017-10-23 20:50:22
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https://forskning.ruc.dk/da/publications/discrete-dynamics-versus-analytic-dynamics
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# Discrete dynamics versus analytic dynamics
Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review
### Resumé
For discrete classical Molecular dynamics obtained by the “Verlet” algorithm (VA) with the time increment h there exists a shadow Hamiltonian H˜ with energy E˜(h) , for which the discrete particle positions lie on the analytic trajectories for H˜ . Here, we proof that there, independent of such an analytic analogy, exists an exact hidden energy invariance E * for VA dynamics. The fact that the discrete VA dynamics has the same invariances as Newtonian dynamics raises the question, which of the formulations that are correct, or alternatively, the most appropriate formulation of classical dynamics. In this context the relation between the discrete VA dynamics and the (general) discrete dynamics investigated by Lee [Phys. Lett. B122, 217 (1983)] is presented and discussed.
Originalsprog Engelsk Journal of Chemical Physics 140 04 5 0021-9606 https://doi.org/10.1063/1.4862173 Udgivet - 2014
### Citer dette
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title = "Discrete dynamics versus analytic dynamics",
abstract = "For discrete classical Molecular dynamics obtained by the “Verlet” algorithm (VA) with the time increment h there exists a shadow Hamiltonian H˜ with energy E˜(h) , for which the discrete particle positions lie on the analytic trajectories for H˜ . Here, we proof that there, independent of such an analytic analogy, exists an exact hidden energy invariance E * for VA dynamics. The fact that the discrete VA dynamics has the same invariances as Newtonian dynamics raises the question, which of the formulations that are correct, or alternatively, the most appropriate formulation of classical dynamics. In this context the relation between the discrete VA dynamics and the (general) discrete dynamics investigated by Lee [Phys. Lett. B122, 217 (1983)] is presented and discussed.",
author = "S{\o}ren Toxv{\ae}rd",
year = "2014",
doi = "10.1063/1.4862173",
language = "English",
volume = "140",
journal = "Journal of Chemical Physics",
issn = "0021-9606",
publisher = "American Institute of Physics",
number = "04",
}
I: Journal of Chemical Physics, Bind 140, Nr. 04, 2014.
Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review
TY - JOUR
T1 - Discrete dynamics versus analytic dynamics
AU - Toxværd, Søren
PY - 2014
Y1 - 2014
N2 - For discrete classical Molecular dynamics obtained by the “Verlet” algorithm (VA) with the time increment h there exists a shadow Hamiltonian H˜ with energy E˜(h) , for which the discrete particle positions lie on the analytic trajectories for H˜ . Here, we proof that there, independent of such an analytic analogy, exists an exact hidden energy invariance E * for VA dynamics. The fact that the discrete VA dynamics has the same invariances as Newtonian dynamics raises the question, which of the formulations that are correct, or alternatively, the most appropriate formulation of classical dynamics. In this context the relation between the discrete VA dynamics and the (general) discrete dynamics investigated by Lee [Phys. Lett. B122, 217 (1983)] is presented and discussed.
AB - For discrete classical Molecular dynamics obtained by the “Verlet” algorithm (VA) with the time increment h there exists a shadow Hamiltonian H˜ with energy E˜(h) , for which the discrete particle positions lie on the analytic trajectories for H˜ . Here, we proof that there, independent of such an analytic analogy, exists an exact hidden energy invariance E * for VA dynamics. The fact that the discrete VA dynamics has the same invariances as Newtonian dynamics raises the question, which of the formulations that are correct, or alternatively, the most appropriate formulation of classical dynamics. In this context the relation between the discrete VA dynamics and the (general) discrete dynamics investigated by Lee [Phys. Lett. B122, 217 (1983)] is presented and discussed.
U2 - 10.1063/1.4862173
DO - 10.1063/1.4862173
M3 - Journal article
VL - 140
JO - Journal of Chemical Physics
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SN - 0021-9606
IS - 04
ER -
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2019-09-17 18:51:31
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http://www-old.newton.ac.uk/programmes/SAS/seminars/2012070314001.html
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# SAS
## Seminar
### On the inversion of computable functions
Hoyrup, M (INRIA Paris - Rocquencourt)
Tuesday 03 July 2012, 14:00-15:00
Seminar Room 1, Newton Institute
#### Abstract
Ergodic shift-invariant measures inherit many effective properties of the uniform measure: for instance, the frequency of $1$'s in a typical sequence converge effectively, hence it converges at every Schnorr random sequence; the convergence is robust to small violations of randomness [1]; every Martin-Löf random sequence has a tail in every effective closed set of positive measure [2]. These properties are generally not satisfied by a non-ergodic measure, unless its (unique) decomposition into a combination of ergodic measures is effective. V'yugin [3] constructed a computable non-ergodic measure whose decomposition is not effective. This measure is a countable combination of ergodic measures. What happens for finite combinations? Is there a finitely but non-effectively decomposable measure?
We prove that the answer is positive: there exist two non-computable ergodic measures $P$ and $Q$ such that $P+Q$ is computable. Moreover, the set of pairs $(P,Q)$ such that neither $P$ nor $Q$ is computable from $P+Q$ is large in the sense of Baire category.
This result can be generalized into a theorem about the inversion of computable functions, which gives sufficient conditions on a one-to-one computable function $f$ that entail the existence of a non-computable $x$ such that $f(x)$ is computable.
We also establish a stronger result ensuring the existence of a sufficiently generic'' $x$ such that $f(x)$ is computable, in the spirit of Ingrassia's notion of $p$-genericity [4].
[1] Vladimir V. V'yugin. "Non-robustness property of the individual ergodic theorem." Problems of Information Transmission, 37(2):27–39, 2001.
[2] Laurent Bienvenu, Adam Day, Ilya Mezhirov, and Alexander Shen. "Ergodic-type characterizations of algorithmic randomness." In Computability in Europe (CIE 2010), volume 6158 of LNCS, pages 49–58. Springer, 2010.
[3] Vladimir V. V'yugin. "Effective convergence in probability and an ergodic theorem for individual random sequences." SIAM Theory of Probability and Its Applications, 42(1):39–50, 1997.
[4] M.A. Ingrassia. P-genericity for Recursively Enumerable Sets. University of Illinois at Urbana-Champaign, 1981.
#### Video
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2014-08-31 02:45:40
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https://electronics.stackexchange.com/questions/364074/does-using-electronics-with-poor-power-factor-affect-the-pulse-power-capability
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# Does using electronics with poor power factor affect the pulse power capability of a generator rated for a certain KVA?
The situation is such:
There is a multi-pole, three-phase, permanent magnet generator (rated for 100 KVA, let's say) that feeds into a simple diode rectifier with a capacitive/DC Voltage load (something like a battery).
Without any active correction, the power factor of this circuitry is going to be pretty poor (maybe down around 0.5).
Now I understand that poor power factor is going to give us much higher $I^{2}R$ losses in our wiring, diodes, etc. What I don't understand is the impact on the generator when running high power for short times with poor power factor.
I have two questions:
1. Can a multi-pole, 3 phase, permanent magnet generator even supply the power equivalent of its rated KVA at all with such power factor? (i.e. can I ever get 100kW out of this generator this way?)
2. Supposing I can, how long can I do that for? My limited understanding is that this will produce a lot of heat in the generator and cause it to fail if used this way continuously, but could still, in theory, provide this power. If I am to run it at 100kW at poor power factor for only a couple of minutes or less, am I not damaging the generator?
It is not not just the power factor that is the issue, it is also the distorted waveform causing poor power factor. It the power factor was low without the waveform being distorted, the power would be reduced, but only because the generator would be producing less power with the same current (W = V X A X Pf X SqRt3). If you know the power factor, you can easily calculate how much power the generator can produce without exceeding the maximum current rating. Check to see if the generator has a kVA, current or minimum power factor rating in addition to the power rating. With waveform distortion, some additional heating may be produced with the same current. However that would be no different than loading a transformer with a rectifier. I believe that some derating may be required for that.
The time that you can safely "overload" the generator depends on the time and amount of additional heating. Only the generator manufacturer can tell you.
• Thanks for your answer. This is all hypothetical, so no generator spec to check. I am still curious as to whether or not "overloading" is even possible. If the engine is spinning at max RPM, and the generator produces a certain KVA, there is no theoretical way to extract all of that power with poor power factor, right?
– Nino
Mar 22 '18 at 23:05
• To be more clear, essentially what I am asking is if the engine provides a constant 100kW of power, can the generator supply 100kW of average power that involves peaks above 100kW by some fashion of stored energy (i.e. does a generator have implicit energy storage)?
– Nino
Mar 22 '18 at 23:20
• The engine will only respond to the power demand. It will provide the torque required to deliver the power and not be much affected by excess current that is due to low power factor rather than a power overload. Excess current will cause increased losses in the generator that will increase the generator temperature. However the normal generator losses are only about 5%. If the losses increase by 40%, to 7%, that overheat the generator pretty quickly, but only cause a 2% overload on the engine. Mar 22 '18 at 23:25
Perhaps it would help to understand what a poor power factor actually is, and what happens in the wires.
If you have a device with a power factor of 0.8, it means that during each AC cycle, it is only using 80% of the power it takes in, with the remaining 20% being sent back to the source at some point during that same AC cycle.
This means an 80 kW device will draw 100 KVA, but then return 20 KVA back to the generator, each cycle. It averages out at 80 kW (with 0.8 PF) but it is measured as 100 KVA as that is the peak power draw, and it means you need to deliver 100 KVA to run this 80 kW device, even though 20 KVA of it is only "borrowed" and returned almost immediately.
As COVID-19 is currently in progress, an analogy would be that a person goes shopping and buys 100 toilet rolls, then takes them home and realises that they only have enough space to fit 80 rolls. The surplus 20 rolls get returned to the shop, but the shop can't resell them for hygiene reasons so has to throw them out. Thus the toilet paper factory had to produce 100 toilet rolls (100 KVA) even though the person only ended up using 80 rolls (80 kW). This person has a toilet paper factor (TPF) of 0.8.
So when you say can you ever get 100 kW out of the generator with a poor power factor, well the answer is you already are, but your device then takes some of that power and sends it back to the generator every cycle - two steps forward and one step back, so to speak. You are asking whether the toilet paper hoarder can ever get 100 toilet paper rolls from the shop, but that's already happening, it's just that some of them always get returned unused.
The generator probably doesn't know what to do with this returned power - it has just generated and sent out 100 KVA and suddenly 20 KVA is coming back in again, but it doesn't have any way of storing that power to send it back on the next cycle.
I am not 100% sure what actually happens inside a generator - probably it will momentarily switch to a motor with that incoming power, causing the rotation speed to increase, which then has to be compensated to avoid going out of the 50/60 Hz range. But moments later the load is back, so the generator may run rough, have increased vibration, etc. With inverters, the returned energy will typically be disposed of as waste heat (making the inverter run hotter and wasting power), which is why inverters have both a kW and a KVA rating. The difference between the two figures is how much energy they can dispose of without overheating, due to powering devices with poor power factors.
I said above that the generator has no way of storing the power that gets returned to it in each cycle, but you may have thought that something like a capacitor could be used to store this energy. This is indeed possible, and it's why inductive devices like motors can use capacitors to improve their power factor. The trick is that the required capacitance varies depending on the load, which is why "active power factor correction" circuits exist. These are usually designed to work on the appliance end rather than the generator end however, because you want the power lines between the generator and the appliance to also have an improved power factor - otherwise they transmit less overall power, if a chunk of the power they send out is returned back again (just like all those trucks delivering toilet rolls that end up being returned and thrown away). Whether it's electricity or trucks full of toilet paper, it's better to send out only what will be used, instead of sending more trucks (using larger wires) to deliver things that will end up being returned unused.
Hopefully from this you can see that if you want to run your poor power factor motor at 100 kW (without correcting its power factor), then you're going to have to run your generator past its spec. If the power factor is 0.8, then 100 kW divided by 0.8 gives you 125 KVA. This means you'd have to run your 100 KVA generator at 125 KVA in order to get the motor running at 100 kW (complete with 25 KVA being returned unused due to the poor power factor). Running the generator at 25% over its maximum rating is going to produce more heat than it is designed to cope with, so while it will work for a short while while it's cold and warming up, if you leave it running too long without extra cooling, it will overheat. Not to mention that the wires/cables/windings etc. are now probably a little too thin for the larger amount of power, so even more of it will be lost as heat due to wire resistance, which might mean you have to push the generator up to say 130 KVA to run your load at 100 kW. And then figure out how to get rid of 30 KVA (30 kW) of heat so the generator doesn't get cooked.
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2021-12-01 05:43:22
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https://ftp.aimsciences.org/article/doi/10.3934/dcdss.2013.6.1599
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# American Institute of Mathematical Sciences
December 2013, 6(6): 1599-1608. doi: 10.3934/dcdss.2013.6.1599
## Prolegomena to studies on dynamic materials and their space-time homogenization
1 Université Pierre et Marie Curie, Institut Jean Le Rond d'Alembert, UMR CNRS 7190, Case 162, Tour 55, 4 place Jussieu, 75252 Paris Cedex 05, France, France
Received June 2012 Revised September 2012 Published April 2013
This short contribution aims at introducing the notion of dynamic materials (as initiated by Blekhman and Lurie) and the corresponding allied techniques of homogenization and asymptotic analysis. Main role is played by the canonical conservation laws of energy and wave momentum - the latter most often ignored in the field of continuum mechanics - as follows from an application of the celebrated theorem of E. Noether.
Citation: Gerard A. Maugin, Martine Rousseau. Prolegomena to studies on dynamic materials and their space-time homogenization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1599-1608. doi: 10.3934/dcdss.2013.6.1599
##### References:
show all references
##### References:
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2019 Impact Factor: 1.233
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2021-04-23 14:38:30
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https://math24.net/indefinite-integral-basic-rules-integration.html
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# The Indefinite Integral and Basic Rules of Integration
## Antiderivatives and the Indefinite Integral
Let a function f (x) be defined on some interval I. The function F (x) is called an antiderivative of f (x), if
$F^\prime\left( x \right) = f\left( x \right)$
for all x in the interval I.
There is an infinite number of antiderivatives of a function f (x), all differing only by a constant C:
$\left( {F\left( x \right) + C} \right)^\prime = F^\prime\left( x \right) + C^\prime = f\left( x \right) + 0 = f\left( x \right).$
The set of all antiderivatives for a function f (x) is called the indefinite integral of f (x) and is denoted as
${\int} {{f\left( x \right)}{dx}} = F\left( x \right) + C,\;\;\text{if}\;\;F^\prime\left( x \right) = f\left( x \right).$
In this definition, the is called the integral symbol, f (x) is called the integrand, x is called the variable of integration, dx is called the differential of the variable x, and C is called the constant of integration.
## Indefinite Integral of Some Common Functions
Integration is the reverse process of differentiation, so the table of basic integrals follows from the table of derivatives.
It is supposed here that $$a,$$ $$p\left( {p \ne 1} \right),$$ $$C$$ are real constants, $$b$$ is the base of the exponential function $$\left( {b \ne 1, b \gt 0} \right).$$
## Properties of the Indefinite Integral
1. If $$a$$ is some constant, then
$\int {af\left( x \right)dx} = a\int {f\left( x \right)dx},$
i.e. the constant coefficient can be carried outside the integral sign.
2. For functions $$f\left( x \right)$$ and $$g\left( x \right),$$
$\int {\left[ {f\left( x \right) \pm g\left( x \right)} \right]dx} = \int {f\left( x \right)dx} \pm \int {g\left( x \right)dx} ,$
i.e. the indefinite integral of the sum (difference) equals to the sum (difference) of the integrals.
Calculation of integrals using the linear properties of indefinite integrals and the table of basic integrals is called direct integration.
## Solved Problems
Click or tap a problem to see the solution.
### Example 1
Evaluate the indefinite integral $\int {\left( {3{x^2} - 6x + 2\cos x} \right)dx} .$
### Example 2
Find the indefinite integral $\int {\left( {1 + x} \right)\left( {1 + 2x} \right)dx}.$
### Example 3
Find the indefinite integral $\int {\left( {\frac{1}{{{x^2}}} - \frac{1}{{{x^3}}}} \right)dx}.$
### Example 4
Calculate $\int {\left( {\sqrt x + \sqrt[3]{x}} \right)dx}.$
### Example 5
Find the indefinite integral $\int {\frac{{x + 1}}{{\sqrt x }} dx}.$
### Example 6
Find the indefinite integral $\int {{{\left( {x + \sqrt x } \right)}^2}dx}.$
### Example 7
Calculate the integral $\int {\left( {\frac{3}{{\sqrt[3]{x}}} + \frac{2}{{\sqrt x }}} \right)dx}.$
### Example 8
Find the indefinite integral $\int {\left( {\sqrt[3]{x} + {e^3}} \right)dx}.$
### Example 1.
Evaluate the indefinite integral $\int {\left( {3{x^2} - 6x + 2\cos x} \right)dx} .$
Solution.
Applying the properties $$1$$ and $$2,$$ we have
$I = \int {\left( {3{x^2} - 6x + 2\cos x} \right)dx} = \int {3{x^2}dx} - \int {6xdx} + \int {2\cos xdx} = 3{\int {{x^2}dx}} - 6{\int {xdx}} + 2{\int {\cos xdx}} .$
All three integrals can be evaluated using the integration table. This yields:
$I = 3 \cdot {\frac{{{x^3}}}{3}} - 6 \cdot {\frac{{{x^2}}}{2}} + 2 \cdot {\sin x} + C = {x^3} - 3{x^2} + 2\sin x + C.$
### Example 2.
Find the indefinite integral $\int {\left( {1 + x} \right)\left( {1 + 2x} \right)dx}.$
Solution.
We can simplify the integrand:
$\left( {1 + x} \right)\left( {1 + 2x} \right) = 1 + x + 2x + 2{x^2} = 2{x^2} + 3x + 1.$
Then the integral is given by
$\int {\left( {1 + x} \right)\left( {1 + 2x} \right)dx} = \int {\left( {2{x^2} + 3x + 1} \right)dx} = \int {2{x^2}dx} + \int {3xdx} + \int {1dx} = 2\int {{x^2}dx} + 3\int {xdx} + \int {dx} = 2 \cdot \frac{{{x^3}}}{3} + 3 \cdot \frac{{{x^2}}}{2} + x + C = \frac{{2{x^3}}}{3} + \frac{{3{x^2}}}{2} + x + C.$
### Example 3.
Find the indefinite integral $\int {\left( {\frac{1}{{{x^2}}} - \frac{1}{{{x^3}}}} \right)dx}.$
Solution.
By the sum rule,
$I = \int {\left( {\frac{1}{{{x^2}}} - \frac{1}{{{x^3}}}} \right)dx} = \int {\frac{{dx}}{{{x^2}}}} - \int {\frac{{dx}}{{{x^3}}}} .$
The integrands in both integrals are power functions, so we have
$I = \int {{x^{ - 2}}dx} - \int {{x^{ - 3}}dx} = \frac{{{x^{ - 1}}}}{{\left( { - 1} \right)}} - \frac{{{x^{ - 2}}}}{{\left( { - 2} \right)}} + C = - \frac{1}{x} + \frac{1}{{2{x^2}}} + C.$
### Example 4.
Calculate $\int {\left( {\sqrt x + \sqrt[3]{x}} \right)dx}.$
Solution.
$\int {\left( {\sqrt x + \sqrt[3]{x}} \right)dx} = \int {\sqrt x dx} + \int {\sqrt[3]{x}dx} = \int {{x^{\frac{1}{2}}}dx} + \int {{x^{\frac{1}{3}}}dx} = \frac{{{x^{\frac{1}{2} + 1}}}}{{\frac{1}{2} + 1}} + \frac{{{x^{\frac{1}{3} + 1}}}}{{\frac{1}{3} + 1}} + C = \frac{{2{x^{\frac{3}{2}}}}}{3} + \frac{{3{x^{\frac{4}{3}}}}}{4} = \frac{{2\sqrt {{x^3}} }}{3} + \frac{{3\sqrt[3]{{{x^4}}}}}{4} + C.$
### Example 5.
Find the indefinite integral $\int {\frac{{x + 1}}{{\sqrt x }} dx}.$
Solution.
We write the integrals as the sum of two integrals and calculate them separately:
$\int {\frac{{x + 1}}{{\sqrt x }}dx} = \int {\left( {\frac{x}{{\sqrt x }} + \frac{1}{{\sqrt x }}} \right)dx} = \int {\left( {\sqrt x + \frac{1}{{\sqrt x }}} \right)dx} = \int {\sqrt x dx} + \int {\frac{{dx}}{{\sqrt x }}} = \frac{{{x^{\frac{3}{2}}}}}{{\frac{3}{2}}} + 2\sqrt x + C = \frac{{2\sqrt {{x^3}} }}{3} + 2\sqrt x + C.$
### Example 6.
Find the indefinite integral $\int {{{\left( {x + \sqrt x } \right)}^2}dx}.$
Solution.
Expand the square in the integrand:
$I = \int {{{\left( {x + \sqrt x } \right)}^2}dx} = \int {\left( {{x^2} + 2x\sqrt x + {{\left( {\sqrt x } \right)}^2}} \right)dx} = \int {\left( {{x^2} + 2{x^{\frac{3}{2}}} + x} \right)dx} .$
Using the basic properties of integrals, we have
$I = \int {\left( {{x^2} + 2{x^{\frac{3}{2}}} + x} \right)dx} = \int {{x^2}dx} + 2\int {{x^{\frac{3}{2}}}dx} + \int {xdx} .$
The last expression contains only table integrals. Then
$I = \frac{{{x^3}}}{3} + 2 \cdot \frac{{{x^{\frac{5}{2}}}}}{{\frac{5}{2}}} + \frac{{{x^2}}}{2} + C = \frac{{{x^3}}}{3} + \frac{{4{x^{\frac{5}{2}}}}}{5} + \frac{{{x^2}}}{2} + C = \frac{{{x^3}}}{3} + \frac{{4\sqrt {{x^5}} }}{5} + \frac{{{x^2}}}{2} + C.$
### Example 7.
Calculate the integral $\int {\left( {\frac{3}{{\sqrt[3]{x}}} + \frac{2}{{\sqrt x }}} \right)dx}.$
Solution.
Using the power rule for integrals, we have
$\int {\left( {\frac{3}{{\sqrt[3]{x}}} + \frac{2}{{\sqrt x }}} \right)dx} = \int {\frac{{3dx}}{{\sqrt[3]{x}}}} + \int {\frac{{2dx}}{{\sqrt x }}} = 3\int {{x^{ - \frac{1}{3}}}dx} + 2\int {{x^{ - \frac{1}{2}}}dx} = 3 \cdot \frac{{{x^{ - \frac{1}{3} + 1}}}}{{ - \frac{1}{3} + 1}} + 2 \cdot \frac{{{x^{ - \frac{1}{2} + 1}}}}{{ - \frac{1}{2} + 1}} + C = \frac{{9{x^{\frac{2}{3}}}}}{2} + 4{x^{\frac{1}{2}}} + C = \frac{{9\sqrt[3]{{{x^2}}}}}{2} + 4\sqrt x + C.$
### Example 8.
Find the indefinite integral $\int {\left( {\sqrt[3]{x} + {e^3}} \right)dx}.$
Solution.
Using the basic properties of the integrals, we can write:
$I = \int {\left( {\sqrt[3]{x} + {e^3}} \right)dx} = \int {\left( {{x^{\frac{1}{3}}} + {e^3}} \right)dx} = \int {{x^{\frac{1}{3}}}dx} + \int {{e^3}dx} = \int {{x^{\frac{1}{3}}}dx} + {e^3}\int {dx} .$
As you can see, we have table integrals. In the second integral, $${e^3}$$ is a constant, so it can be pulled through the integral sign. This yields:
$I = \int {{x^{\frac{1}{3}}}dx} + {e^3}\int {dx} = \frac{{{x^{\frac{4}{3}}}}}{{\frac{4}{3}}} + {e^3}x + C = \frac{{3\sqrt[3]{{{x^4}}}}}{4} + {e^3}x + C.$
See more problems on Page 2.
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2022-05-19 04:56:54
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https://cyrrion.com/article-10-1007-s10800-014-0665-3/
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Kiến thức
# Heat generated during electrochemical double-layer capacitor “self-discharge”
• Short Communication
• Open Access
• Published: 24 January 2014
# Heat generated during electrochemical double-layer capacitor “self-discharge”
Journal of Applied Electrochemistry
volume 44pages 551–554 (2014)
• 2279 Accesses
• 2 Citations
• Metrics details
## Abstract
Eight commercial 10F electrochemical double-layer capacitors (EDLCs) were connected together and placed in a container filled with mineral oil. The whole system was placed into a Dewar container. Temperature variation and heat exchanged between the test EDLC and the environment during its charging, discharging, and “self-discharge” were measured, together with voltage U changes. Charge separation during charging was equivalent to a transition into a more ordered system, which results in entropy decrease, while discharging caused entropy increase (the Peltier–Seebeck effect). Consequently, a number of charging/discharging cycles led to a corresponding series of entropy and temperature changes. The final shape of temperature versus time curve during charging/discharging cycles was due to overlapping of irreversible Joule–Lenz and reversible Peltier heats. When charged EDLC was kept under the open-circuit condition, measured heat flow was negligible in comparison to energy loss calculated from potential drop, assuming that energy E accumulated is proportional at any time to voltage to the second power (i.e., E ~ U 2). The result was interpreted assuming that the EDLC “self-discharge” phenomenon is not associated with energy loss by the device, but rather with charge redistribution between EDLC particles characterized by different time constants.
## Introduction
The energy E accumulated by an electrochemical double-layer capacitor (EDLC) is usually calculated according to Eq. (
1
) valid for dielectric and electrolytic capacitors:
$$E = frac{1}{2}CU^{2} ,$$
(1)
where C is the capacitance and U is potential difference between electrodes (voltage). A spontaneous voltage drop between EDLC electrodes, when it is kept under the open-circuit condition, is commonly called “self-discharge” and interpreted, according to Eq. (
1
), as a relatively fast energy loss [
1
,
2
]. The law of energy preservation suggests that the voltage drop can generate energy exchange with the environment. Recent papers describe a model of charge redistribution in porous electrodes as the reason for EDLC voltage changes [
3
5
]. The model does not assume energy dissipation, in contrast to leakage current and faradaic mechanisms. In addition, it has been recently shown that during EDLC operation, equations valid for dielectric and electrolytic capacitors do not hold in the case of EDLCs [
6
,
7
]. Therefore, the assumption that the energetic state of the EDLC is proportional at any time to the voltage to the second power may not be valid. All these indicate that voltage changes called “self-discharge” may not reflect the energetic state of the device. If voltage drop is due to energy loss, then it can be converted inside EDLC into reagents free energy (faradaic mechanism) or exchanged with the environment in the form of work, electromagnetic radiation, or heat. To our knowledge, there are no reports about the contribution of faradaic processes or electromagnetic radiation to EDLC fast voltage drop. Moreover, EDLCs cannot work under the open-circuit condition. Consequently, heat exchange seems to be the main process responsible for the “self-discharge” understood as energy loss. Heat generation during EDLC charging and discharging was observed [
8
], while the corresponding effect during its storage has not been reported under open-circuit conditions. The general aim of this study was to measure experimentally the level of heat exchange between the environment and an EDLC kept under the open-circuit condition, in comparison to that assumed from the “self-discharge” curve (according to the Eq. (
1
)).
## Experimental
10F commercial capacitors (PC10) were obtained from Maxwell Technologies. Each cell dimension was 29.6 mm × 23.6 mm × 3.5 mm, surface area: 17.70 cm2 with a mass of 6.6 g. Two sets of four PC10 EDLCs combined in parallel (with capacitance of 40F) were connected in a series (eight PC10 devices, with total nominal capacitance of 20 F). The total surface and mass of eight EDLCs were 141.60 cm2 and 52.8 g. Ultracapacitors were placed in a polypropylene container filled with 80.0 g of mineral oil (Finavestan A360B, Total Oil Australia Ltd.). High-resistivity Kanthal-D wire (Kanthal, Fe–Cr–Al alloy, 25.3 cm long, diameter 0.16 mm, resistance 67.1 Ω m−1), was placed in the system (132.8 g of eight capacitors and oil). Temperature was measured with an NTC chip TT2-10KC3-10 thermistor (Tewa, Poland). In the narrow temperature range of 293–298 K, the R = f(T) characteristic was linear with the temperature coefficient of 499 Ω K−1. The whole system was placed into a Dewar container. Charging of capacitors and calibration of the system with the resistive wire was performed with the Atlas-Sollich 0461MBI electrochemical system. Resistance of the thermistor was measured with an ME-32 universal meter (Metex, Korea) with an accuracy of 10 Ω (which is equivalent to ±0.02 K). Impedance spectrum was taken with the use of a G750 frequency response analyzer (Gamry, USA) at a frequency range of 100 kHz–10 mHz, at the open-circuit potential and amplitude of 10 mV.
## Results and discussion
Heat capacity of the system (capacitors, mineral oil, the thermistor, resistive wire, and the container) was determined from the calibration curve shown in Fig.
1
. A current I = 0.12 A flowing through the wire of resistance R = 17.14 Ω during time of t = 600 s generated the Joule–Lenz heat of Q cal = RI 2t = 148.1 J. The calibration procedure was performed three times. Heat capacity of the system detected from the slope of calibration curves was c p = 1.43 ± 0.05 J g−1 K−1. Then the EDLC was charged from 0 to 5 V with a current of 2 A (within ca. 47 s) hence, accumulated energy, calculated from Eq. (
1
), was 250 J. Heat generated during the EDLC charging and discharging was not linear but had a shape shown in Fig.
2
a. This is due to the fact that electric charge separation (charging) is equivalent to a transition into a more ordered system, which results in entropy decrease, while discharging causes entropy increase (the Peltier–Seebeck effect, Fig.
2
b). Consequently, a number of charging/discharging cycles lead to a corresponding series of entropy and temperature changes. The final shape of the T = f(t) curve, shown in Fig.
2
a, is due to overlapping of irreversible Joule–Lenz and reversible Peltier heats. The temperature recorded at t = 46 s was T = 296.48 K, while the corresponding value for t = 425 s was T = 299.44 K. Therefore, the temperature change with time was ΔTt = 0.36 K 47 s−1. The Joule–Lenz heat produced during one charging or discharging cycle (47 s) calculated from series resistance R s = 0.080 Ω (measured with EIS, Fig.
3
) was ca. ( Q_{{{text{R}}_{text{s}} }} ) = R sI 2t = 0.08 Ω × 4 A2 × 47 s = 15.04 J (P = 0.32 W). The corresponding value estimated from temperature changes (Fig.
2
) was Q J–L = mc p ΔT = 132.8 g × 1.43 J g−1 K−1 × 0.36 K = 68.4 J (P J–L = 1.46 W). The difference in both heats (or Joule–Lenz power) is due to the fact that, in addition to the series resistance, the corresponding resistance of the electrolyte in the pores should also be taken into account [
7
]. The Peltier heat may be calculated from the heat difference between discharging (( Q_{text{Pelt}}^{text{ch}} )) and charging (( Q_{text{Pelt}}^{text{disch}} )) cycles:
$$Q^{text{ch}} = Q_{text{J {-} L}} + Q_{text{Pelt}} quad {text{and}}quad Q^{text{disch}} = Q_{text{J {-} L}} – Q_{text{Pelt}} .$$
(2)
Slopes of the discharging and charging curves shown in Fig.
2
b were −0.0016 and 0.0171 K s−1, respectively. This leads to the Peltier heat (referred to each 47 s charging or discharging step) of Q Pelt = 83.50 J (P Pelt = 1.78 W).
Figure
4
shows a typical U = f(t) “self-discharge” curve of the EDLC under study after switching it off into the open-circuit condition after galvanostatic charging. At the beginning, U decreases from its initial value ( U_{text{o}}^{2} ) relatively fast and slows down with time. At the same time, temperature changes were measured with the thermistor. Measurements of potential and temperature changes were performed 10 times. Heat flow detected from temperature changes was calculated as Q flow = mc p ΔT. This value may be compared to the potential heat calculated from Eq. (
1
), assuming that during the potential decay, energy E is exchanged exclusively in the form of heat. In other words, temperature and voltage changes were converted to the same (energy) units. If the temperature change reflected energy loss due to voltage drop, then both c pΔT = f(t) and 1/2C(U 2( U_{text{o}}^{2} )) = f(t) values should be present at the same curve. However, experimental data (an example shown in Fig.
5
) did not support this assumption. It can be seen in Fig.
5
that energy measured from temperature changes during the “self-discharge” was approximately constant (ca.: 4 J during 180 s ≈ 1 J during 47 s). However, the corresponding value calculated from voltage changes via Eq. (
1
) was much higher (ca. 13 J during 180 s ≈ 3.39 J during 47 s). Such a result was obtained in all 10 experiments (different charging current and final voltage). Therefore, the spontaneous voltage drop between EDLC electrodes, when it is kept under the open-circuit condition, commonly called the “self-discharge,” cannot be interpreted as a relatively fast energy. However, the model of charge redistribution between EDLC particles characterized by different time constants can be applied to explain this phenomenon.
## Conclusions
When an EDLC is kept under the open-circuit condition, measured heat flow is negligible in comparison to energy loss calculated from potential drop, assuming that energy accumulated is proportional at any time to voltage to the second power. The result suggests that the EDLC “self-discharge” phenomenon is not associated with energy loss by the device.
## References
1. 1.
Diab Y, Venet P, Gualous H, Rojat G (2009) Self-discharge characterization and modeling of electrochemical capacitor used for power electronics applications. IEEE Trans Power Electron 24:510–517
Article
2. 2.
Wang G, Zhang L, Zhang J (2012) A review of electrode materials for electrochemical supercapacitors. Chem Soc Rev 41:797–828
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3. 3.
Black J, Andreas HA (2009) Effects of charge redistribution on self-discharge of electrochemical capacitors. Electrochim Acta 54:3568–3574
Article
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4. 4.
Kaus M, Kowal J, Sauer DU (2010) Modelling the effects of charge redistribution during self-discharge of supercapacitors. Electrochim Acta 55:7516–7523
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5. 5.
Kowal J, Avaroglu E, Chamekh F, Senfelds A, Thien T, Wijaya D, Sauer DU (2011) Detailed analysis of the self-discharge of supercapacitors. J Power Sources 196:573–579
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6. 6.
Lewandowski A, Jakobczyk P, Galinski M (2012) Capacitance of electrochemical double layer capacitors. Electrochim Acta 86:225–231
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7. 7.
Lewandowski A, Jakobczyk P, Galinski M, Biegun M (2013) Self-discharge of electrochemical double layer capacitors. Phys Chem Chem Phys 15:8692–8699
Article
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8. 8.
Shiffer J, Linzen D, Sauer DU (2006) Heat generation in double layer capacitors. J Power Sources 160:765–772
Article
CAS
## Acknowledgments
Support of Grant 31-254/13 DS PB is gratefully acknowledged.
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## Author information
Authors
### Corresponding author
Correspondence to
Andrzej Lewandowski
.
## Rights and permissions
Reprints and Permissions
Lewandowski, A., Jakobczyk, P., Gnat, M. et al. Heat generated during electrochemical double-layer capacitor “self-discharge”. J Appl Electrochem 44, 551–554 (2014). https://doi.org/10.1007/s10800-014-0665-3
• Accepted: 10 January 2014
• Published: 24 January 2014
• Issue Date: May 2014
• DOI:
https://doi.org/10.1007/s10800-014-0665-3
### Keywords
• Electrochemical double-layer capacitor
• Self-discharge
• Heat exchange
Chuyên mục: Kiến thức
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2021-10-15 19:40:52
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https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_10A_Problems/Problem_23&diff=prev&oldid=28744
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# Difference between revisions of "2005 AMC 10A Problems/Problem 23"
## Problem
Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$. Let $D$ and $E$ be points on the circle such that $DC \perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$?
$\mathrm{(A) \ } \frac{1}{6}\qquad \mathrm{(B) \ } \frac{1}{4}\qquad \mathrm{(C) \ } \frac{1}{3}\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \frac{2}{3}$. The area of
## Solution
$AC$ is $\frac{1}{3}$ of diameter and $CO$ is $\frac{1}{2}$ - $\frac{1}{3}$ = $\frac{1}{6}$.
$OD$ is the radius of the circle, so using the Pythagorean theorem height $CD$ of $\triangle ADB$ is $\sqrt{(\frac{1}{2})^2-(\frac{1}{6})^2$ (Error compiling LaTeX. ! Missing } inserted.) = $\frac{\sqrt{2}}{3}$.
Area of the $\triangle DCO$ is $\frac{1}{2}\cdot\frac{1}{6}\cdot\frac{\sqrt{2}}{3}$ = $\frac{\sqrt{2}}{36}$.
The height of $\triangle DCE$ can be found using the area of $\triangle DCO$ and $DO$ as base. Hence the height of $\triangle DCE$ is $\frac{\frac{\sqrt{2}}{36}}{\frac{1}{2}\cdot\frac{1}{2}}$ = $\frac{\sqrt{2}}{9}$. The diameter is the base for both the triangles $\triangle DCE$ and $\triangle ABD$. Hence, the ratio of area of $\triangle DCE$ to the area of $\triangle ABD$ is $\frac{\frac{\sqrt{2}}{36}}{\frac{\sqrt{2}}{9}}$ is $\frac{1}{3} \Rightarrow C$
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2021-04-22 03:37:04
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http://openmx.ssri.psu.edu/thread/4189
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# Error in runHelper NLOPT unrecognized error -1
17 posts / 0 new
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Joined: 10/29/2015 - 17:22
Error in runHelper NLOPT unrecognized error -1
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While running simulations with the development version of OpenMx, I ran into an interesting error:
Error in runHelper(model, frontendStart, intervals, silent, suppressWarnings, :
NLOPT unrecognized error -1; please report to developers
It is somewhat difficult to replicate and it took some time to put together the attached example. As you may see in the attached, I read in the same dataset and fit the same model 10 times, obtaining likelihood-based CIs each time for a single parameter. The error only shows up for some replications.
Thought you all should know!
In the meantime, I have no problem rolling back to a more stable version, provided that the boundAdj argument is available for mxCI (even if in this particular example it's not necessary)
Thank you!
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Joined: 05/24/2012 - 00:35
unrecognized error
Nice to see that you are trying boundAdj. This option is only available with the latest code.
I just pushed a similar simulation,
inst/models/enormous/wu-neale-2012-lgc.R
This is how I recommend you structure your simulations. Use set.seed(result[rx,'rep']) inside the replications loop to ensure that every replication can be replicated. That way you can point to a specific replication that triggers NLOPT unrecognized error.
As you suspected, NLOPT unrecognized error -1 is only reported by SLSQP. You will not see this error with other optimizers. However, if you are interested in the boundary adjusted CIs then you will need to use SLSQP.
Hm, looking at your script, I think you need to add lbound=0 for the latent factor covariance. But wait, this parameter cannot be less than 0 so the boundary adjustment is not needed. I will email you my draft manuscript to explain better. I suspect you can avoid optimizer failures by estimating the cholesky factor of the correlation matrix. inst/models/passing/bivCorM.R is an example of how to do this.
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Joined: 05/24/2012 - 00:35
negative covariance
Oh, perhaps the covariance can be negative so the boundary adjustment is needed.
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Joined: 10/29/2015 - 17:22
I see, thank you! I put this
I see, thank you!
I put this together quickly, and so didn't notice that perhaps I could also put bounds on the factor correlations (-1 to 1 or some such if that's possible).
So, this is a known problem for SLSQP? Do we know how frequently (and under what conditions) this occurs? and why?
I also noticed that the OpenMx version number seems odd to me, even though I got the latest version from GitHub and compiled from source yesterday in order to put this together: 2.6.8.198 [GIT v2.6.8-198-gacf5469]
I thought the most stable release was 2.6.9?
I will have a follow-up question regarding implementation of the underlying algorithm for likelihood-based CIs - in particular whether Wu & Neale is always used (as opposed to the Neale & Miller algorithm), and where to poke around in the source code for its actual implementation. For the most part, I can "trick" OpenMx into giving me what I want for a recent manuscript, but my manual implementation of Wu & Neale fails for trivial cases even if I use Pek & Wu code as a model. But, I still have to debug to ensure I haven't done something incorrectly. Perhaps that issue is best placed in a separate thread anyway.
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Joined: 05/24/2012 - 00:35
Regarding the bounds on
Regarding the bounds on factor correlations, note that Wu & Neale will only be attempted if there is a lower bound or an upper bound, but not both.
Regarding the version, 2.6.8 and 2.6.9 are almost exactly the same. It outputs the incorrect version because I forgot to rebase on top of 2.6.9 when we did the release. 2.6.8.X is the latest.
"So, this is a known problem for SLSQP? Do we know how frequently (and under what conditions) this occurs? and why?" -- Yes, it is a known problem. It seems to occur when the parameter vector is close to boundary of the feasible set.
The implementation is in ComputeGD.cpp starting around line 310. You can get details about which algorithm was used by using summary(model, verbose=TRUE) after the CIs are run. I provide an example in the draft manuscript I sent over.
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Joined: 10/29/2015 - 17:22
Wait, so if one asks for CIs,
Wait, so if one asks for CIs, using mxCI for example, and there are no actual boundaries on the parameter(s) of interest, Neale & Miller is used? Wu and Neale is not the default?
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Joined: 05/24/2012 - 00:35
default
Yes. Also, the algorithm used is reported in summary(model, verbose=TRUE)
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Joined: 09/07/2010 - 17:14
Is that the case?
Hi,
I think OpenMx has moved from Neale and Miller (1994) to Wu and Neale (2012). By Wu and Neale (2012) I don't mean the algorithm for a bounded parameter, but the algorithm for a parameter without any bound. The help file for the function mxComputeConfidenceInterval cites Pek and Wu (2015) rather than Wu and Neale (2012). The help file for the function mxCI says "The likelihood-based confidence intervals returned using MxCI are obtained by increasing or decreasing the value of each parameter until the -2 log likelihood of the model increases by an amount corresponding to the requested interval", which is a description of Wu and Neale algorithm.
To be specific, when looking for the CI for a parameter "a" without bounds, does the program:
1) minimize (wrt "a") the function:
F_L= (F+3.84- F_a)^2+a, where F is the -2F_{ML}, and
F_U= (F+3.84- F_a)^2-a,
or
2) maximize and minimize the parameter "a" subject to the constraint that
F_a-F <3.84 ?
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Joined: 05/24/2012 - 00:35
constraint type
You're asking whether OpenMx uses an equality or inequality constraint? It depends on the optimizer. SLSQP prefers an inequality constraint. The other optimizers use a penalty function that is added to the fit.
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Joined: 09/07/2010 - 17:14
Did you mean...
Hi, thanks for attending to my question. I appreciate it a lot.
My question concerns the two different algorithms that could be used for calculating a CI for a parameter without any artificial bound. They do differ in whether an inequality constraint is imposed, but they are fundamentally different algorithms and minimize different functions. Algorithm 1 (#1 in my last post) was proposed by Neale and Miller (1994), it minimizes the function
(F+3.84- F_a)^2 +/- a
This algorithm does not find the CI as defined by profile likelihood CI. This was acknowledged in their paper.
Algorithm 2 (#2 in my last post) was proposed by Wu and Neale and later reviewed by Pek and Wu. Although in that paper we tackled parameters with bounds, but the algorithm was proposed for parameters with no bound. It maximize and minimize the parameter "a" subject to the constraint that F_a-F <3.84. This gives the profile likelihood CI as it is defined.
When you said "SLSQP prefers an inequality constraint", did you mean Algorithm 2 (b/c algorithm 1 has no constraint at all)? When you said "the other optimizers use a penalty function that is added to the fit", did you mean Algorithm 1?
By the way, SLSQP is the default optimizer. Am I right?
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Joined: 05/24/2012 - 00:35
algortihms
As you have described them, algorithm 1 & 2 both arrive at the same MLE. If one gives the profile likelihood CI then the other one does too.
> When you said "SLSQP prefers an inequality constraint", did you mean Algorithm 2 (b/c algorithm 1 has no constraint at all)? When you said "the other optimizers use a penalty function that is added to the fit", did you mean Algorithm 1?
This is clarified in Pritikin, Rappaport, & Neale (in-press). See Equations 1-4.
> By the way, SLSQP is the default optimizer. Am I right?
It was, but CSOLNP is the default in OpenMx v2.7.4
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Joined: 09/07/2010 - 17:14
Thanks
Thanks for clarification. But the two algorithms technically do not give the same result. Neale and Miller (1994) commented right below their Eq. 1 that their algorithm has in a bias (the last term of Eq. 1). Wu and Neale algorithm does not have this problem.
May I know why SLSQP is no longer the default?
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Joined: 05/24/2012 - 00:35
constraints
> But the two algorithms technically do not give the same result. Neale and Miller (1994) commented right below their Eq. 1 that their algorithm has in a bias (the last term of Eq. 1).
I think it is not the form of the constraint that matters but whether the constraint is encoded as a penalty function (which will have bias) or as a separate constraint (to a constraint aware optimizer).
> May I know why SLSQP is no longer the default?
Some believe that CSOLNP performs better in most circumstances.
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Joined: 09/07/2010 - 17:14
You are absolutely right.
You are absolutely right. With CSOLNP, would the CI be found with the penalty function or as a separate constraint?
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Joined: 01/24/2014 - 12:15
penalty
CSOLNP and NPSOL both by default use the "penalty" formulation (I believe--Joshua will correct me if I'm wrong). Only SLSQP by default uses the "separate constraint" formulation, since it is unambiguously the best of the three gradient-descent optimizers at handling nonlinear constraints.
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Joined: 09/07/2010 - 17:14
Thanks
Thank you to you and Josh for picking up this discussion. I appreciate it.
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Joined: 09/07/2010 - 17:14
I revised your code and it works. See below. The key is to set bounds of parameters to restrict the search of the CI algorithm. Note that the bounds I mentioned here has nothing to do with bounds of the parameters discussed in Wu and Neale (2012). Their introduction is just to make the search successful.
The reason that your version did not work out is because you did not specify appropriate bounds for the parameters. The search for CI is very delicate and prone to failure. You need to bound the search within some reasonable range. In your model, the parameter cov is now bound between -0.4 and 0.9; two loadings are bound to be positive to avoid sign change. As long as the search does not touch the artificial bounds, it is OK. If one of them is touched, set a slightly more lenient bound.
Here is the revised code.
for (i in 1:100)
{
print(i)
colnames(dat)<-paste("X",1:8,sep="")
data <- mxData( observed=dat, type="raw" )
resVars <- mxPath( from=paste("X",1:8,sep=""), arrows=2,
free=TRUE, values=rep(1,8), labels=paste("e",1:8,sep="") )
latVars <- mxPath( from=c("F1","F2"), arrows=2,connect="unique.pairs",
free=c(FALSE,TRUE,FALSE), values=c(1,0,1),
labels =c("varF1","cov","varF2") )
facLoads1 <- mxPath( from="F1", to=paste("X",1:4,sep=""), arrows=1,
free=rep(TRUE,4), values=rep(1,4),labels =paste("l",1:4,sep="") )
facLoads2 <- mxPath( from="F2", to=paste("X",5:8,sep=""), arrows=1,
free=rep(TRUE,4), values=rep(1,4),
labels =paste("l",5:8,sep="") )
means <- mxPath( from="one", to=c(paste("X",1:8,sep=""),"F1","F2"), arrows=1,
free=c(rep(TRUE,8),FALSE,FALSE), values=c(rep(1,8),0,0),
labels =c(paste("meanx",1:8,sep=""),NA,NA) )
mxpindx<-"cov"
CIModel <- mxModel("CIModel", type="RAM",
manifestVars=paste("X",1:8,sep=""), latentVars=c("F1","F2"),
}
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2017-06-28 19:10:06
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https://www.vedantu.com/physics/dielectric-constant
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Dielectric Constant
What Is Dielectric Constant?
Many materials possess a tremendous property to hold their electrical charge for long intervals and in large quantities as well. Such property of materials refers to dielectric property. Many students get confused with the term dielectric constant, property, and often ask what is relative permittivity. This guide is beneficial to understand the meaning and factors affecting the dielectric constant.
What are Dielectric Materials?
Dielectric materials have weak electrical conductivity but possess the ability to store an electrical charge. There are several dielectric materials, including vacuum, air, and more. The values of dielectric constants of some dielectric materials include:
• For air- 1.00059
• For glass- 3.8-14.5
• For paper- 3.6
• For vacuum- 1.00
• For PVC- 4.0
What Is Meant By Dielectric Constant?
The dielectric constant of any substance refers to the relative permittivity of the dielectric substance. It is the proportion of the permittivity of the material to the permittivity of the free space. Mathematically, the dielectric constant can be expressed as:
K = $\frac{\epsilon }{\epsilon_{0} }$
where K refers to Dielectric constant
$\epsilon$ refers to the permittivity of the substance
$\epsilon_{0}$ is defined as the permittivity of the free space
Dielectric Constant Theory
Many students ask what the meaning of dielectric constant is. In order to understand the meaning of dielectric constant, it’s crucial to first understand the theory behind it. Dielectric constant serves as the major factor required to describe a capacitor. A capacitor is an electronic device built by inserting a dielectric insulating plate in-between the metal conducting plates. It’s the layer made from a dielectric material that decides if a capacitor can store a high charge or not. That’s why it is essential to choose the best dielectric material depending on the dielectric property.
From the dielectric constant formula:
K = $\frac{\epsilon }{\epsilon_{0} }$, we get the value of relative permittivity of free space is always greater than or equal to relative permittivity of substance. Hence, the value of the dielectric constant (K) is always either equal to greater than 1.
What does a High Dielectric Constant Mean?
The high value of the dielectric constant means the value of capacitance can be maximised. It can be seen from the capacitance formula in the parallel plate capacitor:
C = Kϵ0 A/d
where K refers to the dielectric constant
C refers to the capacitance of a parallel plate capacitor.
A refers to the area of parallel conducting plates
$\epsilon_{0}$ is defined as permittivity of free space
d refers to the separation between parallel conducting plates
Hence, from this formula, it’s clear that there are two ways to increase the value of capacitance. The first is to decrease the separation between parallel conducting plates, and the second is to enhance the dielectric constant value.
What Do You Mean by Dielectric Constant Equation?
According to the dielectric constant equation:
K = $\frac{\epsilon }{\epsilon_{0} }$
K is the ratio of two entities with the same dimension. Hence, the dielectric constant is a unitless and dimensionless quantity.
What Factors Affect the Value of the Dielectric Constant?
After knowing what is dielectric constant, there come different factors that affect the dielectric constant value such as:
• Temperature: The arrangement of molecules in the dielectric material is tough at the low temperature. However, with the increase in temperature, the dipole moment increases, and hence there is a rise in the dielectric constant value. The temperature at which the dielectric constant starts increasing refers to the transition temperature. Moreover, if the temperature goes above the transition temperature, the dielectric constant will start decreasing.
• Heating Effect: The students asking what is dielectric property can understand from the fact that on heating any dielectric material, there is a dielectric loss. It is due to the dielectric property of the material due to which whenever there is any movement of molecules inside the material, there is the dissipation of energy. When the dielectric material absorbs electrical energy, it dissipates energy in the form of heat.
• Applied Voltage: The value of the dielectric constant decreases in the presence of a direct current voltage. However, the dielectric constant value increases when an alternating current voltage is applied.
• Frequency: The frequency of the applied voltage serves as one of the critical factors that affect the dielectric constant to a great extent. Whenever there is an increase in the frequency of the external voltage applied, the dielectric constant value becomes non-linear.
• Humidity and Moisture: When there is an increase in humidity or moisture, there is a decrease in the strength of dielectric material.
The Coulomb force present between the two-point charges in a material is affected by material property; this process is termed as relative permittivity of a material. The factor relative permittivity is used to decrease the charges present between the electric field relative to the vacuum.
While comparing the relative permittivity of two similar capacitors, the relative permittivity can be easily defined as the ratio between the capacitance of the capacitor of a material to the capacitance of that of a vacuum.
Relative permittivity is widely known as dielectric constant also.
The historic name for the relative permittivity is the dielectric constant. Because of its ambiguity, some older records used it widely as absolute permittivity instead of the term dielectric constant. This term is deprecated by the standard organizations. The permittivity is considered as either a static property or a frequency-dependent variant.
Applications
Relative permittivity has a huge number of applications too. Some of the major applications which are common are described in detail below:
Energy:
Relative permittivity in the case of energy is used as an essential piece of information that is further used in designing Capacitors. The following material might be expected to introduce capacitance into a circuit. The materials which are having high relative permittivity when placed in an electric field reduces the magnitude of that particular field within the volume of the dielectric. The following process which is described above is used to increase the capacitance of a capacitor design. In printed circuit boards (PCBs) there exists a layer beneath the attached conductor's which act as a dielectric.
Communication:
These are majorly used in RF transmission lines. Polyethene is highly used in between the centre conductor and outside shield, in the case of a coaxial cable. When placed inside waveguides which are also used to form filters. The best example of dielectric waveguides is optical fibres. This waveguide consists of doped impurities which further helps in order to control the precise value of relative permittivity within the cross-sections. Being an appropriate factor to control the refractive index of the material, these are highly used in optical modes of transmission. However, technically the relative permittivity matters because they are not operated in the electrostatic limit while considering the above-mentioned case.
Environment:
Relative permittivity plays a vital role in changes in environmental factors. The air is highly affected by the changes in temperature, humidity and barometric pressure. Due to the changes in the relative permittivity, changes in the capacitance occur. These changes can be measured through sensors. As barometric pressure is always fairly stable, most of the changes occurring in the environment are due to the effect of temperature and humidity only.
Taking into consideration capacitance and the measured temperature along with using the engineering formulas the relative humidity can be obtained.
Chemistry:
The measure of the chemical polarity is the relative static permittivity of a solvent. In order to understand the above statement let us consider an example. As we all know that water is highly polar in nature the relative static permittivity of water is 18.10 at 200C. On the other hand, n-hexane is a nonpolar solvent with a relative permittivity of 1.89 at 200C. While dealing with analytical chemistry, this information is highly used in order to design separation, sample preparation and chromatography.
These are some of the major applications highly used on the basis of dielectric constant.
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2022-05-25 23:23:42
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https://homework.cpm.org/category/CC/textbook/ccg/chapter/11/lesson/11.1.1/problem/11-18
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### Home > CCG > Chapter 11 > Lesson 11.1.1 > Problem11-18
11-18.
For each geometric relationship below, determine whether $a$ or $b$ is larger, or if they are equal. Assume that the diagrams are not drawn to scale. If there is not enough information, explain what information is missing.
1. How does the central angle relate to the inscribed angle?
1. Find the measure of the last angle.
How do angles relate to the lengths of the sides?
Since the angle opposite side '$a$' is bigger, '$a$' must be larger than '$b$'.
1. Use the given information to find the areas.
Which is bigger?
$a = 15.6\text{ units}^2$
$b = 16\text{ units}^2$
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2021-06-18 22:04:12
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https://jeremykun.com/tag/binary-search/
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Binary Search on Graphs
Binary search is one of the most basic algorithms I know. Given a sorted list of comparable items and a target item being sought, binary search looks at the middle of the list, and compares it to the target. If the target is larger, we repeat on the smaller half of the list, and vice versa.
With each comparison the binary search algorithm cuts the search space in half. The result is a guarantee of no more than $\log(n)$ comparisons, for a total runtime of $O(\log n)$. Neat, efficient, useful.
There’s always another angle.
What if we tried to do binary search on a graph? Most graph search algorithms, like breadth- or depth-first search, take linear time, and they were invented by some pretty smart cookies. So if binary search on a graph is going to make any sense, it’ll have to use more information beyond what a normal search algorithm has access to.
For binary search on a list, it’s the fact that the list is sorted, and we can compare against the sought item to guide our search. But really, the key piece of information isn’t related to the comparability of the items. It’s that we can eliminate half of the search space at every step. The “compare against the target” step can be thought of a black box that replies to queries of the form, “Is this the thing I’m looking for?” with responses of the form, “Yes,” or, “No, but look over here instead.”
As long as the answers to your queries are sufficiently helpful, meaning they allow you to cut out large portions of your search space at each step, then you probably have a good algorithm on your hands. Indeed, there’s a natural model for graphs, defined in a 2015 paper of Emamjomeh-Zadeh, Kempe, and Singhal that goes as follows.
You’re given as input an undirected, weighted graph $G = (V,E)$, with weights $w_e$ for $e \in E$. You can see the entire graph, and you may ask questions of the form, “Is vertex $v$ the target?” Responses will be one of two things:
• Yes (you win!)
• No, but $e = (v, w)$ is an edge out of $v$ on a shortest path from $v$ to the true target.
Your goal is to find the target vertex with the minimum number of queries.
Obviously this only works if $G$ is connected, but slight variations of everything in this post work for disconnected graphs. (The same is not true in general for directed graphs)
When the graph is a line, this “reduces” to binary search in the sense that the same basic idea of binary search works: start in the middle of the graph, and the edge you get in response to a query will tell you in which half of the graph to continue.
And if we make this example only slightly more complicated, the generalization should become obvious:
Here, we again start at the “center vertex,” and the response to our query will eliminate one of the two halves. But then how should we pick the next vertex, now that we no longer have a linear order to rely on? It should be clear, choose the “center vertex” of whichever half we end up in. This choice can be formalized into a rule that works even when there’s not such obvious symmetry, and it turns out to always be the right choice.
Definition: median of a weighted graph $G$ with respect to a subset of vertices $S \subset V$ is a vertex $v \in V$ (not necessarily in $S$) which minimizes the sum of distances to vertices in $S$. More formally, it minimizes
$\Phi_S(v) = \sum_{u \in S} d(v, u)$,
where $d(u,v)$ is the sum of the edge weights along a shortest path from $v$ to $u$.
And so generalizing binary search to this query-model on a graph results in the following algorithm, which whittles down the search space by querying the median at every step.
Algorithm: Binary search on graphs. Input is a graph $G = (V,E)$.
• Start with a set of candidates $S = V$.
• While we haven’t found the target and $|S| > 1$:
• Query the median $v$ of $S$, and stop if you’ve found the target.
• Otherwise, let $e = (v, w)$ be the response edge, and compute the set of all vertices $x \in V$ for which $e$ is on a shortest path from $v$ to $x$. Call this set $T$.
• Replace $S$ with $S \cap T$.
• Output the only remaining vertex in $S$
Indeed, as we’ll see momentarily, a python implementation is about as simple. The meat of the work is in computing the median and the set $T$, both of which are slight variants of Dijkstra’s algorithm for computing shortest paths.
The theorem, which is straightforward and well written by Emamjomeh-Zadeh et al. (only about a half page on page 5), is that this algorithm requires only $O(\log(n))$ queries, just like binary search.
Before we dive into an implementation, there’s a catch. Even though we are guaranteed only $\log(n)$ many queries, because of our Dijkstra’s algorithm implementation, we’re definitely not going to get a logarithmic time algorithm. So in what situation would this be useful?
Here’s where we use the “theory” trick of making up a fanciful problem and only later finding applications for it (which, honestly, has been quite successful in computer science). In this scenario we’re treating the query mechanism as a black box. It’s natural to imagine that the queries are expensive, and a resource we want to optimize for. As an example the authors bring up in a followup paper, the graph might be the set of clusterings of a dataset, and the query involves a human looking at the data and responding that a cluster should be split, or that two clusters should be joined. Of course, for clustering the underlying graph is too large to process, so the median-finding algorithm needs to be implicit. But the essential point is clear: sometimes the query is the most expensive part of the algorithm.
Alright, now let’s implement it! The complete code is on Github as always.
Always be implementing
We start with a slight variation of Dijkstra’s algorithm. Here we’re given as input a single “starting” vertex, and we produce as output a list of all shortest paths from the start to all possible destination vertices.
from collections import defaultdict
from collections import namedtuple
Edge = namedtuple('Edge', ('source', 'target', 'weight'))
class Graph:
# A bare-bones implementation of a weighted, undirected graph
def __init__(self, vertices, edges=tuple()):
self.vertices = vertices
self.incident_edges = defaultdict(list)
for edge in edges:
edge[0],
edge[1],
1 if len(edge) == 2 else edge[2] # optional weight
)
self.incident_edges[u].append(Edge(u, v, weight))
self.incident_edges[v].append(Edge(v, u, weight))
def edge(self, u, v):
return [e for e in self.incident_edges[u] if e.target == v][0]
And then, since most of the work in Dijkstra’s algorithm is tracking information that you build up as you search the graph, we define the “output” data structure, a dictionary of edge weights paired with back-pointers for the discovered shortest paths.
class DijkstraOutput:
def __init__(self, graph, start):
self.start = start
self.graph = graph
# the smallest distance from the start to the destination v
self.distance_from_start = {v: math.inf for v in graph.vertices}
self.distance_from_start[start] = 0
# a list of predecessor edges for each destination
# to track a list of possibly many shortest paths
self.predecessor_edges = {v: [] for v in graph.vertices}
def found_shorter_path(self, vertex, edge, new_distance):
# update the solution with a newly found shorter path
self.distance_from_start[vertex] = new_distance
if new_distance < self.distance_from_start[vertex]:
self.predecessor_edges[vertex] = [edge]
else: # tie for multiple shortest paths
self.predecessor_edges[vertex].append(edge)
def path_to_destination_contains_edge(self, destination, edge):
predecessors = self.predecessor_edges[destination]
if edge in predecessors:
return True
return any(self.path_to_destination_contains_edge(e.source, edge)
for e in predecessors)
def sum_of_distances(self, subset=None):
subset = subset or self.graph.vertices
return sum(self.distance_from_start[x] for x in subset)
The actual Dijkstra algorithm then just does a “breadth-first” (priority-queue-guided) search through $G$, updating the metadata as it finds shorter paths.
def single_source_shortest_paths(graph, start):
'''
Compute the shortest paths and distances from the start vertex to all
possible destination vertices. Return an instance of DijkstraOutput.
'''
output = DijkstraOutput(graph, start)
visit_queue = [(0, start)]
while len(visit_queue) > 0:
priority, current = heapq.heappop(visit_queue)
for incident_edge in graph.incident_edges[current]:
v = incident_edge.target
weight = incident_edge.weight
distance_from_current = output.distance_from_start[current] + weight
if distance_from_current <= output.distance_from_start[v]:
output.found_shorter_path(v, incident_edge, distance_from_current)
heapq.heappush(visit_queue, (distance_from_current, v))
return output
Finally, we implement the median-finding and $T$-computing subroutines:
def possible_targets(graph, start, edge):
'''
Given an undirected graph G = (V,E), an input vertex v in V, and an edge e
incident to v, compute the set of vertices w such that e is on a shortest path from
v to w.
'''
dijkstra_output = dijkstra.single_source_shortest_paths(graph, start)
return set(v for v in graph.vertices
if dijkstra_output.path_to_destination_contains_edge(v, edge))
def find_median(graph, vertices):
'''
Compute as output a vertex in the input graph which minimizes the sum of distances
to the input set of vertices
'''
best_dijkstra_run = min(
(single_source_shortest_paths(graph, v) for v in graph.vertices),
key=lambda run: run.sum_of_distances(vertices)
)
return best_dijkstra_run.start
And then the core algorithm
QueryResult = namedtuple('QueryResult', ('found_target', 'feedback_edge'))
def binary_search(graph, query):
'''
Find a target node in a graph, with queries of the form "Is x the target?"
and responses either "You found the target!" or "Here is an edge on a shortest
path to the target."
'''
candidate_nodes = set(x for x in graph.vertices) # copy
while len(candidate_nodes) > 1:
median = find_median(graph, candidate_nodes)
query_result = query(median)
if query_result.found_target:
return median
else:
edge = query_result.feedback_edge
legal_targets = possible_targets(graph, median, edge)
candidate_nodes = candidate_nodes.intersection(legal_targets)
return candidate_nodes.pop()
Here’s an example of running it on the example graph we used earlier in the post:
'''
Graph looks like this tree, with uniform weights
a k
b j
cfghi
d l
e m
'''
G = Graph(['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i',
'j', 'k', 'l', 'm'],
[
('a', 'b'),
('b', 'c'),
('c', 'd'),
('d', 'e'),
('c', 'f'),
('f', 'g'),
('g', 'h'),
('h', 'i'),
('i', 'j'),
('j', 'k'),
('i', 'l'),
('l', 'm'),
])
def simple_query(v):
ans = input("is '%s' the target? [y/N] " % v)
if ans and ans.lower()[0] == 'y':
return QueryResult(True, None)
else:
print("Please input a vertex on the shortest path between"
" '%s' and the target. The graph is: " % v)
for w in G.incident_edges:
print("%s: %s" % (w, G.incident_edges[w]))
target = None
while target not in G.vertices:
target = input("Input neighboring vertex of '%s': " % v)
return QueryResult(
False,
G.edge(v, target)
)
output = binary_search(G, simple_query)
print("Found target: %s" % output)
The query function just prints out a reminder of the graph and asks the user to answer the query with a yes/no and a relevant edge if the answer is no.
An example run:
is 'g' the target? [y/N] n
Please input a vertex on the shortest path between 'g' and the target. The graph is:
e: [Edge(source='e', target='d', weight=1)]
i: [Edge(source='i', target='h', weight=1), Edge(source='i', target='j', weight=1), Edge(source='i', target='l', weight=1)]
g: [Edge(source='g', target='f', weight=1), Edge(source='g', target='h', weight=1)]
l: [Edge(source='l', target='i', weight=1), Edge(source='l', target='m', weight=1)]
k: [Edge(source='k', target='j', weight=1)]
j: [Edge(source='j', target='i', weight=1), Edge(source='j', target='k', weight=1)]
c: [Edge(source='c', target='b', weight=1), Edge(source='c', target='d', weight=1), Edge(source='c', target='f', weight=1)]
f: [Edge(source='f', target='c', weight=1), Edge(source='f', target='g', weight=1)]
m: [Edge(source='m', target='l', weight=1)]
d: [Edge(source='d', target='c', weight=1), Edge(source='d', target='e', weight=1)]
h: [Edge(source='h', target='g', weight=1), Edge(source='h', target='i', weight=1)]
b: [Edge(source='b', target='a', weight=1), Edge(source='b', target='c', weight=1)]
a: [Edge(source='a', target='b', weight=1)]
Input neighboring vertex of 'g': f
is 'c' the target? [y/N] n
Please input a vertex on the shortest path between 'c' and the target. The graph is:
[...]
Input neighboring vertex of 'c': d
is 'd' the target? [y/N] n
Please input a vertex on the shortest path between 'd' and the target. The graph is:
[...]
Input neighboring vertex of 'd': e
Found target: e
A likely story
The binary search we implemented in this post is pretty minimal. In fact, the more interesting part of the work of Emamjomeh-Zadeh et al. is the part where the response to the query can be wrong with some unknown probability.
In this case, there can be many shortest paths that are valid responses to a query, in addition to all the invalid responses. In particular, this rules out the strategy of asking the same query multiple times and taking the majority response. If the error rate is 1/3, and there are two shortest paths to the target, you can get into a situation in which you see three responses equally often and can’t choose which one is the liar.
Instead, the technique Emamjomeh-Zadeh et al. use is based on the Multiplicative Weights Update Algorithm (it strikes again!). Each query gives a multiplicative increase (or decrease) on the set of nodes that are consistent targets under the assumption that query response is correct. There are a few extra details and some postprocessing to avoid unlikely outcomes, but that’s the basic idea. Implementing it would be an excellent exercise for readers interested in diving deeper into a recent research paper (or to flex their math muscles).
But even deeper, this model of “query and get advice on how to improve” is a classic learning model first formally studied by Dana Angluin (my academic grand-advisor). In her model, one wants to design an algorithm to learn a classifier. The allowed queries are membership and equivalence queries. A membership is essentially, “What’s its label of this element?” and an equivalence query has the form, “Is this the right classifier?” If the answer is no, a mislabeled example is provided.
This is different from the usual machine learning assumption, because the learning algorithm gets to construct an example it wants to get more information about, instead of simply relying on a randomly generated subset of data. The goal is to minimize the number of queries before the target hypothesis is learned exactly. And indeed, as we saw in this post, if you have a little extra time to analyze the problem space, you can craft queries that extract quite a lot of information.
Indeed, the model we presented here for binary search on graphs is the natural analogue of an equivalence query for a search problem: instead of a mislabeled counterexample, you get a nudge in the right direction toward the target. Pretty neat!
There are a few directions we could take from here: (1) implement the Multiplicative Weights version of the algorithm, (2) apply this technique to a problem like ranking or clustering, or (3) cover theoretical learning models like membership and equivalence queries in more detail. What interests you?
Until next time!
Well Orderings and Search
Binary Search
Binary search is perhaps the first and most basic nontrivial algorithm a student learns. For the mathematicians out there, binary search is a fast procedure to determine whether a sorted list contains a particular element. Here is a pseudocode implementation:
# Binary Search:
# Given a list L, sorted via the total order <, and a sought
# element x, return true iff L contains x.
function binarySearch(L, x, <):
# base case
if(length(L) == 1):
return L[0] == x
middleIndex = floor(length(L) / 2)
if (L[middleIndex] == x):
return true
# inductive step, with ellipsis notation meaning slices of L
# from the beginning and to the end, respectively
if (x < L[middleIndex]):
return binarySort(L[...middleIndex-1], x, <)
else:
return binarySort(L[middleIndex+1...], x, <)
Colloquially, this is the optimal strategy in a game of “guess the number,” where the guesser is told if her guess is correct, too high, or too low. Try the middle number in the range of possible numbers. If the guess is too high, try the number which is 1/4th in the ordering, otherwise try 3/4ths, continuing this process until the number is guessed. This algorithm is obviously made for recursion (and for those advanced programmers, we resign to hope our working language supports tail-call optimization).
Binary search’s runtime is rather easy to analyze. At each step of the algorithm, we either finish, or cut the problem size in half. By no stretch of the imagination, binary search runs in $O(\log n)$ where $n$ is the length of the input list. In other words, in at worst $\log_2 n$ steps, we will reduce the input list to have size 1, and can definitively say whether the list as a whole contains the sought element.
Notice that the success of the algorithm depends on the fact that the list is sorted, but the specific total order $<$ is irrelevant. We will investigate this idea further, but first we need some deeper maths.
Well and Total Orders
For those of us who aren’t set theorists, it isn’t fair to talk about total orders and well orders without defining them. So here comes a definition:
Definition: A strict total order $<$ is a relation on a set $S$ with the following statements holding for all $a, b, c \in S$:
• It is never the case that $a < a$. (anti-reflexivity)
• Exactly one of the following are true: $a < b, b < a,$ or $a = b$. (trichotomy)
• If $a < b$ and $b < c$, then $a < c$. (transitivity)
These conditions create an environment in which sorting is possible: we need to be able to compare any two elements (trichotomy), we need to be able to inspect two elements at a time and know that our analysis generalizes to the whole list (transitivity), and we can’t break the world of strictness (anti-reflexivity).
Aside: The use of a strict total order as opposed to a non-strict total order is irrelevant, because each strict total order corresponds bijectively to a non-strict total order. Hence, there are two equivalent formulations of sorting with respect to strict and non-strict total orders, and we may choose one arbitrarily.
Now, we may elevate a total order $<$ to a well order if every non-empty subset $R \subset S$ has a least element with respect to $<$. We computer scientists only sort finite lists, so every total order is automatically a well order. However, the reader may be interested in the mathematical need for such a distinction, so here it is:
Consider the integers $\mathbb{Z}$ with the standard ordering $<$. While $\mathbb{Z}$ itself has no smallest element, neither does any subset which has infinitely many negative numbers, such as the evens or odds. More generally, any open interval in the real numbers $\mathbb{R}$ obviously doesn’t have a least element with respect to the natural order. In contrast, we rely on the crucial fact that the set of natural numbers $\mathbb{N}$ is well-ordered to apply mathematical induction.
Interestingly enough, a theorem due to Ernst Zermelo states that every set can be well ordered, and it is equivalent to the Axiom of Choice. While many people have a hard time visualizing a well-ordering of the real numbers $\mathbb{R}$, we simply resign (as mystical as it is) to admit there is one out there somewhere, though we may never know what it is.
As another aside, it turns out that we only need one of the inequalities in $(<, \leq, >, \geq)$ and the standard logical operations and (infix &&), or (infix ||), and not (prefix !) in order to define the other three (and indeed, to define $=, \neq$ as well). This is a computer science trick that comes in handy, as we’ll see later, so here is the described equivalence. Given $<$, we define the remaining operations as follows:
• $x > y$ is equivalent to $y < x$
• $x \geq y$ is equivalent to $!(x < y)$
• $x \leq y$ is equivalent to $!(y < x)$
• $x = y$ is equivalent to $!(x < y) \textup{ and } !(y < x)$
• $x \neq y$ is equivalent to $x < y \textup{ or } y < x$
So if we are interested in sorting a set via some procedure, all we need from the user is the $<$ operation, and then we may compare any way our heart desires.
Defining a New Well Order
Consider a deck of cards which is initially sorted (with some arbitrary ordering on the suits), and then is “cut” at some arbitrary point and the bottom part is placed on top of the deck. We may simplify this “cut” operation to a list of numbers, say ten, and provide the following example of a cut:
(5,6,9,10,11,13,1,3,3,4)
To pick a standard working language, we say the “cut point” of this list is 5, not 4.
We have a few (naive) options to search through cut data: we may sort it with respect to the natural total order on $\mathbb{N}$, and then search through it; we may stick the elements of the list into a hash set (a constant-time lookup table), and then query existence that way; or we may traverse the list element-by element looking for a particular value.
The problem with the first two methods, though they determine existence, is that they don’t allow us to know where the value is in the list. If this it not important, and we are searching many times (compared to the size of the list) on many different values, then a has table would be the correct choice. However, if we are searching only a few times, and need to know where the value is hidden, all three of the above approaches are slow and inelegant.
Enter well orders. You may have noticed that a cut list of numbers has a very simple well ordering in terms of the natural order on $\mathbb{N}$. Verbally, if the two numbers are separated across the cut point, then the larger number is in fact the smaller number. Otherwise, we may appeal to the regular ordering. Here it is in pseudocode:
function lessThan(x, y):
if (y < cutPoint <= x):
return true
else if (x < cutPoint <= y):
return false
else:
return x < y
And we may compress these if statements into a single condition:
function lessThan(cutPoint, x, y):
y < cutPoint <= x || (!(x < cutPoint <= y) && x < y)
function makeCutOrdering(cutPoint):
lambda x,y: lessThan(cutPoint, x, y)
So we have found that a cut list of numbers is in fact well ordered with respect to this new relation. Forget about preprocessing the data, we can just do a binary search using this new ordering! Here’s a Mathematica script that does just that. Here we assume constant-time list length calculations, and fast slicing (which Mathematica has). Note that list indexing and slicing has double square bracket [ [ ] ] syntax, while function application is single square brackets [ ]. It should look very similar to the earlier pseudocode for binary search.
eq[x_, y_, lt_] := !lt[x,y] && !lt[y,x];
(* base case *)
BinarySearch[{oneElt_}, sought_, lt_] := eq[oneElt, sought, lt];
(* inductive step *)
BinarySearch[lst_List, sought_, lt_] :=
Module[{size = Length[lst], midVal, midNdx},
midNdx = Floor[size/2];
midVal = lst[[midNdx]];
If[eq[midVal, sought, lt],
True,
If[lt[sought, midVal],
BinarySearch[lst[[ ;; midNdx - 1]], sought, lt]
BinarySearch[lst[[midNdx + 1 ;; ]], sought, lt]
]
]
];
Notice that if we use the standard $<$ (in Mathematica, the function Less), then the BinarySearch function reverts to a standard binary search. Marvelous! Now we have a reusable piece of code that searches through any well-ordered set, provided we provide the correct well order.
The lesson to take from this is know your data! If your input list is not sorted, but still structured in some way, then there’s a good chance it is sorted with respect to a non-natural total order. For example, many operating systems order filenames which end in numbers oddly (e.g. “file1”, “file11”, “file12”, “file2”), and in certain places in the world, financial calendars are ordered differently (In Australia, the fiscal year starts in July). So take advantage of that, and you’ll never need to write binary search again.
|
2018-12-16 22:30:59
|
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|
https://quant.stackexchange.com/questions/25327/how-to-create-time-series-with-lagged-in-r
|
# How to create time series with lagged in R [closed]
Would anyone else advise me, how to create time series with lagged in R. I would the result is the difference with lagged, there is a function Delt() but the result is the percentage change. Please advise how to do that. Thank you
getSymbols("^GSPC")
DeltLagGSPC<-Delt(Cl(GSPC),k=1:5)
update
I would calculate the computed result with the successive difference. The output format of Delt() is same as I expected, but the result is the percentage change with lagged. I would calculate the difference with lagged. Is there any function, same parameters as Delt() but calculate the difference change.
## closed as off-topic by Bob Jansen♦Apr 9 '16 at 16:45
• This question does not appear to be about quantitative finance within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.
• did you have a look at the function "lag" ? Furthermore this is at most a question for stackoverflow ... – Ric Apr 9 '16 at 16:16
• That, or implement shift or get it from data.table. You can do a few things but I agree it doesn't fit here. On StackOverflow it would be a duplicate, so closing. – Bob Jansen Apr 9 '16 at 16:45
• I'm voting to close this question as off-topic because belongs on StackOverflow where it would be a duplicate – Bob Jansen Apr 9 '16 at 16:45
Look at the function "lag" and if you want a lag function that does not depend on some time series structure of the object then you can use this one:
shift<-function(x,shift_by){
stopifnot(is.numeric(shift_by))
stopifnot(is.numeric(x))
if (length(shift_by)>1)
return(sapply(shift_by,shift, x=x))
out<-NULL
abs_shift_by=abs(shift_by)
if (shift_by > 0 )
out<-c(tail(x,-abs_shift_by),rep(NA,abs_shift_by))
else if (shift_by < 0 )
|
2019-08-25 17:47:21
|
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|
https://gmatclub.com/forum/the-table-shows-the-results-of-a-poll-which-asked-drivers-97818.html
|
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# The table shows the results of a poll which asked drivers
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The table shows the results of a poll which asked drivers how many accidents they had had over the previous 5 years. What is the median number of accidents per driver?
A. 0.5
B. 1
C. 1.5
D. 2
E. 4
[Reveal] Spoiler: OA
_________________
-------------------------------------------------------------------------------------------
Amar
http://amarnaik.wordpress.com
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Joined: 19 Jul 2009
Posts: 52
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Schools: kellogg, booth, stern, ann arbor
Re: median number of accidents per driver? [#permalink]
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i actually messed up the question the first time cause i didn't add correctly. in either case, i got C also.
in order to find the median we must put the values in order from lowest to highest values.
6 5 4 3 2 1 0 = number of accidents
-----------------------------
1 2 2 4 21 13 17 = values representing people
we can now find the median by finding the middle number. there is a total of
1 + 2 + 2 + 4 + 21 + 13 + 17 = 60 people.
middle value will be represented as average in 30th and 31st person.
so 30th person had 2 accidents while 31st person had one. median will thusly be:
2 + 1 / 2 = 1.5
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Paaaaayyy Meeeee!!!!!
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Re: median number of accidents per driver? [#permalink]
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anaik100 wrote:
The table shows the results of a poll which asked drivers how many accidents they had had over the previous 5 years. What is the median number of accidents per driver?
A. 0.5
B. 1
C. 1.5
D. 2
E. 4
If a set has odd number of terms the median of a set is the middle number when arranged in ascending or descending order;
If a set has even number of terms the median of a set is the average of the two middle terms when arranged in ascending or descending order.
There are total of $$17+13+21+4+2+2+1=60$$ drivers (60 terms) so the median number of accidents per driver would be the average of accidents of 30th and 31st drivers (as we have even # of terms).
30th term equals to 1 and 31st term equals to 2 so $$median=\frac{1+2}{2}$$.
Answer: C.
To elaborate more, you can imagine these data points as:
0, ..., 0, 1, ..., 1, 2, ..., 2, 3, ..., 3, 4, ..., 4, 5, ..., 5, 6, ..., 6 --> 17 zeros, 13 ones, 21 twos and so on, total of 60 data points. Median would be the average of 30th and 31st terms: $$median=\frac{1+2}{2}$$.
Hope it's clear.
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Re: median number of accidents per driver? [#permalink]
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27 Sep 2010, 08:54
can we just estimate and don't do the calculation, the frequencies at 1 accident is 13 and 2 accidents is 21 (most concentration of all) , then the average will be arround 1 to 2 then i chose 1.5
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Re: median number of accidents per driver? [#permalink]
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27 Sep 2010, 09:28
Mikko wrote:
can we just estimate and don't do the calculation, the frequencies at 1 accident is 13 and 2 accidents is 21 (most concentration of all) , then the average will be arround 1 to 2 then i chose 1.5
You should be careful with such approximations. For example if it were: 1-14 and 2-20 (instead of 1-13 and 2-21) the answer would be B(1) and not C(1.5).
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Re: The table shows the results of a poll which asked drivers [#permalink]
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18 Jun 2013, 01:41
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Re: The table shows the results of a poll which asked drivers [#permalink]
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06 Aug 2015, 12:31
A quicker way to do this is to note that there are 30 drivers with 0-1 accidents and 30 drivers with 2-6 accidents so you know that your median will fall exactly between 1 and 2. Hence the answer of 1.5.
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Re: The table shows the results of a poll which asked drivers [#permalink]
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09 Apr 2017, 08:50
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Re: The table shows the results of a poll which asked drivers [#permalink]
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18 Apr 2017, 16:12
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anaik100 wrote:
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The table shows the results of a poll which asked drivers how many accidents they had had over the previous 5 years. What is the median number of accidents per driver?
A. 0.5
B. 1
C. 1.5
D. 2
E. 4
To determine where the median falls in a data set, we can use the following expression in which n = the total number of data points in the set (here, n = 60 since 60 drivers were surveyed):
median position = (n + 1)/2
(60 + 1)/2 = 61/2 = 30.5
Thus, the median is halfway between the 30th and 31st data points, which is the same as the average of the 30th data point and the 31st data point when the data are sorted from least to greatest.
Looking at the chart, we see that the 30th data point is 1 accident and the 31st data point is 2 accidents. Thus, the median is (1+2)/2 = 1.5.
Answer: C
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Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.
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2017-06-28 21:26:05
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http://teledynelecroy.com/doc/using-histograms-to-determine-ddr2-readwrite-timing-limits
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Characterizing DDR2 memory devices requires the ability to isolate read and write access cycles. In LAB_WM776 LeCroy’s WaveScan was used to isolate DDR2 Read/Write operations by filtering waveform data using hold time. In read operations the data changes are synchronous with the strobe transition. During write operations the strobe transition precedes the data transition. Using parameter histograms it is possible to investigate the differences. Figure 1 shows a histogram of the hold time parameter for a 500 µs acquisition of the data and strobe lines of a DDR2 memory. The write operations have a hold time of approximately 750 ps. Read operations, which can have both inphase and out of phase timing show hold times below 500 ps and in the range of 1.2 to 2.4 ns. Based on this data it is possible to setup WaveScan to isolate the read and write operations.
Figure 1:
The histogram of hold time shows the distribution of hold times for DDR2 data and strobe lines for both read and write operations.
Figure 2 shows the WaveScan setup for capturing the read operations where the strobe transition occurs within 600 ps of the data transition. Figure 3 shows the WaveScan setup for capturing read cycles with hold times in the range of 1.2 to 2.4 ns. This range must be set as a limit to exclude holds times that exceed 2.4 ns that can occur at the boundaries between read and write operations. These boundary conditions should be excluded.
Figure 2:
WaveScan setup to capture read cycles with hold time less than 600 ps
Figure 3:
WaveScan Setup to capture Read cycles with hold times in the range of 1.2 to 2.4 ns
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2017-10-21 17:31:50
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https://motls.blogspot.com/2008/02/missile-defense-probably-destroys.html
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## Thursday, February 21, 2008 ... //
### Missile defense probably destroys a hostile spy satellite
An interceptor designed for missile defense was ordered to be launched by Robert Gates personally and used to hit a dying U.S. spy satellite:
The New York Times
Most likely, the dangerous fuel tank with 1,000 pounds of hydrazine (N2H4, derived from NH3 but similar to H2O) has been ruptured. We will know today. Don't pick the debris.
If true, and even if not quite true, it shows how real and useful the technology behind the missile defense system can be, even for peaceful purposes - in this case, it is useful to protect the U.S. spying know-how against wise guys elsewhere.
I think it is a good idea for the U.S. or the democratic world to build such a system. At the same moment, it is also clear that some countries that are effectively weakened by such a system may feel a kind of dissatisfaction.
The missile defense system has been criticized by arguments that I have always considered completely bizarre - for example by the comment that it can't be 100% reliable so it is useless and won't change any decisions. Well, nothing in the real world is 100% reliable so the people whose decisions are only influenced by 100% reliable gadgets and information and who are only impressed by 100% success are clearly detached from the real world completely.
In the process of evolution of life much like in the process of development of new technologies, the initial versions of organs or devices were less reliable and less efficient and they could be improved later, by incremental additions and adjustments. The missile defense system would follow the path that has been tested millions of times.
And that's the memo.
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2019-07-24 06:52:15
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http://documenta.sagemath.org/vol-18/25.html
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#### DOCUMENTA MATHEMATICA, Vol. 18 (2013), 749-783
Reza Taleb
An Equivariant Main Conjecture in Iwasawa Theory and the Coates-Sinnott Conjecture
We formulate and prove an Equivariant Main Conjecture (EMC) for {\it all} prime numbers $p$ under the assumptions $\mu = 0$ and the validity of the 2-adic Main Conjecture in Iwasawa theory \cite{Wi}. This equivariant version coincides with the version, which Ritter and Weiss formulated and proved for odd $p$ under the assumption $\mu=0$ in \cite{RW2}. Our proof combines the approach of Ritter and Weiss with ideas and techniques used by Greither and Popescu in \cite{GP2} in a recent proof of an equivalent formulation of the above EMC under the same assumptions ($p$ odd and $\mu=0$) as in \cite{RW2}. As an application of the EMC we prove the Coates-Sinnott Conjecture, again assuming $\mu=0$ and the 2-adic Main Conjecture.
2010 Mathematics Subject Classification: 11R23, 11R42, 14F42, 11R70, 11R33, 11R34
Keywords and Phrases: Iwasawa theory, global and p-adic L-functions, motivic cohomology, algebraic K-theory, Fitting ideals
Full text: dvi.gz 56 k, dvi 146 k, ps.gz 349 k, pdf 310 k.
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2017-06-27 03:44:12
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http://math.stackexchange.com/tags/conic-sections/new
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# Tag Info
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For my convenience, I have changed the notation a little bit --- $a = r$ and $\alpha = R$. Fact:- The equation of the tangent pair from P(X, Y) to the circle $x^2 + y^2 = r^2$ is $$(X^2 + Y^2 – r^2)(x^2 + y^2 – r^2) = (xX + yY – r^2)^2$$ As seen from the figure, we have the following two equations $$(1) … ((k + R)^2 – r^2)(x^2 + y^2 – r^2) = x(k + R) – ... 1 OK, I looked at the paper. I think the passage you are asking about is very confusingly written. What they appear to be saying is that for this specific value, \alpha A +B is the (unique) degenerate conic in the pencil. So the plural "conics" in the first line and the pronoun "they" later on are incorrect. So to sum up: 1) the pencil is \lambda A +B ... 1 C is made from the two matrices A and B each defining a conic. Let \alpha=0 and you get the one, let \alpha=\infty and you get the other. The whole pencil is described by varying \alpha. (If you don't feel comfortable with one conic appearing only in the limit \alpha\to\infty, consider \lambda A+(1-\lambda) B instead.) Assuming the conics ... 2 If you are looking for the two points of the ellipse where the slope of the normal lines equal -4 as well as the equations of the two lines tangent to the ellipse at those two points, then proceed as follows: As you correctly deduce,$$\dfrac{dy}{dx}=-\dfrac{9x}{4y}$$Therefore$$-\dfrac{9x}{4y}=\dfrac{1}{4}$$at the points in question, so they lie on ... 0 Obs: I'm assuming you want to know which points int the ellipse have a normal line with slope -4 You seem to be confusing the normal line with the tangent line. You have the equation$$ \frac{x^2}{16}+\frac{y^2}{36}=1\Rightarrow y=\pm \frac{3}{2}\sqrt{16-x^2}$$So taking the derivative, we have two situations:$$y'=\mp \frac{3x}{2\sqrt{16-x^2}}$$Now, ... 0 For one curve/surface, we can parametrize it differently. For example, consider an ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. It can be parametrized as \langle a\cos(t),b\sin(t)\rangle, or as \langle a\cos(t),-b\sin(t)\rangle. You can verify that the first parametrization is counter-clockwise, and the second one is clockwise. There is no principle,... 1 THIS IS A COMMENT NOT AN ANSWER. In http://debart.pagesperso-orange.fr/seconde/contruc_cercle.html there are ten problems of contact concerning circles which are solved in a quite interesting way. The text is in French and the problem 10 is the Apollonius’s consisting in finding a circle tangent to three given circles of which eight solutions are ... 2 Here is an outline of the a procedure. Assume first that you want to find a circle that encloses all three others and touches them from the outside. Observe that if you increase the radii of all the three circles by the same amount, then the center of the surrounding circle will stay at the same point. Only its radius will increase (by the same amount). ... 0 If your angle is \theta you know (y-k)/(x-h) = \tan{\theta} or x-h = (y-k)/\tan{\theta}. Plug this into your equation for your ellipse to get \displaystyle \frac{(y-k)^2}{(a\tan{\theta})^2}+\frac{(y-k)^2}{b^2}=1 So there you have an equation that you can solve readily for y-k. Just be careful about which quandrant of the plane you are in. 2 u = \frac xa\\ v = \frac yb\\ \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1 \to u^2 + v^2 = 1\\ xy = 1 \to uv = \frac {1}{ab} uv = k touches u^2 + v^2 = 1 when k =\frac 12 For any ellipse that touches the hyperbola, ab = 2, and its area = 2 \pi 1 Consider the situation using transformations of functions. We know the vertex of y=x^2 is located at (0,0). To translate the parabola horizontally h units, we modify the input (x) value like y=(x-h)^2. Thus, the vertex is now at (h,0). The a-value is a vertical stretch or compression, and also determines whether the parabola is reflected ... 1 Consider the graph of the parabola y=ax^2. Its vertex is clearly at (0,0). Now, if you replace x with x-h in any equation, its graph gets shifted to the right by a distance of h. Similarly, replacing y by y-k shifts the graph up by k. If we make both of these substitutions in the above equation of the parabola, its vertex gets shifted to (h,... 1 Since (x-h)^2 is always nonnegative, for any value of x, it's at its smallest when the thing in parens is zero, i.e., when x - h. If a is positive, this will be where the parabola has its minimum, i.e., x = h is the x-coord of the vertex. (If a is negative, this is the location of the max, but the same result applies: x = h is the x-coord ... 3 This should give you an ellipse around point P with axis vectors \mathbf{u} and \mathbf{v}$$\mathbf{r} = P + \mathbf{u} \cos(t) + \mathbf{v} \sin(t)$$0 The slope of the asymptotes is \pm \frac{b}{a}=\pm 2 \implies b=2a All hyperbola with an equation like \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 are centered on the origin. The hyperbola does contain the point (1,1) too, so \frac{1}{a^2}-\frac{1}{b^2}=1 \frac{1}{a^2}-\frac{1}{4a^2}=1 \frac{3}{4a^2}=1 a^2=\frac{3}{4} and b^2=4a^2=4\frac{3}{4}=3 0 An equation of the hyperbola centred at (0,0), with the coordinate axes as axes of symmetry is:$$\textsf{product of the equations of the asymptotes = constant}.$$Here, you have \;y^2-4x^2=1^2-4\cdot1^2=-3, or, in normalised form$$\frac43x^2-\frac13y^2=1.$$0 The asymptotes of the hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 are y = \pm \frac{bx}{a}. Since (1, 1) lies on your curve then \frac{1}{a^2} - \frac{1}{b^2} = 1 \iff b^2 -a^2 = a^2b^2. But you know that \frac{b}{a} = 2\iff b = 2a. Plug this into the above equation to get (a\neq 0)$$3a^2 = 4a^4 \Rightarrow a^2 = \frac{3}{4}$$and b^2 = ... 0 If you draw the line FG, and draw the angle \alpha at F, then you can draw the axes and essentially have the ellipse. So the interesting part is finding a match with \beta. Let's look at this problem algebraically for a moment. If H=(x,y), then one can compute the normal of the tangent direction as a linear function in these variables. So assume ... 0 first of all "Sorry for my english" .......I do a problem in my blog about a rotation of a parabola,wich is rotate about the focus and the equation of the guideline is ax+by+c=0$$$$firts cleared y.$$y=\frac{-a}{b}x-c$$we calculated the slope of the perpendicular straight line.$$m_{2}=-\frac{1}{m_{1}}=\frac{-1}{-a/b}=\frac{b}{a}$$we have$$y=\frac{b}...
0
If you use calculus then the problem is very easy. You don't need those geometric argument but instead you require some algebraic manipulations. Consider the parabola $y^{2} = 4ax$ with focus at $(a, 0)$ and directrix $x = -a$. A line perpendicular to the directrix is of the form $y = k$ which meets the parabola in point $P = (k^{2}/4a, k)$. The slope of ...
1
If you rotate the original parabola by any angle other than a multiple of $\pi$, the resulting curve cannot be expressed in the form $y=f(x)$. If you start with the equation $$ax^2+bx+c-y=0\tag{1}$$ instead, though, and apply the transformations to that, you’ll end up with a quadratic in $x$ and $y$ that you can then solve for $y$ using your favorite method (...
0
Let $a,b$ be the major and minor axes of the ellipse. Let $x_0,y_0$ be the coordinates of the center on the ellipse. Let $x_1,y_1$ be the coordinates of the one point on the ellipse. First, we write the equation of the ellipse, those major axis is parallel to the $x$ axis of the coordinate system (which is at this point completely arbitrary). $$\frac{(x-... 1 In projective geometry, a circle is just a special case of a conic, namely one passing through the ideal circle points (1,\pm i,0). So if you allow for complex transformations, then you can take a pair of conics and map two points of intersection to these special points to obtain a configuration with two circles. For more than two conics, though, this only ... 1 The equation of an ellipsoid centered at the origin has the form q_A(\mathbf{v}) = \mathbf{v}^T A \mathbf{v} = C where C > 0 and A is a symmetric matrix whose eigenvalues are all positive. If the eigenvalues of A are \lambda_i with a corresponding orthonormal basis of eigenvectors v_i (so Av_i = \lambda_i v_i) then the semi-principal axes of ... 0 Well you are describing a curve with polar equation: r = a \sin(kt) + b. where a,b,k are some constants, t is the angle from some axis (usually x-axis) and r is the distance from origin. In your question you chose k = 2, but that does not give you an ellipse. For example if you pick (a,b) = (2,5) you get a peanut shape. You can use Graph ... 0 I suspect this is hard to do in general. The points on the ellipse can be parameterized by an angle t (the angle from the center of the ellipse to the point, as measured counterclockwise from the positive x-direction) as$$P(t)=\left(X_e+a\cos(t),Y_e+b\sin(t)\right)$$for 0\le t<2\pi. If you know the specific values of all the constants, you could ... 0 In General formula generic for Pythagorean triples looks a little different.$$x^2+y^2=az^2$$If the number can be represented as a sum of squares. a=t^2+k^2 The solution has the form:$$x=-tp^2+2kps+ts^2y=kp^2+2tps-ks^2z=p^2+s^2$$1 Yes. Well...relatively easy if you're willing to allow me to use Sylvester's law of inertia, which says that every symmetric matrix over the reals is diagonalizable. The matrix A can be replaced by (A + A^t)/2 without changing the ellipse, so we may assume A is symmetric. Write A = Q^t D Q for some orthogonal matrix Q and diagonal matrix D. ... 0 In general I'd write a projective transformation in homogeneous coordinates as$$\begin{pmatrix}x\\y\\z\end{pmatrix}\mapsto \begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}\cdot \begin{pmatrix}x\\y\\z\end{pmatrix}$$or, in inhomogeneous coordinates, as$$\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}(ax+by+c)/(gx+hy+i)\\(dx+...
1
Let the focal chord cut the parabola at $A,A'$ and let $C$ be the midpoint of $A, A'$. By definition, the distance from both $A, A'$ to the focus $(a,0)$ is equal to the respective distances to the directrix $x=-a$. Let these distances be $m,n$ respectively. The radius of the circle with $AA'$ as diameter has radius $\frac {m+n}2$, as $AA'=m+n$. As $C$ ...
1
The equation of the circle is given by $$\left(x-\frac{x_1+x_2}{2}\right)^2+\left(y-\frac{y_1+y_2}{2}\right)^2=\frac 14\left((x_1-x_2)^2+(y_1-y_2)^2\right)$$ From what you've got, you can write it as $$\left(x-\left(a+\frac{2a}{m^2}\right)\right)^2+\left(y-\frac{2a}{m}\right)^2=4a^2\left(\frac{1}{m^2}+1\right)^2$$ Now, set $x=-a$ which is the equation of the ...
1
HINT: Using this OR this, WLOG the endpoints of one diameter the relevant circle can be written as $$(at^2,2at), (a/t^2,-2a/t)$$ Use this, to form the equation of the circle Now, as the equation of directrix is $x+a=0,$ put $x=-a$ in the equation of the circle
1
Justification of the formula: After centering (translation to let the linear terms vanish), the equation becomes $$Ax^2+Bxy+Cy^2+F'=0.$$ Then you apply a rotation of angle $\theta$ to let the mixed term $Bxy$ vanish from the quadratic terms, $$A(x\cos\theta-y\sin\theta)^2+B(x\cos\theta-y\sin\theta)(x\sin\theta+y\cos\theta)+C(x\sin\theta+y\cos\theta)^2.$$ ...
1
There're two principal axes in general, so \begin{align*} \theta &=\frac{1}{2} \tan^{-1} \frac{B}{A-C}+\frac{n\pi}{2} \\ &= \tan^{-1} \left( \frac{C-A}{B} \color{red}{\pm} \frac{\sqrt{(A-C)^{2}+B^{2}}}{B} \: \right) \\ \end{align*} The centre is given by $$(h,k)= \left( \frac{2CD-BE}{B^2-4AC}, \frac{2AE-BD}{B^2-4AC} \right)$$ ...
1
Although it sounds like a question, for calculation did you use atan2 function or atan function? Quadrant placement is also important.
Top 50 recent answers are included
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2016-07-24 14:59:12
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http://jas.ia.ac.cn/en/article/2020/6
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A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
## Vol. 7, No. 6, 2020
Display Method:
2020, 7(6): 1465-1477. doi: 10.1109/JAS.2020.1003201
Abstract(1307) HTML (581) PDF(116)
Abstract:
With the increasing presence of robots in our daily life, there is a strong need and demand for the strategies to acquire a high quality interaction between robots and users by enabling robots to understand users’ mood, intention, and other aspects. During human-human interaction, personality traits have an important influence on human behavior, decision, mood, and many others. Therefore, we propose an efficient computational framework to endow the robot with the capability of under-standing the user’s personality traits based on the user’s nonverbal communication cues represented by three visual features including the head motion, gaze, and body motion energy, and three vocal features including voice pitch, voice energy, and mel-frequency cepstral coefficient (MFCC). We used the Pepper robot in this study as a communication robot to interact with each participant by asking questions, and meanwhile, the robot extracts the nonverbal features from each participant’s habitual behavior using its on-board sensors. On the other hand, each participant’s personality traits are evaluated with a questionnaire. We then train the ridge regression and linear support vector machine (SVM) classifiers using the nonverbal features and personality trait labels from a questionnaire and evaluate the performance of the classifiers. We have verified the validity of the proposed models that showed promising binary classification performance on recognizing each of the Big Five personality traits of the participants based on individual differences in nonverbal communication cues.
2020, 7(6): 1478-1488. doi: 10.1109/JAS.2020.1003351
Abstract(1374) HTML (546) PDF(111)
Abstract:
Pneumatic muscle actuators (PMAs) are compliant and suitable for robotic devices that have been shown to be effective in assisting patients with neurologic injuries, such as strokes, spinal cord injuries, etc., to accomplish rehabilitation tasks. However, because PMAs have nonlinearities, hysteresis, and uncertainties, etc., complex mechanisms are rarely involved in the study of PMA-driven robotic systems. In this paper, we use nonlinear model predictive control (NMPC) and an extension of the echo state network called an echo state Gaussian process (ESGP) to design a tracking controller for a PMA-driven lower limb exoskeleton. The dynamics of the system include the PMA actuation and mechanism of the leg orthoses; thus, the system is represented by two nonlinear uncertain subsystems. To facilitate the design of the controller, joint angles of leg orthoses are forecasted based on the universal approximation ability of the ESGP. A gradient descent algorithm is employed to solve the optimization problem and generate the control signal. The stability of the closed-loop system is guaranteed when the ESGP is capable of approximating system dynamics. Simulations and experiments are conducted to verify the approximation ability of the ESGP and achieve gait pattern training with four healthy subjects.
2020, 7(6): 1489-1497. doi: 10.1109/JAS.2020.1003180
Abstract(1139) HTML (584) PDF(107)
Abstract:
In order to improve detection system robustness and reliability, multi-sensors fusion is used in modern air combat. In this paper, a data fusion method based on reinforcement learning is developed for multi-sensors. Initially, the cubic B-spline interpolation is used to solve time alignment problems of multi-source data. Then, the reinforcement learning based data fusion (RLBDF) method is proposed to obtain the fusion results. With the case that the priori knowledge of target is obtained, the fusion accuracy reinforcement is realized by the error between fused value and actual value. Furthermore, the Fisher information is instead used as the reward if the priori knowledge is unable to be obtained. Simulations results verify that the developed method is feasible and effective for the multi-sensors data fusion in air combat.
2020, 7(6): 1498-1510. doi: 10.1109/JAS.2020.1003354
Abstract(878) HTML (519) PDF(36)
Abstract:
Group role assignment (GRA) is originally a complex problem in role-based collaboration (RBC). The solution to GRA provides modelling techniques for more complex problems. GRA with constraints (GRA+) is categorized as a class of complex assignment problems. At present, there are few generally efficient solutions to this category of problems. Each special problem case requires a specific solution. Group multi-role assignment (GMRA) and GRA with conflicting agents on roles (GRACAR) are two problem cases in GRA+. The contributions of this paper include: 1) The formalization of a new problem of GRA+, called group multi-role assignment with conflicting roles and agents (GMAC), which is an extension to the combination of GMRA and GRACAR; 2) A practical solution based on an optimization platform; 3) A sufficient condition, used in planning, for solving GMAC problems; and 4) A clear presentation of the benefits in avoiding conflicts when dealing with GMAC. The proposed methods are verified by experiments, simulations, proofs and analysis.
2020, 7(6): 1511-1527. doi: 10.1109/JAS.2020.1003312
Abstract(876) HTML (418) PDF(53)
Abstract:
Location estimation of underwater sensor networks (USNs) has become a critical technology, due to its fundamental role in the sensing, communication and control of ocean volume. However, the asynchronous clock, security attack and mobility characteristics of underwater environment make localization much more challenging as compared with terrestrial sensor networks. This paper is concerned with a privacy-preserving asynchronous localization issue for USNs. Particularly, a hybrid network architecture that includes surface buoys, anchor nodes, active sensor nodes and ordinary sensor nodes is constructed. Then, an asynchronous localization protocol is provided, through which two privacy-preserving localization algorithms are designed to estimate the locations of active and ordinary sensor nodes. It is worth mentioning that, the proposed localization algorithms reveal disguised positions to the network, while they do not adopt any homomorphic encryption technique. More importantly, they can eliminate the effect of asynchronous clock, i.e., clock skew and offset. The performance analyses for the privacy-preserving asynchronous localization algorithms are also presented. Finally, simulation and experiment results reveal that the proposed localization approach can avoid the leakage of position information, while the location accuracy can be significantly enhanced as compared with the other works.
2020, 7(6): 1528-1541. doi: 10.1109/JAS.2019.1911648
Abstract(1292) HTML (836) PDF(64)
Abstract:
The passwords for unlocking the mobile devices are relatively simple, easier to be stolen, which causes serious potential security problems. An important research direction of identity authentication is to establish user behavior models to authenticate users. In this paper, a mobile terminal APP browsing behavioral authentication system architecture which synthesizes multiple factors is designed. This architecture is suitable for users using the mobile terminal APP in the daily life. The architecture includes data acquisition, data processing, feature extraction, and sub model training. We can use this architecture for continuous authentication when the user uses APP at the mobile terminal.
2020, 7(6): 1542-1554. doi: 10.1109/JAS.2019.1911729
Abstract(1640) HTML (904) PDF(85)
Abstract:
In this paper, a kind of lateral stability control strategy is put forward about the four wheel independent drive electric vehicle. The design of control system adopts hierarchical structure. Unlike the previous control strategy, this paper introduces a method which is the combination of sliding mode control and optimal allocation algorithm. According to the driver’s operation commands (steering angle and speed), the steady state responses of the sideslip angle and yaw rate are obtained. Based on this, the reference model is built. Upper controller adopts the sliding mode control principle to obtain the desired yawing moment demand. Lower controller is designed to satisfy the desired yawing moment demand by optimal allocation of the tire longitudinal forces. Firstly, the optimization goal is built to minimize the actuator cost. Secondly, the weighted least-square method is used to design the tire longitudinal forces optimization distribution strategy under the constraint conditions of actuator and the friction oval. Beyond that, when the optimal allocation algorithm is not applied, a method of axial load ratio distribution is adopted. Finally, CarSim associated with Simulink simulation experiments are designed under the conditions of different velocities and different pavements. The simulation results show that the control strategy designed in this paper has a good following effect comparing with the reference model and the sideslip angle $\,\beta$ is controlled within a small rang at the same time. Beyond that, based on the optimal distribution mode, the electromagnetic torque phase of each wheel can follow the trend of the vertical force of the tire, which shows the effectiveness of the optimal distribution algorithm.
2020, 7(6): 1555-1564. doi: 10.1109/JAS.2019.1911627
Abstract(1398) HTML (853) PDF(61)
Abstract:
The stabilization problem of distributed proportional-integral-derivative (PID) controllers for general first-order multi-agent systems with time delay is investigated in the paper. The closed-loop multi-input multi-output (MIMO) framework in frequency domain is firstly introduced for the multi-agent system. Based on the matrix theory, the whole system is decoupled into several subsystems with respect to the eigenvalues of the Laplacian matrix. Considering that the eigenvalues may be complex numbers, the consensus problem of the multi-agent system is transformed into the stabilizing problem of all the subsystems with complex coefficients. For each subsystem with complex coefficients, the range of admissible proportional gains is analytically determined. Then, the stabilizing region in the space of integral gain and derivative gain for a given proportional gain value is also obtained in an analytical form. The entire stabilizing set can be determined by sweeping proportional gain in the allowable range. The proposed method is conducted for general first-order multi-agent systems under arbitrary topology including undirected and directed graph topology. Besides, the results in the paper provide the basis for the design of distributed PID controllers satisfying different performance criteria. The simulation examples are presented to check the validity of the proposed control strategy.
2020, 7(6): 1565-1574. doi: 10.1109/JAS.2020.1003357
Abstract(732) HTML (417) PDF(48)
Abstract:
In recent years, reconstructing a sparse map from a simultaneous localization and mapping (SLAM) system on a conventional CPU has undergone remarkable progress. However, obtaining a dense map from the system often requires a high-performance GPU to accelerate computation. This paper proposes a dense mapping approach which can remove outliers and obtain a clean 3D model using a CPU in real-time. The dense mapping approach processes keyframes and establishes data association by using multi-threading technology. The outliers are removed by changing detections of associated vertices between keyframes. The implicit surface data of inliers is represented by a truncated signed distance function and fused with an adaptive weight. A global hash table and a local hash table are used to store and retrieve surface data for data-reuse. Experiment results show that the proposed approach can precisely remove the outliers in scene and obtain a dense 3D map with a better visual effect in real-time.
2020, 7(6): 1575-1584. doi: 10.1109/JAS.2020.1003360
Abstract(769) HTML (405) PDF(24)
Abstract:
In this paper, we propose an improved torque sensorless speed control method for electric assisted bicycle, this method considers the coordinate conversion. A low-pass filter is designed in disturbance observer to estimate and compensate the variable disturbance during cycling. A DC motor provides assisted power driving, the assistance method is based on the real-time wheel angular velocity and coordinate system transformation. The effect of observer is proved, and the proposed method guarantees stability under disturbances. It is also compared to the existing methods and their performances are illustrated through simulations. The proposed method improves the performance both in rapidity and stability.
2020, 7(6): 1585-1592. doi: 10.1109/JAS.2019.1911636
Abstract(1239) HTML (862) PDF(33)
Abstract:
The Möller algorithm is a self-stabilizing minor component analysis algorithm. This research document involves the study of the convergence and dynamic characteristics of the Möller algorithm using the deterministic discrete time (DDT) methodology. Unlike other analysis methodologies, the DDT methodology is capable of serving the distinct time characteristic and having no constraint conditions. Through analyzing the dynamic characteristics of the weight vector, several convergence conditions are drawn, which are beneficial for its application. The performing computer simulations and real applications demonstrate the correctness of the analysis’s conclusions.
2020, 7(6): 1593-1603. doi: 10.1109/JAS.2019.1911720
Abstract(1061) HTML (531) PDF(25)
Abstract:
In a passive ultra-high frequency (UHF) radio frequency identification (RFID) system, the recovery of collided tag signals on a physical layer can enhance identification efficiency. However, frequency drift is very common in UHF RFID systems, and will have an influence on the recovery on the physical layer. To address the problem of recovery with the frequency drift, this paper adopts a radial basis function (RBF) network to separate the collision signals, and decode the signals via FM0 to recovery collided RFID tags. Numerical results show that the method in this paper has better performance of symbol error rate (SER) and separation efficiency compared to conventional methods when frequency drift occurs.
2020, 7(6): 1604-1626. doi: 10.1109/JAS.2020.1003363
Abstract(837) HTML (411) PDF(35)
Abstract:
In the cyber-physical environment, the clock synchronization algorithm is required to have better expansion for network scale. In this paper, a new measurement model of observability under the equivalent transformation of minimum mean square error (MMSE) is constructed based on basic measurement unit (BMU), which can realize the scaled expansion of MMSE measurement. Based on the state updating equation of absolute clock and the decoupled measurement model of MMSE-like equivalence, which is proposed to calculate the positive definite invariant set by using the theoretical-practical Luenberger observer as the synthetical observer, the local noncooperative optimal control problem is built, and the clock synchronization system driven by the ideal state of local clock can reach the exponential convergence for synchronization performance. Different from the problem of general linear system regulators, the state estimation error and state control error are analyzed in the established affine system based on the set-theory-in-control to achieve the quantification of state deviation caused by noise interference. Based on the BMU for isomorphic state map, the synchronization performance of clock states between multiple sets of representative nodes is evaluated, and the scale of evaluated system can be still expanded. After the synchronization is completed, the state of perturbation system remains in the maximum range of measurement accuracy, and the state of nominal system can be stabilized at the ideal state for local clock and realizes the exponential convergence of the clock synchronization system.
2020, 7(6): 1627-1637. doi: 10.1109/JAS.2018.7511192
Abstract(628) HTML (373) PDF(12)
Abstract:
Negative selection algorithm (NSA) is one of the classic artificial immune algorithm widely used in anomaly detection. However, there are still unsolved shortcomings of NSA that limit its further applications. For example, the nonself-detector generation efficiency is low; a large number of nonself-detector is needed for precise detection; low detection rate with various application data sets. Aiming at those problems, a novel radius adaptive based on center-optimized hybrid detector generation algorithm (RACO-HDG) is put forward. To our best knowledge, radius adaptive based on center optimization is first time analyzed and proposed as an efficient mechanism to improve both detector generation and detection rate without significant computation complexity. RACO-HDG works efficiently in three phases. At first, a small number of self-detectors are generated, different from typical NSAs with a large number of self-sample are generated. Nonself-detectors will be generated from those initial small number of self-detectors to make hybrid detection of self-detectors and nonself-detectors possible. Secondly, without any prior knowledge of the data sets or manual setting, the nonself-detector radius threshold is self-adaptive by optimizing the nonself-detector center and the generation mechanism. In this way, the number of abnormal detectors is decreased sharply, while the coverage area of the nonself-detector is increased otherwise, leading to higher detection performances of RACO-HDG. Finally, hybrid detection algorithm is proposed with both self-detectors and nonself-detectors work together to increase detection rate as expected. Abundant simulations and application results show that the proposed RACO-HDG has higher detection rate, lower false alarm rate and higher detection efficiency compared with other excellent algorithms.
2020, 7(6): 1638-1648. doi: 10.1109/JAS.2019.1911531
Abstract(1040) HTML (539) PDF(76)
Abstract:
This paper investigates the sliding mode control (SMC) problem for a class of discrete-time nonlinear networked Markovian jump systems (MJSs) in the presence of probabilistic denial-of-service (DoS) attacks. The communication network via which the data is propagated is unsafe and the malicious adversary can attack the system during state feedback. By considering random Denial-of-Service attacks, a new sliding mode variable is designed, which takes into account the distribution information of the probabilistic attacks. Then, by resorting to Lyapunov theory and stochastic analysis methods, sufficient conditions are established for the existence of the desired sliding mode controller, guaranteeing both reachability of the designed sliding surface and stability of the resulting sliding motion. Finally, a simulation example is given to demonstrate the effectiveness of the proposed sliding mode control algorithm.
2020, 7(6): 1649-1661. doi: 10.1109/JAS.2020.1003423
Abstract(905) HTML (414) PDF(44)
Abstract:
In this paper, we present a tensor least square based model for sand/sandstorm removal in images. The main contributions of this paper are as follows. First, an important intrinsic natural feature of outdoor scenes free of sand/sandstorm is found that the outlines in RGB channels are somewise similar, which discloses the physical validation using the tensor instead of the matrix. Second, a tensor least square optimization model is presented for the decomposition of edge-preserving base layers and details. This model not only decomposes the color image (taken as an inseparable indivisibility) in X, Y directions, but also in Z direction, which meets the statistical feature of natural scenes and can physically disclose the intrinsic color information. The model’s advantages are twofold: one is the decomposition of edge-preserving base layers and details that can be employed for contrast enhancement without artificial halos, and the other one is the color driving ability that makes the enhanced images as close to natural images as possible via the inherent color structure. Thirdly, the tensor least square optimization model based image enhancement scheme is discussed for the sandstorm weather images. Finally, the experiments and comparisons with the state-of-the-art methods on real degraded images under sandstorm weather are shown to verify our method’s efficiency.
2020, 7(6): 1662-1674. doi: 10.1109/JAS.2020.1003426
Abstract(805) HTML (376) PDF(59)
Abstract:
This paper studies the problem of optimal parallel tracking control for continuous-time general nonlinear systems. Unlike existing optimal state feedback control, the control input of the optimal parallel control is introduced into the feedback system. However, due to the introduction of control input into the feedback system, the optimal state feedback control methods can not be applied directly. To address this problem, an augmented system and an augmented performance index function are proposed firstly. Thus, the general nonlinear system is transformed into an affine nonlinear system. The difference between the optimal parallel control and the optimal state feedback control is analyzed theoretically. It is proven that the optimal parallel control with the augmented performance index function can be seen as the suboptimal state feedback control with the traditional performance index function. Moreover, an adaptive dynamic programming (ADP) technique is utilized to implement the optimal parallel tracking control using a critic neural network (NN) to approximate the value function online. The stability analysis of the closed-loop system is performed using the Lyapunov theory, and the tracking error and NN weights errors are uniformly ultimately bounded (UUB). Also, the optimal parallel controller guarantees the continuity of the control input under the circumstance that there are finite jump discontinuities in the reference signals. Finally, the effectiveness of the developed optimal parallel control method is verified in two cases.
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2022-05-28 09:44:54
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https://leanprover-community.github.io/archive/stream/113489-new-members/topic/names.20of.20theorems.20in.203.2E6.html
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## Stream: new members
### Topic: names of theorems in 3.6
#### Michael Beeson (Jul 13 2020 at 07:35):
section 3.6 of Theorem Proving in Lean has a bunch of propositional identities and says "you can use them in your proofs".
So where can I find a list of names of these theorems, as I will need their names to use them, right?
For example I have h: P->Q and h2 : P', where P' is almost the same as P but two conjuncts somewhere deep inside
have the wrong order. So I want to rewrite it using p \and q = q \and p.
#### Rob Lewis (Jul 13 2020 at 07:42):
You can use rw [and_comm] at h2. If there are other ands around you might need to give it more information, e.g. rw [and_comm p q].
#### Jasmin Blanchette (Jul 13 2020 at 07:47):
That solves the specific example Michael gave, but not the general problem that the identities are not named.
#### Rob Lewis (Jul 13 2020 at 07:53):
I don't know why they're not named there. Maybe guessing them is good practice to learn the mathlib naming conventions? You can find a lot of them with e.g. #find _ ∧ _ ↔ _ ∧ _.
#### Kenny Lau (Jul 13 2020 at 08:16):
https://github.com/leanprover-community/leanprover-community.github.io/pull/93
#### Michael Beeson (Jul 13 2020 at 19:11):
I tried rw [and_comm] at h2 as suggested. It seems you do need to supply arguments to and_comm, which has a universally quantified type. The conjunction I want to commute is inside a quantifier so I used simp_rw. But one of the propositions I want to commute mentions the quantified variable u. And I get the error message "unknown variable u". What now?
#### Jalex Stark (Jul 13 2020 at 19:15):
i don't have advice, but I probably would if you posted code
#### Jalex Stark (Jul 13 2020 at 19:16):
when I get in situations where I want to do things like rw add_comm, I try to reduce to a subproblem that tauto can solve by itself
#### Michael Beeson (Jul 13 2020 at 19:29):
tauto is not available as I am working with intuitionistic logic. There is ifinish but it usually fails or times out. I understand that I should post a working example if I want specific advice. Maybe I will do that later. For now I'm just going to tear the formulas down and get on with the proof.
#### Jalex Stark (Jul 13 2020 at 19:43):
i think tauto is intuitionistic
#### Jalex Stark (Jul 13 2020 at 19:43):
and tauto! is not
#### Jalex Stark (Jul 13 2020 at 19:43):
tactic#tautology says that tauto! uses LEM and implies that tauto does not, but maybe you're worried about something else
#### Jalex Stark (Jul 13 2020 at 19:46):
if "tear the formulas down" means "get rid of the quantifiers with e.g. intro, function application", then I agree that's a fine way to proceed
Last updated: May 14 2021 at 06:16 UTC
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2021-05-14 07:22:23
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https://search.r-project.org/CRAN/refmans/corpustools/html/tCorpus_data.html
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tCorpus_data {corpustools} R Documentation
## Methods and functions for viewing, modifying and subsetting tCorpus data
### Details
Get data
$get() Get (by default deep copy) token data, with the possibility to select columns and subset. Instead of copying you can also access the token data with tc$tokens $get_meta() Get meta data, with the possibility to select columns and subset. Like tokens, you can also access meta data with tc$meta get_dtm() Create a document term matrix get_dfm() Create a document term matrix, using the Quanteda dfm format $context() Get a context vector. Currently supports documents or globally unique sentences. Modify The token and meta data can be modified with the set* and delete* methods. All modifications are performed by reference. $set() Modify the token data by setting the values of one (existing or new) column. $set_meta() The set method for the document meta data$set_levels() Change the levels of factor columns. $set_meta_levels() Change the levels of factor columns in the meta data$set_name() Modify column names of token data. $set_meta_name() Delete columns in the meta data$delete_columns() Delete columns. $delete_meta_columns() Delete columns in the meta data Modifying is restricted in certain ways to ensure that the data always meets the assumptions required for tCorpus methods. tCorpus automatically tests whether assumptions are violated, so you don't have to think about this yourself. The most important limitations are that you cannot subset or append the data. For subsetting, you can use the tCorpus$subset method, and to add data to a tcorpus you can use the merge_tcorpora function.
subset() Modify the token and/or meta data using the subset function. A subset expression can be specified for both the token data (subset) and the document meta data (subset_meta). subset_query() Subset the tCorpus based on a query, as used in search_contexts $subset() Like subset, but as an R6 method that changes the tCorpus by reference$subset_query() Like subset_query, but as an R6 method that changes the tCorpus by reference
$n The number of tokens (i.e. rows in the data)$n_meta The number of documents (i.e. rows in the document meta data) $names The names of the token data columns$names_meta The names of the document meta data columns
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2022-05-26 13:37:30
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https://dsp.stackexchange.com/questions/14724/transmission-line-information-loss/14728
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# Transmission Line Information Loss
Suppose we have a communication channel that has a bit-error rate of 50% and the bits are flipped at random such that the receiver has no efficient way to tell which bits have been flipped and there are no error correction bits transmitted with the signal. How do you calculate the information loss (in bits) imposed by the channel? I know there a Shannon formula for doing this but I don't know what it is! Could someone solve this for me with a specific example? Thank you.
• I don't have the formula at hand, but I can tell you that the information rate is 0. This is easy to see once you realize that all output bit streams are equally likely regardless of the input bit stream. – Jim Clay Mar 1 '14 at 2:43
• And is this homework? – Jim Clay Mar 1 '14 at 2:43
• Whether or not any "error correction bits" are transmitted with the signal, the channel capacity is $0$ as Jim Clay has already pointed out. If you want a specific formula that evaluates to $0$, then calculate the capacity from $$C = 1 - H(p) = 1 + p\log_2 p + (1-p)\log_2(1-p)$$ where $p$ is the crossover probability, $\frac{1}{2}$ in this case. – Dilip Sarwate Mar 1 '14 at 3:32
• @JimClay Hi Jim and thank you and Dilip for the insightful comments, and no this is not homework, I am just a beginner in information theory. – William Hird Mar 1 '14 at 4:28
A binary input - binary output channel which flips bits with equal probabilities is known as a Binary Symmetric Channel.
(Image Credit: Wikipedia - Binary Symmetric Channel)
The channel capacity $C$ is the maximum (actually, a supremum) mutual information between the channel inputs and the channel outputs over input distributions. You can't get information through a channel more than its capacity. Intuitively, this says that by designing the way you put inputs into the channel well, you can maximize the amount of stuff coming out of the channel that you can recover (and this limit is the capacity).
Mathematically, this is $$C = \sup_{p(x)} I(X;Y)$$ where $p(x)$ is probability distributions on the input $X$ and $Y$ is the output of the channel.
In this case, you can write the input of the channel as $P(X=0) = q$ and $P(X=1) = 1-q$. With this distribution, you can see that $P(Y=0) = P(X=0 \text{ and no crossover}) + P(X = 1 \text { and crossover}) = q(1-p) + (1-q)p$ and $P(Y=1) = 1-P(Y=0)$. The mutual information between $X$ and $Y$ is given by $I(X;Y)=\sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(x)p(y)}$ where $p(x,y)$ is the joint distribution of $x$ and $y$, and $p(x)$, $p(y)$ are the distributions of $x$ and $y$ respectively. By writing down $I(X;Y)$ and then maximizing over $q$, you will see $C = 1 - H(p)$ where $H(p)$ is the binary entropy function $-p \log_2 p - (1-p) \log_2 (1-p)$. Then, in your case, $p = \frac{1}{2}$ where $H(p) = 1$, so the capacity is $0$.
If you're interested in learning more about this, a few good books to look (at very different levels of difficulty):
• Elements of Information Theory (Second Edition), by Tom Cover and Joy Thomas (Easy, Probably Best Starting Point)
• Information Theory and Reliable Communication, by Robert Gallegar (A bit harder)
• Information Theory: Coding Theorems for Discrete Memoryless Channels by Czisar and Korner (Pretty hard)
• Information Theory by Robert B. Ash (Dover book, Easy)
And an unrelated book which I love, but couldn't keep myself from mentioning, is David Mackay's Information Theory, Inference and Learning Algorithms (free on his website), even though it doesn't go over channel capacity.
• Nice answer :-) – William Hird Mar 1 '14 at 14:31
the information capacity through a channel is, from Shannon,
$$I = \int_0^B \log_2 \left( 1 + \frac{S(f)}{N(f)} \right) df$$
where $I$ is the rate of information passed through the channel in bits per unit time, $B$ is the one-sided bandwidth, $S(f)$ is the power spectrum of the signal and $N(f)$ is the power spectrum of the noise.
• -1 This answer is not useful. – Dilip Sarwate Mar 1 '14 at 3:33
• The channel considered in this problem is a binary symmetric channel, for which this does not apply. – Batman Mar 1 '14 at 5:52
• both of you guys are mistaken, technically. if you make some assumptions (like $S(f)/N(f)$ is constant with $f$, at least for $0<f<B$), you can derive $S/N$ from the problem statement. that equation is the sole governing relationship. you down-arrows speak more about your useful answers than about mine. – robert bristow-johnson Mar 1 '14 at 16:58
• Unfortunately, comments cannot be downvoted, but the answerer's response above is even more nonsensical and useless (and deserving of a downvote) than the original answer. – Dilip Sarwate Mar 1 '14 at 22:47
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2019-07-17 21:51:52
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https://nukephysik101.wordpress.com/2012/07/20/parity-2/
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long time ago, the post on parity is quit a mess and confusing. now, i have a better understanding and try to do it once again.
here, i will clarify 2 things:
1. what is parity.
2. what will happen if parity is hold or violated.
the answer of 1st question is
Parity is a transform that reverse the space dimension.
it sound simple. the tricky part is, there are 2 kinds of transform: active or passive. in short, active transform is change of vector. while pass is change of coordinate. To avoid confusion and make me and reader thinking to be the same, i use active transform in here.
in 1-D space, the parity is just reverse of the direction. in 2-D space, parity is equal a rotation of 180 degree on the plane, thus, it is not really a parity but as a special case for rotation. in 3-D space, parity is just like changing our left-hand to right hand and vice versa. many people will say that, it is same as MIRROR transform, which is a plan inverse. But it is not true. the correct description is a mirror transform + 180 rotation around the normal of the mirror plan. Thus, if we only concern about the shape instead of orientation, a mirror transform is ok. Nevertheless, when we talking current loop, the direction is important that, the magnet is a mirrored inside the mirror. and we knew that all magnetic field is either by current loop or spin. in 4-D, well, i don’t know.
To answer the 2nd question, we have to know what is parity conservation. the parity conservation is that, thing keep the same under parity transform. i.e.
$P \Psi(x) = \Psi(-x) = \pm \Psi(x)$
the plus sign means EVEN parity, and minus sign means ODD parity. people knowing some maths will know that, it is corresponding to even function and odd function.
Lets look at the parity of electromagnetic phenomena. lets there is a current loop, rotating at left hand. the B field will be pointing down. under MIRROR (not parity) by building an identical current loop which rotates in right hand, then the magnetic field pointing upward. Wait! is it mean magnetic field violated parity? NO, because it is the pseudovector nature of magnetic field, which, keep the direction perpendicular to the plan of mirror and change sign on parallel to the mirror plan. and as i said before, a parity is mirror + 180 rotation. thus, the parity is conserved. (polar vector has opposites properties that, under mirror, the component parallel to mirror plan is kept but the component perpendicular reversed. well this is what mirror transform was defined)
Now, we found that, all pseudovector are even parity and polar vector are odd parity. (although i don’t have a prove, i believe this is strongly related to spin1/2 and spin 1, and every pseudovector is formed by polar vector. )
Thus, a parity conservation mean that, all physics are same under parity. The other thing to say so, is, the mirrored world is same after we rotated it back.
$P=MR=RM \Rightarrow RP=PR=M$
in order to understood what is parity conserved or violated, we study elastics scattering. if we only have polar vector, a parity violation means the space has preference direction, i.e. the symmetry broken by a pseudovector field. but that contradict our assumption of “only polar vector”. thus, it is impossible to have parity violation for polar vectors. (i am not sure it is a prove or not)
A famous example of parity violation in nature is the experiment on Cobalt-60 nuclei, which will undergoes beta decay.
*** at the beginning, i was very confused. one major confusion is that, the active transform or passive transform and the mirror world. i found that, inside the mirror, we should NOT use “left-hand rule” instead of “right-hand rule”. if we do a mirror transform, and use “left-hand rule” , the transform will be cancelled. because, changing the “handiness” is same as passive transform.
another difficulty is that, a mirror transform is not parity transform, but most popular science mixed them up. a Major difference between mirror and parity is: there are 3 difference reflection plans, but parity is a “point inversion”. when you select difference plan of reflection, your current loop will keep the same or change direction, that make it sometimes keep orientation, but sometimes not.
that difficulty raised because lack of understanding the proper transform of pesudo vector under mirror transform. we the help of the picture, we can now easily imagine parity as mirror.
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2017-09-25 04:29:14
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https://itprospt.com/num/10759696/011-012-413-021-02-023-031-032-033411-2031-012-2032-013
|
3
# 011 012 413 021 02 023 031 032 033411 2031 012 2032 013 2as3 021 022 023 C = 031 032 033012 -Sar 033C13 3a2 031011 012 5011 013 021 022 023 021 022 5031 023 011 01...
## Question
###### 011 012 413 021 02 023 031 032 033411 2031 012 2032 013 2as3 021 022 023 C = 031 032 033012 -Sar 033C13 3a2 031011 012 5011 013 021 022 023 021 022 5031 023 011 012 413 031 032 5a31 03 - 031 032 033 Given |A =5- Find the determinants of the following matrices011 012 013021 (22 023 a11 012 C131. [Bl2 ICl -3.|Dl =4.IF|5.|G] =6A|7.IA-3021
011 012 413 021 02 023 031 032 033 411 2031 012 2032 013 2as3 021 022 023 C = 031 032 033 012 -Sar 033 C13 3a2 031 011 012 5011 013 021 022 023 021 022 5031 023 011 012 413 031 032 5a31 03 - 031 032 033 Given |A =5- Find the determinants of the following matrices 011 012 013 021 (22 023 a11 012 C13 1. [Bl 2 ICl - 3.|Dl = 4.IF| 5.|G] = 6A| 7.IA -3021
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2022-06-25 19:57:11
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https://brilliant.org/problems/dfractextfractiontextfraction-3/
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$$\dfrac{\text{Fraction}}{\text{Fraction}}$$
Logic Level 1
$\large \dfrac{\frac{\bigcirc}{\bigcirc}}{\frac{\bigcirc}{\bigcirc}} + \frac{\frac{\bigcirc}{\bigcirc}}{\frac{\bigcirc}{\bigcirc}} = \text{Integer}$
Only 1-10 distinct digits are allowed to use. Can we make the equation true?
×
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2018-07-16 17:45:15
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https://socratic.org/questions/how-do-you-find-the-equation-of-the-line-given-m-3-3-2
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# How do you find the equation of the line given m=3, (3,2)?
Feb 19, 2017
$y = 3 x - 7$
#### Explanation:
The equation of a line in $\textcolor{b l u e}{\text{point-slope form}}$ is.
$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y - {y}_{1} = m \left(x - {x}_{1}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where m represents the slope and $\left({x}_{1} , {y}_{1}\right) \text{ a point on the line}$
$\text{here "m=3" and } \left({x}_{1} , {y}_{1}\right) = \left(3 , 2\right)$
substitute these values into the equation.
$y - 2 = 3 \left(x - 3\right) \leftarrow \textcolor{red}{\text{ in point-slope form}}$
distributing and simplifying gives an alternative version of the equation.
$y - 2 = 3 x - 9$
$y = 3 x - 9 + 2$
$\Rightarrow y = 3 x - 7 \leftarrow \textcolor{red}{\text{ slope-intercept form}}$
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2022-01-22 18:44:32
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http://math.stackexchange.com/questions/516219/finding-out-the-area-of-a-triangle-if-the-coordinates-of-the-three-vertices-are?answertab=oldest
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# finding out the area of a triangle if the coordinates of the three vertices are given
What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in x-y plane?
One approach is to find the length of each side from the coordinates given and then apply the heron's formula.Is this the best way possible?
Is it possible to compare the area of triangles with their coordinates provided without actually calculating their respective areas?
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What you are looking for is the shoelace formula:
$$\frac{1}{2} \big| (x_A - x_C) (y_B - y_A) - (x_A - x_B) (y_C - y_A) \big|$$
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Heron's formula is inefficient; there is in fact a direct formula. If the triangle has one vertex at the origin, and the other two vertices are $(a,b)$ and $(c,d)$, the formula for its area is $$A = \frac{\left| ad - bc \right|}{2}$$
To get a formula where the vertices can be anywhere, just subtract the coordinates of the third vertex from the coordinates of the other two (translating the triangle) and then use the above formula.
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if $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are the vertices of a triangle then its area is given by
$$|\frac12(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2))|$$
-
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2014-10-30 13:29:10
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http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=zvmmf&paperid=10488&option_lang=eng
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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki
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Zh. Vychisl. Mat. Mat. Fiz.: Year: Volume: Issue: Page: Find
Zh. Vychisl. Mat. Mat. Fiz., 2016, Volume 56, Number 11, Pages 1902–1930 (Mi zvmmf10488)
Convergence of a family of solutions to a Fujita-type equation in domains with cavities
S. V. Pikulin
Dorodnicyn Computing Center, Federal Research Center "Computer Science and Control", Russian Academy of Sciences, Moscow, Russia
Abstract: The Dirichlet problem for a Fujita-type equation, i.e., a second-order quasilinear uniformly elliptic equation is considered in domains $\Omega_\varepsilon$ with spherical or cylindrical cavities of characteristic size $\varepsilon$. The form of the function in the condition on the cavities' boundaries depends on $\varepsilon$. For $\varepsilon$ tending to zero and the number of cavities increasing simultaneously, sufficient conditions are established for the convergence of the family of solutions $\{u_\varepsilon(x)\}$ of this problem to the solution $u(x)$ of a similar problem in the domain $\Omega$ with no cavities with the same boundary conditions imposed on the common part of the boundaries $\partial\Omega$ and $\partial\Omega_\varepsilon$. Convergence rate estimates are given.
Key words: convergence of a family of solutions, nonlinear Fujita-type equation, domains with spherical or cylindrical cavities, convergence rate estimates for solutions.
Funding Agency Grant Number Russian Foundation for Basic Research 13-01-00923_à Russian Academy of Sciences - Federal Agency for Scientific Organizations 3
DOI: https://doi.org/10.7868/S0044466916110119
Full text: PDF file (474 kB)
References: PDF file HTML file
English version:
Computational Mathematics and Mathematical Physics, 2016, 56:11, 1872–1900
Bibliographic databases:
UDC: 519.63
Citation: S. V. Pikulin, “Convergence of a family of solutions to a Fujita-type equation in domains with cavities”, Zh. Vychisl. Mat. Mat. Fiz., 56:11 (2016), 1902–1930; Comput. Math. Math. Phys., 56:11 (2016), 1872–1900
Citation in format AMSBIB
\Bibitem{Pik16} \by S.~V.~Pikulin \paper Convergence of a family of solutions to a Fujita-type equation in domains with cavities \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2016 \vol 56 \issue 11 \pages 1902--1930 \mathnet{http://mi.mathnet.ru/zvmmf10488} \crossref{https://doi.org/10.7868/S0044466916110119} \elib{https://elibrary.ru/item.asp?id=27148427} \transl \jour Comput. Math. Math. Phys. \yr 2016 \vol 56 \issue 11 \pages 1872--1900 \crossref{https://doi.org/10.1134/S0965542516110099} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000389803600005} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85000415071}
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2021-11-27 18:34:31
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https://www.emathhelp.net/notes/calculus-1/lhopitals-rule/indeterminate-forms-of-type-infinity-by-infinity/
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# Indeterminate Forms of Type $\frac{\infty}{\infty}$
## Related calculator: Limit Calculator
Similarly there are limit of functions that represent indeterminate form of type $\frac{\infty}{\infty}$, but can't be calculated using algebraic manipulations.
However, there is corresponding L'Hopital's Rule that allows to handle indeterminate form of type $\frac{\infty}{\infty}$.
Second L'Hopital’s Rule. Suppose ${f{{\left({x}\right)}}}$ and ${g{{\left({x}\right)}}}$ are differentiable on ${\left({a},{b}\right]}$ and ${g{'}}{\left({x}\right)}\ne{0}$ on ${\left({a},{b}\right]}$. If $\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}=\infty$ and $\lim_{{{x}\to{a}}}{g{{\left({x}\right)}}}=\infty$, then $\lim_{{{x}\to{a}}}\frac{{{f{{\left({x}\right)}}}}}{{{g{{\left({x}\right)}}}}}=\lim_{{{x}\to{a}}}\frac{{{f{'}}{\left({x}\right)}}}{{{g{'}}{\left({x}\right)}}}$ if the limit on the right side exists (or is $\infty$ or $-\infty$ ).
It is especially important to verify the conditions before using L'Hopital's Rule.
Second L'Hopital’s Rule is also valid for one-sided limits and for limits at infinity or negative infinity; that is, "${x}\to{a}$" can be replaced by any of the following symbols: ${x}\to{{a}}^{+}$, ${x}\to{{a}}^{{-{}}}$, ${x}\to\infty$, ${x}\to-\infty$.
Example 1. Find $\lim_{{{x}\to\infty}}\frac{{{\ln{{\left({x}\right)}}}}}{{\sqrt{{{x}}}}}$.
Since $\lim_{{{x}\to\infty}}{\left({\ln{{\left({x}\right)}}}\right)}=\infty$ and $\lim_{{{x}\to\infty}}\sqrt{{{x}}}=\infty$ then we can apply L'Hopital's Rule:
$\lim_{{{x}\to\infty}}\frac{{{\ln{{\left({x}\right)}}}}}{{\sqrt{{{x}}}}}=\lim_{{{x}\to\infty}}\frac{{{\left({\ln{{\left({x}\right)}}}\right)}'}}{{{\left(\sqrt{{{x}}}\right)}'}}=\lim_{{{x}\to\infty}}\frac{{\frac{{1}}{{x}}}}{{\frac{{1}}{{{2}\sqrt{{{x}}}}}}}=\lim_{{{x}\to\infty}}\frac{{2}}{{\sqrt{{{x}}}}}={0}$.
Sometimes we need to apply L'Hopital's rule more than once.
Example 2. Find $\lim_{{{x}\to\infty}}\frac{{{{e}}^{{x}}}}{{{{x}}^{{2}}}}$.
Since $\lim_{{{x}\to\infty}}{{e}}^{{x}}=\infty$ and $\lim_{{{x}\to\infty}}{{x}}^{{2}}=\infty$ then we can use L'Hopital's Rule:
$\lim_{{{x}\to\infty}}\frac{{{{e}}^{{x}}}}{{{{x}}^{{2}}}}=\lim_{{{x}\to\infty}}\frac{{{\left({{e}}^{{x}}\right)}'}}{{{\left({{x}}^{{2}}\right)}'}}=\lim_{{{x}\to\infty}}\frac{{{{e}}^{{x}}}}{{{2}{x}}}$.
Since ${{e}}^{{x}}\to\infty$ and ${2}{x}\to\infty$ as ${x}\to\infty$ then we still have indeterminate form of type $\frac{{\infty}}{{\infty}}$ and we apply L'Hopital's rule once more:
$\lim_{{{x}\to\infty}}\frac{{{{e}}^{{x}}}}{{{2}{x}}}=\lim_{{{x}\to\infty}}\frac{{{\left({{e}}^{{x}}\right)}'}}{{{\left({2}{x}\right)}'}}=\lim_{{{x}\to\infty}}\frac{{{{e}}^{{x}}}}{{2}}=\infty$.
Example 3. Find $\lim_{{{x}\to\infty}}\frac{{{1}+\frac{{1}}{{x}}}}{{{x}+{1}}}$.
If we blindly attempt to apply L'Hopital's Rule, we will get that $\lim_{{{x}\to\infty}}\frac{{{\left({1}+\frac{{1}}{{x}}\right)}'}}{{{\left({x}+{1}\right)}'}}=\lim_{{{x}\to\infty}}\frac{{{1}-\frac{{1}}{{{x}}^{{2}}}}}{{{1}}}={1}$.
THIS IS WRONG! We can't apply L'Hopital's rule because $\lim_{{{x}\to\infty}}{\left({1}+\frac{{1}}{{x}}\right)}={1}$ and we don't have indeterminate form.
In fact $\lim_{{{x}\to\infty}}\frac{{{1}+\frac{{1}}{{x}}}}{{{x}+{1}}}=\lim_{{{x}\to\infty}}\frac{{\frac{{{x}+{1}}}{{x}}}}{{{x}+{1}}}=\lim_{{{x}\to\infty}}\frac{{1}}{{x}}={0}$.
Example 3 shows what can go wrong if you use L'Hopital's Rule without thinking checking conditions of theorem.
Now let's see what will be if we ignore condition that limit of ratio of derivatives should exist.
Example 4. Calculate $\lim_{{{x}\to\infty}}\frac{{{x}+{\sin{{\left({x}\right)}}}}}{{x}}$.
We have indeterminate form here, so can apply L'Hopital's Rule:
$\lim_{{{x}\to\infty}}\frac{{{\left({x}+{\sin{{\left({x}\right)}}}\right)}'}}{{{x}'}}=\lim_{{{x}\to\infty}}\frac{{{1}+{\cos{{\left({x}\right)}}}}}{{1}}$.
Since ${\cos{{\left({x}\right)}}}$ oscillates infinitely many times as ${x}\to\infty$ then $\lim_{{{x}\to\infty}}{\cos{{\left({x}\right)}}}$ doesn't exist. Therefore $\lim_{{{x}\to\infty}}{\left({1}+{\cos{{\left({x}\right)}}}\right)}$ doesn't exist.
However, initial limit exist: $\lim_{{{x}\to\infty}}\frac{{{x}+{\sin{{\left({x}\right)}}}}}{{x}}=\lim_{{{x}\to\infty}}{\left({1}+\frac{{\sin{{\left({x}\right)}}}}{{x}}\right)}={1}$.
So, we need to be sure that limit of ratio of derivative exists, otherwise L'Hopital's Rule is inapplicable.
Other limits can be found using L'Hopital's Rule but are more easily found by other methods. So when evaluating any limit, you should consider other methods before using L'Hopital's Rule.
Example 5. Find $\lim_{{{x}\to\infty}}\frac{{{{x}}^{{2}}-{4}}}{{{2}{{x}}^{{2}}-{2}}}$.
Applying L'Hopital's Rule gives $\lim_{{{x}\to\infty}}\frac{{{{x}}^{{2}}-{4}}}{{{2}{{x}}^{{2}}-{2}}}=\lim_{{{x}\to\infty}}\frac{{{\left({{x}}^{{2}}-{4}\right)}'}}{{{\left({2}{{x}}^{{2}}-{2}\right)}'}}=\lim_{{{x}\to\infty}}\frac{{{2}{x}}}{{{4}{x}}}=\lim_{{{x}\to\infty}}\frac{{1}}{{2}}=\frac{{1}}{{2}}$.
But it is more natural to use algebraic manipulations: $\lim_{{{x}\to\infty}}\frac{{{{x}}^{{2}}{\left({1}-\frac{{4}}{{{x}}^{{2}}}\right)}}}{{{{x}}^{{2}}{\left({2}-\frac{{2}}{{{x}}^{{2}}}\right)}}}=\lim_{{{x}\to\infty}}\frac{{{1}-\frac{{4}}{{{x}}^{{2}}}}}{{{2}-\frac{{2}}{{{x}}^{{2}}}}}=\frac{{1}}{{2}}$.
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2022-07-01 05:25:11
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