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# zbMATH — the first resource for mathematics

## Yang, Chunli

Compute Distance To:
 Author ID: yang.chunli Published as: Yang, C-L; Yang, C.; Yang, C. L.; Yang, Chun-Li; Yang, Chunli
 Documents Indexed: 25 Publications since 1992, including 1 Book
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#### Co-Authors

 0 single-authored 2 Chen, Shangdi 2 Wang, Licheng 1 Gao, Bo 1 Jin, Zhengping 1 Li, Lixiang 1 Lu, Xiuhua 1 Shi, Yanfeng 1 Wen, Qiaoyan 1 Yang, Yixian

#### Serials

 1 Ars Combinatoria 1 Mathematical Problems in Engineering 1 Journal of Applied Mathematics 1 Frontiers of Computer Science

#### Fields

 4 Information and communication theory, circuits (94-XX) 2 Computer science (68-XX)

#### Citations contained in zbMATH

16 Publications have been cited 496 times in 494 Documents Cited by Year
ARPACK user’s guide. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. Zbl 0901.65021
Lehoucq, R. B.; Sorensen, D. C.; Yang, C.
1998
A truncated RQ iteration for large scale eigenvalue calculations. Zbl 0918.65023
Sorensen, D. C.; Yang, C.
1998
Joule heating effect on electroosmotic flow and mass species transport in a microcapillary. Zbl 1075.76062
Tang, G. Y.; Yang, C.; Chai, J. C.; Gong, H. Q.
2004
Parallel machine scheduling with common due windows. Zbl 1230.90090
Huang, R-H; Yang, C-L; Huang, H-T
2010
Recursive integral method with Cayley transformation. Zbl 07031739
Huang, R.; Sun, J.; Yang, C.
2018
A synthesis of tortuosity and dispersion in effective thermal conductivity of porous media. Zbl 1194.80080
Yang, C.; Nakayama, A.
2010
False data injection attack on consensus-based distributed estimation. Zbl 1367.93019
Lei, L.; Yang, W.; Yang, C.; Shi, H. B.
2017
A network-based approach for increasing discrimination in data envelopment analysis. Zbl 1171.91377
Liu, J. S.; Lu, W-M; Yang, C.; Chuang, M.
2009
A stochastic scaled boundary finite element method. Zbl 1439.74445
Long, X. Y.; Jiang, C.; Yang, C.; Han, X.; Gao, W.; Liu, J.
2016
STP-LWE: A variant of learning with error for a flexible encryption. Zbl 1407.94106
Gao, Bo; Shi, Yanfeng; Yang, Chunli; Li, Lixiang; Wang, Licheng; Yang, Yixian
2014
A lattice-based signcryption scheme without random oracles. Zbl 1360.94322
Lu, Xiuhua; Wen, Qiaoyan; Jin, Zhengping; Wang, Licheng; Yang, Chunli
2014
ComPASS: a tool for distributed parallel finite volume discretizations on general unstructured polyhedral meshes. Zbl 1329.65197
Dalissier, E.; Guichard, C.; Havé, P.; Masson, R.; Yang, C.
2013
A new construction of multisender authentication codes from symplectic geometry over finite fields. Zbl 1289.94063
Chen, Shang-Di; Yang, Chun-Li
2012
Enhancing heat transfer in the core flow by using porous medium insert in a tube. Zbl 1183.80032
Huang, Z. F.; Nakayama, A.; Yang, K.; Yang, C.; Liu, Wei
2010
An algorithm for minimizing flow time and maximum earliness on a single machine. Zbl 1171.90399
Huang, R-H; Yang, C-L
2009
A model for promotion rate control in hierarchical manpower systems. Zbl 0849.90082
Glen, J. J.; Yang, C. L.
1996
Recursive integral method with Cayley transformation. Zbl 07031739
Huang, R.; Sun, J.; Yang, C.
2018
False data injection attack on consensus-based distributed estimation. Zbl 1367.93019
Lei, L.; Yang, W.; Yang, C.; Shi, H. B.
2017
A stochastic scaled boundary finite element method. Zbl 1439.74445
Long, X. Y.; Jiang, C.; Yang, C.; Han, X.; Gao, W.; Liu, J.
2016
STP-LWE: A variant of learning with error for a flexible encryption. Zbl 1407.94106
Gao, Bo; Shi, Yanfeng; Yang, Chunli; Li, Lixiang; Wang, Licheng; Yang, Yixian
2014
A lattice-based signcryption scheme without random oracles. Zbl 1360.94322
Lu, Xiuhua; Wen, Qiaoyan; Jin, Zhengping; Wang, Licheng; Yang, Chunli
2014
ComPASS: a tool for distributed parallel finite volume discretizations on general unstructured polyhedral meshes. Zbl 1329.65197
Dalissier, E.; Guichard, C.; Havé, P.; Masson, R.; Yang, C.
2013
A new construction of multisender authentication codes from symplectic geometry over finite fields. Zbl 1289.94063
Chen, Shang-Di; Yang, Chun-Li
2012
Parallel machine scheduling with common due windows. Zbl 1230.90090
Huang, R-H; Yang, C-L; Huang, H-T
2010
A synthesis of tortuosity and dispersion in effective thermal conductivity of porous media. Zbl 1194.80080
Yang, C.; Nakayama, A.
2010
Enhancing heat transfer in the core flow by using porous medium insert in a tube. Zbl 1183.80032
Huang, Z. F.; Nakayama, A.; Yang, K.; Yang, C.; Liu, Wei
2010
A network-based approach for increasing discrimination in data envelopment analysis. Zbl 1171.91377
Liu, J. S.; Lu, W-M; Yang, C.; Chuang, M.
2009
An algorithm for minimizing flow time and maximum earliness on a single machine. Zbl 1171.90399
Huang, R-H; Yang, C-L
2009
Joule heating effect on electroosmotic flow and mass species transport in a microcapillary. Zbl 1075.76062
Tang, G. Y.; Yang, C.; Chai, J. C.; Gong, H. Q.
2004
ARPACK user’s guide. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. Zbl 0901.65021
Lehoucq, R. B.; Sorensen, D. C.; Yang, C.
1998
A truncated RQ iteration for large scale eigenvalue calculations. Zbl 0918.65023
Sorensen, D. C.; Yang, C.
1998
A model for promotion rate control in hierarchical manpower systems. Zbl 0849.90082
Glen, J. J.; Yang, C. L.
1996
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#### Cited by 1,020 Authors

 11 Giani, Stefano 9 Kressner, Daniel 8 Mehrmann, Volker 8 Zhou, Yunkai 7 Guglielmi, Nicola 7 Schmid, Peter Joseph 6 Salinger, Andrew G. 6 Schlatter, Philipp 6 Sipp, Denis 6 Sorensen, Danny C. 5 Bangerth, Wolfgang 5 Gattringer, Christof 5 Gedicke, Joscha 5 Heister, Timo 5 Heltai, Luca 5 Henningson, Dan S. 5 Kronbichler, Martin 5 Lehoucq, Richard B. 5 Lin, Wen-Wei 5 Maier, Matthias Sebastian 5 Marquet, Olivier 5 Robinet, Jean-Christophe 5 Saad, Yousef 5 Turcksin, Bruno 4 Alizard, Frédéric 4 Arndt, Daniel 4 Benner, Peter 4 Bergamaschi, Luca 4 Buchot, Jean-Marie 4 Davydov, Denis 4 Dijkstra, Henk A. 4 Hall, Edward J. C. 4 Lang, Christian B. 4 Lubich, Christian 4 Meerbergen, Karl 4 Międlar, Agnieszka 4 Net, Marta 4 Pelteret, Jean-Paul 4 Pini, Giorgio 4 Román, Jose E. 4 Tantet, Alexis 4 Voss, Heinrich 4 Zhang, Lei-Hong 3 Arbenz, Peter 3 Bagheri, Shervin 3 Bai, Zhaojun 3 Bhuruth, Muddun 3 Bonito, Andrea 3 Cadou, Jean-Marc 3 Carini, Marco 3 Carstensen, Carsten 3 Chekroun, Mickaël D. 3 Chomaz, Jean-Marc 3 Daescu, Dacian N. 3 Daube, Olivier 3 Dookhitram, Kumar 3 Embree, Mark 3 Engström, Christian 3 Giannakis, Dimitrios 3 Göckeler, Meinulf 3 Guermond, Jean-Luc 3 Guevel, Y. 3 Harbrecht, Helmut 3 Hetmaniuk, Ulrich L. 3 Hill, Antony A. 3 Houston, Paul 3 Huang, Tsungming 3 Huang, Weiqiang 3 Jacquin, Laurent 3 Juniper, Matthew P. 3 Le Quéré, Patrick 3 Lu, Ding 3 Lu, Linzhang 3 Nataf, Frédéric 3 Navon, Ionel Michael 3 Neelin, J. David 3 Nichols, Joseph W. 3 Niu, Qiang 3 Pawlowski, Roger P. 3 Peters, Michael D. 3 Phipps, Eric T. 3 Rakow, Paul E. L. 3 Reichel, Lothar 3 Rowley, Clarence W. 3 Sánchez, José Luíz 3 Shadid, John N. 3 Sheard, Gregory John 3 Simoncini, Valeria 3 Stathopoulos, Andreas 3 Stoll, Martin 3 Sun, Jiguang 3 Tisseur, Françoise 3 Treyssède, Fabien 3 Tuminaro, Raymond S. 3 Wang, Bofu 3 Wells, David L. 3 Wen, Zaiwen 3 Ye, Qiang 3 Zhang, Yin 2 Adachi, Satoru ...and 920 more Authors
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#### Cited in 141 Serials

 45 Journal of Computational Physics 42 Journal of Fluid Mechanics 23 Journal of Computational and Applied Mathematics 22 Linear Algebra and its Applications 19 SIAM Journal on Scientific Computing 17 Computer Methods in Applied Mechanics and Engineering 15 Numerical Linear Algebra with Applications 13 Applied Numerical Mathematics 12 International Journal for Numerical Methods in Fluids 12 Numerische Mathematik 12 SIAM Journal on Matrix Analysis and Applications 11 Computer Physics Communications 11 International Journal for Numerical Methods in Engineering 11 Journal of Scientific Computing 9 Computers and Fluids 9 Computers & Mathematics with Applications 7 BIT 6 Applied Mathematics and Computation 6 Physics of Fluids 5 International Journal of Heat and Mass Transfer 5 European Journal of Operational Research 5 Optimization Methods & Software 5 Journal of Numerical Mathematics 4 Mathematical Programming. Series A. Series B 4 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 3 Journal of Statistical Physics 3 Mathematics of Computation 3 Automatica 3 Mathematical and Computer Modelling 3 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 3 Numerical Algorithms 3 Annals of Physics 3 Computational Optimization and Applications 3 Communications in Numerical Methods in Engineering 3 European Journal of Mechanics. B. Fluids 3 Computational Geosciences 3 Multiscale Modeling & Simulation 2 Inverse Problems 2 Nuclear Physics. B 2 Physics Letters. A 2 Physics Letters. B 2 Wave Motion 2 Computing 2 Meccanica 2 Applied Mathematical Modelling 2 Computational Statistics and Data Analysis 2 SIAM Journal on Optimization 2 Journal of Nonlinear Science 2 Nuclear Physics, B, Proceedings Supplements 2 Journal of Mathematical Chemistry 2 Mathematical Problems in Engineering 2 International Journal of Computational Fluid Dynamics 2 Journal of Computational Acoustics 2 GAMM-Mitteilungen 2 S$$\vec{\text{e}}$$MA Journal 1 Communications in Mathematical Physics 1 Geophysical and Astrophysical Fluid Dynamics 1 International Journal of Control 1 Journal of Mathematical Analysis and Applications 1 Letters in Mathematical Physics 1 Linear and Multilinear Algebra 1 Mathematical Biosciences 1 Mathematical Methods in the Applied Sciences 1 Nonlinearity 1 Physica A 1 Molecular Simulation 1 Theoretical and Computational Fluid Dynamics 1 Fortschritte der Physik 1 ACM Transactions on Mathematical Software 1 Applied Mathematics and Optimization 1 Calcolo 1 Journal of the American Statistical Association 1 Journal of Applied Probability 1 Mathematics and Computers in Simulation 1 Mathematika 1 Mechanics Research Communications 1 SIAM Journal on Numerical Analysis 1 Studies in Applied Mathematics 1 Systems & Control Letters 1 Applied Mathematics and Mechanics. (English Edition) 1 Mitteilungen der Gesellschaft für Angewandte Mathematik und Mechanik 1 Computer Aided Geometric Design 1 Parallel Computing 1 Acta Mathematicae Applicatae Sinica. English Series 1 Journal of Symbolic Computation 1 ACM Transactions on Graphics 1 Computational Mechanics 1 Journal of Economic Dynamics & Control 1 Applied Mathematics Letters 1 Japan Journal of Industrial and Applied Mathematics 1 Computational Geometry 1 Journal of Global Optimization 1 Computational Statistics 1 International Journal of Computer Mathematics 1 SIAM Journal on Applied Mathematics 1 SIAM Review 1 Applicable Algebra in Engineering, Communication and Computing 1 Archive of Applied Mechanics 1 International Journal of Robust and Nonlinear Control 1 Journal of Mathematical Imaging and Vision ...and 41 more Serials
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#### Cited in 38 Fields

 293 Numerical analysis (65-XX) 129 Fluid mechanics (76-XX) 73 Partial differential equations (35-XX) 47 Linear and multilinear algebra; matrix theory (15-XX) 39 Mechanics of deformable solids (74-XX) 38 Optics, electromagnetic theory (78-XX) 35 Operations research, mathematical programming (90-XX) 32 Quantum theory (81-XX) 25 Computer science (68-XX) 24 Statistical mechanics, structure of matter (82-XX) 20 Dynamical systems and ergodic theory (37-XX) 19 Statistics (62-XX) 18 Classical thermodynamics, heat transfer (80-XX) 18 Systems theory; control (93-XX) 15 Probability theory and stochastic processes (60-XX) 11 Operator theory (47-XX) 10 Ordinary differential equations (34-XX) 8 Geophysics (86-XX) 8 Biology and other natural sciences (92-XX) 7 Combinatorics (05-XX) 7 Calculus of variations and optimal control; optimization (49-XX) 6 Mechanics of particles and systems (70-XX) 4 Approximations and expansions (41-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 4 Information and communication theory, circuits (94-XX) 3 Global analysis, analysis on manifolds (58-XX) 2 History and biography (01-XX) 2 Measure and integration (28-XX) 2 Integral equations (45-XX) 1 General and overarching topics; collections (00-XX) 1 Number theory (11-XX) 1 Group theory and generalizations (20-XX) 1 Functions of a complex variable (30-XX) 1 Potential theory (31-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Difference and functional equations (39-XX) 1 Geometry (51-XX) 1 Relativity and gravitational theory (83-XX)


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# How to prove that such measure does not exist on an infinite product space?

A previous (already submitted) exercise from my coursework was,

Prove that there does not exist a probability measure $$\mathbb{P}$$ on the infinite product space $$(\mathbb{R}^{[0,\infty]}, > \mathcal{B}^{[0, \infty)})$$ with the property that a.s $$\xi \in \mathbb{R}^{[0, \infty]}$$, the function $$t \rightarrow \xi_t$$ is continous.

I could not come up with a good starting approach to solve this one, after revising and understanding what each of these terms mean. I was thinking about applying $$\pi - \lambda$$ type metatheorem, because that can be used usually to prove that some property holds a.s. However, here the converse needs to be proved, which I find not straightforward.

I would really appreciate, some solution or hints to this question.

Let $$C=\{\omega\in\mathbb{R}^{[0,\infty)}:\omega\text{ is continuous}\}$$. The only $$\mathcal{B}^{[0,\infty)}$$-measurable subset of $$C$$ is $$\emptyset$$. To get started proving this, first show that if $$E\in\mathcal{B}^{[0,\infty)}$$, then there exists a strictly increasing sequence $$\{t_n\}_{n=1}^\infty$$ in $$[0,\infty)$$ and a set $$A\in\mathcal{B}(\mathbb{R}^\infty)$$ such that $$E = \{\omega: (\omega(t_1), \omega(t_2), \ldots) \in A\}.$$ (Show that the collection of such sets is a $$\sigma$$-algebra that contains the cylinder sets, and use the fact that the cylinder sets generate $$\mathcal{B}^{[0,\infty)}$$.)

With this fact in hand, now suppose there exists $$P$$ such that $$t\mapsto\omega(t)$$ is continuous a.s. This means there exists $$N\in\mathcal{B}^{[0,\infty)}$$ such that $$P(N)=0$$ and, for all $$\omega\in N^c$$, we have that $$t\mapsto\omega(t)$$ is continuous. This implies that $$N^c\subset C$$. By the above, $$N^c=\emptyset$$, so $$P(N^c)=0$$, which gives $$P(N)=1$$, a contradiction.

EDIT:

This answer assumes that $$\mathcal{B}^{[0,\infty)}$$ is meant to denote the product $$\sigma$$-algebra, $$\mathcal{B}^{[0,\infty)} = \bigotimes_{t\in[0,\infty)} \mathcal{B},$$ where $$\mathcal{B}$$ is the Borel $$\sigma$$-algebra on $$\mathbb{R}$$.

An alternative way to define a $$\sigma$$-algebra on $$\mathbb{R}^{[0,\infty)}$$ is to endow $$\mathbb{R}^{[0,\infty)}$$ with the product topology (or the topology of pointwise convergence) and then use the Borel $$\sigma$$-algebra corresponding to this topology (i.e. the smallest $$\sigma$$-algebra on $$\mathbb{R}^{[0,\infty)}$$ that contains the sets which are open in the product topology.) This latter approach produces a strictly larger $$\sigma$$-algebra. For example, this latter $$\sigma$$-algebra contains singletons, whereas $$\mathcal{B}^{[0,\infty)}$$ does not. See https://math.stackexchange.com/a/248587/11867, for example. As indicated in a now-deleted answer, if we take the latter approach, then the result is not true.

EDIT 2:

Here are some clarifications. A cylinder set is a set of the form $$F = \{\omega\in\mathbb{R}: (\omega(t_1),\ldots,\omega(t_n)) \in B\},$$ for some $$n\in\mathbb{N}$$ and some $$B\in\mathcal{B}(\mathbb{R}^n)$$. Also $$\mathbb{R}^\infty = \{(t_1,t_2,\ldots):t_j\in\mathbb{R}\},$$ and $$\mathcal{B}(\mathbb{R}^\infty) = \bigotimes_{n\in\mathbb{N}}\mathcal{B},$$ which is the $$\sigma$$-algebra generated by sets of the form $$A_1\times A_2\times \cdots,$$ where $$A_n\in\mathcal{B}$$ for each $$n$$. If $$A\in\mathcal{B}(\mathbb{R}^n)$$, then $$A\times\mathbb{R}\times\mathbb{R}\times\cdots = \{(t_1,t_2,\ldots):(t_1,\ldots,t_n)\in A\} \in \mathcal{B}(\mathbb{R}^\infty).$$

• Thank you for your answer. I'm confused when you define E, and say that is a $\sigma$-algebra, because to my understanding E are the cylinder sets, which collection can be used to generate $\mathcal{B}^{[0,\infty)}$, If E would be a $\sigma$ algebra, then the $\sigma$-algebra generated would be itself again, which implies E is Borel. I'm also a bit confused about the difference in $\mathcal{B}(R^{\infty})$ and $\mathcal{B}^{[0,\infty)}$. – boomkin May 2 at 9:51
• A cylinder set uses only finitely many $t$ values, whereas $E$ uses a countably infinite number of $t$ values. Regarding the difference between those two $\sigma$-algebras, it is too complicated to explain further in a comment, but it is not relevant to your question. I only included it because of another answer and comment which is now deleted. If you want to know more, you should probably post a separate question about it. – Jason Swanson May 2 at 11:43
• It is used in your proof stating $A \in \mathcal{B}(\mathbb{R}^{\infty})$, that is the only reason I'm worried about it. – boomkin May 2 at 12:32
• Sorry, I was reading your comment on my phone, so had to parse the tex mentally and misread it. I have added an edit in order to clarify this for you. – Jason Swanson May 2 at 12:49

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March 25, 2016

In my earlier post I talked about how the GAN training procedure can be modified slightly so it minimises $KL[Q|P]$ divergence. I have also talked about how GANs can be understood as

• in the D-step, performing direct log-probability-ratio estimation from data, obtaining an estimate of $\log\frac{Q(x)}{P(x)}$ then
• in the M-step using these estimates to minimise a divergence

From this perspective, training a binary classifier discriminatively is only one way to obtain estimates of the log-probability ratios in question, but I mentioned there are other ways.

This post is basically just having some fun with this idea: I show a different iterative algorithm that has similar flavour. Instead of using a binary classifier, I'm going to use binary pairwise preference learning as the adversarial problem.

As before, we have a generator $G$ and a discriminator $D$ that try to "fight against each other". However, the discriminator now solves a different task: instead of binary classification it performs a two-alternative forced choice (2AFC) task.

#### 2AFC Discriminator

The preference network $D$ receives two inputs, $x_1$ and $x_2$, one of which is a real datapont sampled from $P$, the other one is a synthetic one generated from $Q$ (using the generator $G$). $D$ has to guess which one of its inputs is real. Let's assume that $D$ takes the following symmetric and separable form:

$$D(x_1,x_2;\psi) = \Phi[s(x_1;\psi) - s(x_2;\psi)],$$

where $\Phi$ is the logistic sigmoid, and $s$ is a trainable function (e.g. deep neural network) which I will call the preference function.

The parameters $\psi$ of $D$ are updated using the following objective:

$$\mathcal{L}_{D}(\psi) = -\mathbb{E}_{x_1\sim P}\mathbb{E}_{x_2\sim Q} \log(D(x_1,x_2;\psi))$$

This is similar to a standard binary classification loss, but not quite the same (if you don't understand the objective, note that the symmetry $D(x_1,x_2) = 1 - D(x_2,x_1)$ and try to derive it yourself from the binary classification loss)

#### Bayes optimal behaviour

Let's consider what happens if we fix $Q$, and optimise $D$ until convergence. At this point we can assume that $D$ approximately implements the Bayes optimal behaviour, which in this case will be the following:

$$\Phi[s(x_1) - s(x_2)] \approx \frac{P(x_1)Q(x_2)}{P(x_1)Q(x_2) + P(x_2)Q(x_1)}$$

Working backwards from here, we get an expression for the preference function:

\begin{align}
\Phi[s(x_1) - s(x_2)] &\approx \frac{P(x_1)Q(x_2)}{P(x_1)Q(x_2) + P(x_2)Q(x_1)} \\
s(x_1) - s(x_2) &\approx \Phi^{-1}\left[\frac{P(x_1)Q(x_2)}{P(x_1)Q(x_2) + P(x_2)Q(x_1)}\right]\\
s(x_1) - s(x_2) &\approx \log\left[\frac{\frac{P(x_1)Q(x_2)}{P(x_1)Q(x_2) + P(x_2)Q(x_1)}}{1 - \frac{P(x_1)Q(x_2)}{P(x_1)Q(x_2) + P(x_2)Q(x_1)}} \right]\\
s(x_1) - s(x_2) &\approx \log\left[\frac{P(x_1)Q(x_2)}{P(x_2)Q(x_1)} \right]\\
s(x_1) - s(x_2) &\approx \log\frac{P(x_1)}{Q(x_1)} - \log\frac{P(x_2)}{Q(x_2)},
\end{align}

Therefore, we can say that when $D$ converged, the preference function $s$ approximates the log-probability ratio between $P$ vs $Q$:

$$s(x) \approx \log\frac{P(x)}{Q(x)}$$

and therefore the loss for the generator becomes

$$\mathcal{L}_{G}(\theta) \approx - \mathbb{E}_{x\sim Q_{\theta}} \log\frac{P(x)}{Q(x)} = KL[Q|P].$$

### Is this useful?

I don't know, maybe, most likely not. But that's besides the point I'm trying to make. I just wanted to highlight the arbitrariness of using binary classification in the GAN procedure. One can imagine a whole class of generative modelling procedures based on the same underlying idea of direct probability ratio estimation, and it is unclear (to me) which one method will work best.


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# Continued Fractions

1. Apr 28, 2012

### Charles49

Suppose we can write a real number x as a continued fraction like this

x=a0+1/(a1+1/(a2+1/(a2+...=[a0; a1, a2, a3, a4, ... an...].

Is there a binary operation f(i,x) so that f(i,x)=ai? I was wondering if there was a formula which gives the ith item in the sequence of integers which is connected to every x in the context of this expansion.

Every rational number has a unique continued fraction expansion so I think this is a valid question. Moreover, every irrational number has a unique, infinite continued fraction expansion.

My first guess was to combine the inputs a, b in the Euclidean algorithm from which the continued fraction expansion arises but I don't know how to extract the ith item in the sequence of quotients. Any thoughts?

2. Apr 29, 2012

3. May 1, 2012

### Charles49

Thanks a lot for answering my question. It's a big step in the right direction.

My question was how to compute the continued fraction expansion of a fraction. For example, using the Euclidean algorithm, we have
$$\frac{7}{10}=0+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3}}}$$
Therefore, $$a_0=0, a_1=1, a_2=3, a_3=3, a_4=0, a_5=0, a_6=0,\dots$$

Is there a way to compute $$a_n$$ without using the euclidean algorithm?


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### 相关文章

 Subaru/COMICS Study on Silicate Dust Processing around Young Low-Mass StarsWe have obtained 8-13 μm spectra of 30 young (1-10 Myr) low-masspre-main-sequence stars using COMICS on the 8.2 m Subaru Telescope toexamine dust evolution in protoplanetary disks. Most spectra showsilicate emission features of various strengths and shapes, indicativeof dust processing during the different stages of protoplanetary diskevolution. We have analyzed the observed silicate emission featuresusing a simple model previously applied to the more massive and luminousHerbig Ae/Be systems. We determined the feature strength and shape andderived the composition and typical size of the silicate dust grains. Weconfirm the previously reported dependency of the silicate featurestrength and shape on the grain size of the amorphous silicate dust. Weexamine the relation between the derived dust properties and stellar andcircumstellar disk parameters, such as systemic age, luminosity ofHα (LHα), disk mass, and opacity power-law index β atradio wavelengths. A possible relation is found between silicate featurestrength (grain size indicator) and the LHα, which may be anindicator of accretion activity. It implies that the turbulence inducedby accretion activity may be important for grain size evolution in thedisk. No clear correlation between the crystallinity and thestellar/disk parameters is found. We find that on average 5%-20% in massof the silicate dust grains is in crystalline form, irrespective ofsystemic age. This latter finding supports the idea that crystallinesilicate is formed at an early evolutionary phase, probably at theprotostellar phase, and is remaining during the later stages.Based on data collected at the Subaru Telescope, which is operated bythe National Astronomical Observatory of Japan. CHARM2: An updated Catalog of High Angular Resolution MeasurementsWe present an update of the Catalog of High Angular ResolutionMeasurements (CHARM, Richichi & Percheron \cite{CHARM}, A&A,386, 492), which includes results available until July 2004. CHARM2 is acompilation of direct measurements by high angular resolution methods,as well as indirect estimates of stellar diameters. Its main goal is toprovide a reference list of sources which can be used for calibrationand verification observations with long-baseline optical and near-IRinterferometers. Single and binary stars are included, as are complexobjects from circumstellar shells to extragalactic sources. The presentupdate provides an increase of almost a factor of two over the previousedition. Additionally, it includes several corrections and improvements,as well as a cross-check with the valuable public release observationsof the ESO Very Large Telescope Interferometer (VLTI). A total of 8231entries for 3238 unique sources are now present in CHARM2. Thisrepresents an increase of a factor of 3.4 and 2.0, respectively, overthe contents of the previous version of CHARM.The catalog is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/431/773 Local kinematics of K and M giants from CORAVEL/Hipparcos/Tycho-2 data. Revisiting the concept of superclustersThe availability of the Hipparcos Catalogue has triggered many kinematicand dynamical studies of the solar neighbourhood. Nevertheless, thosestudies generally lacked the third component of the space velocities,i.e., the radial velocities. This work presents the kinematic analysisof 5952 K and 739 M giants in the solar neighbourhood which includes forthe first time radial velocity data from a large survey performed withthe CORAVEL spectrovelocimeter. It also uses proper motions from theTycho-2 catalogue, which are expected to be more accurate than theHipparcos ones. An important by-product of this study is the observedfraction of only 5.7% of spectroscopic binaries among M giants ascompared to 13.7% for K giants. After excluding the binaries for whichno center-of-mass velocity could be estimated, 5311 K and 719 M giantsremain in the final sample. The UV-plane constructed from these datafor the stars with precise parallaxes (σπ/π≤20%) reveals a rich small-scale structure, with several clumpscorresponding to the Hercules stream, the Sirius moving group, and theHyades and Pleiades superclusters. A maximum-likelihood method, based ona Bayesian approach, has been applied to the data, in order to make fulluse of all the available stars (not only those with precise parallaxes)and to derive the kinematic properties of these subgroups. Isochrones inthe Hertzsprung-Russell diagram reveal a very wide range of ages forstars belonging to these groups. These groups are most probably relatedto the dynamical perturbation by transient spiral waves (as recentlymodelled by De Simone et al. \cite{Simone2004}) rather than to clusterremnants. A possible explanation for the presence of younggroup/clusters in the same area of the UV-plane is that they have beenput there by the spiral wave associated with their formation, while thekinematics of the older stars of our sample has also been disturbed bythe same wave. The emerging picture is thus one of dynamical streamspervading the solar neighbourhood and travelling in the Galaxy withsimilar space velocities. The term dynamical stream is more appropriatethan the traditional term supercluster since it involves stars ofdifferent ages, not born at the same place nor at the same time. Theposition of those streams in the UV-plane is responsible for the vertexdeviation of 16.2o ± 5.6o for the wholesample. Our study suggests that the vertex deviation for youngerpopulations could have the same dynamical origin. The underlyingvelocity ellipsoid, extracted by the maximum-likelihood method afterremoval of the streams, is not centered on the value commonly acceptedfor the radial antisolar motion: it is centered on < U > =-2.78±1.07 km s-1. However, the full data set(including the various streams) does yield the usual value for theradial solar motion, when properly accounting for the biases inherent tothis kind of analysis (namely, < U > = -10.25±0.15 kms-1). This discrepancy clearly raises the essential questionof how to derive the solar motion in the presence of dynamicalperturbations altering the kinematics of the solar neighbourhood: doesthere exist in the solar neighbourhood a subset of stars having no netradial motion which can be used as a reference against which to measurethe solar motion?Based on observations performed at the Swiss 1m-telescope at OHP,France, and on data from the ESA Hipparcos astrometry satellite.Full Table \ref{taba1} is only available in electronic form at the CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/430/165} High-Precision Near-Infrared Photometry of a Large Sample of Bright Stars Visible from the Northern HemisphereWe present the results of 8 yr of infrared photometric monitoring of alarge sample of stars visible from Teide Observatory (Tenerife, CanaryIslands). The final archive is made up of 10,949 photometric measuresthrough a standard InSb single-channel photometer system, principally inJHK, although some stars have measures in L'. The core of this list ofstars is the standard-star list developed for the Carlos SánchezTelescope. A total of 298 stars have been observed on at least twooccasions on a system carefully linked to the zero point defined byVega. We present high-precision photometry for these stars. The medianuncertainty in magnitude for stars with a minimum of four observationsand thus reliable statistics ranges from 0.0038 mag in J to 0.0033 magin K. Many of these stars are faint enough to be observable with arraydetectors (42 are K>8) and thus to permit a linkage of the bright andfaint infrared photometric systems. We also present photometry of anadditional 25 stars for which the original measures are no longeravailable, plus photometry in L' and/or M of 36 stars from the mainlist. We calculate the mean infrared colors of main-sequence stars fromA0 V to K5 V and show that the locus of the H-K color is linearlycorrelated with J-H. The rms dispersion in the correlation between J-Hand H-K is 0.0073 mag. We use the relationship to interpolate colors forall subclasses from A0 V to K5 V. We find that K and M main-sequence andgiant stars can be separated on the color-color diagram withhigh-precision near-infrared photometry and thus that photometry canallow us to identify potential mistakes in luminosity classclassification. A catalogue of calibrator stars for long baseline stellar interferometryLong baseline stellar interferometry shares with other techniques theneed for calibrator stars in order to correct for instrumental andatmospheric effects. We present a catalogue of 374 stars carefullyselected to be used for that purpose in the near infrared. Owing toseveral convergent criteria with the work of Cohen et al.(\cite{cohen99}), this catalogue is in essence a subset of theirself-consistent all-sky network of spectro-photometric calibrator stars.For every star, we provide the angular limb-darkened diameter, uniformdisc angular diameters in the J, H and K bands, the Johnson photometryand other useful parameters. Most stars are type III giants withspectral types K or M0, magnitudes V=3-7 and K=0-3. Their angularlimb-darkened diameters range from 1 to 3 mas with a median uncertaintyas low as 1.2%. The median distance from a given point on the sky to theclosest reference is 5.2degr , whereas this distance never exceeds16.4degr for any celestial location. The catalogue is only available inelectronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/393/183 CHARM: A Catalog of High Angular Resolution MeasurementsThe Catalog of High Angular Resolution Measurements (CHARM) includesmost of the measurements obtained by the techniques of lunaroccultations and long-baseline interferometry at visual and infraredwavelengths, which have appeared in the literature or have otherwisebeen made public until mid-2001. A total of 2432 measurements of 1625sources are included, along with extensive auxiliary information. Inparticular, visual and infrared photometry is included for almost allthe sources. This has been partly extracted from currently availablecatalogs, and partly obtained specifically for CHARM. The main aim is toprovide a compilation of sources which could be used as calibrators orfor science verification purposes by the new generation of largeground-based facilities such as the ESO Very Large Interferometer andthe Keck Interferometer. The Catalog is available in electronic form atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/386/492, and from theauthors on CD-Rom. Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition - Comments and statisticsThe Catalogue, available at the Centre de Données Stellaires deStrasbourg, consists of 13 573 records concerning the results obtainedfrom different methods for 7778 stars, reported in the literature. Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521 Sixth Catalogue of Fundamental Stars (FK6). Part III. Additional fundamental stars with direct solutionsThe FK6 is a suitable combination of the results of the HIPPARCOSastrometry satellite with ground-based data, measured over a longinterval of time and summarized mainly in the FK5. Part III of the FK6(abbreviated FK6(III)) contains additional fundamental stars with directsolutions. Such direct solutions are appropriate for single stars or forobjects which can be treated like single stars. Part III of the FK6contains in total 3272 stars. Their ground-based data stem from thebright extension of the FK5 (735 stars), from the catalogue of remainingSup stars (RSup, 732 stars), and from the faint extension of the FK5(1805 stars). From the 3272 stars in Part III, we have selected 1928objects as "astrometrically excellent stars", since their instantaneousproper motions and their mean (time-averaged) ones do not differsignificantly. Hence most of the astrometrically excellent stars arewell-behaving "single-star candidates" with good astrometric data. Thesestars are most suited for high-precision astrometry. On the other hand,354 of the stars in Part III are Δμ binaries in the sense ofWielen et al. (1999). Many of them are newly discovered probablebinaries with no other hitherto known indication of binarity. The FK6gives, besides the classical "single-star mode" solutions (SI mode),other solutions which take into account the fact that hidden astrometricbinaries among "apparently single-stars" introduce sizable "cosmicerrors" into the quasi-instantaneously measured HIPPARCOS proper motionsand positions. The FK6 gives, in addition to the SI mode, the "long-termprediction (LTP) mode" and the "short-term prediction (STP) mode". TheseLTP and STP modes are on average the most precise solutions forapparently single stars, depending on the epoch difference with respectto the HIPPARCOS epoch of about 1991. The typical mean error of anFK6(III) proper motion in the single-star mode is 0.59 mas/year. This isa factor of 1.34 better than the typical HIPPARCOS errors for thesestars of 0.79 mas/year. In the long-term prediction mode, in whichcosmic errors are taken into account, the FK6(III) proper motions have atypical mean error of 0.93 mas/year, which is by a factor of about 2better than the corresponding error for the HIPPARCOS values of 1.83mas/year (cosmic errors included). Spectral Irradiance Calibration in the Infrared. X. A Self-Consistent Radiometric All-Sky Network of Absolutely Calibrated Stellar SpectraWe start from our six absolutely calibrated continuous stellar spectrafrom 1.2 to 35 μm for K0, K1.5, K3, K5, and M0 giants. These wereconstructed as far as possible from actual observed spectral fragmentstaken from the ground, the Kuiper Airborne Observatory, and the IRAS LowResolution Spectrometer, and all have a common calibration pedigree.From these we spawn 422 calibrated spectral templates'' for stars withspectral types in the ranges G9.5-K3.5 III and K4.5-M0.5 III. Wenormalize each template by photometry for the individual stars usingpublished and/or newly secured near- and mid-infrared photometryobtained through fully characterized, absolutely calibrated,combinations of filter passband, detector radiance response, and meanterrestrial atmospheric transmission. These templates continue ourongoing effort to provide an all-sky network of absolutely calibrated,spectrally continuous, stellar standards for general infrared usage, allwith a common, traceable calibration heritage. The wavelength coverageis ideal for calibration of many existing and proposed ground-based,airborne, and satellite sensors, particularly low- tomoderate-resolution spectrometers. We analyze the statistics of probableuncertainties, in the normalization of these templates to actualphotometry, that quantify the confidence with which we can assert thatthese templates truly represent the individual stars. Each calibratedtemplate provides an angular diameter for that star. These radiometricangular diameters compare very favorably with those directly observedacross the range from 1.6 to 21 mas. A catalog of rotational and radial velocities for evolved starsRotational and radial velocities have been measured for about 2000evolved stars of luminosity classes IV, III, II and Ib covering thespectral region F, G and K. The survey was carried out with the CORAVELspectrometer. The precision for the radial velocities is better than0.30 km s-1, whereas for the rotational velocity measurementsthe uncertainties are typically 1.0 km s-1 for subgiants andgiants and 2.0 km s-1 for class II giants and Ib supergiants.These data will add constraints to studies of the rotational behaviourof evolved stars as well as solid informations concerning the presenceof external rotational brakes, tidal interactions in evolved binarysystems and on the link between rotation, chemical abundance and stellaractivity. In this paper we present the rotational velocity v sin i andthe mean radial velocity for the stars of luminosity classes IV, III andII. Based on observations collected at the Haute--Provence Observatory,Saint--Michel, France and at the European Southern Observatory, LaSilla, Chile. Table \ref{tab5} also available in electronic form at CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html Catalogs of temperatures and [Fe/H] averages for evolved G and K starsA catalog of mean values of [Fe/H] for evolved G and K stars isdescribed. The zero point for the catalog entries has been establishedby using differential analyses. Literature sources for those entries areincluded in the catalog. The mean values are given with rms errors andnumbers of degrees of freedom, and a simple example of the use of thesestatistical data is given. For a number of the stars with entries in thecatalog, temperatures have been determined. A separate catalogcontaining those data is briefly described. Catalog only available atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html Towards a fundamental calibration of stellar parameters of A, F, G, K dwarfs and giantsI report on the implementation of the empirical surface brightnesstechnique using the near-infrared Johnson broadband { (V-K)} colour assuitable sampling observable aimed at providing accurate effectivetemperatures of 537 dwarfs and giants of A-F-G-K spectral-type selectedfor a flux calibration of the Infrared Space Observatory (ISO). Thesurface brightness-colour correlation is carefully calibrated using aset of high-precision angular diameters measured by moderninterferometry techniques. The stellar sizes predicted by thiscorrelation are then combined with the bolometric flux measurementsavailable for a subset of 327 ISO standard stars in order to determineone-dimensional { (T, V-K)} temperature scales of dwarfs and giants. Theresulting very tight relationships show an intrinsic scatter induced byobservational photometry and bolometric flux measurements well below thetarget accuracy of +/- 1 % required for temperature determinations ofthe ISO standards. Major improvements related to the actual directcalibration are the high-precision broadband { K} magnitudes obtainedfor this purpose and the use of Hipparcos parallaxes for dereddeningphotometric data. The temperature scale of F-G-K dwarfs shows thesmallest random errors closely consistent with those affecting theobservational photometry alone, indicating a negligible contributionfrom the component due to the bolometric flux measurements despite thewide range in metallicity for these stars. A more detailed analysisusing a subset of selected dwarfs with large metallicity gradientsstrongly supports the actual bolometric fluxes as being practicallyunaffected by the metallicity of field stars, in contrast with recentresults claiming somewhat significant effects. The temperature scale ofF-G-K giants is affected by random errors much larger than those ofdwarfs, indicating that most of the relevant component of the scattercomes from the bolometric flux measurements. Since the giants have smallmetallicities, only gravity effects become likely responsible for theincreased level of scatter. The empirical stellar temperatures withsmall model-dependent corrections are compared with the semiempiricaldata by the Infrared Flux Method (IRFM) using the large sample of 327comparison stars. One major achievement is that all empirical andsemiempirical temperature estimates of F-G-K giants and dwarfs are foundto be closely consistent between each other to within +/- 1 %. However,there is also evidence for somewhat significant differential effects.These include an average systematic shift of (2.33 +/- 0.13) % affectingthe A-type stars, the semiempirical estimates being too low by thisamount, and an additional component of scatter as significant as +/- 1 %affecting all the comparison stars. The systematic effect confirms theresults from other investigations and indicates that previousdiscrepancies in applying the IRFM to A-type stars are not yet removedby using new LTE line-blanketed model atmospheres along with the updatedabsolute flux calibration, whereas the additional random component isfound to disappear in a broadband version of the IRFM using an infraredreference flux derived from wide rather than narrow band photometricdata. Table 1 and 2 are only available in the electronic form of thispaper The Tokyo PMC catalog 90-93: Catalog of positions of 6649 stars observed in 1990 through 1993 with Tokyo photoelectric meridian circleThe sixth annual catalog of the Tokyo Photoelectric Meridian Circle(PMC) is presented for 6649 stars which were observed at least two timesin January 1990 through March 1993. The mean positions of the starsobserved are given in the catalog at the corresponding mean epochs ofobservations of individual stars. The coordinates of the catalog arebased on the FK5 system, and referred to the equinox and equator ofJ2000.0. The mean local deviations of the observed positions from theFK5 catalog positions are constructed for the basic FK5 stars to comparewith those of the Tokyo PMC Catalog 89 and preliminary Hipparcos resultsof H30. Classification and Identification of IRAS Sources with Low-Resolution SpectraIRAS low-resolution spectra were extracted for 11,224 IRAS sources.These spectra were classified into astrophysical classes, based on thepresence of emission and absorption features and on the shape of thecontinuum. Counterparts of these IRAS sources in existing optical andinfrared catalogs are identified, and their optical spectral types arelisted if they are known. The correlations between thephotospheric/optical and circumstellar/infrared classification arediscussed. A catalogue of [Fe/H] determinations: 1996 editionA fifth Edition of the Catalogue of [Fe/H] determinations is presentedherewith. It contains 5946 determinations for 3247 stars, including 751stars in 84 associations, clusters or galaxies. The literature iscomplete up to December 1995. The 700 bibliographical referencescorrespond to [Fe/H] determinations obtained from high resolutionspectroscopic observations and detailed analyses, most of them carriedout with the help of model-atmospheres. The Catalogue is made up ofthree formatted files: File 1: field stars, File 2: stars in galacticassociations and clusters, and stars in SMC, LMC, M33, File 3: numberedlist of bibliographical references The three files are only available inelectronic form at the Centre de Donnees Stellaires in Strasbourg, viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5), or viahttp://cdsweb.u-strasbg.fr/Abstract.html Vitesses radiales. Catalogue WEB: Wilson Evans Batten. Subtittle: Radial velocities: The Wilson-Evans-Batten catalogue.We give a common version of the two catalogues of Mean Radial Velocitiesby Wilson (1963) and Evans (1978) to which we have added the catalogueof spectroscopic binary systems (Batten et al. 1989). For each star,when possible, we give: 1) an acronym to enter SIMBAD (Set ofIdentifications Measurements and Bibliography for Astronomical Data) ofthe CDS (Centre de Donnees Astronomiques de Strasbourg). 2) the numberHIC of the HIPPARCOS catalogue (Turon 1992). 3) the CCDM number(Catalogue des Composantes des etoiles Doubles et Multiples) byDommanget & Nys (1994). For the cluster stars, a precise study hasbeen done, on the identificator numbers. Numerous remarks point out theproblems we have had to deal with. Fifth fundamental catalogue. Part 2: The FK5 extension - new fundamental starsThe mean positions and proper motions for 3117 new fundamental starsessentially in the magnitude range about 4.5 to 9.5 are given in thisFK5 extension. Mean apparent visual magnitude is 7.2 and is on average2.5 magnitudes fainter then the basic FK5 which has a mean magnitude of4.7. (The basic FK5 gives the mean positions and proper motions for theclassical 1535 fundamental stars). The following are discussed: theobservational material, reduction of observations, star selection, andthe system for the FK5 extension. An explanation and description of thecatalog are given. The catalog of 3117 fundamental stars for the equinoxand epoch J2000.0 and B1950.0 is presented. The parallaxes and radialvelocities for 22 extension stars with large forecasting effects aregiven. Catalogs used in the compilation of the FK5 fundamental catalogare listed. A critical appraisal of published values of (Fe/H) for K II-IV stars'Primary' (Fe/H) averages are presented for 373 evolved K stars ofluminosity classes II-IV and (Fe/H) values beween -0.9 and +0.21 dex.The data define a 'consensus' zero point with a precision of + or -0.018 dex and have rms errors per datum which are typically 0.08-0.16dex. The primary data base makes recalibration possible for the large(Fe/H) catalogs of Hansen and Kjaergaard (1971) and Brown et al. (1989).A set of (Fe/H) standard stars and a new DDO calibration are given whichhave rms of 0.07 dex or less for the standard star data. For normal Kgiants, CN-based values of (Fe/H) turn out to be more precise than manyhigh-dispersion results. Some zero-point errors in the latter are alsofound and new examples of continuum-placement problems appear. Thushigh-dispersion results are not invariably superior to photometricmetallicities. A review of high-dispersion and related work onsupermetallicity in K III-IV star is also given. High-resolution spectroscopic survey of 671 GK giants. I - Stellar atmosphere parameters and abundancesA high-resolution spectroscopic survey of 671 G and K field giants isdescribed. Broad-band Johnson colors have been calibrated againstrecent, accurate effective temperature, T(eff), measurements for starsin the range 3900-6000 K. A table of polynomial coefficients for 10color-T(eff) relations is presented. Stellar atmosphere parameters,including T(eff), log g, Fe/H, and microturbulent velocity, are computedfor each star, using the high-resolution spectra and various publishedphotometric catalogs. For each star, elemental abundances for a varietyof species have been computed using a LTE spectrum synthesis program andthe adopted atmosphere parameters. Third preliminary catalogue of stars observed with the photoelectric astrolabe of the Beijing Astronomical Observatory.Not Available Spectroscopy of metal-rich M31 globular clustersIntegrated CCD spectra in the 3850-4200 A region were obtained for ninemetal-rich globular clusters in M31. The spectra are analyzed using thereddening-free spectral-classification system of Rose (1984). The dataare combined with data obtained by Rose and Tripicco (1986) for ninemetal-rich Galactic globular clusters and with spectra of 72 stars withwell-determined atmospheric parameters. It is found that the 4000 Alight from metal rich M31 globulars is dominated by dwarfs, while giantsand dwarfs contribute equally to the light from the mean of themetal-rich Galactic globulars. The contribution of early-type stars tothe 400 A light from M31 clusters is less than 5 percent in most cases.Also, the 3883 and 4216 CN bands in the M31 globulars are very strongfor their integrated spectral types. The results suggest that themetal-rich M31 globular clusters are much younger than Galacticmetal-rich clusters like 47 Tuc or M71. E. W. Fick Observatory stellar radial velocity measurements. I - 1976-1984Stellar radial velocity observations made with the large vacuumhigh-dispersion photoelectric radial velocity spectrometer at FickObservatory are reported. This includes nearly 2000 late-type starsobserved during 585 nights. Gradual modifications to this instrumentover its first eight years of operation have reduced the observationalerror for high-quality dip observations to + or - 0.8 km/s. Transformation equations and other aids for VRI photometryTransformations among VRI systems are commonly beset by Paschen-jumpeffects, for which fully satisfactory allowance has not previously beenmade. This paper describes two new techniques which are based on thework of Gutierrez-Moreno, and which allow fully for the effects of thePaschen jump. Values of E(V-R)/E(B-V) and E(R-I)/E(B-V) are also givenfor the Cousins system for a wide range of temperatures. These and thenew techniques contribute to a set of new transformation relations whichapply for most VRI systems; the status of the remaining systems isreviewed, and future work needed for them is described. Two majorsources of Cousins VRI data underlie the new relations; the consistencyof these sources is reviewed and found to be generally satisfactory,although more work on this question is needed. Finally, three tables oftransformed standard-star and other data are given for the Cousins andJohnson systems, and a description of ways to reproduce the latter ispresented. IRAS catalogues and atlases - Atlas of low-resolution spectraPlots of all 5425 spectra in the IRAS catalogue of low-resolutionspectra are presented. The catalogue contains the average spectra ofmost IRAS poiont sources with 12 micron flux densities above 10 Jy. Catalogue of the energy distribution data in spectra of stars in the uniform spectrophotometric system.Not Available Classification of 831 two-micron sky survey sources south of +5 degrees.Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1975AJ.....80.1011H&db_key=AST Strong-Cyanogen Stars: Photometry and KinematicsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1971ApJ...165..561J&db_key=AST Determination of atmospheric parameters for G and K giants by means of photoelectric indices.Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1971A&A....15..123H&db_key=AST Catalog of Indidual Radial Velocities, 0h-12h, Measured by Astronomers of the Mount Wilson ObservatoryAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1970ApJS...19..387A&db_key=AST Etude entre 3600 et 6000 angstroems des etoiles G et K par photometrie photoelectrique a 11 bandes passantes.Not Available

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### 下列团体成员

#### 观测天体数据

 星座: 獵戶座 右阿森松: 05h11m41.60s 赤纬: +16Â°02'44.0" 视星: 5.18 距离: 98.522 天文距离 右阿森松适当运动: 4 赤纬适当运动: 7 B-T magnitude: 7.197 V-T magnitude: 5.348


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# zbMATH — the first resource for mathematics

Well-posedness criteria in optimization with application to the calculus of variations. (English) Zbl 0841.49005
In this paper, for the global optimization problem $$(X, J)$$, to minimize the proper extended real-valued function $$J: X\to (- \infty, \infty)$$ over the given subset $$X$$ of a normed linear space equipped with the strong convergence, well-posedness criteria are derived. The given problem is embedded into a smoothly parametrized family $$(X, I(., p))$$ of minimization problems, where $$p$$ is a parameter belonging to a given Banach space $$L$$, and $$p^*$$ is the parameter value to which the given unperturbed problem corresponds, i.e., $$I(x, p^*)= J(x)$$ $$\forall x\in X$$. Defining the value function $$V(p)= \inf\{I(x, p)\mid x\in X\}$$ the author gives the following definition of well-posedness.
$$(X, J)$$ is well-posed with respect to the embedding iff $$V(p)> -\infty$$, $$\forall p\in L$$, and there exists a unique $$x^*= \arg\min(X, J)$$ and for every sequence $$p_n\to p^*$$ and every sequence $$x_n\in X$$ such that $$I(x_n, p_n)- V(p_n)\to 0$$ as $$n\to \infty$$ we have $$x_n\to x^*$$ in $$X$$.
This definition is stronger than the Tikhonov well-posedness. In the following, the defined well-posedness is related under suitable conditions to the differentiability properties of $$V$$ at $$p^*$$. These abstract results are applied to one-dimensional problems of the calculus of variations.

##### MSC:
 49J27 Existence theories for problems in abstract spaces 49K99 Optimality conditions 90C99 Mathematical programming
##### Keywords:
global optimization problem; well-posedness criteria
Full Text:
##### References:
 [1] Asplund, E.; Rockafellar, R.T., Gradients of convex functions, Trans. am. math. soc., 139, 443-467, (1969) · Zbl 0181.41901 [2] Fitzpatrick, S., Metric projections and the differentiability of distance functions, Bull. aust. math. soc., 22, 291-312, (1980) · Zbl 0437.46012 [3] Fleming, W.H., The Cauchy problem for a nonlinear first order partial differential equation, J. diff. eqns, 5, 515-530, (1969) · Zbl 0172.13901 [4] Fleming, W.H.; Rishel, R., () [5] Clarke, F.H.; Loewen, P.D., The value function in optimal control: sensitivity, controllability and time-optimality, SIAM J. control optim., 24, 243-263, (1986) · Zbl 0601.49020 [6] Clarke, F.H., () [7] Ekeland, I.; Lebourg, G., Generic Fréchet differentiability and perturbed optimization problems in Banach spaces, Trans. am. math. soc., 224, 193-216, (1976) · Zbl 0313.46017 [8] Georgiev, P.G., Well posed optimization problems depending on a parameter and differentiability of the optimal value function, () [9] Henry, D., Nonuniqueness of solutions in the calculus of variations: a geometric approach, SIAM J. control. optim., 18, 627-639, (1980) · Zbl 0454.49008 [10] Fleming, W.H.; Soner, H.M., () [11] Dontchev, A.L.; Zolezzi, T., () [12] Kuratowski, C., () [13] Ekeland, I., Nonconvex minimization problems, Bull. am. math. soc., 1, 443-474, (1979) · Zbl 0441.49011 [14] Kutznetzov, N.N.; Siskin, A.A., On a many dimensional problem in the theory of quasilinear equations, Zh. vychisl. mat. fiz., 4, 192-205, (1964) [15] Cesari, L., () [16] Marcellini, P., (), 16-57
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Compare coefficients across groups in panel data set

I want to test for differences in (regression) coefficients across two groups. I understand, that this is possible by including interaction terms and performing a Chow-Test.

But what do you do if you can't identify the groups in your panel dataset anymore? Have a look at this example. This may clarify my problem:

Imagine 2 groups of people. Over 1 month, you record how much everybody spend on ice-cream (4 sorts), how they rated the ice-cream and what size of ice-cream they got. Your dependent variable is rating, your independent variables are cost and size.

   +-------------------------------------------------+
| day | ice-cream | rating | cost  | size  | group|
|-----+-----------+--------+-------+--------------+
1. |  1  |  vanilla  |  2     |  3    |  4    | 1    |
2. |  1  |  choco    |  1     |  4    |  3    | 0    |
3. |  1  |  vanilla  |  2     |  2    |  1    | 1    |
4. |  1  |  hazelnut |  9     |  2    |  1    | 0    |
5. |  2  |  hazelnut |  2     |  1    |  1    | 0    |
6. |  2  |  hazelnut |  5     |  1    |  1    | 0    |
7. |  2  |  vanilla  |  6     |  5    |  2    | 1    |
8. |  3  |  berry    |  1     |  6    |  3    | 0    |
9. |  3  |  berry    |  6     |  4    |  2    | 0    |
10. |  3  |  berry    |  7     |  3    |  4    | 1    |
11. |  3  |  berry    |  5     |  2    |  1    | 1    |
12. |  3  |  vanilla  |  4     |  1    |  2    | 0    |
13. |  3  |  vanilla  |  4     |  5    |  2    | 0    |
etc |  .  |  ......   |  .     |  .    |  ...  | .    |
+-------------------------------------------------+

Not caring about groups at all, I could just aggregate the data to a day-icecream level (because I want to consider fixed effects at the ice-cream level). My panel dataset would look like this:

    +------------------------------------------+
| day | ice-cream | rating | cost  | size  |
|-----+-----------+--------+-------+-------+
1. |  1  |  vanilla  |  2     |  2.5  |  2.5  |
2. |  1  |  choco    |  1     |  4    |  3    |
3. |  1  |  hazelnut |  9     |  2    |  1    |
4. |  2  |  hazelnut |  3.5   |  1    |  1    |
5. |  2  |  vanilla  |  6     |  5    |  2    |
6. |  3  |  berry    |  4.75  |  3.7  |  2.5  |
7. |  3  |  vanilla  |  4     |  3    |  2    |
etc |  .  |  ......   |  .     |  .    |  ...  |
+------------------------------------------+

However, this doesn't allow me to calculate a regression model for each group. So, I followed a different path: I split the raw data set according to groups (one set for each group) and aggregated to the daily level afterwards. The two panel datasets looks like this:

Group 1

    +-------------------------------------------------+
| day | ice-cream | rating | cost  | size  | group|
|-----+-----------+--------+-------+--------------+
1. |  1  |  vanilla  |  1.5   |  2.5  |  2.5  | 1    |
2. |  2  |  vanilla  |  6     |  5    |  2    | 1    |
3. |  3  |  berry    |  7     |  3    |  4    | 1    |
4. |  3  |  vanilla  |  6     |  2.5  |  2.5  | 1    |
etc |  .  |  ......   |  .     |  .    |  ...  | .    |
+-------------------------------------------------+

Group 2

    +-------------------------------------------------+
| day | ice-cream | rating | cost  | size  | group|
|-----+-----------+--------+-------+--------------+
1. |  1  |  choco    |  1     |  4    |  3    | 0    |
2. |  1  |  hazelnut |  9     |  2    |  1    | 0    |
3. |  2  |  hazelnut |  3.5   |  1    |  1    | 0    |
4. |  3  |  berry    |  3.5   |  5    |  2.5  | 0    |
5. |  3  |  vanilla  |  4     |  3    |  2    | 0    |
etc |  .  |  ......   |  .     |  .    |  ...  | .    |
+-------------------------------------------------+

For each dataset, I run a fixed-effects regression. This leaves me with two model outputs which I want to compare and see whether the coefficients are different.

Any ideas how to test coefficients across models in this case?

• You have already answered your own question. Just re-stack your datasets, noting the group number in each dataset before you do it. Then estimate your model, with the appropriate group interaction term. – D L Dahly Dec 3 '13 at 16:31
• @DLDahly, stacking both datasets leaves me with more than one timevariable per unit (ice-cream), which is not allowed in a panel dataset. Do I need to recode the time variable? (e.g. vanilla on day 3 would occur twice in a stacked dataset). Thanks! – SPi Dec 5 '13 at 10:00
• Can anyone help? I'm stuck here but I feel like DLDahly's comment points to the right direction. Still, I can't figure out how to 'restock' the datasets correcty to panel format.. – SPi Dec 6 '13 at 8:38
• Instead of splitting the dataset and then aggregating, why not aggregrate by group within the full dataset. That said, I suspect that you don't really need to aggregate/collapse your data at all. Can you specify the hypothesis or research question you are trying to test or model? – D L Dahly Dec 6 '13 at 9:40
• Imagine the ice-cream to be different stocks. Thus, the baseline model (without accounting for different groups of people) is based on a panel dataset with one stock per day (fixed-effects at the stock-level). The IVs stem from the people's behavior (ice-cream spendings and size in this example). Next I want to test whether the regression coefficients I get for the IVs are different for the two groups of people..the ice-cream illustrates the problem quite realistically.. Thanks a lot for your help! – SPi Dec 6 '13 at 16:14


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Article

# A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT.

Department of Mathematics, University Innsbruck, 6020 Innsbruck, Austria.
(Impact Factor: 3.54). 11/2009; 28(11):1727-35. DOI: 10.1109/TMI.2009.2022623
Source: arXiv

ABSTRACT Fourier reconstruction algorithms significantly outperform conventional backprojection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction.

### Full-text

Available from: Markus Haltmeier, Apr 25, 2014
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##### Article: The spherical mean Radon transform with centers on cylindrical surfaces
[Hide abstract]
ABSTRACT: The spherical mean Radon transform maps a function in an $n$-dimensional space to its integrals (or averages) over spherical surfaces. Recovering a function from its spherical mean Radon transform with centers of spheres of integration restricted to some hypersurface is at the heart of several modern imaging technologies, including SONAR, ultrasound imaging, and photo- and thermoacoustic tomography. In this paper we study the inversion of the spherical mean Radon transform with centers restricted to a cylindrical domain of the form $B \times \mathbb{R}^{n_2}$, where $B$ is any bounded domain in $\mathbb{R}^{n_1}$. As our main results we derive explicit inversion formulas of the backprojection type for that transform. For that purpose we show that the spherical mean Radon transform with centers on a cylindrical surface can be decomposed into two lower dimensional spherical mean Radon transforms. For a cylindrical center set in $\mathbb{R}^3$, we demonstrate that the derived inversion formulas can be implemented by filtered-backprojection type algorithms only requiring $\mathcal O(N^{4/3})$ floating point operations, where $N$ is the total number of unknowns to be recovered. We finally present some results of our numerical simulations performed in Matlab, showing that our implementations accurately recover a 3D image of millions of unknowns in a few minutes on an ordinary notebook.
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##### Article: Computational Realization of a Non-Equidistant Grid Sampling in Photoacoustics with a Non-Uniform FFT
[Hide abstract]
ABSTRACT: To obtain the initial pressure from the collected data on a planar sensor arrangement in Photoacoustic tomography, there exists an exact analytic frequency domain reconstruction formula. An efficient realization of this formula needs to cope with the evaluation of the datas Fourier transform on a non-equispaced mesh. In this paper, we use the non-uniform fast Fourier transform to handle this issue and show its feasibility in 3D experiments. This is done in comparison to the standard approach that uses polynomial interpolation. Moreover, we investigate the effect and the utility of flexible sensor location on the quality of photoacoustic image reconstruction. The computational realization is accomplished by the use of a multi-dimensional non-uniform fast Fourier algorithm, where non-uniform data sampling is performed both in frequency and spatial domain. We show that with appropriate sampling the imaging quality can be significantly improved. Reconstructions with synthetic and real data show the superiority of this method.
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##### Article: Inversion of circular means and the wave equation on convex planardomains
[Hide abstract]
ABSTRACT: We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary $\partial \Om$ of a smooth convex bounded domain $\Om \subset \R^2$. As a main result we establish back-projection type inversion formulas that recover any initial data with support in $\Om$ modulo an explicitly computed smoothing integral operator $\K_\Om$. For circular and elliptical domains the operator $\K_\Om$ is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on $\partial \Om$. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography.
Computers & Mathematics with Applications 04/2013; 65(7). DOI:10.1016/j.camwa.2013.01.036 · 2.00 Impact Factor


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 Title Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations Author(s) Nicolas T. Courtois, Alexander Klimov, Jacques Patarin, Adi Shamir Type Book, Chapter in Book, Conference Proceeding Abstract The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. This problem is NP-hard over any field. When the number of equations m is the same as the number of unknowns n the best known algorithms are exhaustive search for small fields, and a Gröbner base algorithm for large fields. Gröbner base algorithms have large exponential complexity and cannot solve in practice systems with n ≥ 15. Kipnis and Shamir [9] have recently introduced a new algorithm called relinearization . The exact complexity of this algorithm is not known, but for sufficiently overdefined systems it was expected to run in polynomial time. In this paper we analyze the theoretical and practical aspects of relinearization. We ran a large number of experiments for various values of n and m, and analysed which systems of equations were actually solvable. We show that many of the equations generated by relinearization are linearly dependent, and thus relinearization is less efficient that one could expect. We then develop an improved algorithm called XL which is both simpler and more powerful than relinearization. For all 0 < ε ≤ 1/2, and m ≥ εn 2, XL and relinearization are expected to run in polynomial time of approximately $$n^{\mathcal{O}(1/\sqrt \varepsilon )}$$ . Moreover, we provide strong evidence that relinearization and XL can solve randomly generated systems of polynomial equations in subexponential time when m exceeds n by a number that increases slowly with n. Length 16 URL http://dx.doi.org/10.1007/3-540-45539-6_27 Language English Series Lecture Notes in Computer Science Volume 1807 Pages 392-407 Publisher Springer Berlin / Heidelberg Year 2000 Note 10.1007/3-540-45539-6_27 Editor Preneel, Bart Translation No Refereed No Book Advances in Cryptology — EUROCRYPT 2000
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# Difference between revisions of "2014 AMC 12A Problems/Problem 17"

## Problem

A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?

$[asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy]$

$\textbf{(A) }2+2\sqrt 7\qquad \textbf{(B) }3+2\sqrt 5\qquad \textbf{(C) }4+2\sqrt 7\qquad \textbf{(D) }4\sqrt 5\qquad \textbf{(E) }4\sqrt 7\qquad$

## Solution 1

Let $A$ be the point in the same plane as the centers of the top spheres equidistant from said centers. Let $B$ be the analogous point for the bottom spheres, and let $C$ be the midpoint of $\overline{AB}$ and the center of the large sphere. Let $D$ and $E$ be the points at which line $AB$ intersects the top of the box and the bottom, respectively.

Let $O$ be the center of any of the top spheres (you choose!). We have $AO=1\cdot\sqrt{2}$, and $CO=3$, so $AC=\sqrt{3^2-\sqrt2^2}=\sqrt{7}$. Similarly, $BC=\sqrt{7}$. $\overline{AD}$ and $\overline{BE}$ are clearly equal to the radius of the small spheres, $1$. Thus the total height is $AD+AC+BC+BE=2+2\sqrt7$, or $\boxed{\textbf{(A)}}$.

## Solution 2

Let $A$ be the center of the large sphere and $C$ be the center of any small sphere. Let $D$ be a vertex of the rectangular prism closest to point $C$. Let $F$ be the point on the edge of the prism such that $\overline{DF}$ and $\overline{AF}$ are perpendicular. Let points $B$ and point $E$ lie on $\overline{AF}$ and $\overline{DF}$ respectively such that $\overline{CE}$ and $\overline{CB}$ are perpendicular at $C$.

$AC$ is the radii of the spheres, so $AC=2+1=3$. $CE$ is the shortest length between the center of a small sphere and the edge of the prism, so $CE=\sqrt{2}$. Similarly, $AF=2\sqrt{2}$. Since $CEFB$ is a rectangle, $BF=CE=\sqrt{2}$. Since $AF=2\sqrt{2}$, $AB=AF - BF = \sqrt{2}$. Then, $BC=\sqrt{3^2-\sqrt2^2}=\sqrt{7}=EF$. $DE$ is the length from $C$ to the top of the prism or $1$. Thus, $DF=DE+EF=1+\sqrt{7}$. The prism is symmetrical, so $h=2DF=\boxed{\textbf{(A)}}$

(Solution by BJHHar)

## Solution 3

take a cross section and see that h is made up of two radii of the circle plus some radical expression. the only choice satisfying this condition is (a)

 2014 AMC 12A (Problems • Answer Key • Resources) Preceded byProblem 16 Followed byProblem 18 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions


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2022/01/03

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Make the method random

Can someone give me a little test problem? An easy one please, I'm new to this lol.
Make the method random, so I have to try all three. Sub,Fac,Conj

LCKurtz
Homework Helper
Gold Member

Can someone give me a little test problem? An easy one please, I'm new to this lol.
Make the method random, so I have to try all three. Sub,Fac,Conj

$$lim_{x\rightarrow 4}\frac {2x-8}{\sqrt x - 2}$$

$$lim_{x\rightarrow 4}\frac {2x-8}{\sqrt x - 2}$$

Uhh, I doubt I got it right lol butttt

0/2?

If I didn't get it right, can you show me how to do it please? I'm trying to learn calculus by myself... lol I know. I just really want to learn it.

LCKurtz
Homework Helper
Gold Member

No, that isn't right. And even though it's wrong, you would normally simplify 0/2 wouldn't you?

I'll give you a hint. Try rationalizing the denominator.

No, that isn't right. And even though it's wrong, you would normally simplify 0/2 wouldn't you?

I'll give you a hint. Try rationalizing the denominator.

lol, Like I said.. Can you show me how to do it.

I seriously just kinda started learning this stuff by myself. I can grasp it, if someone can show me the steps. If I can see the steps, I can understand it. Unless there are certain rules of course that go along with it but those are technically steps.

I don't see how I can rationalize it.

Haha, that kind of problem haunted me for a long time.

It is specifically because of those problems I appreciate the beauty of the number 1 and the amazing cleverness of it. It also illuminated all that complex multiplication I memorized without a thought to the why

Anyway, www.khanacademy.org has some amazing videos on limits, you should watch them all to get a grasp of what you're doing.

I'll quote something from my book that I think was just a great real life description of limits;

Let us imagine a traveller on a train due in New York at 5:17pm.
He must be there on time, so he continually looks at his watch and checks with the timetable.
He notices:
5.01pm the train is 10 miles out.
5:10pm, the train is 1 mile out.
5:17 exactly the train pulls up at its platform in Grand Central Terminal.

As the distance between the train station and the passenger approaches zero, the time reaches the maximum time allowed: 5:17


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# The Thom space of a Whitney sum of vector bundles

Let $\xi$ and $\eta$ be vector bundles over the same base space $X$. Their Whitney sum is a bundle $\xi\oplus\eta$ over $X$. I read somewhere (without proof) that its Thom space is given by $$T(\xi\oplus\eta) = T(\eta|_{D(\xi)})/T(\eta|_{S(\xi)}),$$ where the restrictions are really pullbacks of $\eta$ under the projection maps $D(\xi),S(\xi)\to X$.

Can anyone provide a proof, or a reference to a proof?

• $D(\xi\oplus\eta)$ can be interpreted as $D(\eta|_{D(\xi)})$, and the sphere bundle is the union of $S(\eta|_{D(\xi)})$ and $D(\eta|_{S(\xi)})$. All more or less by the definitions. – Alex Degtyarev Apr 21 '16 at 11:48
• @Alex: You are right, this does appear to be an easier way to see it, thanks. – Mark Grant Apr 22 '16 at 13:21

The Thom space is the homotopy cofiber of the Unit Sphere Projection: $$\mathbb{S}_\xi X \to X \to T(\xi)$$ while the unit sphere in a sumspace $\mathbb{S}[U\oplus V]$ is a join $$\begin{array}{c} \mathbb{S}[U] \times \mathbb{S}[V] & \rightarrow & \mathbb{S}[U] \\ \downarrow & & \downarrow \\ \mathbb{S}[V] & \rightarrow & \mathbb{S}[U\oplus V] \end{array}$$ (homotopy-pushout). By one of "Mather's" Cube Theorems (it's a kind of distrutivity) this fiberwise pushout description integrates to a homotopy pushout of bundles: $$\begin{array}{c} \mathbb{S}_\xi \times_X \mathbb{S}_\eta &\rightarrow & \mathbb{S}_\xi \\ \downarrow & & \downarrow \\ \mathbb{S}_\eta & \rightarrow & \mathbb{S}_{\xi\oplus \eta} \end{array}$$ The top and the left maps are both, as it happens, sphere fibrations, the ones relevant to your question. Consider the composable arrows $$\mathbb{S}_\xi \times_X \mathbb{S}_\eta \to \mathbb{S}_\xi \to X$$ and form the following small squares as homotopy pushouts: $$\begin{array}{c} \mathbb{S}_\xi \times_X \mathbb{S}_\eta &\rightarrow & \mathbb{S}_\xi & \rightarrow & \mathbb{S}_{\xi\oplus\eta} & \to & X \rlap{\simeq D\eta} \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ * & \to & T(\eta|_{S\xi}) & \to & T(\eta|_{S\xi}) \vee T(\xi|_{S\eta}) & \to & W \\ & & \downarrow & & \downarrow & & \downarrow \\ & & * & \to & T(\xi|_{S\eta}) & \to & T(\xi) \rlap{\simeq T(\xi|_{D\eta})}\\ && && \downarrow & & \downarrow \\ & & & & * & \to & T(\xi\oplus\eta) \end{array}$$ where $W$ doesn't matter for the rest of the argument.

Now, I've sneakily applied the homotopy pushout pasting lemma, several times: to identify the $-\vee-$ in the second row (which gives the Thom space below it) and to identify the other Thom spaces we want. That says the homotopy type we want is the homotopy cofiber of a map of Thom spaces; and the particular fine result you want is: the inclusion $T(\xi|_{S\eta}) \to T(\xi|_{D\eta})$ is a cofibration and represents the homotopy class of the pushout structure map indicated by the diagram.

• Thanks, this is great! I think I've convinced my self that the cofibre of $\mathbb{S}_\xi \times_X \mathbb{S}_\eta \to \mathbb{S}_{\xi\oplus\eta}$ is $T(\eta|_{S\xi})\vee T(\xi|_{S\eta})$. Now I just have to convince myself that the pushout of $\mathbb{S}_{\xi\oplus\eta}\to D\eta$ and $\mathbb{S}_{\xi\oplus\eta}\to T(\xi|_{S\eta})$ is $T(\xi|_{D\eta})$, and I'll have fully understood your argument. – Mark Grant Apr 22 '16 at 13:08
• (This is the kind of proof I was trying to come up with, and failing.) – Mark Grant Apr 22 '16 at 13:09
• As I say, the pasting lemma is your friend. – Jesse C. McKeown Apr 22 '16 at 17:00
• I'm not sure I know what you mean by "the pasting lemma". I assume you mean the following: in a diagram consisting of two squares side by side sharing a common arrow, if any two of the `squares' are homotopy pushouts, then so is the third. Please let me know if its something else! – Mark Grant Apr 23 '16 at 8:41
• @MarkGrant, it's not true that if any two squares are pushouts the third also is. That's 3 cases, two of which are true. Say the horizontal arrows point to the right and the squares share a vertical side. If the two little squares are pushouts so is the outer square (rectangle?). And if the outer square and the square on the left are pushouts so is the square on the right. The other case doesn't hold. And for pullbacks it's swapped: outer and right imply left; but with left and right swapped it would be false. (Oh, and to answer your question: it is what Jesse means by pasting lemma.) – Omar Antolín-Camarena Apr 23 '16 at 17:45

I think Statement (3.6) in the following paper of Becker and Schultz

Becker, J.C.; Schultz, R.E. Equivariant function spaces and stable homotopy theory. I. Comment. Math. Helv. 49, 1-34 (1974).

is in the line of what you are after.


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One thing I really like about Astro is the frontmatter.

We can put any JavaScript in there, and one of the most useful things we can do is to fetch data at build time.

We do so using fetch(), the Fetch API.

The nice thing about Astro is that it makes it possible to use top-level await in the frontmatter, so we don’t have to do anything fancy like an IIFE or calling an async function in order to use fetch(), which is a promise-based API.

In other words, you can use await directly:

---
const res = await fetch('https://api.sampleapis.com/coffee/hot')
const data = await res.json()
---

{data.map(item => (

{item.title}

))}

It’s worth mentioning again that this network request is made at build time, so it’s done just once when the site is deployed.

And of course when it’s deployed again.

So any time we make a change in the API, if you want the site to show the new data, you have to trigger a rebuild which is something all platforms offer via webhooks.


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1. ## Vectors

1. Find two unit vectors that make an angle of 60* with v = <3,4>

2. Find the cross product a X b for a = i + (e^t)k +(e^-t)k, b = 2i + (e^t)j - (e^-t)k

3. Use the scalar triple product to determine whether the points A(1,3,2), B(3,-1,6),C(5,2,0), and D(3,6,-4) lie on the same plane.

1. ?

2. I set it up as the matrices way..
| i j k |
| 1 e^t e^-t |
| 2 e^t -e^-t |

and came out with -2i - (-e^-t - 2e^-t)j + (-e^-t-2e^t)k

3)
| 1 3 2 |
| 3 -1 6 |
| 5 2 0 |
|3 6 -4 |

is this the way to do this problem? I end up with like
|0|
|-4| if i do this.

can someone help me with these? I think i am doing 2 and 3 incorrectly and 1 i do not know how.

thanks.

2. Hello, xfyz!

I'm still working on #1 . . . don't know why it's giving me trouble.

2. Find the cross product $\displaystyle a \times b$ for: .$\displaystyle a \:= \:i + e^tj + e^{-t}k,\;\;b \:= \:2i + e^tj - e^{-t}k$

I set up the matrix: .$\displaystyle \begin{vmatrix} i & j & k \\ 1 & e^t & e^{\text{-}t} \\ 2 & e^t & \text{-}e^{\text{-}t} \end{vmatrix}$

and came out with: .$\displaystyle -2i - \underbrace{(-e^{\text{-}t} - 2e^{\text{-}t})}_{\text{Combine!}}j + \underbrace{(-e^{\text{-}t} - 2e^t)}_{\text{Wrong!}}k$

We have: .$\displaystyle \begin{vmatrix}i & j & k \\ 1 & e^t & e^{\text{-}t} \\ 2 & e^t & \text{-}e^{\text{-}t} \end{vmatrix}$

. . $\displaystyle = \;i(\text{-}1-1) -j(\text{-}e^{\text{-}t}-2e^{\text{-}t}) + k(e^t - 2e^t)$

. . $\displaystyle = \;{\color{blue}-2i + 3e^{-t}j - e^tk}$

3. Use the scalar triple product to determine whether the points:
$\displaystyle A(1,3,2),\;B(3,-1,6),\;C(5,2,0),\;D(3,6,-4)$ lie on the same plane.
It took me a minute to remember what the scalar triple product is for.

Given three vectors $\displaystyle \vec u,\:\vec v,\:\vec w$, the scalar triple product, $\displaystyle u\cdot (v \times w)$ gives us
the volume of the parallelepiped ("slanted box") determined by the vectors.

Let: .$\displaystyle \begin{array}{ccccc}\vec{u} & = & \overrightarrow{AB} & = & \langle2,\text{-}4,4\rangle \\ \vec v & = & \overrightarrow{AC} & = & \langle 4,\text{-}1,\text{-}2\rangle \\ \vec{w} & = & \overrightarrow{AD} & = & \langle2,3,\text{-}6\rangle \end{array}$

The three vectors have a common point, $\displaystyle A.$
If the vectors point in three "different" directions (like the legs of a tripod),
. . the parallelepiped will have a volume (greater than zero).
If the vectors are in the same plane, then the "box" will have volume zero.

Scalar triple product: .$\displaystyle \begin{vmatrix}2 & \text{-}4 & 4 \\ 4 & \text{-}1 & \text{-}2 \\ 2 & 3 & \text{-}6 \end{vmatrix} \;=\;2(6+6) + 4(-24+4) + 4(12+2)$

. . $\displaystyle = \;2(12) + 4(-20) + 4(14) \;=\;24 - 80 + 56 \;=\;{\color{blue}0}$ . . . . There!

3. Originally Posted by xfyz
1. Find two unit vectors that make an angle of 60* with v = <3,4>
For vectors $\displaystyle u=(u_1,u_2)$ and $\displaystyle v=(v_1,v_2)$, the scalar product u.v can be calculated in two ways. It is equal to $\displaystyle u_1v_1+u_2v_2$ and it is also equal to $\displaystyle |u||v|\cos\theta$, where θ is the angle between u and v.

If $\displaystyle u=(x,y),\;v=(3,4)$, θ = 60° and u is a unit vector, then $\displaystyle u.v = 3x+4y$ (from the first of those two formulas), and $\displaystyle u.v = 5\times{\textstyle\frac12}$ (from the second one, because |u|=1, $\displaystyle |v|=\sqrt{3^2+4^2}=5$ and cos(60°)=1/2). Also, $\displaystyle x^2+y^2=1$ because u is a unit vector.

Thus $\displaystyle 3x+4y = 5/2$, from which $\displaystyle y = (5-6x)/8$. Substitute that value for y into the equation $\displaystyle x^2+y^2=1$ and you'll get a quadratic equation for x ...


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# Proving existence of a triangle of area $\leq \frac{1}{8}$

Problem: Given any set $S$ of $9$ points within a unit square, show that there always exist $3$ distinct points in $S$ such that the area of the triangle formed by these $3$ points is less than or equal to $\frac{1}{8}$

I am supposed to do this using the pigeonhole principle. I used the usual technique of partitioning the square to conclude that there is a triangle with area less than $\frac{1}{4}$ but I am unable to improve the bound.

• You need a sharper estimate of the area of the largest triangle in a $0.5\times 0.5$ square. – André Nicolas Jun 5 '16 at 18:12
First divide the unit square into four squares with sides $s$ of length $\frac{1}{2}$.
Now given $9$ points and $4$ squares, by the pigeon hole principle at least one square $S$ which contains three of those points. Those three points form a triangle whose area $A$ is bounded by half the area of $S$ (see here for a proof). Since the sides $s$ of $S$ have length $\frac{1}{2}$, we have that $$A\leq \frac{1}{2}s^2= \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}=\frac{1}{8}$$


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# Solve the Following System of Linear Equation Graphically and Shade the Region Between the Two Lines And X-axis: 2x + 3y = 12 X − Y = 1 - Mathematics

Solve the following system of linear equation graphically and shade the region between the two lines and x-axis:

2x + 3y = 12

x − y = 1

#### Solution

The given equation is

2x + 3y = 12  ......(i)

x − y = 1  .......(ii)

Putting x = 0 in equation (i) we get

=> 2xx0 + 3y = 12

=> y = 4

Putting y = 0 in equation (i) we get

=> 2x + 3 xx 0 = 12

=> x = 6

x = 6 , y = 0

Use the following table to draw the graph.

 x 0 6 y 4 0

Draw the graph by plotting the A(0, 4), B(6,0) two points from table

x - y = 1 ....(ii)

Putting x = 0 in equation (ii) we get

=> 0 - y = 1

=> y = -1

x = 0, y = -1

Putting y = 0 in equation (ii) we get

=> x - 0 = 1

=> x = 1

x = 1, y = 0

Use the following table to draw the graph.

 x 0 1 y -1 0

Draw the graph by plotting the two points C(0,-1), D(1,0) from table.

The two lines intersect at P(3,2). The region enclosed by the lines represented by the given equations and x−axis are shown in the above figure

Hence, x = 3 and y = 2 is the solution

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 3 Pair of Linear Equations in Two Variables
Exercise 3.2 | Q 24.1 | Page 30


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# 卷积神经网络（CNN）从入门到精通——一个过来人的总结

## 基础入门

### 粗略了解

CNN笔记：通俗理解卷积神经网络https://www.2cto.com/kf/201607/522441.html

Visualizing and Understanding Convolutional Networks中文笔记http://www.gageet.com/2014/10235.php

### 基本实践

tensorflow官网http://www.tensorflow.org/

pytorch 官网http://pytorch.org/

tensorflow

TensorFlow 如何入门？https://www.zhihu.com/question/49909565

TensorFlow入门http://hacker.duanshishi.com/?p=1639

pytorch

PyTorch深度学习：60分钟入门(Translation)https://zhuanlan.zhihu.com/p/25572330

## 进阶学习

### 实践深入

dropout，lrn这些过去常用的模块最近已经用得越来越少了，就不赘述了，有关正则化，推荐BatchNorm https://www.zhihu.com/question/38102762， 思想简单，效果好

## 细化研究

### Understanding / Generalization / Transfer

Distilling the knowledge in a neural network (2015), G. Hinton et al. http://arxiv.org/pdf/1503.02531

Deep neural networks are easily fooled: High confidence predictions for unrecognizable images (2015), A. Nguyen et al. http://arxiv.org/pdf/1412.1897

How transferable are features in deep neural networks? (2014), J. Yosinski et al.http://papers.nips.cc/paper/5347-how-transferable-are-features-in-deep-neural-networks.pdf

CNN features off-the-Shelf: An astounding baseline for recognition (2014), A. Razavian et al. http://www.cv-foundation.org//openaccess/content_cvpr_workshops_2014/W15/papers/Razavian_CNN_Features_Off-the-Shelf_2014_CVPR_paper.pdf

Learning and transferring mid-Level image representations using convolutional neural networks (2014), M. Oquab et al. http://www.cv-foundation.org/openaccess/content_cvpr_2014/papers/Oquab_Learning_and_Transferring_2014_CVPR_paper.pdf

Visualizing and understanding convolutional networks (2014), M. Zeiler and R. Fergus http://arxiv.org/pdf/1311.2901

Decaf: A deep convolutional activation feature for generic visual recognition (2014), J. Donahue et al. http://arxiv.org/pdf/1310.1531

### Optimization / Training Techniques

Training very deep networks (2015), R. Srivastava et al.http://papers.nips.cc/paper/5850-training-very-deep-networks.pdf

Batch normalization: Accelerating deep network training by reducing internal covariate shift (2015), S. Loffe and C. Szegedy http://arxiv.org/pdf/1502.03167

Delving deep into rectifiers: Surpassing human-level performance on imagenet classification (2015), K. He et al. http://www.cv-foundation.org/openaccess/content_iccv_2015/papers/He_Delving_Deep_into_ICCV_2015_paper.pdf

Dropout: A simple way to prevent neural networks from overfitting (2014), N. Srivastava et al. http://jmlr.org/papers/volume15/srivastava14a/srivastava14a.pdf

Adam: A method for stochastic optimization (2014), D. Kingma and J. Bahttp://arxiv.org/pdf/1412.6980

Improving neural networks by preventing co-adaptation of feature detectors (2012), G. Hinton et al. http://arxiv.org/pdf/1207.0580.pdf

Random search for hyper-parameter optimization (2012) J. Bergstra and Y. Bengiohttp://www.jmlr.org/papers/volume13/bergstra12a/bergstra12a

### Convolutional Neural Network Models

Rethinking the inception architecture for computer vision (2016), C. Szegedy et al. http://www.cv-foundation.org/openaccess/content_cvpr_2016/papers/Szegedy_Rethinking_the_Inception_CVPR_2016_paper.pdf

Inception-v4, inception-resnet and the impact of residual connections on learning (2016), C. Szegedy et al.http://arxiv.org/pdf/1602.07261

Identity Mappings in Deep Residual Networks (2016), K. He et al. https://arxiv.org/pdf/1603.05027v2.pdf

Deep residual learning for image recognition (2016), K. He et al. http://arxiv.org/pdf/1512.03385

Spatial transformer network (2015), M. Jaderberg et al., http://papers.nips.cc/paper/5854-spatial-transformer-networks.pdf

Going deeper with convolutions (2015), C. Szegedy et al.http://www.cv-foundation.org/openaccess/content_cvpr_2015/papers/Szegedy_Going_Deeper_With_2015_CVPR_paper.pdf

Very deep convolutional networks for large-scale image recognition (2014), K. Simonyan and A. Zisserman http://arxiv.org/pdf/1409.1556

Return of the devil in the details: delving deep into convolutional nets (2014), K. Chatfield et al. http://arxiv.org/pdf/1405.3531

OverFeat: Integrated recognition, localization and detection using convolutional networks (2013), P. Sermanet et al.http://arxiv.org/pdf/1312.6229

Maxout networks (2013), I. Goodfellow et al. http://arxiv.org/pdf/1302.4389v4

Network in network (2013), M. Lin et al. http://arxiv.org/pdf/1312.4400

ImageNet classification with deep convolutional neural networks (2012), A. Krizhevsky et al.http://papers.nips.cc/paper/4824-imagenet-classification-with-deep-convolutional-neural-networks.pdf

### Image: Segmentation / Object Detection

You only look once: Unified, real-time object detection (2016), J. Redmon et al.http://www.cv-foundation.org/openaccess/content_cvpr_2016/papers/Redmon_You_Only_Look_CVPR_2016_paper.pdf

Fully convolutional networks for semantic segmentation (2015), J. Long et al. http://www.cv-foundation.org/openaccess/content_cvpr_2015/papers/Long_Fully_Convolutional_Networks_2015_CVPR_paper.pdf

Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks (2015), S. Ren et al.http://papers.nips.cc/paper/5638-faster-r-cnn-towards-real-time-object-detection-with-region-proposal-networks.pdf

Fast R-CNN (2015), R. Girshick http://www.cv-foundation.org/openaccess/content_iccv_2015/papers/Girshick_Fast_R-CNN_ICCV_2015_paper.pdf

Rich feature hierarchies for accurate object detection and semantic segmentation (2014), R. Girshick et al.http://www.cv-foundation.org/openaccess/content_cvpr_2014/papers/Girshick_Rich_Feature_Hierarchies_2014_CVPR_paper.pdf

Spatial pyramid pooling in deep convolutional networks for visual recognition (2014), K. He et al. http://arxiv.org/pdf/1406.4729

Semantic image segmentation with deep convolutional nets and fully connected CRFs, L. Chen et al. https://arxiv.org/pdf/1412.7062

Learning hierarchical features for scene labeling (2013), C. Farabet et al. https://hal-enpc.archives-ouvertes.fr/docs/00/74/20/77/PDF/farabet-pami-13.pdf

### Image / Video / Etc

Image Super-Resolution Using Deep Convolutional Networks (2016), C. Dong et al. https://arxiv.org/pdf/1501.00092v3.pdf

A neural algorithm of artistic style (2015), L. Gatys et al. https://arxiv.org/pdf/1508.06576

Deep visual-semantic alignments for generating image descriptions (2015), A. Karpathy and L. Fei-Feihttp://www.cv-foundation.org/openaccess/content_cvpr_2015/papers/Karpathy_Deep_Visual-Semantic_Alignments_2015_CVPR_paper.pdf

Show, attend and tell: Neural image caption generation with visual attention (2015), K. Xu et al. http://arxiv.org/pdf/1502.03044

Show and tell: A neural image caption generator (2015), O. Vinyals et al. http://www.cv-foundation.org/openaccess/content_cvpr_2015/papers/Vinyals_Show_and_Tell_2015_CVPR_paper.pdf

Long-term recurrent convolutional networks for visual recognition and description (2015), J. Donahue et al.http://www.cv-foundation.org/openaccess/content_cvpr_2015/papers/Donahue_Long-Term_Recurrent_Convolutional_2015_CVPR_paper.pdf

VQA: Visual question answering (2015), S. Antol et al.http://www.cv-foundation.org/openaccess/content_iccv_2015/papers/Antol_VQA_Visual_Question_ICCV_2015_paper.pdf

DeepFace: Closing the gap to human-level performance in face verification (2014), Y. Taigman et al.http://www.cv-foundation.org/openaccess/content_cvpr_2014/papers/Taigman_DeepFace_Closing_the_2014_CVPR_paper.pdf

Large-scale video classification with convolutional neural networks (2014), A. Karpathy et al. http://vision.stanford.edu/pdf/karpathy14.pdf

Two-stream convolutional networks for action recognition in videos (2014), K. Simonyan et al. http://papers.nips.cc/paper/5353-two-stream-convolutional-networks-for-action-recognition-in-videos.pdf

3D convolutional neural networks for human action recognition (2013), S. Ji et al.http://machinelearning.wustl.edu/mlpapers/paper_files/icml2010_JiXYY10.pdf

### VIP内容

http://cea.ceaj.org/CN/abstract/abstract39601.shtml

### 最新论文

Prostate cancer is one of the most prevalent cancers worldwide. One of the key factors in reducing its mortality is based on early detection. The computer-aided diagnosis systems for this task are based on the glandular structural analysis in histology images. Hence, accurate gland detection and segmentation is crucial for a successful prediction. The methodological basis of this work is a prostate gland segmentation based on U-Net convolutional neural network architectures modified with residual and multi-resolution blocks, trained using data augmentation techniques. The residual configuration outperforms in the test subset the previous state-of-the-art approaches in an image-level comparison, reaching an average Dice Index of 0.77.

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text:
## Elementary Linear Algebra 7th Edition

Eq. 1: $\frac{1}{2}x - \frac{1}{3}y$ = 1 Eq. 2: $-2x + \frac{4}{3}y$ = -4 If we multiply equation 1 by -4, the equation will become: -4($\frac{1}{2}x - \frac{1}{3}y$) = -4(1) $\frac{-4}{2}x - \frac{-4}{3}y$ = -4 $-2x + \frac{4}{3}y$ = -4 Equations 1 and 2 are the same. Their graphs will coincide, meaning that all points on one line overlay all points on the other, so every point will be a solution. Plugging values in for both equations will return the same answer for every point in both equations. Therefore, there are infinitely many solutions.