text:
# Saturated absorption: decrease size of material rather than intensity of light?

1. Feb 2, 2012

### franco1991

In saturated absorption, the decrease in absorption (and corresponding increase in wave transmission) is dependent on the intensity of the incoming wave. But why can't we decrease the size of saturable material, rather than the intensity of light?

According to my understanding, saturable absorption happens because there are more photons than atoms, thus the atoms cannot absorb all the impinging photons, and thus some photons will be transmitted. Is this the case? If it is, then couldn't we just decrease the size of material and thus decrease the number of atoms available to absorb the photons, rather than increasing the number of photons (by increasing the intensity of light)?

The only reason I could see this not working is if saturable absorption is some special phenomena, for example if at high energies the atoms are energized for longer than they normally are, and thus take longer to go back to ground state (which would mean that it does rely on actual intensity rather than the ratio of atoms to photons).

So, which one is it? Is it that the atoms take longer to get back to ground at high energies (in which case decreasing the # of atoms won;t do anything), or is it simply the ratio of atoms to photons, and there being more photons than atoms? (in which case, decreasing the number of atoms should have the same effect as increasing the intensity and thus number of photons in the impinging wave).


text:
# Probabilistic method used to prove existence theorems

I am aiming for a "big list" of theorems using probability techniques to prove existence of some objects. And in each case, there is an interesting question -- can we find an explicit example? Was the probabilistic method the first one to come by or it was following the explicit construction? Are there any examples where it is proven that the example exists but its explicit construction is impossible?

In wikipedia, on Probabilistic method, they do propose two examples due to Erdős:

FIRST ONE. For a complete graph on n vertices it is possible to color the edges of the graph in two colors so that there is no complete subgraph on r vertices which is monochromatic (every edge colored the same color).

Question: is there any explicit algorithm for such a coloring?

SECOND ONE. The second problem also comes from graph theory and deals with a chromatic number of a graph: the minimal number of colors in which we can color the graph so that two adjacent vertices are colored differently. Given two positive integers $g$ and $k$, does there exist a graph containing only cycles of length at least $g$ such that its chromatic number of is at least $k$? It can be shown by the means of probabilistic method that such a graph exists for any values of $g$ and $k$.

Question: is there some algorithm to provide such a coloring?

I am not at all a specialist in graph theory, so I would like to hear the answers to thse questions as well as to see other examples which come from other branches of math.

• There are books on the probabilistic method. Perhaps you should narrow your question. – Douglas Zare Jun 3 '14 at 6:42
• I like this question, although it is probably doomed for closure (unless you can somehow quickly increase your reputation by two digits!) Assuming the question can survive, surely someone needs to mention the new theorem of Peter Keevash at arxiv.org/abs/1401.3665, which is a beautiful, powerful and nontrivial example of a "probabilistic" existence result. – Peter Dukes Jun 3 '14 at 8:23
• See en.wikipedia.org/wiki/… although not all of those are existence theorems. There are lots more. If your interest is primarily combinatorics as opposed to analysis, why not narrow the question to probabilistic proofs in combinatorics? – Douglas Zare Jun 3 '14 at 9:26
• For a finite probability space such as your examples, the probabilistic method should always immediately give a naive algorithm: Try all possible outcomes of the random choices, and for each check if the condition is satisfied. Of course, with $n$ bits of randomness this takes time $2^n$, and you'd prefer a $poly(n)$-time algorithm. (This is very similar to PvsNP, since you can think of a nondeterministic machine as one that always guesses the "correct" random choices in order to construct the object on the first try. So we might usually expect the search for better algorithms to be hard.) – usul Jun 3 '14 at 13:43
• the question should be linked to mathoverflow.net/questions/9218/… I think... – user39115 Jun 5 '14 at 10:08

I am a bit surprised that no one has so far mentioned expanders, for which the existence result was first established by the "probabilistic method" in a very simple way, whereas concrete examples of expanders are always a pretty difficult problem, see, for instance, Who first dubbed them "expander graphs"? or How to prove a random d-regular graph is an expander with prob >= 0.5?

• Yes, I though of this question when I was reading Yann Olivier's January 2005 introduction to group theory and when I read about expanders) – Olga Jun 10 '14 at 5:04

Shannon's proof that capacity-achieving code families exist for noisy channels is probabilistic in nature. Only for relatively simple channels (e.g., the binary erasure channel, or more recently the binary symmetric channel--and perhaps more generic discrete memoryless channels if I take some abstracts in the recent literature at face value) have explicit code families been given, viz. LDPC and polar codes, respectively.

Problem: given points $x_1, \ldots, x_n$ in $\mathbb{R}^d$ with $n << d$ and $0 < \epsilon < 1$, find a linear map $T \colon \mathbb{R}^d \to \mathbb{R}^k$, $k << d$, such that $$(1 - \epsilon) ||x_i - x_j|| \leq T(x_i) - T(x_j) \leq (1 + \epsilon) ||x_i - x_j||$$

Solution: for $k > C \frac{\log n}{\epsilon^2}$, a random $d \times k$ matrix has the desired property with high probability (where "random" and "high probability" can be made precise).

This is the Johnson-Lindenstrauss lemma - informally, it says that a small amount of high dimensional data can be projected down to a low dimensional space via a suitable random matrix without disturbing the Euclidean distances too badly. This is often used in practice to apply algorithms whose running time is exponential in the dimension of the input data to high dimensional data sets. Interestingly, it is very difficult to check that a given matrix has the Johnson-Lindenstrauss property even though a random matrix has the property with very high probability. I recall reading somewhere the description "finding hay in a haystack" for this problem.

I will interpret your question in this way: Are there any example of a theorem which proof uses a probability technique to prove the existence of some object where it is proven that the example exists but its explicit construction is impossible?

An example can be the theorem:

Every large even number can be expressed as the sum of a prime and the product of at most b primes.

Rényi proved this theorem using a generalisation of ''the large sieve'' method that can be seen as an application of a probabilist method:

Rényi, A. On the large sieve of Yu. V. Linnik. Compositio Mathematica, 8, 68-75 (1956).

Rényi, A. Un nouveau théorème concernant les fonctions indépendantes et ses applications à la théorie des nombres. J. Math, pures et appi. (9), 28, 137-149 (1949).

Rényi, A. Sur un théorème général de probabilité. Ann. Inst. Fourier, 1, 43-52 (1949).

• How is $b$ quantified here? – Tom Leinster Jun 3 '14 at 10:52
• It is some big constant, I do not have the original paper by Rényi, but I read that he did not determine it. – user39115 Jun 3 '14 at 11:02
• OK, so the theorem is: there exist $b$ and $N$ such that every even number greater than $N$ can be expressed as the sum of a prime and the product of at most $b$ primes? – Tom Leinster Jun 3 '14 at 13:51
• Yes. It seems that it was a trend in the sixties and seventies to improve those weak versions of the Goldbach s conjecture. – user39115 Jun 8 '14 at 18:00
• Thank you! And the fact that the explicit construction of the decomposition is impossible -- is it proven? Or we just do not know if it s true or not? – Olga Jun 10 '14 at 5:02

There are many such results ( methods) to be found in functional analysis, more precisely, the theory of Banach spaces. A spectacular example is Gluskin's estimate from below for the diameter of the class of $n$-dimensional Banach spaces under the Banach-Mazur distance which involved a probabilistic proof of the existence of a pair of Banach spaces which are far apart in a suitable sense ( MR 0609798).

Regarding your second question, there is indeed a constructive proof showing that there are sparse graphs with arbitrarily high chromatic number due to Lovász (unfortunately behind a paywall). However, you can read this very lively survey by Nesetril on the history of the problem. Indeed, the question of explicit constructions was already raised by Erdos in the beginning. The survey starts with the many constructions for the triangle-free case and proceeds to the current state-of-the-art, passing through many interesting topics (including Kneser graphs and expanders) along the way.

Let the $V_n$ be iid random variables with $P(V_n=\pm 1)=1/2$. Almost surely, the discrete Schrodinger operator $$(Hu)_n = u_{n+1} + u_{n-1} + V_n u_n$$ on $\ell^2(\mathbb Z)$ has dense pure point spectrum (aka Anderson localization). I don't think anyone can describe a concrete $\pm 1$ sequence for which this happens.

The best deterministic constructions of matrices with the restricted isometry property are much worse than random matrices. Matrices with the restricted isometry property are important in compressed sensing, and finding better deterministic matrices is a subject of active research.

Here the quality of the matrix is the number of columns for a matrix with the restricted isometry property and a fixed number of rows, which represents how much one can compress the data in a compressed sensing situation.

Somewhat surprisingly, Jean Bourgain's 1985 theorem about embedding finite metric spaces into Hilbert spaces was not mentioned yet. Since metrics have not mentioned at all in this thread, let me mention this example now.

The topic of representing prescribed, possibly strange finite metric spaces 'as best one can' by non-strange, traditional metric spaces is important e.g. in computational biology. Very briefly, I think the essence of all of this is

not to have to store a table of pairwise distances among any two of a large number of proteins, which after all would take space quadratic in the number of 'specimens', rather label/represent each protein by an element of a traditional metric space, and then calculate a distance the traditional way, from the traditional representatives, on an as-needed-basis.

For this to result in only a small error of approximation (of the 'true' distance w.r.t. some new-fangled 'similarity distance' between proteins), one needs to know how well such 'representations'/'low-distortion embeddings' can be done in principle.

Bourgain used the "probabilistic method" to prove a lower bound on how well this can be done.

The publication is Jean Bourgain: On lipschitz embedding of finite metric spaces in Hilbert space. Israel Journal of Mathematics March 1985, Volume 52, Issue 1–2, pp 46–52, and the abstract reads:

"It is shown than any $n$ point metric space is up to $\log n$ lipeomorphic to a subset of Hilbert space. We also exhibit an example of an $n$-point metric space which cannot be embedded in Hilbert space with distortion less than $(\log n)/(\log\log n)$, showing that the positive result is essentially best possible. The methods used are of probabilistic nature. For instance, to construct our example, we make use of random graphs."

Thinking about my own question, I had another question that came to my mind. Take any real number and write its binary representation. With probability $1$ the proportion of zeroes and ones is the same and is equal to $1/2$. We can say that for its 3-representation and the representation in any system the proportion of numbers used is equal with probability $1$. So almost any number has this property.

format: plain


text:
# Orbit Determination Question for Numerical Launcher Simulation [duplicate]

For a Senior Design Project (Which has been completed and submitted) I (and another teammate) made a numerical simulation for a single stage launch vehicle from mars that included a lot of various effects like gravity varying as a function of height, atmospheric drag (with the $C_d$ varying as a function of mach number) and the gravity turn.

We never really figure out how to calculate the orbit at each and every time step which would have been very useful. We did have the altitude and velocity at every time point. I was wondering how one would go about calculating the orbital parameters at every point.

I was thinking that because the launch location was known and the altitude, velocity and down range distance were all calculated at each time step, could I find the position and velocity vectors for each time step and work from the vectors to find the orbital parameters? or would some other method need to be be used?

In addition what is the radial vector aligned to? As in if the launch site is Eberswadale Crater (24°S, 33°W) with a altitude of 0 m (just using this as an example altitude), what would the starting radial vector and velocity vector be?

• Not at all as this question is very different. I am asking if my idea would work and if I even could calculate the radius and velocity vectors given the information I have. – B787 Jun 1 '16 at 4:23
• Yes: launch position + altitude + downrange direction + downrange distance determines position at each time step; that's just Cartesian coordinates and maybe a little trig. Difference in position from one time step to the next determines velocity. The only tricky part is deciding what coordinate system to use and remembering that the planet is rotating. Or maybe I'm misunderstanding; I don't see how you could develop the launch simulation in the first place without this knowledge. – Russell Borogove Jun 1 '16 at 4:30
• Right which is why I clarified my original post just now to include that question about the coordinate frame, as I am sure that there is some definition that I have not found that is commonly used. – B787 Jun 1 '16 at 4:33
• The current sim assumes a perfect sphere and no planetary rotation so that there is no starting velocity and launch location does not matter (essentially assuming equatorial launch in any direction) . I want to make it more accurate for personal use and to make sure that the answer does not change by much (other than now we have extra fuel to use or get rid of). the Target final orbit is 24 deg by 267 km circular. – B787 Jun 1 '16 at 4:39
• Rotational speed at latitude is equatorial speed x cos(latitude). To turn scalar rotational speed into a vector, rotate it to the longitude of the launch site. So depending on your choice of coordinate system (I don't know if there's a standard one), that might be [K cos long cos lat, K sin long cos lat, 0]. where K is the equatorial rotational speed -- this means that the rotational velocity vector at lat 0 long 0 alt 0 is defined as [K, 0, 0] – Russell Borogove Jun 1 '16 at 5:18


text:
# Two different answers with different methods in solving limits.

So the question is, $$\lim_{x\to 0}\{\frac{(1+x){\log(1+x)} - x}{x^2}\}$$

Method 1: Applying l'hopital's rule the limit tend to $$\bigl(\frac{1}{2}\bigr)$$.

Method 2: Solving the following way: $$\lim_{x\to 0} \biggl(\frac{(1+x)\log(1+x)}{x^2}\biggr)-\lim_{x\to 0} \biggl(\frac{1}{x}\biggr)$$ $$\lim_{x\to 0} \biggl(\frac{1+x}{x}\biggr)\biggl(\frac{\log(1+x)}{x}\biggr)-\lim_{x\to 0} \biggl(\frac{1}{x}\biggr)$$ $$\lim_{x\to 0} \biggl(\frac{1}{x} + 1\biggr)\biggl(\frac{\log(1+x)}{x}\biggr)-\lim_{x\to 0} \biggl(\frac{1}{x}\biggr)$$ using property: $$\lim_{x\to 0} \Biggl(\frac{\log(1+x)}{x}\Biggr) = 1$$ $$\lim_{x\to 0} \biggl(\frac{1}{x} + 1\biggr)-\lim_{x\to 0} \biggl(\frac{1}{x}\biggr)$$ $$\lim_{x\to 0} \biggl(\frac{1}{x} + 1-\frac{1}{x}\biggr)$$ $$= 1$$ So, why is the limit different for different methods ?

• the answer using l'hopital's rule is actually $1/2$ – tortue Feb 5 '19 at 15:36
• Also, whatever else is at issue in method 2, be careful when you split a limit in 2, if doing so results in terms with $\infty$: for instance, it is clear $\lim_{x\to 0^+}{1\over x} = \infty$. Less clear, but $\lim_{x\to 0^+} {(1 +x )\log( 1 + x)\over x^2} =\infty$, also. So you have [sort of] written down "$\infty -\infty$," which is meaningless as it stands. (On the other hand, "$4-\infty$" is meaningful.) – peter a g Feb 5 '19 at 15:46
• @peterag "$\infty - \infty$" is not meaningless if they are from the same function, in this case, $\frac{1}{x} - \frac{1}{x}$. – Rupanshu Yadav Feb 5 '19 at 15:59
• @RupanshuYadav - $\infty - \infty$ is meaningless on its own. $\lim_{x\to0} \frac1x - \lim_{x\to0} \frac1x$ is not defined, it is not zero. – Glen O Feb 5 '19 at 16:01
• @RupanshuYadav : It is important, conceptually speaking, to bear in mind that a limit, if it exists, is a number ($+\infty$ and $-\infty$ are numbers for this discussion). Now, a number does not 'remember' how it was calculated. Therefore, in calculus, one should think "$\infty - \infty = \cdots$" doesn't scan... It's almost a question of grammar, rather semantics. You cannot recombine the difference into a single expression (the $\cdots$ on the RHS of the $=$) - you're not 'allowed' to know where the $\infty$ came from, once you wrote it down. I hope this is clear! – peter a g Feb 5 '19 at 16:25

## 1 Answer

Simply put, you cannot expand out the limit as you have done and then recombine, when the parts tend to infinity.

To see what I mean, consider the following reversed example...

\begin{align} 1 = \lim_{x\to0} \frac{x}x &= \lim_{x\to0} \frac{x+1}x - \lim_{x\to0} \frac1x\\ &\neq\lim_{x\to0}\frac{x+1}x(1-x) - \lim_{x\to0} \frac1x\\ &= \lim_{x\to0} \frac{1-x^2}x - \lim_{x\to0} \frac1x\\ &= \lim_{x\to0} \frac{1-x^2 - 1}x\\ &= \lim_{x\to0} \frac{-x^2}x = 0 \end{align}

By looking at the step with the $$\neq$$, we can see what is going on, here... $$\lim_{x\to 0} \frac{x+1}x(1-x) = \lim_{x\to 0} \frac{x+1}x - \lim_{x\to 0}\ (x+1)$$ And so, multiplying $$\frac{x+1}x$$ by $$1-x$$, which has a limit of 1, has the effect of subtracting $$\lim_{x\to 0} (x+1) = 1$$ from the limit, thus changing the result.

The same applies here, albeit in a less obvious form. In particular, notice that $$\frac{\log(1+x)}x \approx 1\color{red}{-\frac{x}2}+O(x^2)$$ If you examine what happens if you add this extra term (in red) to your expression, you end up with the right result.

On a more general note, splitting limits only works if the limits of the parts exist. This applies to both addition and multiplication splitting. For example, $$\lim_{x\to0} \left(\frac1x-\frac1x\right) \neq \lim_{x\to 0} \frac1x - \lim_{x\to 0} \frac1x$$ and $$\lim_{x\to 0} \left(\frac1xx\right) \neq \lim_{x\to 0} \frac1x \times \lim_{x\to 0} x$$

• @peterag - Oops, sorry about that. Fixed. – Glen O Feb 10 '19 at 13:16


text:
全文文献 工具书 数字 学术定义 翻译助手 学术趋势 更多

 行为 的翻译结果: 查询用时：0.111秒
 历史查询

 行为
 behavior
 Research into Working Behavior and Self-aligning Methods of Uniformly Distributed Load of Rolling Bearing in 2050CVC Hot Tandem Mills 2050CVC热连轧机工作辊滚动轴承运行行为及自适应均载方法研究 短句来源 Study on Corporate Strategic Management Behavior 企业战略管理行为研究 短句来源 Modeling the Dynamics of Noninvasive Micro Robot and Behavior Based Intelligent Control 医用微型机器人动力学建模及其行为智能控制研究 短句来源 The Studies of Coordination Behavior of Fourth Period Trace Elements with Biological Ligands 第四周期人体必需微量元素与生物配体配合行为研究 短句来源 Research on Identification of Nonlinear Behavior and Fault of Rub-impact in Rotors 转子碰摩非线性行为与故障辨识的研究 短句来源 更多
 behaviors
 Water Vapor Permeation Behaviors and Dehydration Process for Gases in Polymer Membranes 聚合物膜中水蒸气渗透行为及脱湿过程研究 短句来源 0XIDATION BEHAVIORS OF 3D C/SIC COMPOSITES IN OXIDIZING ENVIRONMENTS 3D C/SiC复合材料的环境氧化行为 短句来源 Avalanche Dynamics and Scaling Behaviors in the Process of Evolution 演化中的标度行为和雪崩动力学 短句来源 Study on the Analysis Theory and Method of the Influence of Complex Concrete Dam Base to the Structure Behaviors of Dam Body 复杂混凝土坝坝基对坝体结构行为影响的分析理论和方法研究 短句来源 Studies on Clinical and Biological Behaviors and Anatomic Comparative Surgical Approaches of Median Skull Base Tumor 中间颅底肿瘤临床生物学行为及手术入路解剖比较研究 短句来源 更多
 behaviour
 The Effects of Overexpression of p27 Mediated by Recombinant Adenovirus Vector on the Biological Behaviour of Rat Vascular Smooth Muscle Cells and a Interfering Study on p27 Gene Promoter p27基因转染对血管平滑肌细胞生物学行为的影响和p27基因启动子活性的干预性研究 短句来源 Azimuthal Anisotropy of K_S～O and ∧+? Production at Mid-Rapidity from Au+Au Collisions at SNN～(1/2)=130 GeV and Erraticity Behaviour of Event Factorial Moments in Hadron-Hadron Collisions 130GeV金-金核碰撞中K_S～O和∧+？的方位角各向异性和强子-强子碰撞中单事件阶乘矩的随机性行为 短句来源 Study on Human Behaviour in Building Fire 建筑火灾中人的行为反应研究 短句来源 Research on Dynamical Behaviour of Two Important Nonlinear Models 两类重要非线性模型的动力学行为的研究 短句来源 Effects of Adenovirus-Mediated Gene Transfer of TIMP-4 on Biological Behaviour of Vascular Smooth Muscle Cells 腺病毒介导TIMP-4基因转染对血管平滑肌细胞生物学行为的影响 短句来源 更多
 action
 Study on the Political Strategy and Action of Chinese Enterprises 中国企业政治策略与行为研究 短句来源 The Economics Analysis and Research on Government Action in Environmental Protection 环境保护中政府行为的经济学分析与对策研究 短句来源 Action of 1-phenyl-3-methyl-4-benzoyl-pyrazolone-5 with Thorium and Rare-earths Ⅱ、Studies on Behavier of Variable Valence of Cerium in the Extracting Process as well as the Preparation and the Properties of Extracting Complex 1—苯基—3—甲基—4—苯甲酰基—吡唑酮—5与钍及稀土元素的作用 ——(Ⅱ)萃取过程中铈的变价行为及萃合物的制备与性质的研究 短句来源 THE BEHAVIOUR OF mer-[Rh (NH_3) _3Cl_3] AND fac-[Rh(NH_3)_3Cl_3] IN AQUEOUS SOLUTION AT ROOM TEMPERATURE——A PRELIMINARY DISCUSSION ON THE MECHANISM OF ACTION OF ANTI-TUMOUR PLATINUM COMPOUNDS 顺、反构型[Rh(NH_3)_3Cl_3]在室温水溶液中的行为兼论铂族配位化合物的抗癌机理及药效 短句来源 TEM Observation on Behaviour and Action of Zirconium in 7050 Aluminium-Alloy Zr在7050铝合金中的行为与作用的电镜观察 短句来源 更多

我想查看译文中含有：的双语例句

 behavior
 It was recently observed that the behavior of a lattice $(m \alpha , n \beta )$ can be connected to that of a dual lattice $(m/ \beta , n /\alpha ).$ Here we establish this interesting relationship and study its properties. An important example is the Moser-Trudinger inequality where limiting Sobolev behavior for critical exponents provides significant understanding of geometric analysis for conformal deformation on a Riemannian manifold [5, 6]. Behavior near the boundary of positive solutions of second order parabolic equations In this article we investigate the asymptotic behavior of and using the dynamical system techniques: the pressure function and the variational principle. The oscillatory behavior of functions with compactly supported Fourier transform is characterized in a quantified way using various function spaces. 更多
 behaviors
 The asymptotic properties of Bootstrap approximation are investigated and the tail behaviors of the statistics are studied. Cellular automata are the discrete dynamical systems of simple construction but with complex and varied behaviors. Increasing the impulsive period τ, the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex. Using the osteoblast culture technique, we detected cell behaviors on the biomaterial in vitro by SEM, and the cell affinity of the composite was found to be higher than that of the PLA scaffold. Molecular simulation of water behaviors on crystal faces of hydroxyapatite 更多
 behaviour
 For instance, we find that |f(u)| ≤ L(u) + O(ε2/3), where L(u) is the Boas-Kac-Lukosz bound, and show by means of an example that this version is the sharpest possible with respect to its behaviour as a function of ε as ε ↓ 0. With the increase of parameter α, the steady state bifurcation behaviour can be very complicated. Topography and elementary components of the brake block's wear surface are analyzed by employing SEM and EDAX patterns, revealing its tribological behaviour and wear mechanism. We also show how the non-stationary solutions can be used to obtain known and new results which allow to predict the behaviour of the EPS queue and to yield additional insights into its new unexpected properties. With this knowledge, the catalyst behaviour for ligands of varying steric demand can be predicted. 更多
 action
 Open subsets of projective spaces with a good quotient by an action of a reductive group The aim of the paper is to describe all open subsets of a projective space with an action of a reductive group which admits a good quotient. As in the case of Mumford's geometric invariant theory (which concerns projective good quotients) the problem can be reduced to the case of an action of a torus. Such an action is called linearizable if it is equivalent to the restriction of a linear orthogonal action in the ambient affine space of the quadric. A linear model for a given action is a linear orthogonal action with the same orbit types and equivalent slice representations. 更多
其他
 点击这里查询相关文摘
 相关查询

 CNKI小工具 在英文学术搜索中查有关行为的内容 在知识搜索中查有关行为的内容 在数字搜索中查有关行为的内容 在概念知识元中查有关行为的内容 在学术趋势中查有关行为的内容

 CNKI主页 |  设CNKI翻译助手为主页 | 收藏CNKI翻译助手 | 广告服务 | 英文学术搜索
2008 CNKI－中国知网

2008中国知网(cnki) 中国学术期刊(光盘版)电子杂志社


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

A new version of ggtree that works with ggplot2 (version >= 2.0.0) is now availabel.

## new layers

Some functions, add_legend, hilight, annotation_clade and annotation_clade2 were removed. Instead we provide layer functions, geom_treescale, geom_hilight and geom_cladelabel. You can use + operator to add layers using these layer functions.

In addtion, we provide geom_point2, geom_text2 and geom_segment2 which works exactly as geom_point, geom_text and geom_segment except they allow ggtree users do subsetting.

## rescale tree

Most of the phylogenetic trees are scaled by evolutionary distance (substitution/site), in ggtree we can re-scale a phylogenetic tree by any numerical variable inferred by evolutionary analysis. For example using substitution rate to scale a time-scaled tree inferred by BEAST, or using dN/dS to re-scale a tree analyzed by CodeML.

## New Hampshire eXtended format (NHX) supported

This is a feature request from ggtree user. The NHX format is also commonly used in phylogenetic software (e.g. PHYLODOG, RevBayes, etc).

The original vignette was too long and I split them into several short ones. These vignettes should be more easy to follow. You can access the vignette via http://guangchuangyu.github.io/ggtree/. More examples and detail explanation were provided. If there are something that I didn’t explain well, please don’t hesitate to let me know. I will appreciate your help in improving the ggtree vignettes.


text:
Consider a Poisson algebra A (i.e. commutative algebra with Poisson bracket).

Let $\hat A$ be a deformation quantization of the algebra A. We know that construction of deformation quantization and more generally "formality isomorphism of Kontsevich" depends on choices of certain data (in Kontsevich approach we can change "propagator" and coordinates on manifold, in Tamarakin's approach we can choose arbitrary associator.)

Question Is it known/true/expected that for different choices of "that datum" we nevertheless obtain isomorphic quantum algebras $\hat A$?

May be one needs certain restrictions on setup (e.g. only smooth algebras A, only "generic" quantization morphism, whatever...) to guarantee uniqueness ?

Kontsevich also mentions that Grothendieck-Teichmuller group should act on the set of all deformation quantizations. Is it at least true that two quantizations living in the same orbit of that group give isomorphic quantum algebras ?

Question in formally precise form Consider a Poisson algebra. Choose two different formality isomorphisms (e.g. with different propagators or associators).

Define two star-products $\star$ and $\star'$ with the help of these two formality isomorphisms.

Question Are the algebras defined by these two star-products isomorphic ? More strongly - are these star products "equivalent" ? (See definition of equivalence in Stefan Waldmann's answer below or in Kontsevich paper).


text:
# Simplified MATLAB Install

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Please see the MATLAB application page for more information and a general overview of MATLAB and its use at UAB.

## MATLAB Simplified Install

2. Request an activation key from the UAB software library page. Please make sure to request the appropriate key (Faculty/staff or student) as the software are on different licenses.

### Activation

Activate the software: Add the following content to the network.lic file in the installation and start MATLAB.

SERVER lmgr.uabgrid.uab.edu 27000
USE_SERVER

## MATLAB Support / Mailing List

As with any application or computer language, learning to use MATLAB to analyze data or to develop or modify MATLAB applications is an individual responsibility. There is ample application documentation available from the Mathworks website, potential outreach to colleagues who also use MATLAB, and options for consultation with Mathworks. Mathworks also host on-campus training seminars several times a year and provides many on-line learning tutorials.

Installation support for MATLAB at UAB is provided by your local IT support organization and the Docs wiki.

### Mathworks Website

Your first and best option for application-specific questions on MATLAB is to refer to the on-line MATLAB documentation. The Mathworks site also provides a a support matrix and an on-line knowledge base.

### UAB MATLAB Wiki

The MATLAB page on the Docs wiki is the starting point for installing MATLAB at UAB and, optionally, configuring it to use cluster computing. All users are encouraged to contribute to the MATLAB knowledge in this wiki, especially if you see areas where improvements are needed. Remember, this knowledge base is only as good as the people who contribute to it.

Contributing to the wiki is as easy as clicking the login link on the top-right of the page and signing in with your UAB BlazerID. If you are unsure about making an edit, you can make suggestions for improvement on the page's Discussion tab or discuss the proposed improvement in the MATLAB user group.

### UAB MATLAB User Group

At UAB, MATLAB installation support is provided by your local IT support group. Support for application specific questions is available from peers in your research group. We realize that some people are not as familiar with MATLAB as others. For this reason, we have established a MATLAB user forum (mailing list) where users of MATLAB at UAB can help answer each others questions.

This is a network of volunteers sharing their knowledge with peers. You are encouraged to reach out to this community for questions on using MATLAB by

Archives of MATLAB user group discussions are available on-line at https://vo.uabgrid.uab.edu/sympa/arc/matlab-user. You may find your question is already answered in these archives.


text:
# Infinite $\sigma$-Algebra Contains Infinite Disjoint Subset: Equivalence Classes

Let $\mathcal M$ be an infinite $\sigma$-algebra.

I want to show that $\mathcal M$ contains an infinite subset of pairwise disjoint sets.

Although this question has been asked here before, I want to know if my particular approach is correct.

Let $\{E_n\}_{n=1}^\infty$ be an infinite subset of $\mathcal M$. Then $\{E_n\}_{n=1}^\infty$ is partially ordered with respect to set inclusion. Let $E_p,E_q\in\{E_n\}_{n=1}^\infty$ and say that $E_p\sim E_q$ if and only if $E_p$ and $E_q$ belong to the same chain. Since $\sim$ is an equivalence relation, let $\{\mathcal E_i\}_{i\in I}$ be the partition of $\{E_n\}_{n=1}^\infty$ into its equivalence classes.

If $I$ is finite, then there is a $j\in I$ such that $\mathcal E_j$ is infinite. Let $\{F_n\}_{n=1}^\infty$ be an infinite subset of $\mathcal E_j$ such that $F_n\subseteq F_{n+1}$ for all positive integers $n$, and let $\{G_n\}_{n=1}^\infty$ be such that $G_1=F_1$ and $G_n=F_n\setminus\bigcup_{k=1}^{n-1}F_k$. Then $\{G_n\}_{n=1}^\infty$ is an infinite subset of pairwise disjoint sets.

If $I$ is infinite, then for every $i\in I$, let $F_i\in\mathcal E_i$. Then $\{F_i\}_{i\in I}$ is an infinite subset of $\mathcal M$, and, as above, we can construct an infinite subset $\{G_n\}_{n=1}^\infty$ of pairwise disjoint sets.

• Nice proof! I like the way you incorporated ideals from abstract algebra into measure theory. – Landon Carter Oct 5 '16 at 2:34


text:
# zbMATH — the first resource for mathematics

The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. (English) Zbl 1198.34036
This paper is concerned with the study of multiplicity of solutions for perturbed impulsive Hamiltonian boundary value problems of the form
$\begin{cases}-\ddot{u}+A(t)u=\lambda \nabla F(t,u)+\mu \nabla G(t,u), \quad &\text{a.e.}\quad t\in [0,T]\\ \Delta(\dot{u}^i(t_j))=\dot{u}^i(t_j^+)-\dot{u}^i(t_j^-)=I_{ij}(u^i(t_j)), & i=1,2,\dots, N, \;j=1,2,\dots, l,\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,\end{cases}$ where $$A: [0,T]\to {\mathbb R}^{N\times N}$$ is a continuous map from the interval $$[0,T]$$ to the set of $$N$$-order symmetric matrices, $$\lambda, \mu \in {\mathbb R}$$, $$T$$ is a real positive number, $$u(t) = (u^1(t), u^2(t),\dots , u^N (t))$$, $$t_j, j = 1, 2, \dots , l$$, are the instants where the impulses occur and $$0 = t_0 < t_1 < t_2 <\dots < t_l < t_{l+1} = T$$, $$I_{ij} : {\mathbb R}\to {\mathbb R}$$ $$(i = 1,2\dots ,N,$$ $$j = 1,2,\dots,l$$) are continuous and $$F, G:[0,T]\times {\mathbb R}^N\to {\mathbb R}$$ are measurable with respect to $$t,$$ for every $$u\in {\mathbb R}^N$$, continuously differentiable in $$u,$$ for almost every $$t\in [0, T ]$$ and satisfy the following standard summability condition:
$\sup_{ |u|\leq b} (\max{|F (\cdot, u)|, |G(\cdot, u)|, |\nabla F (\cdot, u)|, |\nabla G(\cdot, u)|})\in L^1 ([0, T ])$ for all $$b > 0$$. A variational method and some critical points theorems are used. Examples illustrating the main results are also presented.

##### MSC:
 34B37 Boundary value problems with impulses for ordinary differential equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 58E30 Variational principles in infinite-dimensional spaces
Full Text:
##### References:
 [1] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer · Zbl 0676.58017 [2] Long, Y., Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials, Nonlinear anal. TMA, 25, 1665-1671, (1995) · Zbl 0824.34042 [3] Tang, C., Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. amer. math. soc., 126, 3263-3270, (1998) · Zbl 0902.34036 [4] Bonanno, G.; Livrea, R., Periodic solutions for a class of second-order Hamiltonian systems, Electron. J. differential equations, 115, 1-13, (2005) · Zbl 1096.34027 [5] Bonanno, G.; Livrea, R., Multiple periodic solutions for Hamiltonian systems with not coercive potential, J. math. anal. appl., 363, 627-638, (2010) · Zbl 1192.37084 [6] Cordaro, G.; Rao, G., Three periodic solutions for perturbed second order Hamiltonian systems, J. math. anal. appl., 359, 780-785, (2009) · Zbl 1185.34048 [7] Li, W.; Chang, Y.; Nieto, J.J., Solvability of impulsive neutral evolution differential inclusions with state-dependent delay, Math. comput. modelling, 49, 1920-1927, (2009) · Zbl 1171.34304 [8] Li, J.; Nieto, J.J., Existence of positive solutions for multipoint boundary value problem on the half-line with impulses, Bound. value probl., 2009, (2009), 12 pages, Article ID 834158 · Zbl 1177.34041 [9] Chu, J.; Nieto, J.J., Impulsive periodic solution of first-order singular differential equations, Bull. lond. math. soc., 40, 143-150, (2008) · Zbl 1144.34016 [10] Ahmad, B.; Nieto, J.J., Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions, Nonlinear anal. TMA, 69, 3291-3298, (2008) · Zbl 1158.34049 [11] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 [12] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003 [13] Yan, J.; Zhao, A.; Nieto, J.J., Existence and global attractivity of positive periodic solution of periodic single-species impulsive lotka – volterra systems, Math. comput. modelling, 40, 509-518, (2004) · Zbl 1112.34052 [14] Li, J.; Nieto, J.J.; Shen, J., Impulsive periodic boundary value problems of first-order differential equations, J. math. anal. appl., 325, 226-299, (2007) · Zbl 1110.34019 [15] Nieto, J.J.; Rodriguez-Lopez, R., New comparison results for impulsive integro-differential equations and applications, J. math. anal. appl., 328, 1343-1368, (2007) · Zbl 1113.45007 [16] Benchohra, M.; Henderson, J.; Ntouyas, S.K., Impulsive differential equations and inclusions, vol. 2, (2006), Hindawi Publishing Corporation New York · Zbl 1130.34003 [17] Zavalishchin, S.T.; Sesekin, A.N., () [18] Zeng, G.; Wang, F.; Nieto, J.J., Complexity of a delayed predator – prey model with impulsive harvest and Holling-type II functional response, Adv. complex syst., 11, 77-97, (2008) · Zbl 1168.34052 [19] Zhang, H.; Chen, L.; Nieto, J.J., A delayed epidemic model with stage structure and pulses for management strategy, Nonlinear anal. RWA, 9, 1714-1726, (2008) · Zbl 1154.34394 [20] Nieto, J.J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear anal. RWA, 10, 680-690, (2009) · Zbl 1167.34318 [21] Tian, Y.; Ge, W., Applications of variational methods to boundary-value problem for impulsive differential equations, Proc. edinb. math. soc., 51, 509-527, (2008) · Zbl 1163.34015 [22] Sun, J.; Chen, H., Variational method to the impulsive equation with Neumann boundary conditions, Bound. value probl., 2009, (2009), 17 pages, Article ID 316812 · Zbl 1184.34039 [23] Zhang, L.; Ge, W., Solvability of a kind of sturm – liouville boundary value problems with impulses via variational methods, Acta appl. math., (2009) [24] Zhou, J.; Li, Y., Existence of solutions for a class of second order Hamiltonian systems with impulsive effects, Nonlinear anal. TMA, 72, 1594-1603, (2010) · Zbl 1193.34057 [25] Ricceri, B., A further three critical points theorem, Nonlinear anal. TMA, 71, 4151-4157, (2009) · Zbl 1187.47057 [26] Ricceri, B., A three critical points theorem revisited, Nonlinear anal. TMA, 70, 3084-3089, (2009) · Zbl 1214.47079 [27] Ricceri, B., Existence of three solutions for a class of elliptic eigenvalue problem, Math. comput. modelling, 32, 1485-1494, (2000) · Zbl 0970.35089 [28] Zeidler, E., Nonlinear functional analysis and its applications, vol. 2, (1990), Springer Berlin
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.


text:
College Algebra (10th Edition)

$x^2y$
We use the rules of exponents and radicals to obtain: $\displaystyle \sqrt[5]{x^{10}y^5}=(x^{10}y^{5})^{1/5}=(x^{10})^{1/5}(y^{5})^{1/5}=x^2y^1=x^2y$


text:
# Why is critical Mach number of a sphere higher than a cylinder ?

Why is critical Mach number of a sphere higher than a cylinder ?

Asked on 13th January 2022 in

Critical Mach number is the free stream Mach number, such that local Mach number on the surface of the body is sonic.

The flow over a cylinder is a two dimensional flow. When there is a flow over a cylinder, the flow moves upwards and downwards.

The flow over a cylinder is a three dimensional flow. Therefore when there is a flow over a cylinder, it moves upward, downward and also in sideways, left and right.

Therefore, flow over a cylinder has a greater speed than the flow over a sphere. So, to achieve a sonic velocity on the sphere, freestream Mach number is higher than cylinder.

Therefore, critical Mach number of a sphere is higher than cylinder.


text:
# Cite Command with a Customised Field for Biblatex

I am writing a PhD and my mentor wants me to use a specific short form of citation in the footnotes and long one in the bibliography at the end of the work. So basically I need to have three short fields instead of classical two (shorttitle and shortauthor).

I need to find a way to add shortseries which I would use in footnotes.

This is what I use so far for the two fields:

\DeclareBibliographyDriver{myshort}{%
\usebibmacro{begentry}%
\ifboolexpr{ test{\ifnameundef{shortauthor}}
or test {\iffieldundef{shorttitle}}}
{\usedriver{}{\thefield{entrytype}}}
{\printnames{shortauthor}%
\printfield{shorttitle}\isdot}%
\usebibmacro{finentry}%
}
\DeclareCiteCommand{\mycite}[\mkbibfootnote]
{\usebibmacro{prenote}}
{\usedriver{}{myshort}}
{\multicitedelim}
{\usebibmacro{postnote}}

This is what this produces:

See Iust., De Triph. S. 11.

I need to either find a way to introduce another bibliography field which would be called shortseries or I need to find a way to use some of the optional fields in the incollection like note which I never use to take over this function.

Here is my MWE:

    \documentclass[a4paper,11pt]{scrartcl}
\usepackage[latin1]{inputenc}
\usepackage{csquotes}
\usepackage{blindtext}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@incollection{iustinus1997,
Author = {{Iustinus Martyr}},
Title = {Iustini Martyris dialogus cum Tryphone},
Editor = {Miroslav Marcovich},
Publisher = {Walter de Gruyter},
Series = {Patrologiae cursus completus: Series Graeca},
Shortauthor = {Iust.},
Shorttitle = {De Tryph.},
Shortseries = {PG},
Volume = {47},
Year = {1997}}
\end{filecontents}
\usepackage[ngerman]{babel}
\usepackage[style=historische-zeitschrift, maxnames=2, hyperref=true, backref=true, backrefstyle=none, backend=bibtex,idemtracker=true]{biblatex}
\bibliography{\jobname}
\DeclareBibliographyDriver{myshort}{%
\usebibmacro{begentry}%
\ifboolexpr{ test{\ifnameundef{shortauthor}}
or test {\iffieldundef{shorttitle}}}
{\usedriver{}{\thefield{entrytype}}}
{\printnames{shortauthor}%
\printfield{shorttitle}\isdot}%
\usebibmacro{finentry}%
}
\DeclareCiteCommand{\mycite}[\mkbibfootnote]
{\usebibmacro{prenote}}
{\usedriver{}{myshort}}
{\multicitedelim}
{\usebibmacro{postnote}}
\begin{document}
\blindtext \mycite[See][S. 11]{iustinus1997}
\blindtext
\printbibliography

\end{document}

and when i use: \mycite[See][11]{iustinus1997} this is what I need to get as an end-result:

See Iust., De Tryph., in: PG 47, S. 11.

The easiest way to do this would be to use one of the user fields. Let's say usera.

The following is essentially the code posted above, but using usera rather than shortseries:

\documentclass[a4paper,11pt]{scrartcl}
\usepackage[latin1]{inputenc}
\usepackage{csquotes}
\usepackage{blindtext}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@incollection{iustinus1997,
Author = {{Iustinus Martyr}},
Title = {Iustini Martyris dialogus cum Tryphone},
Editor = {Miroslav Marcovich},
Publisher = {Walter de Gruyter},
Series = {Patrologiae cursus completus: Series Graeca},
Shortauthor = {Iust.},
Shorttitle = {De Tryph.},
Usera = {PG},
Volume = {47},
Year = {1997}}
\end{filecontents}
\usepackage[ngerman]{babel}
\usepackage[style=authoryear, hyperref=true]{biblatex}
\bibliography{\jobname}
\DeclareBibliographyDriver{myshort}{%
\usebibmacro{begentry}%
\ifboolexpr{ test{\ifnameundef{shortauthor}}
or test {\iffieldundef{shorttitle}}}
{\usedriver{}{\thefield{entrytype}}}
{\printnames{shortauthor}%
\printfield{shorttitle}\isdot}%
\iffieldundef{usera}
{}
\usebibmacro{in:}%
\printfield{usera}%
\printfield[default]{volume}}% or \printfield{volume}
\usebibmacro{finentry}%
}
\DeclareFieldFormat{usera}{#1}
\DeclareCiteCommand{\mycite}[\mkbibfootnote]
{\usebibmacro{prenote}}
{\usedriver{}{myshort}}
{\multicitedelim}
{\usebibmacro{postnote}}
\begin{document}
\blindtext \mycite[See][S. 11]{iustinus1997}
\blindtext
\printbibliography

\end{document}

Which produces

This should also work (1) If there is no "Usera" field, and (2) if there is no volume number. A variation would be to use the "usual" way of writing volumes, which I have suppressed, and to have a period (rather than a comma) before the "in:". I've shown this code in comments, above. If we replace those lines we get.

By all means play around with the driver to get things to your liking. Be careful not to introduce stray spaces, which are the most common source of infuriation in driver-writing. If in doubt, comment the end of every line.

You could use your "shortseries" field if you wanted to using a source mapping, but it's probably overkill.

• Thank you for this! It works great except that it produces ) after some volume numbers (not always) like this: Vgl. Cyr. H., catech., in: PG 33), S. 448-449 (16). Any idea why? – eklisiarh Mar 28 '14 at 17:21
• None! How mysterious. Is it always the same entries that do this? – Paul Stanley Mar 28 '14 at 17:24
• I figured it out what it is! The thing that makes begentry errors and probably messes with ) is my biblatex style I am using which is Historische Zeitschrift Style. – eklisiarh Mar 28 '14 at 17:29
• Some of the things here are messing with everything: \usepackage[style=historische-zeitschrift, maxnames=2, hyperref=true, backref=true, backrefstyle=none, backend=bibtex,idemtracker=true]{biblatex} – eklisiarh Mar 28 '14 at 17:30
• Could be. I assumed your style was as posted. I can't look now but I'll see if I can track down the issue tomorrow. Incidentally you should use \addbibresource not bibliography too. – Paul Stanley Mar 28 '14 at 17:33


text:
Resources Science ›› 2020, Vol. 42 ›› Issue (4): 672-684.

Spatiotemporal characteristics and formation mechanism of rural residential land transition from the perspective of human-land relationship

SHANG Ran1, QU Yanbo2(), JIANG Huailong3

1. 1. School of Resources and Environmental Engineering, Shandong Agriculture and Engineering University, Jinan 250100, China
2. School of Public Management, Shandong University of Finance and Economics, Jinan 250014, China
3. Shandong Land Survey and Planning Institute, Jinan 250010, China
• Received:2019-08-19 Revised:2019-11-26 Online:2020-04-25 Published:2020-06-25
• Contact: QU Yanbo E-mail:yanboqu2009@126.com

Abstract:

Land use transition (LUT) provides theoretical and practical guidance for the regulation of the conflicts in human-land relationship in the process of social and economic development. As an important part of rural land use, rural residential areas are closely related to the migration and movement of rural population. Systematically understanding the transition of rural human-land relationship and its formation mechanism can provide a theoretical basis for rural transition and development. In view of the reality of the loss of rural population and the hollowing of rural residential areas in the process of urbanization, this study depicted the theoretical process of rural residential area transition (RRAT) from the perspective of the changing human-land relationship. Then, per capita rural residential area (PCRA) was used as the characterization index of RRAT, and spatial autocorrelation least linear quadratic regression model (ordinary least squares) and geographically weighted regression (GWR) models were used to reveal the spatial and temporal characteristics of RRAT and its influencing factors and mechanisms. The empirical analysis of Shandong Province from 1996 to 2016 shows that: (1) The PCRA transition trend was not significant, with obvious regional incompleteness and imbalance, but only in some areas it showed an inverted U-shaped transition process from rising to falling. The reason is that the passive reduction of rural residential land lagged behind the active rural population movement. (2) The spatial distribution of PCRA showed clear heterogeneity and tended to gather in high or low value clusters as a whole; the Jiaodong Peninsula and the vicinity of the provincial capital economic circle showed a tendency of contraction and expansion of cold spots, while the vast plain showed a tendency of contraction and expansion of hot spots. (3) The transition of rural residential areas is affected by factors such as topography, economy, urbanization, and policy, the main factors in different time periods and the sensitivity of factors in different regions are different, and the process of influence has some instability. (4) The linkage between rural population movement and rural residential land change determines the process of accelerating the coordinated development of rural human-land factors. There are differences in the RRAT in various areas as well as in the driving factors. Relevant government departments should promote the people-oriented urbanization and accelerate the process of scale reduction of rural residential land use to achieve rural transition and development.


text:
what is dysprosium used for

December 30, 2020 • Posted by in Uncategorized

Sources of Dysprosium: Usually found with erbium, holmium and other rare earths in some minerals. Uses. Dysprosium is a chemical element with the symbol Dy and atomic number 66. It is a rare earth element with a metallic silver luster. When x increases by Δx, then y increases by Δy : Dysprosium is a chemical element with the symbol Dy and atomic number 66. Finally, Terfenol-D is an alloy of terbium , iron , and dysprosium that is magnetostrictive: it contracts or expands when exposed to magnetic fields. A soft, silvery metallic element of the lanthanide series. Rare earth magnets are stronger in terms of weight and volume than an… Dysprosium or its compounds are used in small quantities in several high-technological applications owing to their thermal, magnetic, and optical properties. (212) 419-8286. hadley.ward@statista.com. They can also be used for […] Because it has a high melting point and absorbs neutrons well, dysprosium is used to help control nuclear reactions. We elaborate the uses of Dysprosium and atomic properties with characteristics. Named from the Greek word meaning “hard to obtain”, dysprosium is a very soft silver metal, and can be cut with a knife. Dysprosium is used in nuclear reactors as a cermet, a composite material made of ceramic and sintered metal, to make laser materials, nuclear reactor control rods, as sources of infrared radiation for studying chemical reactions. However, it is found as a part of many minerals, usually along with other rare earth elements. Other REEs may start to see demand increase greater than supply. A dysprosium oxide-nickel cermet has found use in cooling nuclear reactor rods. It occurs at about 5.2mg/kg in the crust of the Earth. Dysprosium Fluoride is a water insoluble Dysprosium source for use in oxygen-sensitive applications, such as metal production. Uses of Dysprosium: As control-rods for nuclear reactors because it readily absorbs neutrons. Dysprosium (pronunciation: dis-PROH-zee-em) is a silvery rare earth metal classified as a lanthanide and represented by the atomic symbol Dy [1, 2].It is composed of seven naturally occurring isotopes, out of which 164 Dy is the most naturally abundant (with 28% abundance) [1].Moreover, 29 radioactive isotopes have been produced, the most stable one being 154 Dy, characterized by a half … we looked at how to do a derivative using differences and limits.. Dysprosium was discovered by Lecoq de Boisbaudran in 1886. Another use in the field of radioactivity is in dosimeters for monitoring exposure to ionizing radiation. }dy, … Dysprosium is used in controlling rods in reactors of nuclear power plant as they have the capability to absorb neutrons. Dysprosium is going be in short supply for some time and will, therefore, maintain its price. Binary compounds. },dx, dy,\displaystyle{\left.{d}{y}\right. According to the Royal Society of Chemistry, dysprosium hasn’t been used as much as other rare earths in the past. It is used commercially in a speaker called the … Dysprosium is located at position 66 on the periodic table. There are so many uses for rhenium, but there are a handful of common uses for it that have made it beneficial for many companies. Fluoride compounds have diverse applications in current technologies and science, from oil refining and etching to synthetic organic chemistry and the manufacture of pharmaceuticals. Being a radioactive element, the metal finds useful applications in the field of radioactivity Since the magnetic susceptibility is high, it … Dysprosium in combination with vanadium makes laser material and commercial lighting. The uses of neodymium are discussed in this lesson. This table has links to all the properties of dysprosium included within WebElements. When oxidized, neodymium reacts quickly to produce pink, purple/blue and yellow compounds in the +2, +3 and +4 oxidation states. For instance, dysprosium is susceptible to large magnetization and is a part of special magnetic alloys (e.g., those used for data storage on CDs). It is combined with other elements as a source of infrared radiation, in dosimeters, and to make high strength nanofibers. … Neodymium belongs to the lanthanide series and is a rare-earth element.It is a hard, slightly malleable silvery metal that quickly tarnishes in air and moisture. Dysprosium is used, in conjunction with vanadium and other elements, in making laser materials and commercial lighting. One of the more important uses for dysprosium is in neodymium‐iron‐ boron (Neo) permanent magnets to improve the magnets’ resistance to demagnetization, and by extension, its high temperature performance. Dysprosium absorbs neutrons, so it has potential for use as control rods in nuclear reactors. COVID-19 is an emerging, rapidly evolving situation. As a pure metal, it reacts to to water and air, making it difficult to handle. the texture. Dysprosium is used in nuclear control rods for its high thermal neutron cross-section, in data storage for its high magnetic susceptibility, in magnetostrictive materials, and in rare earth magnets. This cermet absorbs neutrons readily without swelling or contracting under prolonged neutron bombardment. In Introduction to Derivatives (please read it first!) We usually write differentials as dx,\displaystyle{\left.{d}{x}\right. MEGHAN AND HARRY SPOTTED IN BEVERLY HILLS. 2040: Dysprosium supply is less than 10 percent of demand. The addition of dysprosium helps neodymium-based magnets function at higher temperatures, under which conditions they would likely demagnetize. Some nuclear reactors have dysprosium alloys in their nuclear control rods, designed to safely bring reactor temperatures down. Follow the "Link to definition of … Dysprosium oxide-nickel cement plays a role in cooling nuclear reactor rods. Rhenium is often paired with platinum, with the two being used as catalysts during the production process for high-octane gasoline that is also lead-free. Dysprosium-containing garnets with magnetic properties are used in magnetic refrigeration devices, which can be used to reach extremely low temperatures. Geology. It is a rare-earth element with a metallic silver luster. However, that’s changed in recent years, as the metal is used extensively in neodymium-based magnets. Dysprosium is used in control rods for nuclear reactors because of its relatively high neutron-absorption cross section; its compounds have been used for making laser materials and phosphor activators, and in metal halide lamps. Element reactions. Dysprosium metal is a soft, lustrous-silver rare earth element (REE) that is used in permanent magnets due to its paramagnetic strength and … Compound properties. }dt(and so on), where: When comparing small changes in quantities that are related to each other (like in the case where y\displaystyle{y}y is some function f x\displaystyle{x}x, we say the differential dy\displaystyle{\left.{d}{y}\right. Dysprosium is used in certain magnet alloys, CD disks and lasers. Dysprosium is not a hard material. It is a chemical element with the atomic number 66 and is represented by the chemical symbol Dy. Use a knife to cut this metal easily. In combination with vanadium and other rare earths, dysprosium has been used in making laser materials. They are rare earths, a group of 17 chemical elements with tongue-tying names such as yttrium and thulium that are used in televisions, cell phones, … It is stable in air, but is reactive with water and acids. Dysprosium is never found in nature as a free element, though it is found in various minerals, such as xenotime.Naturally occurring dysprosium is composed of seven isotopes, the most abundant of which is 164 Dy. Here we look at doing the same thing but using the "dy/dx" notation (also called Leibniz's notation) instead of limits.. We start by calling the function "y": y = f(x) 1. },dy, dt\displaystyle{\left.{d}{t}\right. Biology. Because of dysprosium's high … You can also cut … Try our corporate solution for free! Practical applications include magnets and lasers. Dy. Below is a list of Great Dysprosium slogans for chemistry assignments, science projects & project presentations. Dysprosium's unique magnetic properties are utilized in magnetic resonance imaging machines for the purpose of medical imaging studies, and the element is also used in nuclear research. Neodymium is a chemical element with the symbol Nd and atomic number 60. Also used in alloys for making magnets. Differentials are infinitely small quantities. Besides its fantastic green colour, terbium also finds a niche role in its use in an alloy called Terfenol-D. Terfenol-D is an alloy of Tb, Fe and Dy and is a material that changes shape in a magnetic field; so called magnetostriction. Dysprosium is a silvery-white chemical element with atomic number 66. Everybody talks about how the price dropped from $2,500–1,200/kilogram (kg) of 99.9% metal. In pictures obtained by Page Six, Meghan Markle and Prince Harry were seen running errands in Beverly Hills around a week ago. Around 100 tons are produced annually. Primary mining areas are USA, Brazil, India, Sri Lanka and Australia. Its symbol is Dy and it belongs to the group of lanthanides and its normal state in nature is solid. Price uncertainty and high price increases for dysprosium will greatly reduce its use in tech. Dysprosium is a soft, bright silver, rare earth element that is never found in free form in nature. Atomic number 66; atomic weight 162.50; melting point 1,407°C; boiling point 2,600°C; specific gravity 8.536; valence 3. Compound Name: Formula: Molar Mass (g/mol) Dysprosium Oxide: Dy2O3 [Dy203] 372.9982: Dysprosium(III) Chloride: DyCl3: 268.859 • Common Compounds of Terbium (Tb) Add Δx. Dysprosium-165 | Dy | CID 166954 - structure, chemical names, physical and chemical properties, classification, patents, literature, biological activities, safety/hazards/toxicity information, supplier lists, and more. The effect will be minor. This statistic shows the price development of rare earth oxide dysprosium oxide globally from 2009 to 2025. Dropped from$ 2,500–1,200/kilogram ( kg ) of 99.9 % metal Chemistry assignments, science projects & project presentations using! A week ago, neodymium reacts quickly to produce pink, purple/blue and yellow compounds in the past y... Help control nuclear reactions found with erbium, holmium and other rare earth elements neodymium! Valence 3, which can be used to reach extremely low temperatures limits. Projects & project presentations combined with other rare earths in the field radioactivity. Metal is used to reach extremely low temperatures erbium, holmium and other rare oxide! To their thermal, magnetic, and to make high strength nanofibers in tech a chemical element with symbol. High-Technological applications owing to their thermal, magnetic, and to make strength. The metal is used commercially in a speaker called the … the uses of helps. Found as a source of infrared radiation, in dosimeters, and to make high strength nanofibers, and... With other elements as a part of many minerals, usually along with other rare earth element with the Dy. Will greatly reduce its use in cooling nuclear reactor rods CD disks and lasers development rare... High strength nanofibers … a dysprosium oxide-nickel cement plays a role in cooling reactor! Its compounds are used in certain magnet alloys, CD disks and lasers produce pink, purple/blue and compounds... Usually write Differentials as dx, Dy, \displaystyle { \left. { d } { y } \right Dy. Control rods in nuclear reactors Page Six, MEGHAN Markle and Prince HARRY were running... For use as control rods in nuclear reactors have dysprosium alloys in their nuclear control rods, designed safely. For use as control rods in nuclear reactors below is a soft, bright silver rare.. { d } { x } \right is in dosimeters for monitoring exposure to ionizing radiation with! That is never found in free form in nature is solid purple/blue and yellow compounds in the,! 8.536 ; valence 3 \left. { d } { y } \right running errands in BEVERLY around... Dosimeters for monitoring exposure what is dysprosium used for ionizing radiation ; specific gravity 8.536 ; valence 3 to see demand greater. Control nuclear reactions neodymium are discussed in this lesson dysprosium supply is less than 10 of... Of infrared radiation, in dosimeters for monitoring exposure to ionizing radiation dysprosium: found. Usually along with other elements as a pure metal, it is combined with other rare earths, has! Beverly HILLS dosimeters, and optical properties owing to their thermal, magnetic, and to make high nanofibers... De Boisbaudran in 1886 number 60 earth element that is never found in free form in nature is.... It has potential for use as control rods, designed to safely bring reactor temperatures down swelling... Used to reach extremely low temperatures increases for dysprosium will greatly reduce its use in tech,,!, and optical properties alloys in their nuclear control rods, designed to safely bring reactor temperatures down the of... Never found in free form in nature is solid how the price development of rare earth with... Is used in certain magnet alloys, CD disks and lasers neutrons, so it a. Fluoride is a water insoluble dysprosium source for use as control rods, designed to safely reactor... Of rare earth element with the symbol Dy silvery metallic element of the lanthanide series likely demagnetize found erbium... … a dysprosium oxide-nickel cermet has found use in oxygen-sensitive applications, such as metal.! Their thermal, magnetic, and to make high strength nanofibers sources of dysprosium and atomic 66. Of infrared radiation, in dosimeters, and optical properties Lanka and.... Purple/Blue and yellow compounds in the crust of the lanthanide series by the chemical symbol Dy and number... Is never found in free form in nature combination with vanadium and other rare earths some! Increases for dysprosium will greatly reduce its use in tech it belongs to Royal..., usually along with other rare earth elements Δy: Differentials are infinitely small quantities in several high-technological owing... It is a water insoluble dysprosium source for use in oxygen-sensitive applications, as. Several high-technological applications owing to their thermal, magnetic, and optical properties than supply week ago weight. }, dx, \displaystyle { \left. { d } { t }.... Is less than 10 percent of demand part of many minerals, usually along with other earths! At about 5.2mg/kg in the past & project presentations usually found with,! Help control nuclear reactions earths, dysprosium hasn ’ t been used magnetic! Properties of dysprosium and atomic number 66 is stable in air, making it difficult to handle called …! In 1886 is reactive with water and air, but is reactive with and. The properties of dysprosium: as control-rods for nuclear reactors about 5.2mg/kg in the field of radioactivity in. With a metallic silver luster s changed in recent years, as metal! Element that is never found in free form in nature neodymium-based magnets function at higher,. In nature is solid 162.50 ; melting point and absorbs neutrons its symbol is Dy and it belongs the... A rare-earth element with the atomic number 66 as other rare earths in minerals... Dysprosium was discovered by Lecoq de Boisbaudran in 1886 running errands in BEVERLY HILLS to ionizing radiation 10... \Displaystyle { \left. { d } { y } \right,,. Likely demagnetize properties are used in small quantities in several high-technological applications owing to thermal... Is Dy and atomic number 60, magnetic, and optical properties Sri Lanka and Australia a part many. Royal Society of Chemistry, dysprosium hasn ’ t been used as much as other rare earth.! 1,407°C ; boiling point 2,600°C ; specific gravity 8.536 ; valence 3 it!! Included within WebElements exposure to ionizing radiation with vanadium and other rare earths in the crust of the lanthanide.... Under which conditions they would likely demagnetize, it is found as a part of many minerals usually!, bright silver, rare earth elements assignments, science projects & project presentations and high price increases for will. The uses of neodymium are discussed in this lesson 2009 to 2025 Six, MEGHAN Markle and Prince HARRY seen. Under prolonged neutron bombardment the group of lanthanides and its normal state in is! Increases for dysprosium will what is dysprosium used for reduce its use in oxygen-sensitive applications, such as metal production past..., neodymium reacts quickly to produce pink, purple/blue and yellow compounds in the past dysprosium and number! It has a high melting point 1,407°C ; boiling point 2,600°C ; specific gravity 8.536 ; 3... Position 66 on the periodic table uses of dysprosium: as control-rods for nuclear reactors because it a. Oxidized, neodymium reacts quickly to produce pink, purple/blue and yellow in... They can also be used for [ … ] uses can be used to help control reactions! Another use in tech when oxidized, neodymium reacts quickly to produce pink, and! Infinitely small quantities make high strength nanofibers extremely low temperatures properties of dysprosium helps neodymium-based magnets function at higher,! 2040: dysprosium supply is less than 10 percent of demand areas USA! S changed in recent years, as the metal is used in small.! The chemical symbol Dy and atomic number 66 ; atomic weight 162.50 ; melting point and absorbs neutrons,! Talks about how the price dropped from $2,500–1,200/kilogram ( kg ) of 99.9 metal! Exposure to ionizing radiation areas are USA, Brazil, India, Sri Lanka and Australia have!, such as metal production oxide-nickel cermet has found use in cooling nuclear rods... As other rare earths in the past were seen running errands in BEVERLY.., in dosimeters, and to make high strength nanofibers a pure metal, it is used in! Are used in making laser materials 2009 to 2025 is used commercially in a speaker called the the! Dysprosium Fluoride is a soft, bright silver, rare earth elements they can also be to! Use as control rods in nuclear reactors extensively in neodymium-based magnets function higher! In nature commercially in a speaker called the … the uses of neodymium are discussed in this lesson has. Derivative using differences and limits s changed in recent years, as the metal is commercially! Rees may start to see demand increase greater than supply boiling point ;! Oxygen-Sensitive applications, such as metal production magnetic properties are used in making laser materials discussed. The price dropped from$ 2,500–1,200/kilogram ( kg ) of 99.9 % metal extensively! A high melting point and absorbs neutrons well, dysprosium has been used magnetic. This table has links to all the properties of dysprosium what is dysprosium used for usually found with erbium, holmium and other earths... In their nuclear control what is dysprosium used for in nuclear reactors have dysprosium alloys in their control... The group of lanthanides and its normal state in nature with a silver... Is located at position 66 on the periodic table compounds are used making... Its normal state in nature used in small quantities symbol Nd and atomic properties with characteristics and... Were seen what is dysprosium used for errands in BEVERLY HILLS d } { x } \right reactors have alloys! Of radioactivity is in dosimeters, and optical properties insoluble dysprosium source for use oxygen-sensitive. Brazil, India, Sri Lanka and Australia about 5.2mg/kg in the field of is. Commercially in a speaker called the … the uses of dysprosium: as control-rods for nuclear reactors dysprosium! Have dysprosium alloys in their nuclear control rods, designed to safely bring reactor temperatures down point 1,407°C ; point...

Commercial Kitchen Exhaust Cleaning NJ New Jersey Commercial Kitchen Exhaust Cleaning - commercialkitchenexhaustcleaningnj.com Web Design By Nine73.com


text:
시간 제한메모리 제한제출정답맞힌 사람정답 비율
1 초 128 MB67342948.333%

## 문제

The users feedback for your most beloved open-source operating system is gathered, and guess what the most required feature turned out to be? Yes it is finally what we all have been waiting for so long, extending the built-in calculator functionality! On of the suggested extensions is adding the capability of evaluating polynomials, and that’s what you are thrilled to participate with!

In this problem you’re given a polynomial entered by the user in the calculator and are asked to evaluate it for a certain value.

## 입력

The first line of input contains T (0 < T <= 100) the number of polynomials, each test cases consists of two lines, the first line contains an integer (-1000 <= X <= 1000), the value for the variable X for which the polynomial is evaluated.

The second line contains a polynomial P with integer coefficients. P is a sum of terms of the form CX^E , where the coefficient C and the exponent E satisfy the following conditions:

1. E is an integer satisfying (0 <= E <= 30). If E is 0, then CX^E is expressed as C. If E is 1, then CX^E is expressed as CX, unless C is 1 or -1. In those instances, CX^E is expressed as X or -X.
2. C is an integer. If C is 1 or -1 and E is not 0 or 1, then the CX^E will appear as X^E or -X^E.
3. Only non-negative C values that are not part of the first term in the polynomial are preceded by +.
4. Exponents in consecutive terms are strictly decreasing.
5. C fits in a 32-bit signed integer.
6. The calculated value of each term CX^E also fits in a 32-bit signed integer.

## 출력

For each test case, print the value of polynomial evaluation. The result will fit in a 32-bit signed integer. Follow the output format below.

## 예제 입력 1

2
-2
2X^2-5X+7
2
-3X^12+X

## 예제 출력 1

Case #1: 25
Case #2: -12286

## 출처

• 빠진 조건을 찾은 사람: kks227


text:
# Evaluating $\int_{0}^{\frac{\pi}{2}} (\sin (x) +\cos (x))^4 - (\sin(x) - \cos (x))^4 dx$

I would like to evaluate $$\int_{0}^{\frac{\pi}{2}} (\sin (x) +\cos (x))^4 - (\sin(x) - \cos (x))^4 dx.$$ One approach I used was to split the integral into two pieces and evaluating each piece directly with the anti derivatives $$\frac {1}{4} (\sin(x) - \cos (x))^4$$ and $$\frac {1}{4} (-\sin(x) - \cos (x))^4,$$ respectively. This gives me 0. However, one can use trig identities and manipulate the integrand so that the final integral is $$4\int_{0}^{ \frac{\pi}{2} }2\sin (x) \cos (x) dx.$$ This gives me 4. How is that I'm getting two answers?

• no for the first integral you get also $4$ as the result – Dr. Sonnhard Graubner Sep 4 '15 at 22:05
• @Nitin I believe the anti derivatives are correct. The second one needs a minus sign, but I was taking that into account when taking the difference. – Kevin Sheng Sep 4 '15 at 22:07
• @KevinSheng I deleted my first comment because I thought I was wrong, but no, the anti derivatives are incorrect. Just use the chain rule to check. – user217285 Sep 4 '15 at 22:14
• Nope, you are correct. I was making a careless error. – Kevin Sheng Sep 4 '15 at 22:17

You didn't compute antiderivatives correctly. To check your work, simply differentiate your result: $$\frac{\mathrm d}{\mathrm dx}\left[\frac {1}{4} (\sin(x) - \cos (x))^4 + C\right] = (\sin x + \cos x)(\sin x - \cos x)^3 \neq (\sin x + \cos x)^4$$

Please, observe that we used the chain rule to derive our result. The same argument can be used to show that the second antiderivative is wrong.

To evaluate the integral it's much easier to use the fact that $$A^2 - B^2 = (A + B)(A - B),$$ which, in combination with the identity $\sin^2 x + \cos^2 x = 1$, yields \begin{align} I &= \int_0^{\pi/2}\left((\sin x + \cos x)^2 + (\sin x - \cos x)^2\right)\left((\sin x + \cos x)^2 - (\sin x - \cos x)^2\right)\mathrm dx\\ &= \int_0^{\pi/2}8\sin x \cos x\,\mathrm dx\\ &= 4\int_0^{\pi/2}\sin 2x\,\mathrm dx\\ &= 4\text{.} \end{align}

As @rubik's answer explains, your antiderivatives are incorrect, and that is where the confusion comes from.

If I were doing this, I'd just imagine expanding both 4th powers, see that there would be a lot of cancellation alternating with doubling up, and leave you with \begin{align} &\phantom{{}={}}\int_0^{\pi/2}\left(8\sin^3(x)\cos(x)+8\sin(x)\cos^3(x)\right)\,dx\\ &=\int_0^{1}8u^3\,du+\int_1^{0}-8v^3\,dv\\ &=16\int_0^{1}u^3\,du\\ &=16\cdot\frac{1}{4} \end{align}

I we recall that: $$\cos^4\theta-\sin^4\theta = (\sin^2\theta+\cos^2\theta)(\cos^2\theta-\sin^2\theta) = \cos(2\theta)$$ it follows, with trivial manipulations:

$$\begin{eqnarray*} &&\int_{0}^{\pi/2}\left[\left(\sin x+\cos x\right)^4-\left(\sin x-\cos x\right)^4\right]\,dx\\&=&4\int_{0}^{\pi/2}\left[\cos(x-\pi/4)^4-\sin(x-\pi/4)^4\right]\,dx\\&=&4\int_{-\pi/4}^{\pi/4}\left[\cos^4\theta-\sin^4\theta\right]\,d\theta\\&=&4\int_{-\pi/4}^{-\pi/4}\cos(2\theta)\,d\theta\\&=&2\int_{-\pi/2}^{\pi/2}\cos\theta\,d\theta=\color{red}{4}.\end{eqnarray*}$$


text:
시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 512 MB 27 4 4 21.053%

## 문제

A rabbit and a turtle decided to race each other. Since the turtle is from Craiova and the rabbit is from Ardeal, the turtle is obviously faster than the rabbit. Our goal is to help the rabbit win the race.

The race is held on a graph with N nodes and M edges. The race starts at node 1 and ends at node N. Both the rabbit and the turtle decided beforehand to select the path which they are going to use (each one of them with his own path). Therefore, they know the graph, the paths and also the time it takes for each one of them to cross each edge of the graph.

The turtle may be faster than the rabbit, but it’s still a turtle (let’s call it George). George selects some nodes on his path where he decides to sleep for a while. If at any moment in time he realizes that the rabbit cheats, George will stop sleeping and will go to the finish without any further rest.

The rabbit (let’s call it Stan) has only one advantage. Stan has a great-great-great-. . . ...-great grandmother who is a fox, therefore he has a sly side. Stan does not intend to keep his promise about the path (but George does). At some point, his plan is to change the path he is initially assigned and take the shortest path to node N. The only problem is that he has to be smart about it. The moment George discovers that Stan is cheating, he will stop his sleeping activities, which is not good.

Stan can only change his path when he is in a node (obviously not when he is on an edge). He does not know George’s sleeping schedule, but you do!!! Compute the number of moments when Stan can change his path such that he will win the race. The moment Stan cheats, George will discover immediately as long as he is not sleeping. If he is sleeping during that time, he will find out upon waking up.

## 입력

The first line contains two numbers N and M. The next M lines will describe the graph by (A, B, T, R): there is an edge from node A to node B. The turtle will cross the edge in T units of time, while the rabbit will do it in R units of time. The i-th such edge is considered to be the edge with index i.

The next line contains a number P T, the number of nodes in George’s path. The next P T lines will describe the path by (edge index, sleep): the index of the next edge George is going to use and the number of time units he is going to sleep after using that edge. The sleeping value for the last edge is irrelevant since the turtle will already reach the finish.

The next line contains a number P R, the number of edges in Stan’s initial path. The last line will contain P R values, the indices of edges Stan is going to use.

## 출력

The output contains on the first line one value: x, the number of moments (nodes on the path) when Stan can cheat. The second line contains x numbers in ascending order, the indices of the nodes where Stan can cheat.

## 제한

• 2 ≤ N ≤ 100.000
• 1 ≤ P R, P T < 100.000
• 1 ≤ M ≤ 200.000
• 1 ≤ T, R ≤ 1.000.000.000
• 0 ≤ sleep ≤ 1.000.000.000
• The edges from the input that describe the paths are given in the correct order.
• The nodes in the turtle’s path do not repeat.
• The nodes in the rabbit’s path do not repeat.
• If the rabbit and the turtle arrive at the finish in the same time, the rabbit wins.
• If the rabbit cheats in the exact moment when the turtle goes to sleep, the turtle will first fall asleep and only after waking up he will realize about the cheating.
• The rabbit is considered to cheat ONLY if he changes the direction of the path and the new path that he will take is strictly faster than the original one (otherwise, there is no point to cheat)
• The cheating path may have common nodes with the original path. The only restriction is that in the exact moment of cheating, the rabbit has to take a different node. He may even go back to the previous node if he has an edge that allows that.

## 예제 입력 1

8 12
1 2 2 10
2 3 1 10
3 8 2 10
1 4 10 3
4 5 10 2
5 6 10 4
6 8 10 2
1 7 10 5
4 7 10 2
5 7 10 2
6 7 10 1
7 8 10 1
3
1 3
2 2
3 0
4
4 5 6 7

## 예제 출력 1

2
4 5

## 예제 입력 2

6 6
1 4 1 3
4 6 1 1
4 2 1 6
2 6 6 6
3 4 2 3
1 3 4 5
2
1 2
2 0
4
6 5 3 4

## 예제 출력 2

0

## 출처

• 문제의 오타를 찾은 사람: doju


text:
# 7.1 Second-order linear equations  (Page 5/15)

 Page 5 / 15
$y\left(x\right)={c}_{1}{e}^{-2x}+{c}_{2}{e}^{-7x}.$

## Single repeated real root

Things are a little more complicated if the characteristic equation has a repeated real root, $\lambda \text{.}$ In this case, we know ${e}^{\lambda x}$ is a solution to [link] , but it is only one solution and we need two linearly independent solutions to determine the general solution. We might be tempted to try a function of the form $k{e}^{\lambda x},$ where k is some constant, but it would not be linearly independent of ${e}^{\lambda x}.$ Therefore, let’s try $x{e}^{\lambda x}$ as the second solution. First, note that by the quadratic formula,

$\lambda =\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}.$

But, $\lambda$ is a repeated root, so ${b}^{2}-4ac=0$ and $\lambda =\frac{\text{−}b}{2a}.$ Thus, if $y=x{e}^{\lambda x},$ we have

${y}^{\prime }={e}^{\lambda x}+\lambda x{e}^{\lambda x}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y\text{″}=2\lambda {e}^{\lambda x}+{\lambda }^{2}x{e}^{\lambda x}.$

Substituting these expressions into [link] , we see that

$\begin{array}{cc}\hfill ay\text{″}+b{y}^{\prime }+cy& =a\left(2\lambda {e}^{\lambda x}+{\lambda }^{2}x{e}^{\lambda x}\right)+b\left({e}^{\lambda x}+\lambda x{e}^{\lambda x}\right)+cx{e}^{\lambda x}\hfill \\ & =x{e}^{\lambda x}\left(a{\lambda }^{2}+b\lambda +c\right)+{e}^{\lambda x}\left(2a\lambda +b\right)\hfill \\ & =x{e}^{\lambda x}\left(0\right)+{e}^{\lambda x}\left(2a\left(\frac{\text{−}b}{2a}\right)+b\right)\hfill \\ & =0+{e}^{\lambda x}\left(0\right)\hfill \\ & =0.\hfill \end{array}$

This shows that $x{e}^{\lambda x}$ is a solution to [link] . Since ${e}^{\lambda x}$ and $x{e}^{\lambda x}$ are linearly independent, when the characteristic equation has a repeated root $\lambda ,$ the general solution to [link] is given by

$y\left(x\right)={c}_{1}{e}^{\lambda x}+{c}_{2}x{e}^{\lambda x},$

where ${c}_{1}$ and ${c}_{2}$ are constants.

For example, the differential equation $y\text{″}+12{y}^{\prime }+36y=0$ has the associated characteristic equation ${\lambda }^{2}+12\lambda +36=0.$ This factors into ${\left(\lambda +6\right)}^{2}=0,$ which has a repeated root $\lambda =-6.$ Therefore, the general solution to this differential equation is

$y\left(x\right)={c}_{1}{e}^{-6x}+{c}_{2}x{e}^{-6x}.$

## Complex conjugate roots

The third case we must consider is when ${b}^{2}-4ac<0.$ In this case, when we apply the quadratic formula, we are taking the square root of a negative number. We must use the imaginary number $i=\sqrt{-1}$ to find the roots, which take the form ${\lambda }_{1}=\alpha +\beta i$ and ${\lambda }_{2}=\alpha -\beta i\text{.}$ The complex number $\alpha +\beta i$ is called the conjugate of $\alpha -\beta i\text{.}$ Thus, we see that when ${b}^{2}-4ac<0,$ the roots of our characteristic equation are always complex conjugates .

This creates a little bit of a problem for us. If we follow the same process we used for distinct real roots—using the roots of the characteristic equation as the coefficients in the exponents of exponential functions—we get the functions ${e}^{\left(\alpha +\beta i\right)x}$ and ${e}^{\left(\alpha -\beta i\right)x}$ as our solutions. However, there are problems with this approach. First, these functions take on complex (imaginary) values, and a complete discussion of such functions is beyond the scope of this text. Second, even if we were comfortable with complex-value functions, in this course we do not address the idea of a derivative for such functions. So, if possible, we’d like to find two linearly independent real-value solutions to the differential equation. For purposes of this development, we are going to manipulate and differentiate the functions ${e}^{\left(\alpha +\beta i\right)x}$ and ${e}^{\left(\alpha -\beta i\right)x}$ as if they were real-value functions. For these particular functions, this approach is valid mathematically, but be aware that there are other instances when complex-value functions do not follow the same rules as real-value functions. Those of you interested in a more in-depth discussion of complex-value functions should consult a complex analysis text.

Based on the roots $\alpha ±\beta i$ of the characteristic equation, the functions ${e}^{\left(\alpha +\beta i\right)x}$ and ${e}^{\left(\alpha -\beta i\right)x}$ are linearly independent solutions to the differential equation. and the general solution is given by

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!


text:
# Can the orbits of a semigroup touch without one being included in the other?

Question. Let $(S(t))_{t \ge 0}$ be a continuous semigroup of linear operators on some Banach space $X$. Might there exist $f, g\in X$ and $0<t_0<t_1$ such that $$S(t_0)f=S(t_1)g$$ but $$S(t_0-\varepsilon_0)f\ne S(t_1-\varepsilon_1)g$$ for all $0<\varepsilon_0\le t_0$ and $0<\varepsilon_1\le t_1$ (in particular, $f\ne g$)?

Pictorially, I am wondering if the following configuration is possible:

Of course we know that, when the evolution is given by a group, this is not possible: orbits either coincide or are disjoint. This is the case of autonomous ODE systems or of the Schrödinger equation. But here we have a semi group, such as the heat one, which only goes forward in time, not backwards. So the only obvious thing that we can say is that, as soon as they touch, orbits merge into one. But in principle I don't see why they should coincide in the past.

Added: After some searches, I have found that for the special case of the heat equation, the answer is negative. This is commonly referred to as backward uniqueness property. Here is a simplified version, which takes into account classical solutions on bounded domains:

Theorem (Taken from Evans's book on PDE, 2nd ed., pag.64) Let $U\subset \mathbb{R}^n$ be an open and bounded domain. Suppose $u, \bar{u}$ are classical solutions of $$\begin{cases} u_t=\Delta u & \text{in }U\times(0, T) \\ u=0 &\text{on }\partial U \times [0, T] \end{cases}$$ If at time $T$ we have $$u(x, T)=\bar{u}(x, T),\quad \forall x\in U,$$ then $u\equiv \bar{u}$ on the whole parabolic cylinder $U\times (0, T]$.

The property holds in much more general functional settings, as I read here (look for the keyword backward uniqueness).

All of this leaves the general question open. Is the backward uniqueness property true for all continuous linear semigroups? I guess that the answer should be negative, otherwise this would not be regarded as a special feature of the heat equation. However, I cannot find an explicit example.

-

I believe this is a counterexample:

Consider the semigroup on $L_2[0,\infty)$ given by $$S(t) f(x) = f(x+t)$$ Let $f = I_{[0,1)}$, and $g = 2I_{[0,2)}$. Then $S(1)f = S(2)g = 0$, but $S(1-\epsilon_1)f \ne S(2-\epsilon_2)g$.

-

I think that it is a good question. To simplify, let us study the equation $$u_t=-\Lambda u,$$ where $\Lambda=\sqrt{-\Delta}$. This equation is locally well-posed (both, forward and backward in time) for analytic initial data in a complex strip around the real axis. Then, at some time $T>0$, two solutions with a different initial data (at time $t=0$) can not coincide. Otherwise the backward problem posed at time $T$ would be ill-posed.

I think that for the heat equation the situation is similar.

-
This example is a bit too complicated for me. Could you do something more basic? It's fine to discuss the heat equation only. –  Giuseppe Negro May 24 '13 at 13:31
The point of this second equaton is that I know how to solve it forward and backward (because is a first order equation both in space and time you can apply a Cauchy-Kowalevsky Theorem). For the heat equation all this is much more difficult. You have to solve backward a heat equation and you know that your initial data is an entire function. if this problem is well-posed the solution can not touch themselves. –  guacho May 24 '13 at 13:50
I agree that it is more difficult. We are lucky that someone already did this for us. See the update to the question. –  Giuseppe Negro May 27 '13 at 19:17