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# Unlinking particle system datablocks from python

I have a problem which has been solved on this site before, right here:

Except for the fact that I am setting up my whole scene from python, and having to manually go to every object with a particle effect, click on the users icon, and set the start and end frames by hand, is becoming very tedious, when everything else in the scene has been accomplished by script.

My situation: I have a helicopter from which I am shooting rockets. I am creating the scene by reading an xml file output from another application. I start with a blend file that has one helicopter and one rocket, and then make a duplicate of the helicopter and as many duplicates of the rocket as are called for in the xml file.

The problem: I have a particle system attached to my sample rocket. I need to set the start and end frames of each rocket's particle system according to the time it was fired and the time it hit something and exploded. When I make a duplicate of the rocket, it makes a linked copy of the same particle system, so when I set the start and end frames, all the rockets end up with the same start and end frame.

I have tried every permutation of make_single_user that I can think of, but actually it doesn't even do what I want when I call it through the interface, so I'm not sure if it affects particle systems. In any event it does nothing for me in python.

Anybody know how to do this?

Copy the particle system settings

Here is a simple test script on what I believe you want to achieve. Select object with at least one particle system. Copies the context object, copies the settings of particle system 0, changes the start frame on copy to 22 and sets this to particle system 0 on copy.

import bpy

context = bpy.context
scene = context.scene
ob = context.object
# copy object
obcopy = ob.copy()
# copy particle settings
ps = ob.particle_systems[0].settings.copy()
ps.frame_start = 22
# new settings on copy
obcopy.particle_systems[0].settings = ps

Expanding on @batFINGER answer. If the second object (obcopy) does not have a modifier, then you would need to create a modifier before duplicating the particle settings from the original object (ob):
# Duplicate particle settings


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# TikZ Diagrams in Math Mode

I've attached a picture which shows what I want to be able to do. I have been making simple Hackenbush diagrams as follows:

\begin{tikzpicture}
\node[hackennode] (Bottom Blue) at (0,0) {};
\node[hackennode] (Top Blue) at (0,1) {};
\draw[hackenline,blue] (Bottom Blue) -- (Top Blue);
\end{tikzpicture}

I'd like to be able to use them along with normal mathematical symbols, as the picture shows. Thanks!

(I also don't know how to put the dotted line in, but I'm less concerned about that.)

-
The code posted in questions (and answers) should always compile as is. In particular, you should include the definition of hackennode and hackenline from my answer to your last question –  Caramdir Feb 15 '11 at 2:12

You can simply nest tikzpictures inside math environments. Basically, the only thing you have to consider is to add an appropriate baseline to get vertical centering right. Here is the code, explanation will come afterwards.

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\tikzstyle{hackennode}=[draw,circle,fill=white,inner sep=0,minimum size=4pt]
\tikzstyle{hackenline}=[line width=3pt]

{\Large
$\begin{tikzpicture}[baseline=-0.65ex,scale=0.5] \draw[densely dashed] (-1,-1) -- (1,-1); \node[hackennode] (middle) at ( 0, 0) {}; \node[hackennode] (left) at (-0.5,-1) {}; \node[hackennode] (right) at ( 0.5,-1) {}; \node[hackennode] (top) at ( 0, 1) {}; \draw[hackenline,blue] (left) -- (middle) -- (right); \draw[hackenline,red] (middle) -- (top); \end{tikzpicture} = \left\{ \begin{tikzpicture}[baseline=-0.65ex,scale=0.5] \draw[densely dashed] (-0.5,-1) -- (1,-1); \node[hackennode] (middle) at ( 0, 0) {}; \node[hackennode] (right) at ( 0.5,-1) {}; \node[hackennode] (top) at ( 0, 1) {}; \draw[hackenline,blue] (middle) -- (right); \draw[hackenline,red] (middle) -- (top); \end{tikzpicture} \tikz[baseline=-0.65ex,scale=0.5] \node[inner sep=0] at (0,-1) {,\,}; \begin{tikzpicture}[baseline=-0.65ex,scale=0.5] \draw[densely dashed] (-1,-1) -- (0.5,-1); \node[hackennode] (middle) at ( 0, 0) {}; \node[hackennode] (left) at (-0.5,-1) {}; \node[hackennode] (top) at ( 0, 1) {}; \draw[hackenline,blue] (middle) -- (left); \draw[hackenline,red] (middle) -- (top); \end{tikzpicture} \middle| \begin{tikzpicture}[baseline=-0.65ex,scale=0.5] \draw[densely dashed] (-1,-1) -- (1,-1); \node[hackennode] (middle) at ( 0, 0) {}; \node[hackennode] (left) at (-0.5,-1) {}; \node[hackennode] (right) at ( 0.5,-1) {}; \draw[hackenline,blue] (left) -- (middle) -- (right); \end{tikzpicture} \right\}$
}% End group with \Large
\end{document}

• The baseline option determines which height in the picture corresponds to the baseline in the surrounding text. We want the point (0,0) to be at the height of the equality sign, which is about 0.65ex above the baseline (in Computer Modern).
• The scale property (which doesn't affect line width and node size) is useful to get pictures in exactly the size you want them, without having to fiddle with the coordinates.
• The comma is set using TikZ to easily achieve the correct vertical position.
• I added \Large to get a larger equal sign and comma.
• Dashed lines are achieved with the option (surprise!) dashed. Now guess what densely dashed means.
• In order to have the vertices overpaint the dashed line, I added a fill=white option to hackennode.
• Martin's suggestion to wrap the tikzpictures in \mathord is a good one. You should do that to get better spacing.

Note, that in your picture the math symbols were actually produced with the graphics program and not with the text typesetting program (compare the braces and equal sign). This is of course possible to do with TikZ, but I don't see a good reason to do so.

-

You can use the tikzpictures inside \mathord{ } like normal math symbols. See below for an example. If you need the same diagram several times I would suggest you save it into a macro or box.

% Draws the shown example (but without circles)
\documentclass{standalone}
\usepackage{tikz}
\begin{document}

$\mathord{% \begin{tikzpicture}[baseline={(0,1)}] \draw[dashed] (0,0) -- (2,0); \draw[blue,very thick] (.5,0) -- (1,1) -- (1.5,0); \draw[red,very thick] (1,1) -- (1,2); \end{tikzpicture}% } = \left\{ \mathord{% \begin{tikzpicture}[baseline={(0,1)}] \draw[dashed] (0,0) -- (2,0); \draw[blue,very thick] (.75,0) -- (.5,1) (1.5,1) -- (1.25,0); \draw[red,very thick] (.5,1) -- (.5,2) (1.5,1) -- (1.5,2); \end{tikzpicture}% } \left| \mathord{% \begin{tikzpicture}[baseline={(0,1)}] \draw[dashed] (0,0) -- (2,0); \draw[blue,very thick] (.5,0) -- (1,1) -- (1.5,0); \end{tikzpicture}% } \right. \right\}$

\end{document}

Result:

-

format: plain


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# tf.contrib.nn.sampled_sparse_softmax_loss

tf.contrib.nn.sampled_sparse_softmax_loss(
weights,
biases,
labels,
inputs,
num_sampled,
num_classes,
sampled_values=None,
remove_accidental_hits=True,
partition_strategy='mod',
name='sampled_sparse_softmax_loss'
)

Computes and returns the sampled sparse softmax training loss.

This is a faster way to train a softmax classifier over a huge number of classes.

This operation is for training only. It is generally an underestimate of the full softmax loss.

A common use case is to use this method for training, and calculate the full softmax loss for evaluation or inference. In this case, you must set partition_strategy="div" for the two losses to be consistent, as in the following example:

if mode == "train":
loss = tf.nn.sampled_sparse_softmax_loss(
weights=weights,
biases=biases,
labels=labels,
inputs=inputs,
...,
partition_strategy="div")
elif mode == "eval":
logits = tf.matmul(inputs, tf.transpose(weights))
logits = tf.nn.bias_add(logits, biases)
loss = tf.nn.sparse_softmax_cross_entropy_with_logits(
labels=tf.squeeze(labels),
logits=logits)

Also see Section 3 of Jean et al., 2014 (pdf) for the math.

#### Args:

• weights: A Tensor of shape [num_classes, dim], or a list of Tensor objects whose concatenation along dimension 0 has shape [num_classes, dim]. The (possibly-sharded) class embeddings.
• biases: A Tensor of shape [num_classes]. The class biases.
• labels: A Tensor of type int64 and shape [batch_size, 1]. The index of the single target class for each row of logits. Note that this format differs from the labels argument of nn.sparse_softmax_cross_entropy_with_logits.
• inputs: A Tensor of shape [batch_size, dim]. The forward activations of the input network.
• num_sampled: An int. The number of classes to randomly sample per batch.
• num_classes: An int. The number of possible classes.
• sampled_values: a tuple of (sampled_candidates, true_expected_count, sampled_expected_count) returned by a *_candidate_sampler function. (if None, we default to log_uniform_candidate_sampler)
• remove_accidental_hits: A bool. whether to remove "accidental hits" where a sampled class equals one of the target classes. Default is True.
• partition_strategy: A string specifying the partitioning strategy, relevant if len(weights) > 1. Currently "div" and "mod" are supported. Default is "mod". See tf.nn.embedding_lookup for more details.
• name: A name for the operation (optional).

#### Returns:

A batch_size 1-D tensor of per-example sampled softmax losses.


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# Chapter 9: Basics of hypercontractivity

In 1970, Bonami proved the following central result:

The Hypercontractivity Theorem Let $f : \{-1,1\}^n \to {\mathbb R}$ and let ${1 \leq p \leq q \leq \infty}$. Then $\|\mathrm{T}_\rho f\|_q \leq \|f\|_p$ for $0 \leq \rho \leq \sqrt{\tfrac{p-1}{q-1}}$.

As stated, this theorem may look somewhat opaque. In this chapter we consider some special cases of it which are easier to understand, easier to prove, and which encompass almost all of the theorem’s uses. The proof of the full theorem is deferred to Chapter 10. The special cases in this chapter are the following:

Bonami Lemma Let $f : \{-1,1\}^n \to {\mathbb R}$ have degree $k$. Then $\|f\|_4 \leq \sqrt{3}^k \|f\|_2$

The fundamental idea of this statement is that if ${\boldsymbol{x}} \sim \{-1,1\}^n$ and $f : \{-1,1\}^n \to {\mathbb R}$ has low degree then the random variable $f({\boldsymbol{x}})$ is quite “reasonable”; e.g., it is “nicely” distributed around its mean. The Bonami Lemma has a very easy inductive proof and is already powerful enough to obtain many of the well-known applications of “hypercontractivity”, including the KKL Theorem (proven at the end of this chapter) and the Invariance Principle.

$\mathbf{(2,q)}$-Hypercontractivity Theorem Let $f : \{-1,1\}^n \to {\mathbb R}$ and let $2 \leq q \leq \infty$. Then $\|\mathrm{T}_{1/\sqrt{q-1}} f\|_q \leq \|f\|_2$. As a consequence, if $f$ has degree at most $k$ then $\|f\|_q \leq \sqrt{q-1}^k \|f\|_2$.

This theorem quantifies the extent to which $\mathrm{T}_\rho$ is a “smoothing” operator; equivalently, it gives even more control over the “reasonableness” of low-degree polynomials. Its consequences include a generalization of the Level-$1$ Inequality (from Chapter 5.4) to “Level-$k$ Inequalities”, as well as a Chernoff-like tail bound for low-degree polynomials of random bits.

$\mathbf{(p,2)}$-Hypercontractivity Theorem Let $f : \{-1,1\}^n \to {\mathbb R}$ and let $1 \leq p \leq 2$. Then $\|\mathrm{T}_{\sqrt{p-1}} f\|_2 \leq \|f\|_p$. Equivalently, $\mathbf{Stab}_\rho[f] \leq \|f\|_{1+\rho}^2$ for $0 \leq \rho \leq 1$.

This theorem is actually “equivalent” to the $(2,q)$-Hypercontractivity Theorem by virtue of Hölder’s inequality. When specialized to the case of $f : \{-1,1\}^n \to \{0,1\}$ it gives a precise quantification of the fact that the “noisy hypercube graph” is a “small-set expander”. Qualitatively, this means that if $A \subseteq \{-1,1\}^n$ is “small”, ${\boldsymbol{x}} \sim A$, and $\boldsymbol{y} \sim N_\rho(x)$, then $\boldsymbol{y}$ is very unlikely to be in $A$.

### 2 comments to Chapter 9: Basics of hypercontractivity

• david

Shouldn’t the statement of the Hypercontractivity lemma cover the case $p = q$ as well? (i.e., $p \le q$ instead of $p < q$). It follows from your statemente by continuity, but since you're already using the equality case in the $(2,q)$-Hypercontractivity Theorem it makes sense to include it.

• You’re right, there’s no reason to exclude the $p = q$ case in the statement. Thanks!


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# transformation de fourier simple

Distributions, Analyse de Fourier et Transformation de Laplace - Cours et exercices | Ahmed Lesfari | download | B–OK. Espace de Schwartz pour la transformation de Fourier hypergéométrique . The inverse of Discrete Time Fourier Transform - DTFT is called as the inverse DTFT. Baron Jean Baptiste Joseph Fourier $$\left( 1768-1830 \right)$$ introduced the idea that any periodic function can be represented by a series of sines and cosines which are harmonically related. This website uses cookies to ensure you get the best experience. The "Fast Fourier Transform" (FFT) is an important measurement method in science of audio and acoustics measurement. Fourier Transform is used to analyze the frequency characteristics of various filters. The Python example uses a sine wave with multiple frequencies 1 Hertz, 2 Hertz and 4 Hertz. $\begingroup$ When I was learning about FTs for actual work in signal processing, years ago, I found R. W. Hamming's book Digital Filters and Bracewell's The Fourier Transform and Its Applications good intros to the basics. Asif Khan: 2020-11-14 20:33:22 Hello, I did a fourier series for a function f(x) defined as f(x) = -x -pi x 0, f(x) = 0 0 x pi when i plugged in the results in the calculator I got the same answers for An and Bn when n > 0. QUADRATURE FORMULE QUADRATURE TRANSFORMATION FOURIER RAPIDE ALGORITHME OPTIMAL ESTIMATION ERREUR ANALYSE NUMERIQUE MATHEMATIQUES APPLIQUEES Keyword (en) NUMERICAL ANALYSIS APPLIED MATHEMATICS Keyword (es) ANALYSIS NUMERICA MATEMATICAS APPLICADAS Classification Pascal 001 Exact sciences and technology / 001A … 1996, 167 p., ref : 48 ref. ft De T ω ω π ω − − = = = ∫ ... • DCT is a Fourier-related transform similar to the DFT but using only real numbers • DCT is equivalent to DFT of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function Une durée d'analyse infiniment longue associe par contre à chaque fréquence une valeur spectrale, de … I dusted off an old algorithms book and looked into it, and enjoyed reading about … Free Fourier Series calculator - Find the Fourier series of functions step-by-step. The Fast Fourier Transform (FFT) is one of the most important algorithms in signal processing and data analysis. !f(!) 16.1 Fourier Series The period waveform of function f(t) is repetition over time such that f(t-mT) = f(t) m = 1, 2, 3, ….. (16.1) where T is the period. Cet article décrit un algorithme de transformation de Fourier rapide proposé récemment, qui présente des avantages en complexité de calcul, occupation mémoire et régularité de structure. The Fourier Transform is one of deepest insights ever made. If X is a multidimensional array, then fft(X) treats the values along the first array dimension whose size does not equal 1 as vectors and returns the Fourier transform of each vector. methods to generate Fourier series and the application of Fourier series in ac steady-state circuit analysis. Details about these can be found in any image processing or signal processing textbooks. Every circle rotating translates to a simple sin or cosine wave. A Tutorial on Fourier Analysis Linear Combination In the interval [u 1,u 2] a function Θ(u) can be written as a linear combination: Θ(u) = X∞ i=0 α iψ i(u) where functions ψ i(u) make up a set of simple elementary functions. Dieser zeigt, dass die Fourier 's Annahme für begrenzt Funktionen nicht halten. 318 Chapter 4 Fourier Series and Integrals Zero comes quickly if we integrate cosmxdx = sinmx m π 0 =0−0. and f has period 2π. Le drvelopment des algorithmes principaux (Cooley-Tukey, split-radix FFT, algorithmes des facteurs premiers, et transform6e rapide de Winograd) est drcrit. Fourier Series. A second motivation is the fact that the Fourier transformation for a nite interval (4) is simple and identical with the limit T!1in this case. Answer: f(x) ∼ 4 π ∞ n=0 sin(2n+1)x (2n+1). The Python module numpy.fft has a function ifft() which does the inverse transformation of the DTFT. FFTs are used for fault analysis, quality control, and condition monitoring of machines or systems. Thereafter, Fourier Transform of Array Inputs. If X is a matrix, then fft(X) treats the columns of X as vectors and returns the Fourier transform of each column.. Water, ethanol, n-butanol, n-hexanol, formic acid, acetic acid, citric acid, and water-acetic acid mixtures at … to Applied Math. I use the following conventions: For simplicity the hat of the Fourier transform f^(!) ... once one has realized that simple difference operators play here the role of some elements of the unipotent radical of a parabolic subalgebra. La distribution uσ , ν i . Des intervalles d'analyse finis entraînent une transformation de Fourier discrète, en ce sens que le spectre n'a de signification que pour des valeurs discrètes des fréquences. Find the Fourier series of the functionf deﬁned by f(x)= −1if−π


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# Prof.  Itay Neeman

Department of Mathematics

University of California Los Angeles

Los Angeles, CA 90095-1555

Office: Math Sciences 6334

Phone: (310) 794 5317

Fax: (310) 206 6673

Email:

## Selected invited talks

On higher analogues of the proper forcing axiom: Rutgers MAMLS and Banff (2013), Fields Institute (2012), Oberwolfach (2011).

Forcing with ultrafilters. Logic Colloquium 2009, Sofia, Bulgaria, plenary talk.

Steel forcing in reverse mathematics. CIRM workshop on Set Theory, Luminy, France, 2008, and Steel VIG, UCLA, 2009.

Determinacy and large cardinals. International Congress of Mathematicians, Logic and Foundations of Mathematics section, Madrid, Spain, 2006.

Set theory, infinite games, and strong axioms. Wissenschaftskolleg zu Berlin, Germany, 2005, general audience talk.

Inner models and ultrafilters in L(R). Talk 1; Talk 2; Talk 3. CIRM workshop on Set Theory, Luminy, France, 2004, tutorial.

Determinacy proofs for long games. Talk 1; Talk 2; Talk 3. Logic Colloquium 2001, Vienna, Austria, tutorial.

## Books

The determinacy of long games. De Gruyter Series in Logic and its Applications, Volume 7, Walter de Gruyter and Co., Berlin, November 2004.

## Articles (contact me by email for reprints)

Itay Neeman and John Susice, Chang's Conjecture with $$\Box_{\omega_1,2}$$ from an $$\omega_1$$-Erdos Cardinal.

Itay Neeman and Zach Norwood, Coding along trees and generic absoluteness.

Omer Ben-Neria, Moti Gitik, Itay Neeman, and Spencer Unger, On the powerset of singular cardinals in HOD.

William Chen and Itay Neeman, On the relationship between mutual and tight stationarity.

Thomas Gilton and Itay Neeman, Side conditions and iteration theorems.

Itay Neeman and Zach Norwood, Happy and MAD families in $$L({\mathbb R})$$, J. of Symbolic Logic, vol. 83 (2018), pp.572–597.

Itay Neeman, Two applications of finite side conditions at $$\omega_2$$, Archive for Mathematical Logic, vol. 56 (2017), special issue dedicated to the memory of James Baumgartner, pp. 983–1036.

Itay Neeman and John Steel, Equiconsistencies at subcompact cardinals, Archive for Mathematical Logic, vol. 55 (2016), special issue dedicated to the memory of Richard Laver, pp. 207–238.

Itay Neeman, An inner models proof of the Kechris–Martin Theorem, in Ordinal definability and recursion theory, the Cabal Seminar Vol. III, Kechris, Loewe, Steel editors, pp. 220–242, Cambridge University Press, 2016.

William Chen and Itay Neeman, Square principles with tail-end agreement, Archive for Mathematical Logic, vol. 54 (2015), pp. 439–452.

Itay Neeman, The tree property up to $$\aleph_{\omega+1}$$, J. Symbolic Logic, vol. 79 (2014), pp. 429–459.

Itay Neeman, Forcing with sequences of models of two types, Notre Dame J. Formal Logic, vol. 55 (2014), pp. 265–298.

Itay Neeman and Spencer Unger, Aronszajn trees and the SCH. In Appalachian Set Theory 2006–2012 (Cummings, Schimmerling, eds.), pp. 187–206, LMS Lecture Notes Series 406, 2013.

Itay Neeman, Necessary use of $$\Sigma^1_1$$ induction in a reversal. J. Symbolic Logic, vol. 76 (2011), pp. 561–574.

Andreas Blass, Yuri Gurevich, Michal Moskal, and Itay Neeman, Evidential Authorization. In The future of software engineering (Nanz, ed.), pp. 73–99, Springer, 2011.

Itay Neeman, Ultrafilters and large cardinals. In Ultrafilters across Mathematics (Bergelson, Blass, Di Nasso, Jin, eds.), Contemporary Mathematics Vol. 530, pp. 181–200, AMS, 2010.

Paul Larson, Itay Neeman, and Saharon Shelah, Universally measurable sets in generic extensions. Fund. Math., vol. 208 (2010), pp. 173–192.

Itay Neeman, Aronszajn trees and failure of the singular cardinal hypothesis. J. of Mathematical Logic, vol. 9 (2009, published 2010) 139–157.

Gunter Fuchs, Itay Neeman, and Ralf Schindler, A criterion for coarse iterability. Archive for Mathematical Logic, vol. 49 (2010), pp. 447–467.

Itay Neeman, Determinacy in $$L({\mathbb R})$$. In Handbook of Set Theory (Foreman, Kanamori, eds.), pp. 1887–1950, Springer, 2010.

Yuri Gurevich and Itay Neeman, The logic of infons. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, No. 98 (2009), pp. 150–178, and ACM Trans. Comput. Log. 12 (2011) no. 2, Art. 9. See also technical report MSR-TR-2011-90.

Itay Neeman, Monadic theories of wellorders. In Logic, Methodology and Philosophy of Science Proceedings of the Thirteenth International Congress (Glymour, Wei, Westerstahl, eds.), pp. 108–121, College Publications, 2009.

Itay Neeman, The strength of Jullien's indecomposability theorem. J. of Mathematical Logic, vol. 8 (2008, published June 2009), pp. 93–119.

Yuri Gurevich and Itay Neeman, DKAL: Distributed Knowledge Authorization Language. Proceedings of the 21st IEEE Computer Security Foundations Symposium, pp. 149–162, IEEE Computer Society, 2008. See also technical report MSR-TR-2008-09.

Itay Neeman, Propagation of the scale property using games. In Games, scales, and Suslin cardinals, the Cabal seminar vol. I (Kechris, Loewe, Steel, eds.), pp. 75–89, Lecture Notes in Logic 31, 2008.

Itay Neeman, Monadic definability of ordinals. In Computational Prospects of Infinity, Part II, Presented Talks, pp. 193–205, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 15, World Scientific Publishing, 2008.

Itay Neeman, Hierarchies of forcing axioms II. J. of Symbolic Logic, vol. 73 (2008), pp. 522–542.

Itay Neeman, Finite state automata and monadic definability of singular cardinals. J. of Symbolic Logic, vol. 73 (2008), pp. 412–438.

Itay Neeman and Ernest Schimmerling, Hierarchies of forcing axioms I. J. of Symbolic Logic, vol. 73 (2008), pp. 343–362.

Itay Neeman, Games of length $$\omega_1$$. J. of Mathematical Logic, vol. 7 (2007), pp. 83–124.

Alex Andretta, Greg Hjorth, and Itay Neeman, Effective cardinals of boldface pointclasses. J. of Mathematical Logic, vol. 7 (2007), pp. 35–82.

Moti Gitik, Itay Neeman, and Dima Sinapova, A cardinal preserving extension making the set of points of countable $$V$$ cofinality nonstationary. Archive for Mathematical Logic, vol. 46 (2007), pp. 451–456.

Itay Neeman, Inner models and ultrafilters in $$L({\mathbb R})$$. Bull. of Symbolic Logic, vol. 13 (2007), pp. 31–53.

Itay Neeman, Determinacy and Large Cardinals. Proceedings of the International Congress of Mathematicians, vol. II, Madrid 2006, pp. 27–43, European Math. Society Publishing House, 2007.

Itay Neeman and John Steel, Counterexamples to the unique and cofinal branches hypotheses. J. of Symbolic Logic, vol. 71 (2006) pp. 977–988.

Itay Neeman, Determinacy for games ending at the first admissible relative to the play. J. of Symbolic Logic, vol. 71 (2006), pp. 425–459.

Itay Neeman, Unraveling $$\Pi^1_1$$ sets, revisited. Israel J. of Math., vol. 152 (2006), pp. 181–203.

Itay Neeman, An introduction to proofs of determinacy of long games. Logic Colloquium ’01, pp. 43–86, Lecture Notes in Logic No. 20, Association for Symbolic Logic, Urbana, IL, 2005.

Itay Neeman, The Mitchell order below rank-to-rank. J. of Symbolic Logic, vol. 69 (2004), pp. 1143–1162.

Donald A. Martin, Itay Neeman, and Marco Vervoort, The strength of Blackwell determinacy. J. of Symb. Logic, vol. 68 (2003), pp. 615–636.

Itay Neeman, Optimal proofs of determinacy II. J. of Math. Logic, vol. 2 (2002), pp. 227–258.

Itay Neeman, Inner models in the region of a Woodin limit of Woodin cardinals. Ann. of Pure and Applied Logic, vol. 116 (2002), pp. 67–155.

Alex Andretta, Itay Neeman, and John Steel, The domestic levels of $$K^c$$ are iterable. Israel J. of Math., vol. 125 (2001), pp. 157–201.

Itay Neeman and Jindrich Zapletal, Proper forcing and $$L({\mathbb R})$$. J. of Symb. Logic, vol. 66 (2001), pp. 801–810.

Itay Neeman, Unraveling $$\Pi^1_1$$ sets. Ann. of Pure and Applied Logic, vol. 106 (2000), pp. 151–205.

Itay Neeman and John Steel, A weak Dodd–Jensen lemma. J. of Symb. Logic, vol. 64 (1999), pp. 1285–1294.

Itay Neeman, Games of countable length. In Sets and Proofs, LMS Lecture Note Series 258, pp. 159–196, Cambridge University Press, 1999.

Itay Neeman and Jindrich Zapletal, Proper forcing and absoluteness in $$L({\mathbb R})$$. Commentationes Math. Uni. Carolinea, vol. 39 (1998), pp. 281–301.

Itay Neeman, Optimal proofs of determinacy. Bull. of Symb. Logic, vol. 1 (1995), pp. 327–339.


text:
# How do I calculate two linear regressions in one dataset [duplicate]

First of all, apologies if I am not using the correct terminology to describe the problem. I have a dataset of two variables, which looks like this:

There are clearly two groups of points. I want to separate them so that I can fit a linear regression to each of them.

I can draw a line and decide that everything above it belongs to one group and everything below to the other, but it is a method too crude and the results are not good enough.

Cluster analysis probably can do it, but it seems a bit overkill and I was wondering if there is a simpler way to do it.

A simple data.frame (in R) for testing:

xx <- data.frame(A=c(1,2,3,4,1,1.1,1.2,1.3), B=c(1.1,1.2,1.3,1.4,1,2,3,4))
plot(xx$A, xx$B)

• Have you tried optimization? It's likely to be more effective than usual clustering algorithms algorithm although it is not less overkill. – Pere Jan 17 '18 at 13:23
• @Pere not sure what you mean, tbh – rs028 Jan 17 '18 at 14:11
• We start by deciding that everything above it belongs to one group and everything below to the other, as you said. Then, we keep changing observations to the other group until a measure of correlation within each group gets an optimal value. However, beware that correlation within each group in your graph doesn't seem to be very large at first glance. – Pere Jan 17 '18 at 14:53


text:
Rating: 3.0

1. Google for the constants from the python file (0.4829629131445341, 0.8365163037378077, 0.2241438680420134, -0.12940952255126034])
2. Find a few pages on Daubechies Wavelets and Discrete Wavelet Transforms (the wikipedia article on DWTs has an image with a similar structure to the given one)
3. At this point, we tried dumping the original image to disk, trying to do a single-pass inverse discrete wavelet transform (iwdt), and messing around with switching up some of coefficients/quadrants in the algorithm
a. In some cases, we could make out the flag's VolgaCTF{ header, but the rest of the flag was still too distorted or noisy to read
4. We ended up scripting an inverse wavelet transform that iteratively reconstructed the image: [https://gist.github.com/wwong/6a812908a3a06d884ce4b1547c5aeb54](https://gist.github.com/wwong/6a812908a3a06d884ce4b1547c5aeb54)
1. Starting at the top-left 4px by 4px square: reconstruct the image and replace the input section with its own result (plus some padding for alignment)
1. A 4x4 square was the smallest size we could start reconstructing the image
2. We originally padded the wrong ends of the output, which left us with an image that looked like it had been smeared up and left by a few pixels. Switching up the padding fixed that
2. Keep expanding the area we process until we process the whole image
3. Clamp values to something that sci-kit image will export (any |value| < 1)
4. Dump image to file and get the flag


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# zbMATH — the first resource for mathematics

A multi- dimensional extension of the arcsine law. (Une extension multidimensionnelle de la loi de l’arc sinus.) (French) Zbl 0738.60072
Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 294-314 (1989).
[For the entire collection see Zbl 0722.00030.]
Lévy’s arcsine law of the occupation time of the positive demi-line of the BM has beautiful extensions in this paper. The underlying model is the Walsh process [J. B. Walsh, Astérisque 52-53, 37-45 (1978; Zbl 0385.60063)], i.e. Bessel process of order $$\mu$$ on $$k$$ demi-lines with comon point (point 0) and with a given probability law $$p_ i$$ $$(i\leq k)$$ to select the $$i$$-th demi-line when the process attains 0 (for Lévy’s case: $$\mu=1/2$$, $$k=2$$, $$p_ 1=1/2$$). Let $$A_ i(t)$$, $$U_ i(t)$$ $$(i\leq k)$$ be occupation times on the $$i$$-th demi-line of the Walsh process and Walsh bridge before t, respectively, $$\ell_ t$$, $$\lambda_ t$$ be local times of the Bessel process and the Walsh bridge at 0 before $$t$$, respectively. Laws of $$\{A_ i(u)/(\ell_ u)^{1/\mu}$$ $$(i\leq k)\}$$ and $$\{A_ i(1)$$ $$(i\leq k)$$, $$(\ell_ 1)^{1/\mu}\}$$, expressions of $$E f(u_ i/(\lambda_ 1)^{1/\mu}$$ $$(i\leq k))$$ and $$E f(u_ i\;(i\leq k),(\lambda_ 1)^{-1/\mu})$$ are identified by laws and expectations of functions of $$k$$ independent stable variables. $$E f(A_ i(1)\;(i\leq k))$$, $$E f(u_ i\;(i\leq k))$$ and the Laplace transform of $$\{(t A_ i(t), (i\leq k))\}$$ are calculated explicitly. When $$p_ i\equiv 1/k\;(i\leq k)$$, the asymptotic law of $$\{k^{1/\mu}A_ i(t)\;(i\leq p),\;(\ell_ 1)^{1/\mu}\}$$ for fixed $$p$$ is also identified as $$k\to\infty$$.

##### MSC:
 60J65 Brownian motion 60J99 Markov processes
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Search
Question: Flagging overlapping gene exons (was Re: Question about CSAMA10 ...)
0
7.6 years ago by
Denali
Steve Lianoglou12k wrote:
Howdy, On Tue, Sep 28, 2010 at 12:28 PM, Martin Morgan <mtmorgan at="" fhcrc.org=""> wrote: > On 09/28/2010 08:09 AM, Paul Geeleher wrote: >> I guess my next question is if there is a convenient way of filtering >> out the part of the gene >> that overlaps another one using GenomicRanges? I've been meaning to do something that basically performs this core functionality for a while. I just finished putting together a first cut of it and added it to my GenomicFeatures extensions package: http://github.com/lianos/GenomicFeaturesX I'm also not an IRanges guru, so I'm sure there are better/more-magical ways to do this. I'll need to test this more thoroughly, but it currently looks to do the right thing for the example provided below. The context I'm using it for is to create an "annotated" GRanges object (1 per chromosome) that contains annotated intervals that (i) span the length of the chromosome; and (ii) are annotated with exon information for parts (or all) of exons that exclusively belong to a given gene. I think some example code + results will speak better about what I mean than words can. Note that if you have GenomicRanges installed and the "testthat" library, you can easily play along by sourcing urls straight from github, so let's set up the R environment R> library(testthat) R> library(GenomicRanges) I have a list of GRanges objects (you can use a GRangesList if you like) with exon annotations for genes. lchr1 is an example object for a pseudo "chr1" that only has a few (4) gene annotations on it and is 10,000 bp long. GeneA and GeneC have some exons that overlap, and will become disjoint later. GeneD will cause further "fractionation" if we toss out strand info: R> source("http://github.com/lianos/GenomicFeaturesX/raw/master/inst/t ests/test-annotateChromosome.R") R> lchr1 GRanges with 4 ranges and 1 elementMetadata value seqnames ranges strand | exon.anno <rle> <iranges> <rle> | <character> [1] chr1 [ 10, 30] + | utr5 [2] chr1 [ 40, 60] + | cds [3] chr1 [ 80, 100] + | cds [4] chr1 [120, 140] + | utr3 ... I've written a function annotateChromosome that will calculate the ranges along a chromosome that exclusive belong to a gene as explained above, and assigns each interval the appropriate exon annotation. R> source("http://github.com/lianos/GenomicFeaturesX/raw/master/R/anno tateGenome.R") R> annotateChromosome(lchr1, 'chr1', 1000, stranded=TRUE) GRanges with 14 ranges and 2 elementMetadata values seqnames ranges strand | exon.anno symbol <rle> <iranges> <rle> | <character> <character> [1] chr1 [ 10, 30] + | utr5 GeneA [2] chr1 [ 15, 45] - | cds GeneD [3] chr1 [ 40, 49] + | cds GeneA [4] chr1 [ 50, 100] - | utr5 GeneD [5] chr1 [ 50, 60] + | overlap NA [6] chr1 [ 61, 79] + | cds GeneC [7] chr1 [ 80, 85] + | overlap NA [8] chr1 [ 86, 100] + | cds GeneA [9] chr1 [105, 115] + | cds GeneC [10] chr1 [120, 140] + | utr3 GeneA [11] chr1 [200, 230] + | utr5 GeneB [12] chr1 [250, 280] + | cds GeneB [13] chr1 [300, 330] + | cds GeneB [14] chr1 [400, 430] + | utr3 GeneB You can try setting stranded=FALSE to see how that changes the results. If you look at the function here: http://github.com/lianos/GenomicFeaturesX/blob/master/R/annotateGenome .R The lapply block that generates the "interval.annos" object (~ line 34) does the bookkeeping involved in splitting apart overlapping regions and keeping only the parts of exons that are unique to each gene (look around the call to tabulate(...)). The lapply block that generates "clean.list" (~ line 51) goes back to the original annotated gene list and picks out the appropriate annotations for each now-disjoint exon. That's about it. As I said earlier, this is a first cut at doing this. I'm sure it can be cleaner and done more efficiently, but I thought you'd be interested in the code that does the disjoining stuff (in the context of this function, or just by itself). It's only about 20 lines of code, so ... you might not have to do a rewrite in Python after all :-) [not that I have anything against Python .. I rather like it] -steve ps - the content/validity of those github links will (obviously) change over time. In case they change before you get a change to play with the contents of this message, you can get the code @ the time of this commit from these links: annotateChromosome function: http://github.com/lianos/GenomicFeaturesX/blob/fa69b3c2/R/annotateGeno me.R code to generate the example lchr1 object: http://github.com/lianos/GenomicFeaturesX/blob/fa69b3c2/inst/tests /test-annotateChromosome.R -- Steve Lianoglou Graduate Student: Computational Systems Biology ?| Memorial Sloan-Kettering Cancer Center ?| Weill Medical College of Cornell University Contact Info: http://cbio.mskcc.org/~lianos/contact


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# Consider the combustion of methane: Ch4(g)+2O2(g) --> CO2(g)+2H2O(g), suppose 2.8 moles of methane are allowed to react with 3 moles of oxygen, what is the limiting reactant?

Oct 28, 2015

${O}_{2}$ is the limiting reactant.

#### Explanation:

There are different methods of finding the limiting reactant, here is the simplest to my opinion.

The reaction is:
$C {H}_{4} \left(g\right) + \textcolor{b l u e}{2} {O}_{2} \left(g\right) \to C {O}_{2} \left(g\right) + 2 {H}_{2} O \left(g\right)$

Find the molar ratio between the experimental number of moles and theoretical number of moles of the reactants:

For $C {H}_{4}$: $\frac{2.8 \cancel{m o l}}{1 \cancel{m o l}} = 2.8$.
The $1 \text{ mol}$ is taking from the coefficient of $C {H}_{4}$ from the balanced equation.

For ${O}_{2}$: $\frac{3 \cancel{m o l}}{\textcolor{b l u e}{2 \cancel{m o l}}} = 1.5$.
The $2 \text{ mol}$ is taking from the coefficient of ${O}_{2}$ from the balanced equation.

The reactant that gives the smaller molar ratio is the limiting reactant which is in this case the oxygen ${O}_{2}$

More Explanation:
To consume the $2.8 \text{ moles of } C {H}_{4}$ we need $5.6 \text{ moles of } {O}_{2}$ since the molar ratio is 1:2. We have only $3 \text{ moles of } {O}_{2}$; therefore, ${O}_{2}$ is the limiting reactant.


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Simplicial Matrix-Tree Theorems

# Simplicial Matrix-Tree Theorems

Art M. Duval Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968-0514 Caroline J. Klivans Departments of Mathematics and Computer Science, The University of Chicago, Chicago, IL 60637  and  Jeremy L. Martin Department of Mathematics, University of Kansas, Lawrence, KS 66047
August 14, 2008
###### Abstract.

We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes , extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of . As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of and replacing the entries of the Laplacian with Laurent monomials. When is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.

###### Key words and phrases:
Simplicial complex, spanning tree, tree enumeration, Laplacian, spectra, eigenvalues, shifted complex
###### 2000 Mathematics Subject Classification:
Primary 05A15; Secondary 05E99, 05C05, 05C50, 15A18, 57M15
Second author partially supported by NSF VIGRE grant DMS-0502215. Third author partially supported by an NSA Young Investigators Grant.

## 1. Introduction

### 1.1. The classical Matrix-Tree Theorem

We begin by reviewing the classical case; for a more detailed treatment, see, e.g., [8]. Let be a finite, simple, undirected graph with vertices and edges . A spanning subgraph of is a graph with and ; thus a spanning subgraph may be specified by its edge set. A spanning subgraph  is a spanning tree if (a) is acyclic; (b) is connected; and (c) . It is a fundamental property of spanning trees (the “two-out-of-three theorem”) that any two of these three conditions together imply the third.

The Laplacian of is the symmetric matrix with entries

 Lij=⎧⎨⎩degG(i) if i=j,−1 if i,j are adjacent,0 otherwise,

where is the degree of vertex  (the number of edges having as an endpoint). Equivalently, , where is the (signed) vertex-edge incidence matrix and is its transpose. If we regard as a one-dimensional simplicial complex, then is just the simplicial boundary map from 1-faces to 0-faces, and is the simplicial coboundary map. The matrix is symmetric, hence diagonalizable, so it has real eigenvalues (counting multiplicities). The number of nonzero eigenvalues of is , where is the number of components of .

The Matrix-Tree Theorem, first observed by Kirchhoff [22] in his work on electrical circuits (modern references include [8], [29] and [34, Chapter 5]), expresses the number of spanning trees of in terms of . The theorem has two equivalent formulations.

###### Theorem 1.1 (Classical Matrix-Tree Theorem).

Let be a connected graph with vertices, and let be its Laplacian matrix.

1. If the eigenvalues of are , then

 τ(G)=λ1⋯λn−1n.
2. For , let be the reduced Laplacian obtained from by deleting the row and column. Then

 τ(G)=detLi.

Well-known corollaries of the Matrix-Tree Theorem include Cayley’s formula [9]

 τ(Kn)=nn−2 (1)

where is the complete graph on vertices, and Fiedler and Sedláçek’s formula [16]

 τ(Kn,m)=nm−1mn−1 (2)

where is the complete bipartite graph on vertex sets of sizes and .

The Matrix-Tree Theorem can be refined by introducing an indeterminate for each pair of vertices , setting if do not share a common edge. The weighted Laplacian is then defined as the matrix with entries

 ^Lij=⎧⎪⎨⎪⎩∑nk=1eik if i=j,−eij if i,j are adjacent,0 otherwise.
###### Theorem 1.2 (Weighted Matrix-Tree Theorem).

Let be a graph with vertices, and let be its weighted Laplacian matrix.

1. If the eigenvalues of are , then

 ∑T∈T(G)∏ij∈Teij=^λ1⋯^λn−1n

where is the set of all spanning trees of .

2. For , let be the reduced weighted Laplacian obtained from by deleting the row and column. Then

 ∑T∈T(G)∏ij∈Teij=det^Li.

By making appropriate substitutions for the indeterminates , it is often possible to obtain finer enumerative information than merely the number of spanning trees. For instance, when , introducing indeterminates and setting for all yields the Cayley-Prüfer Theorem, which enumerates spanning trees of by their degree sequences:

 ∑T∈T(G) n∏i=1xdegT(i)i=x1⋯xn(x1+⋯+xn)n−2. (3)

Note that Cayley’s formula (1) can be recovered from the Cayley-Prüfer Theorem by setting .

### 1.2. Simplicial spanning trees and how to count them

To extend the scope of the Matrix-Tree Theorem from graphs to simplicial complexes, we must first say what “spanning tree” means in arbitrary dimension. Kalai [20] proposed a definition that replaces the acyclicity, connectedness, and edge-count conditions with their analogues in simplicial homology. Our definition adapts Kalai’s definition to a more general class of simplicial complexes.111There are many other definitions of “simplicial tree” in the literature, depending on which properties of trees one wishes to extend; see, e.g., [4, 10, 15, 19, 28]. By adopting Kalai’s idea, we choose a definition that lends itself well to enumeration. The closest to ours in spirit is perhaps that of Masbaum and Vaintrob [28], whose main result is a Matrix-Tree-like theorem enumerating a different kind of 2-dimensional tree using Pfaffians rather than Laplacians.

Let be a -dimensional simplicial complex, and let be a subcomplex containing all faces of of dimension . We say that is a simplicial spanning tree of if the following three conditions hold:

 ~Hd(Υ,Z)=0, (4a) |~Hd−1(Υ,Z)|<∞, and (4b) fd(Υ)=fd(Δ)−~βd(Δ)+~βd−1(Δ), (4c)

where denotes reduced simplicial homology (for which see, e.g., [18, §2.1]). (The conditions (4a) and (4b) were introduced by Kalai in [20], while (4c) is more general, as we will explain shortly.) When , the conditions (4a)…(4c) say respectively that is acyclic, connected, and has one fewer edge than it has vertices, recovering the definition of the spanning tree of a graph. Moreover, as we will show in Proposition 3.5, any two of the three conditions together imply the third.

A graph has a spanning tree if and only if is connected. The corresponding condition for a simplicial complex of dimension  is that for all ; that is, has the rational homology type of a wedge of -dimensional spheres. We will call such a complex acyclic in positive codimension, or APC for short. This condition, which we will assume throughout the rest of the introduction, is much weaker than Cohen-Macaulayness (by Reisner’s theorem [30]), and therefore encompasses many complexes of combinatorial interest, including all connected graphs, simplicial spheres, shifted, matroid, and Ferrers complexes, and some chessboard and matching complexes.

For , let be the th simplicial boundary matrix of  (with rows and columns indexed respectively by -dimensional and -dimensional faces of ), and let be its transpose. The (th up-down) Laplacian of is ; this can be regarded either as a square matrix of size or as a linear endomorphism on ()-chains of . Define invariants

 πk=πk(Δ) = product of all nonzero eigenvalues of L, τk=τk(Δ) =∑Υ∈Tk(Δ)|~Hk−1(Υ)|2,

where denotes the set of all -trees of (that is, simplicial spanning trees of the -skeleton of ).

Kalai [20] studied these invariants in the case that is a simplex on vertices, and proved the formula

 τk(Δ)=n(n−2k) (5)

(of which Cayley’s formula (1) is the special case ). Kalai also proved a natural weighted analogue of (5) enumerating simplicial spanning trees by their degree sequences, thus generalizing the Cayley-Prüfer Theorem (3).

Given disjoint vertex sets (“color classes”), the faces of the corresponding complete colorful complex are those sets of vertices with no more than one vertex of each color. Equivalently, is the simplicial join of the 0-dimensional complexes . Adin [1] extended Kalai’s work by proving a combinatorial formula for , which we shall not reproduce here, for every . Note that when , the complex is a complete bipartite graph, and if for all , then is a simplex. Thus both (2) and (5) can be recovered from Adin’s formula.

Kalai’s and Adin’s beautiful formulas inspired us to look for more results about simplicial spanning tree enumeration, and in particular to formulate a simplicial version of the Matrix-Tree Theorem that could be applied to as broad a class of complexes as possible. Our first main result generalizes the Matrix-Tree Theorem to all APC simplicial complexes.

###### Theorem 1.3 (Simplicial Matrix-Tree Theorem).

Let be a -dimensional APC simplicial complex. Then:

1. We have

 πd(Δ)=τd(Δ)τd−1(Δ)|~Hd−2(Δ)|2.
2. Let be the set of facets of a -SST of , and let be the reduced Laplacian obtained by deleting the rows and columns of corresponding to . Then

 τd(Δ)=|~Hd−2(Δ)|2|~Hd−2(ΔU)|2detLU.

We will prove these formulas in Section 4.

In the special case , the number is just the number of spanning trees of the graph , recovering the classical Matrix-Tree Theorem. When , there can exist spanning trees with finite but nontrivial homology groups (the simplest example is the real projective plane). In this case, is greater than the number of spanning trees, because these “torsion trees” contribute more than 1 to the count. This phenomenon was first observed by Bolker [7], and arises also in the study of cyclotomic matroids [26] and cyclotomic polytopes [3].

The Weighted Matrix-Tree Theorem also has a simplicial analogue. Introduce an indeterminate for each facet (maximal face) , and for every set of facets define monomials and . Construct the weighted boundary matrix by multiplying each column of by , where is the facet of corresponding to that column. Let be the product of the nonzero eigenvalues of , and let

 ^τk=^τk(Δ)=∑Υ∈Tk(Δ)|~Hk−1(Υ)|2XΥ.
###### Theorem 1.4 (Weighted Simplicial Matrix-Tree Theorem).

Let be a -dimensional APC simplicial complex. Then:

1. We have222Despite appearances, there are no missing hats on the right-hand side of this formula! Only has been replaced with its weighted analogue; is still just an integer.

 ^πd(Δ)=^τd(Δ)τd−1(Δ)|~Hd−2(Δ)|2.
2. Let be the set of facets of a -SST of , and let be the reduced Laplacian obtained by deleting the rows and columns of corresponding to . Then

 ^τd(Δ)=|~Hd−2(Δ)|2|~Hd−2(ΔU)|2det^LU.

We will prove these formulas in Section 5.

Setting for all in Theorem 1.4 recovers Theorem 1.3. In fact, more is true; setting in the multiset of eigenvalues of the weighted Laplacian (reduced or unreduced) yields the eigenvalues of the corresponding unweighted Laplacian. If the complex is Laplacian integral, that is, its Laplacian matrix has integer eigenvalues, then we can hope to find a combinatorial interpretation of the factorization of furnished by Theorem 1.4. An important class of Laplacian integral simplicial complexes are the shifted complexes.

### 1.3. Results on shifted complexes

Let be integers, and let . A simplicial complex on vertex set is shifted if the following condition holds: whenever are vertices and is a face such that and , then . Equivalently, define the componentwise partial order on finite sets of positive integers as follows: whenever , , and for all . Then a complex is shifted precisely when it is an order ideal with respect to the componentwise partial order. (See [33, chapter 3] for general background on partially ordered sets.)

Shifted complexes were used by Björner and Kalai [5] to characterize the -vectors and Betti numbers of all simplicial complexes. Shifted complexes are also one of a small handful of classes of simplicial complexes whose Laplacian eigenvalues are known to be integral. In particular, Duval and Reiner [13, Thm. 1.1] proved that the Laplacian eigenvalues of a shifted complex on are given by the conjugate of the partition , where is the degree of vertex , that is, the number of facets containing it.

In the second part of the article, Sections 610, we study factorizations of the weighted spanning tree enumerator of under the combinatorial fine weighting

 xF=k+1∏i=1xi,vi

(described in more detail in Section 6), where is a -dimensional face of . Thus the term of corresponding to a particular simplicial spanning tree of contains more precise information than its vertex degrees alone (which can be recovered by further setting for all ).

For integer sets and as above, we call the ordered pair a critical pair of if , , and covers in the componentwise order. That is, for some . The long signature of is the ordered pair , where and . The corresponding -polynomial is defined as

 z(S,T)=1↑XS∑j∈TXS∪j

where for each , and the operator   is defined by . (See Section 8.1 for more details, and Example 1.7 for an example.) The set of critical pairs is especially significant for a shifted family (and by extension, for a shifted complex). Since a shifted family is just an order ideal with respect to the componentwise partial order , the critical pairs identify the frontier between members and non-members of  in the Hasse diagram of . (See Example 1.7 or [23] for more details.)

Thanks to Theorem 1.4, the enumeration of SST’s of a shifted complex reduces to computing the determinant of the reduced combinatorial finely-weighted Laplacian. We show in Section 6 how this computation reduces to the computation of the eigenvalues of the algebraic finely weighted Laplacian. This modification of the combinatorial fine weighting, designed to endow the chain groups of with the structure of an algebraic chain complex, is described in detail in Section 6.3. Its eigenvalues turn out to be precisely the -polynomials associated with critical pairs.

###### Theorem 1.5.

Let be a -dimensional shifted complex, and let . Then the eigenvalues of the algebraic finely weighted up-down Laplacian are precisely , where ranges over all long signatures of critical pairs of -dimensional faces of .

In turn, the -polynomials are the factors of the weighted simplicial spanning tree enumerator .

###### Theorem 1.6.

Let be a -dimensional shifted complex with initial vertex . Then:

 ^τd(Σ) =⎛⎝∏F∈Λd−1X~F⎞⎠⎛⎝∏(S,T)∈¯σ(Δd)z(S,~T)X1,p⎞⎠ =⎛⎝∏F∈Λd−1X~F⎞⎠⎛⎝∏(S,T)∈¯σ(Δd)∑j∈~TXS∪jX~S⎞⎠

where ; ; and .

Theorems 1.5 and 1.6 are proved in Sections 8 and 9, respectively.

###### Example 1.7.

As an example to which we will return repeatedly, consider the equatorial bipyramid, the two-dimensional shifted complex with vertices and facets 123, 124, 125, 134, 135, 234, 235. A geometric realization of is shown in the figure on the left below. The figure on the right illustrates how the facets of  can be regarded as an order ideal. The boldface lines indicate critical pairs.

The Laplacian eigenvalues corresponding to the critical pairs of are as follows:

 Critical pair(125,126)(135,136)(135,145)(235,236)(235,245)Eigenvaluez(12,12345)z(13,12345)z(1,123)z(23,12345)z(2,123)

To show one of these eigenvalues in more detail,

 z(13,12345)=X1,1X2,1X3,3+X1,1X2,2X3,3+X1,1X2,3X3,3+X1,1X2,3X3,4+X1,1X2,3X3,5X2,1X3,3.

The eigenvalues of this complex are explained in more detail in Section 8.4. Its spanning trees are enumerated in Examples 9.1 (fine weighting) and 9.3 (coarse weighting).

We prove Theorem 1.5 by exploiting the recursive structure of shifted complexes. As in [13], we begin by calculating the algebraic finely weighted eigenvalues of a near-cone in terms of the eigenvalues of its link and deletion with respect to its apex (Proposition 7.6). We can then write down a recursive formula (Theorem 8.2) for the nonzero eigenvalues of shifted complexes, thanks to their characterization as iterated near-cones, simultaneously showing that these eigenvalues must be of the form . Finally, we independently establish a recurrence (Corollary 8.8) for the long signatures of critical pairs of a shifted complex, which coincides with the recurrence for the , thus yielding a bijection between nonzero eigenvalues and critical pairs.

Corollary 8.10 shows what the eigenvalues look like in coarse weighting. Passing from weighted to unweighted eigenvalues then easily recovers the Duval-Reiner formula for Laplacian eigenvalues of shifted complexes in terms of degree sequences [13, Thm. 1.1]. Similarly, Corollary 9.2 gives the enumeration of SST’s of a shifted complex in the coarse weighting.

We are also able to show that the finely-weighted eigenvalues (though not the coarsely-weighted eigenvalues) are enough to recover the shifted complex (Corollary 8.9), or, in other words, that one can “hear the shape” of a shifted complex.

Several known results can be obtained as consequences of the general formula of Theorem 1.6.

• The complete -skeleton of a simplex is easily seen to be shifted, and applying Theorem 1.6 to such complexes recovers Kalai’s generalization of the Cayley-Prüfer Theorem.

• The one-dimensional shifted complexes are precisely the threshold graphs, an important class of graphs with many equivalent descriptions (see, e.g., [25]). When , Theorem 1.6 specializes to the weighted spanning tree enumerator for threshold graphs proved by Martin and Reiner [26, Thm. 4] and following from an independent result of Remmel and Williamson [31, Thm. 2.4].

• Thanks to an idea of Richard Ehrenborg, the formula for threshold graphs can be used to recover a theorem of Ehrenborg and van Willigenburg [14], enumerating spanning trees in certain bipartite graphs called Ferrers graphs (which are not in general Laplacian integral).

We discuss these corollaries in Section 10.

Some classes of complexes that we think deserve further study include matroid complexes, matching complexes, chessboard complexes and color-shifted complexes. The first three kinds of complexes are known to be Laplacian integral, by theorems of Kook, Reiner and Stanton [24], Dong and Wachs [11], and Friedman and Hanlon [17] respectively. Every matroid complex is Cohen-Macaulay [32, §III.3], hence APC, while matching complexes and chessboard complexes are APC for certain values of their defining parameters (see [6]). Color-shifted complexes, which are a common generalization of Ferrers graphs and complete colorful complexes, are not in general Laplacian integral; nevertheless, their weighted simplicial spanning tree enumerators seem to have nice factorizations.

It is our pleasure to thank Richard Ehrenborg, Vic Reiner, and Michelle Wachs for many valuable discussions. We also thank Andrew Crites and an anonymous referee for their careful reading of the manuscript.

## 2. Notation and definitions

### 2.1. Simplicial complexes

Let be a finite set. A simplicial complex on is a family of subsets of such that

1. ;

2. If and , then .

The elements of are called vertices of , and the faces that are maximal under inclusion are called facets. Thus a simplicial complex is determined by its set of facets. The dimension of a face is , and the dimension of is the maximum dimension of a face (or facet). The abbreviation indicates that . We say that is pure if all facets have the same dimension; in this case, a ridge is a face of codimension 1, that is, dimension .

We write for the set of -dimensional faces of , and set . The -skeleton of is the subcomplex of all faces of dimension ,

 Δ(i)=⋃−1≤j≤iΔj,

and the pure -skeleton of is the subcomplex generated by the -dimensional faces, that is,

 Δ[i]={F∈Δ:F⊆G for some G∈Δi}.

We assume that the reader is familiar with simplicial homology; see, e.g., [18, §2.1]. Let be a simplicial complex and . Let be a ring (if unspecified, assumed to be ), and let be the simplicial chain group of , i.e., the free -module with basis . We denote the simplicial boundary and coboundary maps respectively by

 ∂Δ,i :Ci(Δ)→Ci−1(Δ), ∂∗Δ,i :Ci−1(Δ)→Ci(Δ),

where we have identified cochains with chains via the natural inner product. We will abbreviate the subscripts in the notation for boundaries and coboundaries whenever no ambiguity can arise. We will often regard (resp. ) as a matrix whose columns and rows (resp. rows and columns) are indexed by and respectively. The (reduced) homology group of is , and the (reduced) Betti number is the rank of the largest free -module summand of .

### 2.2. Combinatorial Laplacians

We adopt the notation of [13] for the Laplacian operators (or, equivalently, matrices) of a simplicial complex. We summarize the notation and mention some fundamental identities here.

We will often work with multisets (of eigenvalues or of vertices), in which each element occurs with some non-negative integer multiplicity. For brevity, we drop curly braces and commas when working with multisets of integers: for instance, 5553 denotes the multiset in which 5 occurs with multiplicity three and 3 occurs with multiplicity one. The cardinality of a multiset is the sum of the multiplicities of its elements; thus . We write to mean that the multisets and differ only in their respective multiplicities of zero; for instance, . Of course, is an equivalence relation. The union operation on multisets is understood to add multiplicities: for instance, .

For , define linear operators , , on the vector space by

 LudΔ,i=∂i+1∂∗i+1 (the \emph{up-down Laplacian}), LduΔ,i=∂∗i∂i (the \emph{down-up Laplacian}), LtotΔ,i=LudΔ,i+LduΔ,i (the \emph{total Laplacian}).

The spectrum of is the multiset of its eigenvalues (including zero); we define and similarly. Since each Laplacian operator is represented by a symmetric matrix, it is diagonalizable, so

 |stoti(Δ)|=|sudi(Δ)|=|sdui(Δ)|=fi(Δ).

The various Laplacian spectra are related by the identities

 sudi(Δ) ∘=sdui+1(Δ), stoti(Δ) ∘=sudi(Δ)∪sdui(Δ)

[13, eqn. (3.6)]. Therefore, each of the three families of multisets

 {stoti(Δ):−1≤i≤dimΔ},{sudi(Δ):−1≤i≤dimΔ},{sdui(Δ):−1≤i≤dimΔ}

determines the other two, and we will feel free to work with whichever one is most convenient in context.

Combinatorial Laplacians and their spectra have been investigated for a number of classes of simplicial complexes. In particular, it is known that chessboard [17], matching [11], matroid [24], and shifted [13] complexes are Laplacian integral, i.e., all their Laplacian eigenvalues are integers. Understanding which complexes are Laplacian integral is an open question. As we will see, Laplacian eigenvalues and spanning tree enumerators are inextricably linked.

## 3. Simplicial spanning trees

In this section, we generalize the notion of a spanning tree to arbitrary dimension using simplicial homology, following Kalai’s idea. Our definition makes sense for any ambient complex that satisfies the relatively mild APC condition.

###### Definition 3.1.

Let be a simplicial complex, and let . A -dimensional simplicial spanning tree (for short, SST or -SST) of is a -dimensional subcomplex such that and

 ~Hk(Υ)=0, (6a) |~Hk−1(Υ)|<∞,and (6b) fk(Υ)=fk(Δ)−~βk(Δ)+~βk−1(Δ). (6c)

We write for the set of all -SST’s of , omitting the subscript if . Note that for all .

A zero-dimensional SST is just a vertex of . If is a 1-dimensional simplicial complex on vertices—that is, a graph—then the definition of 1-SST coincides with the usual definition of a spanning tree of a graph: namely, a subgraph of which is connected, acyclic, and has edges. Next, we give a few examples in higher dimensions.

###### Example 3.2.

If is a simplicial sphere (for instance, the boundary of a simplicial polytope), then deleting any facet of while keeping its ()-skeleton intact produces a -SST. Therefore .

###### Example 3.3.

In dimension , spanning trees need not be -acyclic, merely -acyclic. For example, let be a triangulation of the real projective plane, so that , , and . Then satisfies the conditions of Definition 3.1 and is a 2-SST of itself (in fact, the only such).

###### Example 3.4.

Consider the equatorial bipyramid  of Example 1.7. A 2-SST of can be constructed by removing two facets , provided that contains neither of the vertices 4,5. A simple count shows that there are 15 such pairs , so .

Before proceeding any further, we show that Definition 3.1 satisfies a “two-out-of-three theorem” akin to that for spanning trees of graphs.

###### Proposition 3.5.

Let be a -dimensional subcomplex with . Then any two of the conditions (6a), (6b), (6c) together imply the third.

###### Proof.

First, note that

 fℓ(Υ)=fℓ(Δ)for ℓ≤k−1and% ~βℓ(Υ)=~βℓ(Δ)% for ℓ≤k−2. (7)

Next, we use the standard fact that the Euler characteristic can be calculated as the alternating sum either of the -numbers or of the Betti numbers. Thus

 χ(Υ) =k∑i=0(−1)ifi(Υ) =(−1)kfk(Υ)+k−1∑i=0(−1)ifi(Δ) =(−1)kfk(Υ)+χ(Δ)−(−1)kfk(Δ) (8)

and on the other hand,

 χ(Υ) =k∑i=0(−1)i~βi(Υ) =(−1)k(~βk(Υ)−~βk−1(Υ))+k−2∑i=0(−1)i~βi(Δ) =(−1)k(~βk(Υ)−~βk−1(Υ))+χ(Δ)−(−1)k(~βk(Δ)−~βk−1(Δ)). (9)

Equating (8) and (9) gives

 fk(Υ)−fk(Δ)=~βk(Υ)−~βk−1(Υ)−~βk(Δ)+~βk−1(Δ)

or equivalently

 (fk(Υ)−fk(Δ)+~βk(Δ)−~βk−1(Δ))−~βk(Υ)+~βk−1(Υ)=0.

Since (6a) says that (note that must be free abelian) and (6b) says that , the conclusion follows. ∎

###### Definition 3.6.

A simplicial complex is acyclic in positive codimension, or APC for short, if for all .

Equivalently, a complex is APC if it has the homology type of a wedge of zero or more -dimensional spheres. In particular, any Cohen-Macaulay complex is APC. The converse is very far from true, because, for instance, an APC complex need not even be pure. For our purposes, the APC complexes are the “correct” simplicial analogues of connected graphs for the following reason.

###### Proposition 3.7.

For any simplicial complex , the following are equivalent:

1. is APC.

2. has a -dimensional spanning tree.

3. has a -dimensional spanning tree for every .

###### Proof.

It is trivial that (3) implies (2). To see that (2) implies (1), suppose that has a -dimensional spanning tree . Then for all , so for all . Moreover, in the diagram

 Cd(Δ)∂Δ,d−−→Cd−1(Δ)∂Δ,d−1−−−−→Cd−2(Δ)⋃∥∥Cd(Υ)∂Υ,d−−→Cd−1(Υ)∂Υ,d−1−−−−→Cd−2(Υ)

we have and , so there is a surjection , implying that is APC.

To prove (1) implies (3), it suffices to consider the case , because any skeleton of an APC complex is also APC. We can construct a -SST by the following algorithm. Let . If , then there is some nonzero linear combination of facets of that is mapped to zero by . Let be one of those facets, and let . Then and for , and by the Euler characteristic formula, we have as well. Replacing with and repeating, we eventually arrive at the case , when is a -SST of . ∎

The APC condition is a fairly mild one. For instance, any -acyclic complex is clearly APC (and is its own unique SST), as is any Cohen-Macaulay complex (in particular, any shifted complex).

## 4. Simplicial analogues of the Matrix-Tree Theorem

We now explain how to enumerate simplicial spanning trees of a complex using its Laplacian. Throughout this section, let be an APC simplicial complex on vertex set . For , define

 πk =πk(Δ)=∏0≠λ∈sudk−1(Δ)λ, τk =τk(Δ)=∑Υ∈Tk(Δ)|~Hk−1(Υ)|2.

We are interested in the relationships between these two families of invariants. When , the relationship is given by Theorem 1.1. In the notation just defined, part (1) of that theorem says that , and part (2) says that (i.e., the determinant of the reduced Laplacian obtained from by deleting the row and column corresponding to any vertex ).

The results of this section generalize both parts of the Matrix-Tree Theorem from graphs to all APC complexes . Our arguments are closely based on those used by Kalai [20] and Adin [1] to enumerate SST’s of skeletons of simplices and of complete colorful complexes.

We begin by setting up some notation. Abbreviate , , and . Let be a set of facets of of cardinality , and let be a set of ridges such that . Define

 ΔT=T∪Δ(d−1),¯S=Δ(d−1)∖S,Δ¯S=¯S∪Δ(d−2),

and let be the square submatrix of with rows indexed by and columns indexed by .

###### Proposition 4.1.

The matrix is nonsingular if and only if and .

###### Proof.

We may regard as the top boundary map of the -dimensional relative complex . So is nonsingular if and only if . Consider the long exact sequence

 0→~Hd(Δ¯S)→~Hd(ΔT)→~Hd(Γ)→~Hd−1(Δ¯S)→~Hd−1(ΔT)→~Hd−1(Γ)→⋯ (10)

If , then and cannot both be zero. This proves the “only if” direction.

If , then (since ), so (10) implies . Therefore is a -tree, because it has the correct number of facets. Hence is finite. Then (10) implies that is finite. In fact, it is zero because the top homology group of any complex must be torsion-free. Meanwhile, has the correct number of facets to be a -SST of , proving the “if” direction. ∎

###### Proposition 4.2.

If is nonsingular, then

 |det∂S,T|=|~Hd−1(ΔT)|⋅|~Hd−2(Δ¯S)||~Hd−2(ΔT)|=|~Hd−1(ΔT)|⋅|~Hd−2(Δ¯S)||~Hd−2(Δ)|.
###### Proof.

As before, we interpret as the boundary map of the relative complex . So is a map from to , and is a finite abelian group of order . On the other hand, since has no faces of dimension , its lower boundary maps are all zero, so . Since is finite, the desired result now follows from the piece

 0→~Hd−1(ΔT)→~Hd−1(Γ)→~Hd−2(Δ¯S)→~Hd−2(ΔT)→0 (11)

of the long exact sequence (10). ∎

We can now prove the first version of the Simplicial Matrix-Tree Theorem, relating the quantities and . Abbreviate .

###### Theorem 1.3 (Simplicial Matrix-Tree Theorem).

Let be an APC simplicial complex. Then:

1. We have

 πd(Δ)=τd(Δ)τd−1(Δ)|~Hd−2(Δ)|2.
2. Let be the set of facets of a -SST of , and let denote the reduced Laplacian333A warning: This notation for reduced Laplacians specifies which rows and columns to exclude (in analogy to the notation in the statement of Theorem 1.1), in contrast to the notation for restricted boundary maps, which specifies which rows and columns to include. obtained by deleting the rows and columns of corresponding to . Then

 τd(Δ)=|~Hd−2(Δ)|2|~Hd−2(ΔU)|2detLU.
###### Proof of Theorem 1.3 (1).

The Laplacian is a square matrix with rows and columns, and rank (because is APC). Let be its characteristic polynomial (where is an identity matrix), so that , the product of the nonzero eigenvalues of , is given (up to sign) by the coefficient of in . Equivalently,

 πd =∑S⊂Δd−1|S|=rankLdetLU =∑S⊂Δd−1|S|=fd−~βddetLU (12)

where in each summand. By the Binet-Cauchy formula, we have

 detLU =∑T⊂Δd|T|=|S|(det∂S,T)(det∂∗S,T) =∑T⊂Δd|T|=|S|(det∂S,T)2. (13)

Combining (12) and (13), applying Proposition 4.1, and interchanging the sums, we obtain

 πd=∑T:ΔT∈Td(Δ)  ∑S:Δ¯S∈Td−1(Δ)(det∂S,T)2

and now applying Proposition 4.2 yields

 πd =∑T:ΔT∈Td(Δ)  ∑S:Δ¯S∈Td−1(Δ)(|~Hd−1(ΔT)|⋅|~Hd−2(Δ¯S)||~Hd−2(Δ)|)2 =⎛⎝∑T:ΔT∈Td(Δ)|~Hd−1(ΔT)|2⎞⎠⎛⎜⎝∑S:Δ¯S∈Td−1(Δ)|~Hd−2(Δ¯S)|⎞⎟⎠|~Hd−2(Δ)|2

as desired. ∎

In order to prove the “reduced Laplacian” part of Theorem 1.3, we first check that when we delete the rows of corresponding to a -SST, the resulting reduced Laplacian has the correct size, namely, that of a -SST.

###### Lemma 4.3.

Let be the set of facets of a


text:
22.11 More applications of magnetism

 Page 1 / 12
• Describe some applications of magnetism.

Mass spectrometry

The curved paths followed by charged particles in magnetic fields can be put to use. A charged particle moving perpendicular to a magnetic field travels in a circular path having a radius $r$ .

$r=\frac{\text{mv}}{\text{qB}}$

It was noted that this relationship could be used to measure the mass of charged particles such as ions. A mass spectrometer is a device that measures such masses. Most mass spectrometers use magnetic fields for this purpose, although some of them have extremely sophisticated designs. Since there are five variables in the relationship, there are many possibilities. However, if $v$ , $q$ , and $B$ can be fixed, then the radius of the path $r$ is simply proportional to the mass $m$ of the charged particle. Let us examine one such mass spectrometer that has a relatively simple design. (See [link] .) The process begins with an ion source, a device like an electron gun. The ion source gives ions their charge, accelerates them to some velocity $v$ , and directs a beam of them into the next stage of the spectrometer. This next region is a velocity selector that only allows particles with a particular value of $v$ to get through.

The velocity selector has both an electric field and a magnetic field, perpendicular to one another, producing forces in opposite directions on the ions. Only those ions for which the forces balance travel in a straight line into the next region. If the forces balance, then the electric force $F=\text{qE}$ equals the magnetic force $F=\text{qvB}$ , so that $\text{qE}=\text{qvB}$ . Noting that $q$ cancels, we see that

$v=\frac{E}{B}$

is the velocity particles must have to make it through the velocity selector, and further, that $v$ can be selected by varying $E$ and $B$ . In the final region, there is only a uniform magnetic field, and so the charged particles move in circular arcs with radii proportional to particle mass. The paths also depend on charge $q$ , but since $q$ is in multiples of electron charges, it is easy to determine and to discriminate between ions in different charge states.

Mass spectrometry today is used extensively in chemistry and biology laboratories to identify chemical and biological substances according to their mass-to-charge ratios. In medicine, mass spectrometers are used to measure the concentration of isotopes used as tracers. Usually, biological molecules such as proteins are very large, so they are broken down into smaller fragments before analyzing. Recently, large virus particles have been analyzed as a whole on mass spectrometers. Sometimes a gas chromatograph or high-performance liquid chromatograph provides an initial separation of the large molecules, which are then input into the mass spectrometer.

Cathode ray tubes—crts—and the like

What do non-flat-screen TVs, old computer monitors, x-ray machines, and the 2-mile-long Stanford Linear Accelerator have in common? All of them accelerate electrons, making them different versions of the electron gun. Many of these devices use magnetic fields to steer the accelerated electrons. [link] shows the construction of the type of cathode ray tube (CRT) found in some TVs, oscilloscopes, and old computer monitors. Two pairs of coils are used to steer the electrons, one vertically and the other horizontally, to their desired destination.

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
What is thermal heat all about
why uniform circular motion is called a periodic motion?.
when a train start from A & it returns at same station A . what is its acceleration?
what is distance of A to B of the stations and what is the time taken to reach B from A
BELLO
the information provided is not enough
aliyu
Hmmmm maybe the question is logical
yusuf
where are the parameters for calculation
HENRY
there is enough information to calculate an AVERAGE acceleration
Kwok
mistake, there is enough information to calculate an average velocity
Kwok
~\
Abel
what is the unit of momentum
Abel
wha are the types of radioactivity ?
what are the types of radioactivity
Worku
what is static friction
It is the opposite of kinetic friction
Mark
static fiction is friction between two surfaces in contact an none of sliding over on another, while Kinetic friction is friction between sliding surfaces in contact.
MINDERIUM
I don't get it,if it's static then there will be no friction.
author
It means that static friction is that friction that most be overcome before a body can move
kingsley
static friction is a force that keeps an object from moving, and it's the opposite of kinetic friction.
author
It is a force a body must overcome in order for the body to move.
Eboh
If a particle accelerator explodes what happens
Eboh
why we see the edge effect in case of the field lines of capacitor?
Arnab
what is wave
what is force
Muhammed
force is something which is responsible for the object to change its position
MINDERIUM
more technically it is the product of mass of an object and Acceleration produced in it
MINDERIUM
wave is disturbance in any medium
iqra
energy is distributed in any medium through particles of medium.
iqra
If a particle accelerator explodes what happens
we have to first figure out .... wats a particle accelerator first
Teh
What is surface tension
The resistive force of surface.
iqra
Who can tutor me on simple harmonic motion
on both a string and peldulum?
Anya
spring*
Anya
Yea
yusuf
Do you have a chit-chat contact
yusuf
I dont have social media but i do have an email?
Anya
Which is
yusuf
Where are you chatting from
yusuf
I don't understand the basics of this group
Jimmy
teach him SHM init
Anya
Simple harmonic motion
yusuf
how.an.equipotential.line is two dimension and equipotential surface is three dimension ?
definition of mass of conversion
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
Marisa
how many subject is in physics
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
Praise what level are you
yusuf
If u are doing a levels in your first year you do AS topics therefore you do 5 big topic i.e particles radiation, waves and optics, mechanics,materials, electricity. After that you do A level topics like Specific Harmonic motion circular motion astrophysics depends really
Anya
Yeah basics of physics prin8
yusuf
Heat nd Co for a level
yusuf
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
hallow
Boniface
Boniface
the range of objects and phenomena studied in physics is
Boniface


text:
## video modules

$c=\lambda v$

asalest 2K
Posts: 52
Joined: Wed Sep 30, 2020 9:39 pm

### video modules

Are the "Wave properties of light, classical v quantum mechanics, and photon properties of light" video modules recommended for better understanding of the quantum world?

Tikva Cohen 1K
Posts: 104
Joined: Wed Sep 30, 2020 9:39 pm
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### Re: video modules

he didn't tell us specifically to do those ones like the other modules but since they're on his website I'm sure they would be very helpful otherwise he wouldn't have them there. And there's no harm in checking them out!

Ria Nawathe 1C
Posts: 103
Joined: Wed Sep 30, 2020 9:39 pm
Been upvoted: 3 times

### Re: video modules

They're not mandatory but they are helpful, and the videos are pretty short so it won't take too much time to check them out!

sophie esherick 3H
Posts: 119
Joined: Wed Sep 30, 2020 9:35 pm
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### Re: video modules

The videos are really helpful if you aren't super familiar with quantum mechanics or if you need to refresh your memory from high school chemistry. All the extra videos he has provided for us help a lot, even if you have a pretty good understanding on quantum mechanics already. They were made to help us better understand the topics we will be covering the next week or two in lectures by providing us with a good foundation for understanding quantum mechanics.

Tiao Tan 3C
Posts: 100
Joined: Wed Sep 30, 2020 9:59 pm

### Re: video modules

I personally found these modules helpful!! The post-assessments also helped me after today's lecture. Although they might be easier than what we have on the textbooks, they are definietly helpful if you are new to these concepts.

Jaclyn Dang 3B
Posts: 105
Joined: Wed Sep 30, 2020 10:02 pm

### Re: video modules

As stated before, they aren't mandatory for this class or will affect your grade overall. However, I have found them helpful for review/ as an introduction to the materials being discussed. If you don't have a strong background in chemistry, like me, I highly recommend them. It should be the next four modules posted on the class website.

Ximeng Guo 2K
Posts: 101
Joined: Wed Sep 30, 2020 9:54 pm

### Re: video modules

Yes I recommend taking the modules.

Grace_Remphrey_2J
Posts: 90
Joined: Wed Sep 30, 2020 9:54 pm

### Re: video modules

I found that the modules are very helpful to do before each week's lectures. There are a series of objectives for each week that Dr. Lavelle sent us and you can look at them to see which video to do in preparation for the week!

Simer_Shera_2D
Posts: 108
Joined: Wed Sep 30, 2020 9:58 pm
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### Re: video modules

Although they're not required, I found that watching the lectures ahead of time really helped me understand his current lectures. I'd definitely suggest watching if you have some time!

Winzel Payumo 1B
Posts: 95
Joined: Wed Sep 30, 2020 9:55 pm

### Re: video modules

The modules are not mandatory, but I highly recommend doing them! It really helps you realize what you know and what you need to work on.

BrittneyMyint1D
Posts: 111
Joined: Wed Sep 30, 2020 9:59 pm
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### Re: video modules

They are not mandatory, but I did find them helpful! I believe Professor Lavelle went over wave properties of light in his friday lecture and he kinda touched upon the other video modules in yesterday's lecture, but it's good review :)


text:
# Can there be an advantage to rescaling covariates in a Logit model ?

#### thedreamshaper

##### New Member
Suppose i have some logit model with, say, 7 explanatory variables and i wish to add another variable, x8. x8 is in the interval [1,5] and i would suspect that low values have a negative effect on the probability of y=1 while a higher value would tend to indicate a higher probability of y=1.

Can there be an advantage to rescaling x8 such that it is, say, defined on [-5,5] ?

I have always thought no, but a colleague recently argued that it would make sense.